Misunderstanding basic math concepts, help please?
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Re: Misunderstanding basic math concepts, help please?
It is, perhaps, worth noting that this is not Treatid first thread. People, including Twistar, have *tried* to engage him without resorting to this sort of thing. Good luck with your endeavors, but I suspect you'll have about as much success as the folks in the waterman thread had in translating concepts to and from the "stevean".
Re: Misunderstanding basic math concepts, help please?
JudeMorrigan wrote:It is, perhaps, worth noting that this is not Treatid first thread. People, including Twistar, have *tried* to engage him without resorting to this sort of thing. Good luck with your endeavors, but I suspect you'll have about as much success as the folks in the waterman thread had in translating concepts to and from the "stevean".
Well, I can tell you right now that I don't have any more patience for vague BS than you.
I do think, however, that this thread started with an interesting set of questions which didn't seem too "crackpotty" to me. He actually got me thinking, because I've never before thought about the issues he raised (seems like a layman's halfbaked version of Russel's paradox, basically). I admit not being aware of Treatid's previous threads, but now I'm curious: did they all start as benignly as this one did?
Either way, rest assure that I'm not going to let the OP to derail me into some wild goose chase. My time is precious, and when people willfully waste it, I get annoyed. Very annoyed. I do think he deserves one fair chance, though. Whether he takes that chance, that's up to him.
Anyway, here is my reply to the questions posted in the OP:
Treatid wrote:And here I run into trouble. My thinking is as follows:
Given that axiomatic mathematics as a whole exists; it must belong in one of these two sets:
A. If axiomatic mathematics is not a well formed axiomatic system then we can't be sure what is being described.
B. If axiomatic mathematics is a well formed axiomatic system then it is subject to the Principle of Explosion.
However, since axiomatic mathematics is definitely a thing, and the principle of explosion doesn't apply to axiomatic mathematics as a whole there must be another option that I'm missing. But I can't see what it could be.
There is a third option, which also happens to be true:
Mathematics is not an axiomatic system in itself, but it uses an axiomatic system to give itself structure and meaning.
As the other people here already explained in detail, the axioms that mathematics is using as a foundation are the rules of classical logic. You know, the basic rules of logical deduction.
And on this logical framework, you can build... well, anything you wish, really. You are free to choose the axioms of your system as you see fit, and you have this complete freedom precisely because mathematics isn't an axiomatic system in itself.
By the way, the "principle of explosion" does apply to mathematics as a whole. You just need to remember that whenever someone says "I've proved X" or "X is true", this is actually shorthand for "I've proved X in axiomatic system Y" or "X is true in axiomatic system Y".
So just because, say, 2+2=4 in ordinary arithmetic while 2+2=0 in modulu 4 arithmetic, doesn't mean the principle of explosion is wrong. These are simply two different statements. The "in suchansuch arithmetic" is part of the statement and cannot be omitted.
BTW, I don't think there's any need to delve into the actual mystiques of formal logic in order to understand this. All you need to remember is that formal logic doesn't care how complicated the actual content of your statements is. It always treats your entire statement as a single unit, unless it contains a word which translates directly into a formal logic concept (one of the following: "and", "or", "not", "therefore" and perhaps another one which I forgot the moment).
Here's a simple example from the real world:
If I say "the table to my left is small, and the table to my right is big" then you cannot use the rule of explosion to prove 1=0 from this "contradiction". There's no contradiction at all, because "the table to my left" is a different object from "the table to my right" (even though they both happen to be tables).
And similarly, saying that "2+2=4 in ordinary arithmetic, while 2+2=0 in modulu 4 arithmetic" is not a contradiction. The symbols "2" and "+" in each statement do not refer to the same thing, just like the word "table" referred to two different objects in the previous example.
On the other hand, "2+2=4 in ordinary arithmetic" and "2+2=5 in ordinary arithmetic" cannot both be true. Had both statements been true, than the principle of explosion would indeed render ordinary arithmetic useless. "Luckily" enough, the second statement is false, so we can continue to use arithmetic to count and add without fear.
Re: Misunderstanding basic math concepts, help please?
PsiCubed wrote:gmalivuk wrote:The request that Treatid do some simple worksheetstyle formalism, or at least explain the unwillingness to do so, is a reasonable one, especially since we've all seen how frequently relying on vague intuitions about natural language has led these discussions astray.
I agree that demanding Treatid to explain himself more clearly is not only reasonable, but a requirement to have a fruitful discussion. This thread is, in itself, an excellent example of why this is necessary.
But there's more than one way to do this, and quite clearly Twistar's approach isn't working. And even though I'm a person who is wellversed in mathematics and logic, I don't find it at all surprising that Treatid is refusing to cooperate with Twistar's requests. I myself find these demands to be arrogant and borderline offensive.
It is, perhaps, worth noting that this is not Treatid first thread. People, including Twistar, have *tried* to engage him without resorting to this sort of thing.
Quite right. This is at least the third time Treatid has done this and I've put a lot of work into trying to understand what he's saying and trying to discuss these topics with him. Here is my conclusion. Treatid does not know what rigorous mathematics is. Treatid doesn't know how to do a proof in formal logic. Neither of these are major crimes. But what IS the major crime is when he
1) comes in here thinking that he knows more about logic and mathematics than everyone else and
2) is not willing to listen or learn from anyone.
Sure if you write him a half page post he'll write you one back. Unfortunately he will just skip over all of the salient points and continue on saying exactly whatever he was saying in his previous post. Sometimes he'll make it sound like he's had some major breakthrough but it's just a shift in his thoughts, and never one towards actually learning the subject of formal logic.
I'm done putting up with Treatid and I don't care if I get mean because trust me, I've tried being nice.
To reiterate gmalivuk's point and answer PsiCubed as to why it is so important to me that Treatid work's on the exercises:
Yes, I agree that it is important to effectively communicate technical topics to nontechnical audiences. I recently spent a long time preparing a lecture on quantum mechanics and the interpretations of quantum mechanics for a high school audience which had not yet even taken intro physics, and I think they got what I was trying to convey. The only mathematical symbols in the whole thing were a few "+" symbols.
However, this is not a situation like that for two reasons.
1) Treatid is acting like an expert on the subject. If he wants to be an expert on the subject he better be able to answer questions that a freshman college student who has spent a quarter of a semester sitting in Formal Logic 101 could answer.
2) The topics Treatid is probing into are deeper and more sophisticated than the questions a nonmathematician or layperson would ask or be interested in. As such they need to be addressed with real formality and rigor.
You mention:
Now, I understand that some formality is inescapable when discussing a topic such as this. But it should really be kept to the possible minimum if you are seriously interested in actual communication with nonmathematicians.
What I'm saying is that this bare minimum level of formality for this discussion is a simple 5 line proof in prepositional logic. I'm sure everyone else in the thread would agree with me.
Now, that's not to say I'm not being a dick or in your words "arrogant and borderline offensive". Maybe I am, but I don't care. I'm calling a duck a duck. There is one reason this topic never has gone, isn't going, and never will go anywhere:
Treatid doesn't know what it means to do a proof in formal logic.
And here's the funny thing. I told him EXACTLY how to do a proof in formal logic. Does you remember that part?
Twistar wrote:Here is the structure of a logical proof. There are 3 columns and a variable number of rows.
The 1st column just has the number of the line we are working on.
The 2nd column has a sentence written in whatever language we are working in. It is necessary that this sentence follow the rules of syntax of the given language.
The 3rd column is the justification for writing down the sentence which appears in the 2nd column. The is the meat of the proof.
Notice that for this particular proof the 3rd column has a couple of different things. For the three premises it just says premise. This means we don't need prior justification for writing down those sentences. "premise" is enough justification. However, if you look at line 4 you see that this is a derived sentence because it doesn't say "premise" in column 3, instead it says "1,2 hyp. syll." This means that line 4 was derived FROM lines 1 and 2 using the rule of hypothetical syllogism or "hyp. syll." as it appears on the reference sheet. This line is notable because it is derived ONLY from the premises. However, if you look at the next line, line 5, you see that the this line was derived from lines 3 and 4 again via "hyp. syll." This one is notable because it uses one premise but it ALSO uses the derived formula. If the proof was longer eventually we would get lines which are justified only by derived sentences, but of course it could all be traced back to the premises. Notice that the sentence on line 5 matches the sentence next to "Show:". This means that the proof is complete.
And I would be willing to spend HOURS discussing with Treatid how to do proofs in prepositional logic, how to fill out truth tables, move on to predicate logic etc. That would be thrilling. But that's not what he wants. As far as I can tell all he wants is to be right. He doesn't want to learn anything. That's not what this forum is about. The obvious response is to turn that argument back on me. Maybe there would be some traction to that argument if Treatid was the one who write a proof and I wasn't.
Here's a video lecture where someone is describing (a bit quickly) how to do a particular proof. Maybe that will be a little more clear for Treatid. http://www.youtube.com/watch?v=ZebpxExsY_0&t=22m29s
Anyways, Treatid isn't going to respond to anything I'm saying anymore so have fun everyone.
Here are Treatids old threads:
search.php?keywords=&terms=all&author=Treatid&sc=1&sf=all&sr=topics&sk=t&sd=d&st=0&ch=300&t=0&submit=Search
"Axiomatic mathematics has no foundation" is the first one I think I was involved in. (2014)
"How do axioms specify a new system vs continuing an old one?" (2015)
"Describe an Inconsistent System" (2015)
"Misunderstanding basic math concepts, help please?" (2016)
edit2: He also has threads a lot older than this. I think at that time he was interested in some physics concepts but found he didn't understand enough math so he kept stepping back and stepping back and I think that is how he's come to find himself at mathematical foundations and formal logic.
All of this, over 2 years worth of posting about formal logic, and Treatid can't do a proof in formal logic. If he had spent a few hours watching course lectures online or reading a real textbook he could have learned how to do it. So excuse me for being a little exasperated.
edit: And the point of all of this is that all of these threads boil down to the same issue:
Treatid doesn't know what it means to do a proof in formal logic.
Anyway. This post got heated. To be clear, I'm not insulting Treatid's intelligence. If I'm insulting anything it is his refusal to learn from others.
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:Quite right. This is at least the third time Treatid has done this and I've put a lot of work into trying to understand what he's saying and trying to discuss these topics with him.
Since I've first heard about this history (a few hours ago), I actually went to look on those other threads.
My my... what a mess. But it also served to show me that my hunch was correct and that this thread is different from the others. On this thread, unlike any of the others, Treatid has actually raised good coherent questions. Questions I actually felt compelled to answer, simply because they were interesting in their own right.
It does makes your flippant responses much more understandable, though.
To reiterate gmalivuk's point and answer PsiCubed as to why it is so important to me that Treatid work's on the exercises:
Yes, I agree that it is important to effectively communicate technical topics to nontechnical audiences. I recently spent a long time preparing a lecture on quantum mechanics and the interpretations of quantum mechanics for a high school audience which had not yet even taken intro physics, and I think they got what I was trying to convey. The only mathematical symbols in the whole thing were a few "+" symbols.
However, this is not a situation like that for two reasons.
1) Treatid is acting like an expert on the subject. If he wants to be an expert on the subject he better be able to answer questions that a freshman college student who has spent a quarter of a semester sitting in Formal Logic 101 could answer.
2) The topics Treatid is probing into are deeper and more sophisticated than the questions a nonmathematician or layperson would ask or be interested in. As such they need to be addressed with real formality and rigor.
I strongly disagree with your two points.
I won't argue with #1 because that's an argument which has very little to do with mathematics (and this will likely to get our moderators angry again).
But I will strongly argue against your second point. The questions that Treatid asked can be answered adequetly (and moreorless rigorously) in simple words. They do not require symbolic logic or anything fancy. For an example, note the answer in my previous post (and also note that I've refrained from even hinting at Gödel's theorem in that post, although I was really tempted. When a layman asks a question, you gotta keep things as simple as possible).
There is one reason this topic never has gone, isn't going, and never will go anywhere:
Treatid doesn't know what it means to do a proof in formal logic.
The problem is more like: Treatid started to use formal logic in his posts, when he doesn't know how to do it.
Which is why I won't bother replying to posts in which he does that. Just like I won't respond to posts in Chinese. I'm not going to waste my precious time learning an entire new language  whether it is "Chinese" or "the manipulation of symbols that Treatid is erroneously calling 'formal logic' "  just so I can explain something to some guy on the internet.
That's why I've responded to Treatid in plain English, and why I expect him to do the same. This shouldn't be a problem, if he actually listens to what I'm saying and is actually interested in having a conversation. And if he doesn't... well, in that case, no amount of formal logic will save this thread.
And a final note for the mods:
This is the last post by me on this subject. While I think that discussing these things was important, I also understand that continuing to do so is offtopic and derailing this thread, so I'm stopping right now. If anyone has a comment regarding my actual answer to Treatid, though, I'll be happy to discuss it
Re: Misunderstanding basic math concepts, help please?
PsiCubed wrote:That's why I've responded to Treatid in plain English, and why I expect him to do the same. This shouldn't be a problem, if he actually listens to what I'm saying and is actually interested in having a conversation. And if he doesn't... well, in that case, no amount of formal logic will save this thread.
Good luck.
Re: Misunderstanding basic math concepts, help please?
PsiCubed wrote:I won't argue with #1 because that's an argument which has very little to do with mathematics (and this will likely to get our moderators angry again).
But I will strongly argue against your second point. The questions that Treatid asked can be answered adequetly (and moreorless rigorously) in simple words. They do not require symbolic logic or anything fancy. For an example, note the answer in my previous post (and also note that I've refrained from even hinting at Gödel's theorem in that post, although I was really tempted. When a layman asks a question, you gotta keep things as simple as possible).
Which is why I won't bother replying to posts in which he does that. Just like I won't respond to posts in Chinese. I'm not going to waste my precious time learning an entire new language  whether it is "Chinese" or "the manipulation of symbols that Treatid is erroneously calling 'formal logic' "  just so I can explain something to some guy on the internet.
That's why I've responded to Treatid in plain English, and why I expect him to do the same. This shouldn't be a problem, if he actually listens to what I'm saying and is actually interested in having a conversation. And if he doesn't... well, in that case, no amount of formal logic will save this thread.
And a final note for the mods:
This is the last post by me on this subject. While I think that discussing these things was important, I also understand that continuing to do so is offtopic and derailing this thread, so I'm stopping right now. If anyone has a comment regarding my actual answer to Treatid, though, I'll be happy to discuss it
I do not understand why one thinks one can "disprove" or claim axiomatic mathematics is inconsistent without knowing what axiomatic mathematics is. As far as I can see, treatid seems to be attacking his fantasies of what he thinks axiomatic mathematics is, rather than actual axiomatic mathematics.
Sure, there is no problem with thinking about the foundation of maths in your free time and come up with your own ideas, but if you find some gaps in *your* ideas, it is a good idea to learn about how it is actually done in mathematics and see how mathematicians resolve the problems you came up with. If you've done that step, and are not happy with the resolutions provided, *then* you start to talk and complain.
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:Here are Treatids old threads:
http://forums.xkcd.com/search.php?keywo ... mit=Search
"Axiomatic mathematics has no foundation" is the first one I think I was involved in. (2014)
"How do axioms specify a new system vs continuing an old one?" (2015)
"Describe an Inconsistent System" (2015)
"Misunderstanding basic math concepts, help please?" (2016)
edit2: He also has threads a lot older than this. I think at that time he was interested in some physics concepts but found he didn't understand enough math so he kept stepping back and stepping back and I think that is how he's come to find himself at mathematical foundations and formal logic.
Before this, iirc, he was posting about P=NP. Specifically, that P=NP is true for N=1 (or something similarly indicative of a lack of understanding of what the problem really says). Once those threads were shot down, he shifted to attacking physics and then foundational mathematics, since nothing is true and everything is ambiguous.
And I'll just add the point that Treatid is partially right about something  if we don't find an unambiguous way to communicate with each other, we won't get anywhere. I don't think that mathematics, axiomatic mathematics, etc are ambiguous, but the ways Treatid understands and discusses them certainly are.
Re: Misunderstanding basic math concepts, help please?
Gwydion wrote:P=NP is true for N=1
It's also true for P=0
I absolutely love that one. It's both funny, and a great example of why there's no point in having a discussion unless everybody agrees on the actual meaning of the symbols involved.
(For what its worth, we could all agree to discuss "P=NP" as an equation whose solutions are N=1 and P=0. But such a discussion would be pretty short, not very interesting, and completely unrelated to the famous topic usually referred to by these four characters)
Re: Misunderstanding basic math concepts, help please?
Absolutely, I want to explain myself clearly. Cauchy is doing a brilliant job of looking at what I'm actually saying and pointing out where I'm failing to be concise  or perhaps just wrong. What Cauchy is doing feels constructive for me and for the progress of the thread.
I neglected to check for additional responses after I posted my last post. I missed several pertinent posts  sorry. It wasn't my intention to outright ignore such a clear message.
Given those posts and the subsequent ones: I will attempt to answer one of the questions at the end of this post  I believe question 4 was specifically mentioned and hasn't been covered by anyone else yet. I'll devote the last section of each of my posts to either answering a new question (please specify a preference, Twistar), or correcting flaws in my last answer, (or just trying to learn logic), as appropriate.
Regarding a restatement of my original question... I view axiomatic mathematics as irredeemably flawed. Everyone else doesn't. Obviously I've come to a mistaken conclusion for some reason. But to me, my conclusion appears to be backed by compelling evidence that in turn looks fully justified.
If I'm questioning the fundamental validity of axiomatic mathematics, it seems appropriate to question what I think axiomatic mathematics is. It looks to me as if this has accounted for a significant portion of this thread.
The premises I'm trying to convey are my perception of a fair generalisation of axiomatic mathematics. I'm trying to convey in one place how I think the various facets of axiomatic mathematics fit together. The more clearly I can convey exactly how I'm seeing axiomatic mathematics, the more likely someone can point to a specific flaw in a way that clicks for me.
Premises and deductions
Cauchy, you are, of course, correct that using English semantics is dangerous ground. In tidying up, I found some odds and ends that even I thought looked suspiciously like they'd been swept under 'describe'.
Your reflection of my premises tells me that you are accurately understanding what I'm intending to convey and pointing out the parts where I'm not being clear.
In clarifying, I have, to my knowledge, changed to some degree two of the premises. Previously, y was a set of functions that applied to all z. Now y is the set of all functions whose domain is within (or equal to) z.
The other change is that X is a set. I might not be thinking exactly the same thing as anyone else when thinking of a 'system'. I think specifying a nice, simple set is another opportunity to avoid possible misunderstandings.
Iteration 3:
With the removal of 'describe' and the changes to the premises as noted, I've rewritten all of the deductions.
Attempting to summarise: We have a set that contains every instance of a thing and every instance of every function that applies to any and all of those things. We then introduce recursion by specifying that some q has X as the domain. Therefore both X and all q are members of z. The only thing lacking to really wrap things up is that it hasn't been shown that y = z.
I was wrong
Deduction 7) was simply wrong. I am embarrassed. At the time I wrote that deduction I genuinely thought that it was a logical deduction from the premises. All I needed to do was check that this deduction was valid for this system  but I assumed I had already done that. I hadn't. A related idea in a similar system is not necessarily equivalent to this idea in this system.
So, I make a stupid mistake, then fail to make sure I'm not making a stupid mistake, then proudly declare that mistake as being important and significant in some way.
D'oh!
In slight mitigation, I've been helped into realising that I was making a false assumption. Understanding why I was making that false assumption is illuminating in itself, and allows me to correct the assumption and anything based on that assumption. Learning.
I think there was a point (relating to what structures can be built from "q1 has q2 as its domain, for all possible q") to why I thought that notdeduction was relevant. But for the moment I'll concentrate on what other stupid mistakes trying to rigorously describe my perception of axiomatic mathematics reveals.
...
@Cauchy, My use of 'describe' was, as predicted, on dangerous ground. Clearing out 'describe' has clarified for me what it is I'm getting at. I've taken the opportunity to refine what I'm trying to say as well as how I'm saying it. Which has the consequence that I am, to some degree, shifting the argument under you.
{You're bold doesn't seem to be working at this end.}
2 ) Replaced 'describe.
I think that X being the domain of some q necessarily implies that all elements of X: (y and z (p and q)) are within the domain of q. We have specified that the domain of all q is within z so everything in the domain of any q is in z.
3 ) Replaced 'describe'.
4) Yes. it looks like we agree.
5) The set X contains y and z. It does not contain any instance of !y or !z. Yes, !y and !z could encompass a great many things but none of those things are elements, properties or deductions of X.
6) Replaced 'describe'.
7) I was wrong.
It is apparent at this stage that my attempted use of 'describe' was just confusing.
Notx is the set of all things that are not x. Both y and z are members of that set. y != z in no way conflicts with both y and z being members of the set !x.
In the sense that !x is the set of not x, !x is a collection. In the sense that !x are all the members of the set !x, we are talking about absolutely everything except for x.
!X, !y and !z each have many members. But no element of !X, !y or !z is part of X. Deductions within !X are not deductions of X.
@PsiCubed: Thank you.
The first red flag that comes to mind is that there shouldn't be options as to what axiomatic mathematics is.
I agree that there almost certainly exists some set of axioms for which the third option is true.
However, I think that the option you describe is inconsistent with the axioms of axiomatic mathematics as I'm generalising above.
I don't think that axiomatic mathematics is capable of coexisting with not(axiomatic mathematics) in the way that you describe.
@dalcde: I agree  I need to show exactly what I think axiomatic mathematics is. See the premises above  each of those premises is intended to be a generalisation or statement of the axioms of axiomatic mathematics as I understand them.
Russell's Paradox
I've specified a set that is a member of itself. Very much Russell's Paradox territory.
A set being a member of itself is not by itself sufficient to create a paradox. Even were set X to lead to a paradox, that is only significant if set X is a fair generalisation of axiomatic mathematics.
Formal Logic Worksheet
Since I started on this post, there has been further consensus for this, so I'll dive right in:
1. (~E V P) V U [prem]
2. (~E V P) ⊃ A [prem]
3. U ⊃ B [prem]
4. ~A [prem]
∴ B
5. ~(~E V P) V A [2. cond.]
6. A V ~(~E V P) [5, comm.]
7. ~A ⊃ ~(~E V P) [6, cond.]
8. ~(~E V P) ⊃ U [1, cond.]
9. U ⊃ B ∴ B [3 and done]
Edit  wasn't preserving my 'not's on 6.
I neglected to check for additional responses after I posted my last post. I missed several pertinent posts  sorry. It wasn't my intention to outright ignore such a clear message.
Given those posts and the subsequent ones: I will attempt to answer one of the questions at the end of this post  I believe question 4 was specifically mentioned and hasn't been covered by anyone else yet. I'll devote the last section of each of my posts to either answering a new question (please specify a preference, Twistar), or correcting flaws in my last answer, (or just trying to learn logic), as appropriate.
Regarding a restatement of my original question... I view axiomatic mathematics as irredeemably flawed. Everyone else doesn't. Obviously I've come to a mistaken conclusion for some reason. But to me, my conclusion appears to be backed by compelling evidence that in turn looks fully justified.
If I'm questioning the fundamental validity of axiomatic mathematics, it seems appropriate to question what I think axiomatic mathematics is. It looks to me as if this has accounted for a significant portion of this thread.
The premises I'm trying to convey are my perception of a fair generalisation of axiomatic mathematics. I'm trying to convey in one place how I think the various facets of axiomatic mathematics fit together. The more clearly I can convey exactly how I'm seeing axiomatic mathematics, the more likely someone can point to a specific flaw in a way that clicks for me.
Premises and deductions
Cauchy, you are, of course, correct that using English semantics is dangerous ground. In tidying up, I found some odds and ends that even I thought looked suspiciously like they'd been swept under 'describe'.
Your reflection of my premises tells me that you are accurately understanding what I'm intending to convey and pointing out the parts where I'm not being clear.
In clarifying, I have, to my knowledge, changed to some degree two of the premises. Previously, y was a set of functions that applied to all z. Now y is the set of all functions whose domain is within (or equal to) z.
The other change is that X is a set. I might not be thinking exactly the same thing as anyone else when thinking of a 'system'. I think specifying a nice, simple set is another opportunity to avoid possible misunderstandings.
Iteration 3:
 X is the set of y & z.
 z is a set of every instance of p.
 p are specified to be consistent with the other premises.
 y is the set of all functions q, such that the domain for each q is some subset of z.
 There is at least one q whose domain is the set X.
With the removal of 'describe' and the changes to the premises as noted, I've rewritten all of the deductions.
 X is a member of z.
 All q are members of z.
 For each p there exists some q such that p is in the domain of q.
 notz is explicitly outside the domain of all functions q. No q has any possible notz as its domain.
 This is a closed loop. The only functions that can act on members of z are members of y. Each p only has a subset of z as its domain. notz and noty are explicitly not part of set X.
 For every q_{1} in y, there exists a q_{2} for which q1 is the domain.
 [redacted] *
Attempting to summarise: We have a set that contains every instance of a thing and every instance of every function that applies to any and all of those things. We then introduce recursion by specifying that some q has X as the domain. Therefore both X and all q are members of z. The only thing lacking to really wrap things up is that it hasn't been shown that y = z.
I was wrong
Deduction 7) was simply wrong. I am embarrassed. At the time I wrote that deduction I genuinely thought that it was a logical deduction from the premises. All I needed to do was check that this deduction was valid for this system  but I assumed I had already done that. I hadn't. A related idea in a similar system is not necessarily equivalent to this idea in this system.
So, I make a stupid mistake, then fail to make sure I'm not making a stupid mistake, then proudly declare that mistake as being important and significant in some way.
D'oh!
In slight mitigation, I've been helped into realising that I was making a false assumption. Understanding why I was making that false assumption is illuminating in itself, and allows me to correct the assumption and anything based on that assumption. Learning.
I think there was a point (relating to what structures can be built from "q1 has q2 as its domain, for all possible q") to why I thought that notdeduction was relevant. But for the moment I'll concentrate on what other stupid mistakes trying to rigorously describe my perception of axiomatic mathematics reveals.
...
@Cauchy, My use of 'describe' was, as predicted, on dangerous ground. Clearing out 'describe' has clarified for me what it is I'm getting at. I've taken the opportunity to refine what I'm trying to say as well as how I'm saying it. Which has the consequence that I am, to some degree, shifting the argument under you.
{You're bold doesn't seem to be working at this end.}
2 ) Replaced 'describe.
I think that X being the domain of some q necessarily implies that all elements of X: (y and z (p and q)) are within the domain of q. We have specified that the domain of all q is within z so everything in the domain of any q is in z.
3 ) Replaced 'describe'.
4) Yes. it looks like we agree.
5) I don't see how you get to this at all. Or rather, I'm not sure exactly what you mean by "closed loop", and "impinge". Just because X models the actions of the functions of 'describe' on the elements of z, that doesn't mean that nothing else can model those actions, so I'm not sure what you're driving at.
5) The set X contains y and z. It does not contain any instance of !y or !z. Yes, !y and !z could encompass a great many things but none of those things are elements, properties or deductions of X.
6) Replaced 'describe'.
7) I was wrong.
It is apparent at this stage that my attempted use of 'describe' was just confusing.
If we have three things, x, y, and z, with x != y, x != z, and y != z, then what does notx represent? Certainly, y is not x, but I would balk at the idea of your saying that y is notx. After all, z is not x, but it can't be the case that both y is notx and z is notx, since y != z.
Notx is the set of all things that are not x. Both y and z are members of that set. y != z in no way conflicts with both y and z being members of the set !x.
In the sense that !x is the set of not x, !x is a collection. In the sense that !x are all the members of the set !x, we are talking about absolutely everything except for x.
!X, !y and !z each have many members. But no element of !X, !y or !z is part of X. Deductions within !X are not deductions of X.
@PsiCubed: Thank you.
There is a third option, which also happens to be true:
Mathematics is not an axiomatic system in itself, but it uses an axiomatic system to give itself structure and meaning.
The first red flag that comes to mind is that there shouldn't be options as to what axiomatic mathematics is.
I agree that there almost certainly exists some set of axioms for which the third option is true.
However, I think that the option you describe is inconsistent with the axioms of axiomatic mathematics as I'm generalising above.
I don't think that axiomatic mathematics is capable of coexisting with not(axiomatic mathematics) in the way that you describe.
@dalcde: I agree  I need to show exactly what I think axiomatic mathematics is. See the premises above  each of those premises is intended to be a generalisation or statement of the axioms of axiomatic mathematics as I understand them.
Russell's Paradox
I've specified a set that is a member of itself. Very much Russell's Paradox territory.
A set being a member of itself is not by itself sufficient to create a paradox. Even were set X to lead to a paradox, that is only significant if set X is a fair generalisation of axiomatic mathematics.
Formal Logic Worksheet
Since I started on this post, there has been further consensus for this, so I'll dive right in:
1. (~E V P) V U [prem]
2. (~E V P) ⊃ A [prem]
3. U ⊃ B [prem]
4. ~A [prem]
∴ B
5. ~(~E V P) V A [2. cond.]
6. A V ~(~E V P) [5, comm.]
7. ~A ⊃ ~(~E V P) [6, cond.]
8. ~(~E V P) ⊃ U [1, cond.]
9. U ⊃ B ∴ B [3 and done]
Edit  wasn't preserving my 'not's on 6.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Formal Logic Worksheet
Since I started on this post, there has been further consensus for this, so I'll dive right in:
1. (~E V P) V U [prem]
2. (~E V P) ⊃ A [prem]
3. U ⊃ B [prem]
4. ~A [prem]
∴ B
5. ~(~E V P) V A [2. cond.]
6. A V ~(~E V P) [5, comm.]
7. ~A ⊃ ~(~E V P) [6, cond.]
8. ~(~E V P) ⊃ U [1, cond.]
9. U ⊃ B ∴ B [3 and done]
Edit  wasn't preserving my 'not's on 6.
You skipped a few steps after 8 there, here:
1. (~E V P) V U [prem]
2. (~E V P) ⊃ A [prem]
3. U ⊃ B [prem]
4. ~A [prem]
∴ B
5. ~(~E V P) V A [2. cond.]
6. A V ~(~E V P) [5, comm.]
7. ~A ⊃ ~(~E V P) [6, cond.]
8. ~(~E V P) ⊃ U [1, cond.]
9. ~A ⊃ U [7,8 hyp. syll.]
10. ~A ⊃ B [9,3 hyp. syll.]
11. ∴ B [10, 4 M.P.]
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:The first red flag that comes to mind is that there shouldn't be options as to what axiomatic mathematics is.
It is you who used the phrase "there must be a third option I'm missing" in your opening post. So I told you what you're missing.
However, I think that the option you describe is inconsistent with the axioms of axiomatic mathematics as I'm generalising above.
I don't think that axiomatic mathematics is capable of coexisting with not(axiomatic mathematics) in the way that you describe.
Well then, sorry I've wasted your time with my useless musings. I promise you it won't happen again.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Given those posts and the subsequent ones: I will attempt to answer one of the questions at the end of this post  I believe question 4 was specifically mentioned and hasn't been covered by anyone else yet. I'll devote the last section of each of my posts to either answering a new question (please specify a preference, Twistar), or correcting flaws in my last answer, (or just trying to learn logic), as appropriate.
Formal Logic Worksheet
Since I started on this post, there has been further consensus for this, so I'll dive right in:
1. (~E V P) V U [prem]
2. (~E V P) ⊃ A [prem]
3. U ⊃ B [prem]
4. ~A [prem]
∴ B
5. ~(~E V P) V A [2. cond.]
6. A V ~(~E V P) [5, comm.]
7. ~A ⊃ ~(~E V P) [6, cond.]
8. ~(~E V P) ⊃ U [1, cond.]
9. U ⊃ B ∴ B [3 and done]
Edit  wasn't preserving my 'not's on 6.
Awesome Treatid!! Thank you! This is great. Ok, so what you've done is great. So two little issues that might be related to a larger issue but I'm not really sure.
First, you didn't technically complete the proof. You need a few more lines after Line 8. What you should have is
9. ~A ⊃ U [7,8, Hyp. Syll.]
10. ~A ⊃ B [9,3, Hyp. Syll.]
11. ∴ B [4,10, M.P.]
I think this is a minor issue but I notice two things about it.
1) On this response as well as your previous one you seem to forget to clearly explain the Hypothetical Syllogism steps. I think you think they're like, so obvious they don't need any steps. Let me define X = ~(~E V P). In this proof we had the sentences "~A", "~A ⊃ X" and "X ⊃ B". It is obvious to a human that since we have ~A that implies X which implies B so obviously the proof is done. But being perfectly rigorous, I'm going to say it's important that you write out ALL of the lines of the proof otherwise it's not complete. This would get a 3/4 on an exam. I think the error should have been tipped off when on you're line 9 you wrote [3 and done]. "and done" is not a formal rule of logic which appears on the worksheet.
I have a questions for you about this paragraph.
Q1) Do you think this error occurred because of a misunderstanding you have or do you think it was just a slip of the mind/simple mistake? It's not clear to me which it is.
2) And here's the other issue which is actually starting to look a little bit bigger to me than it was initially. You seem to really like the basic equivalence forms. That's fine, you're allowed to use those for proofs. However, those just allow you to replace one line with another line that says the equivalent thing. However, to complete this proof it is necessary to use what are called the "basic argument forms" on the worksheet. These are rules which require you to use multiple lines to derive a new line. Let's look at how I used it in my line 9.
7. ~A ⊃ ~(~E V P) [6, cond.]
8. ~(~E V P) ⊃ U [1, cond.]
9. ~A ⊃ U [7,8, Hyp. Syll.]
What have I done here? I said I'm using Hyp. Syll. and lines 7 and 8. Let me write the rule for Hyp. Syll. from the worksheet. I'm going to add in a little bit more notation though.
n. p ⊃ q [prem]
m. q ⊃ r [prem]
k. p ⊃ r [n,m, Hyp. Syll.]
So what is going on here? p and q stand for wellformed formulas (WFF)*. This rule says that if on some line with number n. I have the formula "p ⊃ q" and on some other line number m. I have the formula "q ⊃ r" that I am now allowed to write down on some new line numbered k. the formula " p ⊃ r". That is what it means to deduce a new line from previous lines.
So how did I use this in the proof? Well in my case I've made the following replacements and "plugged them into" the rule for Hyp. Syll.: "p = ~A", "q = ~(~E V P)" and "r = U". Do you see how that works? How if you make those exact replacements in the rule at every point where those symbols appear you get exactly what I have in my proof.
edit4: Also, in my case "n = 7", "m = 8" and "k = 9" but it is not necessary that the lines follow in sequential order line that. The only restrictions on the numbers are that k > m and k > n. This just means you can't derive something from lines that don't exist yet.
Anyways, I hope this is a clear description.
What you have done on all of your lines except the last line are all examples of deductions, but they are all single line deductions which are slightly simpler than these 2 line deductions. What you sort of discovered is that this particular proof isn't possible with only single line deductions.
Just so that you can see more examples I'll show you how I would have done this exercise since it is a little different than how you did it and shows more examples of applying the basic argument forms.
1. (~E V P) V U [prem]
2. (~E V P) ⊃ A [prem]
3. U ⊃ B [prem]
4. ~A [prem]
∴ B
5. ~(~E V P) [2,4, M.T.]
6. U [1,5, Disj. Syll.]
7. ∴ B [6,3, M.P.]
It's nice because your way of doing the proof took 11 lines and mine only took 7. That's not to say that your way is wrong, it's just to say that I prefer mine. Either one is correct, that is, either one constitutes a legitimate proof in formal logic.
In any case, maybe you can respond and let me know what you think of what I've said here. Is this idea of using the basic argument forms new to you? Does it make sense? I think to understand what we're talking about here it is important to understand how these basic argument forms are used AS WELL as the basic equivalence forms**
So I understand if you don't want to do anymore of the problems. I only asked you to do one. But if you want I think it would be helpful if you try one or two more so that you can show us that you have an understanding of the basic argument forms as well as get some practice with them. Number 9 looks like a good one. I think it uses some rules which we haven't looked at as much yet. Maybe also another one of your choosing if you're up for it.
And anyways, thanks a lot for looking at the exercise. It restored a lot of my faith in you. If you respond to what I've said here and we all get on board with the basic argument forms I think we'll be in a really good position to have a discussion about the questions you're more interested in.
________________________
edit: Also for the record I want to point out that I didn't see Demki's post until I tried to post mine. I want to point out that we wrote down the EXACT SAME corrections to your proof without having collaborated at all. This is EXACTLY why formal logic is so useful and this is what people mean when they say formal logic is not ambiguous. It is a language so you can write down all sorts of sentences but the rules are very constrained so it is clear when they have been broken.
First, Since Demki and I both agree on the rules we immediately could tell that your line 9 had broken the rules.
Second, Since there are only a few rules (as compared to say English) we both saw a pretty clear path to get from the last sentence you had to the final thing that we are trying to prove.
This is a great demonstration of the power of formal logic.
edit2: Actually I'll point out another thing. While I was responding to your post you actually edited your post to correct an error you made with your "not" symbols. I was going to point it out but you corrected it yourself. This is exactly the same example again of the usefulness of formal logic. It is unambiguous to tell when the rules are not followed. When people say form logic is unambiguous that is what they mean. They mean that anyone looking at a proof (who knows the rules and how to apply them) will be able to tell if a proof is valid or not because they will be able to tell if the rules have been followed or not.
edit3: Typos in my post
___________________________________
*A well formed formula is any syntactically correct string of propositional variables and propositional connectives. This means something like "A & B" or "L" can be a WFF but "A & & B ~ (" is not a WFF. The difference is the latter example is not syntactically correct.
**which it looks like you already have a pretty good grasp on.
Last edited by Twistar on Sun Jun 05, 2016 12:54 am UTC, edited 1 time in total.
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:Sure if you write him a half page post he'll write you one back. Unfortunately he will just skip over all of the salient points and continue on saying exactly whatever he was saying in his previous post.
Very true.
Yet for some odd reason, you continue to try... only to get frustrated again and again.
I've got this kind of response from him once, and believe me: He won't get an opportunity to do it a second time.
And I find it hard to believe that just because he did some problem on some worksheet, an actual fruitful conversation is going to miraculously emerge. But you know something? I'd love to be wrong about this. If you (or Cauchy or anybody else) will be able to go through to him, I'll be very happy.
Re: Misunderstanding basic math concepts, help please?
PsiCubed wrote:Twistar wrote:Sure if you write him a half page post he'll write you one back. Unfortunately he will just skip over all of the salient points and continue on saying exactly whatever he was saying in his previous post.
Very true.
Yet for some odd reason, you continue to try... only to get frustrated again and again.
I've got this kind of response from him once, and believe me: He won't get an opportunity to do it a second time.
And I find it hard to believe that just because he did some problem on some worksheet, an actual fruitful conversation is going to miraculously emerge. But you know something? I'd love to be wrong about this. If you (or Cauchy or anybody else) will be able to go through to him, I'll be very happy.
*Shrug* I keep trying. I definitely lost faith because at the beginning of the thread it seemed like Treatid was willing to learn but then that interest in learning seemed to wain. That coupled to his refusal to listen to multiple people telling him to look at the exercises was enough for me. But him looking at that exercise is in my eyes the biggest development in any of these threads that I linked to.
These walls of text are extremely complicated and there's tons of definition and nuance involved. But a 10 line formal logic proof.. that is something we can all come to understand and agree on. If we can use that a touchstone to come back to I think then we can really have an interesting discussion. We're so close to that point too so I'm excited. Maybe Treatid agrees that we're close. I think it would be the first time we've all been on the same page about something..
Re: Misunderstanding basic math concepts, help please?
Rather than focus on your deductions, we should start by focusing on your "set" X, which I'm not sure is wellformed and therefore anything you deduce about it may not be particularly useful.Treatid wrote:Iteration 3:
 X is the set of y & z.
 z is a set of every instance of p.
 p are specified to be consistent with the other premises.
 y is the set of all functions q, such that the domain for each q is some subset of z.
 There is at least one q whose domain is the set X.
With the removal of 'describe' and the changes to the premises as noted, I've rewritten all of the deductions.
 X is a member of z.
 All q are members of z.
 For each p there exists some q such that p is in the domain of q.
 notz is explicitly outside the domain of all functions q. No q has any possible notz as its domain.
 This is a closed loop. The only functions that can act on members of z are members of y. Each p only has a subset of z as its domain. notz and noty are explicitly not part of set X.
 For every q_{1} in y, there exists a q_{2} for which q1 is the domain.
 [redacted] *
Let me recap: z is a set of statements, which you define as "consistent with the other premises"  in other words, some set of statements for which everything else works (2,3). OK, let's accept that for now. You then define y as a set of functions whose domain is an element or subset of z (4). Sure, reasonable.
Then it gets weird  you create a set X which contains both statements and functions (y and z) (1), and you further specify that for some element q in y, the domain of that function is X (5). In other words, you've stated that there is a function whose domain is not only all the statements in z, but also all the functions in y, including itself (5,1).
I'm not sure what that function might be, or whether it can exist at all. The identity function maybe? Or else the empty set? And if your set y is empty, what does that say about z  that there are no functions that apply to any of its elements? If y only contains an identity function, z is similarly uninteresting.
_____
Now that we've covered that, let's move to your deductions and what they mean. I'll be correcting any language in your deductions that I feel is misleading or incorrect, just for clarity.
1) X is a *subset* of z. This follows from p5, p4. After the above confusion, you're now saying that z must contain not only statements but functions as well. Curiously, it also follows from this that X is z, since X is a subset of z and z is a subset of X (d1, p1).
2) All functions q are in z. This follows from d1 and p1.
3) For all statements p in z, there exists some function q in z for which the domain of q contains p. This is just a restatement of p4, but after d2 moved both the functions and the statements into the same set. This is where the problem comes up that I mentioned before  now that z also contains functions, what functions operate on those?
4) For all functions q in z, the domain of that function is completely within z. Sure, follows.
5) I don't know what a closed loop is, as you define it. Are you trying to say that the set X is closed under the all the functions in y? Because I think that does follow from the above, whatever that ends up meaning.
6) Here there be dragons. Now you're saying that for all functions q1 in y, there exists a function q2 whose domain is another function. What does that mean? Why is this interesting? More importantly, logically you can deduce that for all q1 in z, there exists a function q2 whose domain is q1; I don't see where you've proven that both of these functions have to be in y.
_____
And if X is not a fair generalization of axiomatic mathematics, then what's the point of all this?I've specified a set that is a member of itself. Very much Russell's Paradox territory.
A set being a member of itself is not by itself sufficient to create a paradox. Even were set X to lead to a paradox, that is only significant if set X is a fair generalisation of axiomatic mathematics.
Re: Misunderstanding basic math concepts, help please?
My deepest thanks to everyone in this thread and previous ones.
Proof:
The premises for this proof are the axioms/rules of propositional logic. That is, we assume that all the rules of propositional logic are true. The proof is intended to show that the axioms/rules of propositional logic are inconsistent.
1. A logical statement P is declared true.
2, 3. There exists an object a that implies P.
4, 5. There exists an object b that implies ~P.
1. P [Prem.]
2. a [Prem.]
3. a ⊃ P [Prem.]
4. b [Prem.]
5. b ⊃ ~P [Prem.]
Proof:
The premises for this proof are the axioms/rules of propositional logic. That is, we assume that all the rules of propositional logic are true. The proof is intended to show that the axioms/rules of propositional logic are inconsistent.
1. A logical statement P is declared true.
2, 3. There exists an object a that implies P.
4, 5. There exists an object b that implies ~P.
1. P [Prem.]
2. a [Prem.]
3. a ⊃ P [Prem.]
4. b [Prem.]
5. b ⊃ ~P [Prem.]
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:My deepest thanks to everyone in this thread and previous ones.
Proof:
The premises for this proof are the axioms/rules of propositional logic. That is, we assume that all the rules of propositional logic are true. The proof is intended to show that the axioms/rules of propositional logic are inconsistent.
1. A logical statement P is declared true.
2, 3. There exists an object a that implies P.
4, 5. There exists an object b that implies ~P.
1. P [Prem.]
2. a [Prem.]
3. a ⊃ P [Prem.]
4. b [Prem.]
5. b ⊃ ~P [Prem.]
Well, yes, if you have objects a and b that imply P and ~P respectively, then, if they're both true within a system, then that system is inconsistent. That's neither controversial, nor proof that propositional logic is inconsistent.
In particular, the second and fourth lines of your "proof" do not follow from your stated premises.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:My deepest thanks to everyone in this thread and previous ones.
Proof:
The premises for this proof are the axioms/rules of propositional logic. That is, we assume that all the rules of propositional logic are true. The proof is intended to show that the axioms/rules of propositional logic are inconsistent.
1. A logical statement P is declared true.
2, 3. There exists an object a that implies P.
4, 5. There exists an object b that implies ~P.
1. P [Prem.]
2. a [Prem.]
3. a ⊃ P [Prem.]
4. b [Prem.]
5. b ⊃ ~P [Prem.]
clearly the subsequent lines are:
6. ~P [4,5 M.P.]
7. P & ~P [1,6 Conj.]
8.CONT [7,1,6,4,5]
Here I have created the symbol CONT. The rule for CONT is as follows:
i. q & ~q
j. CONT
When you write down CONT you must write down the line number on which q & ~q appears. You also must write down all of the lines that were necessary to derived q & ~q. So in this case q & ~q appeared on line 7 so I wrote 7. Line seven depended on lines 1 and 6 so I wrote down 1 and 6. line 6 depended on 4 and 5 so I wrote down 4 and 5. 1, 4, and 5 were all premises so I stopped there. Note that in this proof lines 2 and 3 were not necessary to derive a contradiction.
Note that we could also have:
6. ~P [4,5 M.P.]
7. P [2,3 M.P.]
8. P&~P [6,7 Conj.]
9. CONT [8,6,7,4,5,2,3]
Note this proof is one line longer than the last and that it relied on lines 2 and 3 to derive a contradiction but did not use line 1.
All of this isn't super relevant to your main point, I'll get to that. What I'm pointing out is that premises 1, 2, and 3 are redundant with each other. That's not a problem I just wanted to make sure you were aware. The reason you can tell they're redundant is because it's possible to derive premise 1 from premises 2 and 3 as I do on line 7 of the 2nd proof.
__________________________________________________________________
IN ANY CASE, to address your main point (which is now quite clear now that you have written it out formally), you are correct that this set of premises leads to a contradiction. But notice that in all of the worksheet examples it was NOT possible to derive a contradiction. The point is you can only derive a contradiction if your premises allow it. For certain sets of premises (such as those in the exercises) it will not be possible to derive a contradiction. For certain sets of premises (such as those in your example) it is easily possible to derive a contradiction.
A theory is a set of premises + all wffs* that can be derived from those premises.
If it is possible to derive a contradiction (in other words if you are allowed to write down a line that says CONT based on all of the rules of formal logic that have been provided so far) from the premises stated in a theory then we say that the theory is inconsistent. If it is NOT possible to derive a contradiction in the theory (meaning that you can't write down CONT if you're following all the rules on the reference sheet) then we say that the theory is consistent.
The theories on the spreadsheet are all consistent. Try as you might, you won't be able to derive a contradiction. The theory you have put forth is inconsistent.
_____________________________________________________________________
So anyways, I think what you stated in your post all agrees with the definitions posted in this thread. You correctly claim that the premises "P", "a", "b", "a ⊃ P", and "b ⊃ P" are a set of inconsistent premises because it is possible to derive a contradiction from them. But this is just one theory. If I write down a theory with different premises than the ones you have written down then that theory may or may not be inconsistent. All of the ones on the spreadsheet are consistent.
***********************
I guess here's my point. You haven't shown the rules of formal logic are inconsistent. In other words you haven't shown P is inconsistent. You have shown that "P", "a", "b", "a ⊃ P", and "b ⊃ P" are inconsistent when you take them all together. But if you just take P, you haven't shown me a proof that derives a contradiction. You can ONLY derive a contradiction if you add on other premises.
***********************
Just because you can write down an inconsistent theory which INCLUDES P as a premise doesn't mean P by itself is inconsistent. It just means that theory is inconsistent, but it can still be possible to write down a DIFFERENT theory which includes P which is consistent.
Does this make sense?
*see definition in previous post
Edit to response to rmsgrey:
rmsgrey wrote:Treatid wrote:My deepest thanks to everyone in this thread and previous ones.
Proof:
The premises for this proof are the axioms/rules of propositional logic. That is, we assume that all the rules of propositional logic are true. The proof is intended to show that the axioms/rules of propositional logic are inconsistent.
1. A logical statement P is declared true.
2, 3. There exists an object a that implies P.
4, 5. There exists an object b that implies ~P.
1. P [Prem.]
2. a [Prem.]
3. a ⊃ P [Prem.]
4. b [Prem.]
5. b ⊃ ~P [Prem.]
Well, yes, if you have objects a and b that imply P and ~P respectively, then, if they're both true within a system, then that system is inconsistent. That's neither controversial, nor proof that propositional logic is inconsistent.
In particular, the second and fourth lines of your "proof" do not follow from your stated premises.
I think Treatid was trying to say he is taking lines 2, 3, 4, and 5 as premises. At least that is what his lines of logic indicate. My response is summed up as: yes, if you take these premises you can derive a contradiction. If you don't take these premises but take different premises you won't necessarily be able to derive a contradiction anymore.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:My deepest thanks to everyone in this thread and previous ones.
Proof:
The premises for this proof are the axioms/rules of propositional logic. That is, we assume that all the rules of propositional logic are true. The proof is intended to show that the axioms/rules of propositional logic are inconsistent.
1. A logical statement P is declared true.
2, 3. There exists an object a that implies P.
4, 5. There exists an object b that implies ~P.
1. P [Prem.]
2. a [Prem.]
3. a ⊃ P [Prem.]
4. b [Prem.]
5. b ⊃ ~P [Prem.]
One notices that your "proof" doesn't actually contain any deductions. Thus, I don't think you've shown what you think you've shown. Let me restate (without two of your premises, which are unnecessary):
1 Premise: P
2 Premise: b implies ~P
3 Premise: b
__________
4 ~P (2,3, modus ponens)
Now you've shown both P and ~P are true in this system. If you wanted, you could replace premise 1 with your other two premises (a and a implies P) since they can be used to deduce P via modus ponens also.
But what does that mean? Well, it means that your premises are inconsistent with each other. In other words, those three premises can't all simultaneously be true. That should be obvious from their English statements  if P is true (in some consistent system), there can't also be a true statement (in that system) that implies P is false.
You haven't proven axiomatic math, logic, or anything else untrue. You've proven that at least one of your premises is inconsistent with the others and that they can't be simultaneously true, which should be obvious since they were created to be inconsistent.
Edit: twistar beat me to it and did the logic even better, but same idea.
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:Edit to response to rmsgrey:rmsgrey wrote:Treatid wrote:My deepest thanks to everyone in this thread and previous ones.
Proof:
The premises for this proof are the axioms/rules of propositional logic. That is, we assume that all the rules of propositional logic are true. The proof is intended to show that the axioms/rules of propositional logic are inconsistent.
1. A logical statement P is declared true.
2, 3. There exists an object a that implies P.
4, 5. There exists an object b that implies ~P.
1. P [Prem.]
2. a [Prem.]
3. a ⊃ P [Prem.]
4. b [Prem.]
5. b ⊃ ~P [Prem.]
Well, yes, if you have objects a and b that imply P and ~P respectively, then, if they're both true within a system, then that system is inconsistent. That's neither controversial, nor proof that propositional logic is inconsistent.
In particular, the second and fourth lines of your "proof" do not follow from your stated premises.
I think Treatid was trying to say he is taking lines 2, 3, 4, and 5 as premises. At least that is what his lines of logic indicate. My response is summed up as: yes, if you take these premises you can derive a contradiction. If you don't take these premises but take different premises you won't necessarily be able to derive a contradiction anymore.
My point is that he's treating existence and truth as interchangeable concepts, which is only valid if you restrict your context to one where only true sentences exist.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:My deepest thanks to everyone in this thread and previous ones.
Proof:
The premises for this proof are the axioms/rules of propositional logic. That is, we assume that all the rules of propositional logic are true. The proof is intended to show that the axioms/rules of propositional logic are inconsistent.
1. A logical statement P is declared true.
2, 3. There exists an object a that implies P.
4, 5. There exists an object b that implies ~P.
1. P [Prem.]
2. a [Prem.]
3. a ⊃ P [Prem.]
4. b [Prem.]
5. b ⊃ ~P [Prem.]
What do you mean by "object"? My "generous interpretation" reading of your post is that you mean a statement. Then, "There exists an object b that implies ~P." doesn't give you both premise 4 and premise 5 below. You get premise 5, but you don't know that b is true, so you don't have premise 4. Then, you can't conclude ~P (which seems to be your goal, even though you don't provide that part of your proof here). As rmsgrey said, existence and truth are two separate concepts. The statement (a ⊃ ~a) & (~a ⊃ a) exists, in that it's a wellformed formula, but it is not true for any statement a.
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.
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Re: Misunderstanding basic math concepts, help please?
Treatid wrote:My deepest thanks to everyone in this thread and previous ones.
Proof:
The premises for this proof are the axioms/rules of propositional logic. That is, we assume that all the rules of propositional logic are true. The proof is intended to show that the axioms/rules of propositional logic are inconsistent.
1. A logical statement P is declared true.
2, 3. There exists an object a that implies P.
4, 5. There exists an object b that implies ~P.
1. P [Prem.]
2. a [Prem.]
3. a ⊃ P [Prem.]
4. b [Prem.]
5. b ⊃ ~P [Prem.]
Sigh. We've gone *well* past ignorance now. You're either maliciously trolling, or an absolute, unredeemable crackpot at this point. It's quite clear that no attempt to teach you actual logic will make any difference whatsoever, if you present this sort of drivel as meaningful and, even more insultingly, if you imply that something this trivial actually shows something is wrong with the foundation of all mathematics.
You're on my foe list now. Others can continue typing in circles with you if it brings them some perverse pleasure, but I'm done.
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:The premises for this proof are the axioms/rules of propositional logic. That is, we assume that all the rules of propositional logic are true. The proof is intended to show that the axioms/rules of propositional logic are inconsistent.
1. A logical statement P is declared true.
2, 3. There exists an object a that implies P.
4, 5. There exists an object b that implies ~P.
I'll play.
A system which has as axioms 'a', 'b', 'a implies p' and 'b implies ~p' would be inconsistent  but so what? What does that tell us about any system which doesn't have all those axioms?
You're sort of doing the equivalent of: if you wrote down all possible computer statements into a program, you'd end up with something that didn't compile (or would crash, hang, or something worse). And somehow from that concluding that coding is a worthless science...
No. You pick a useful subset of all possible computer statements, place them in the right order and... now you can fly to the moon, or calculate pi to a million decimal places, or explore a virtual world.
Maths is the same: You pick from an infinite number of potential axioms  collectively which would be contradictory  but an infinite number of subsets of which are not  and see what you can deduce.
Depending on the subset you choose, you might get Euclidean geometry, or nonEuclidean geometry, or whatever  each of which is internally consistent  despite having opposing axioms.
It seems to me you think of 'truth' as absolute  whereas the same statement may be true in one system and false in another without contradiction.
Again, it's like you're suggesting that if I set the variable i to the value 10 in one program and to 20 in another, that somehow the software will break or something. Obviously you know it won't break. Nor will maths if I take the axiom of choice to be true for my proof, but someone else takes it to be false for theirs  or, to return to your example, if I take 'a implies p' and someone else takes 'b implies ~p'...
Re: Misunderstanding basic math concepts, help please?
Treatid, imagine for a moment instead of discussing math, we are discussing hockey.
The premises are the rules of the game: So many players on the ice, the size of the rink, how you are allowed to play the puck, etc. If these rules are consistent, then you can play the game in some sensible way. If the rules are inconsistentsay, there's a rule that says you can't touch the puck with your hands, and another rule that says, under the same conditions, you can touch the puck with your hands, then you arrive at a contradiction and nobody knows how to play anymore. This is no fun, so generally we try to make our games with consistent rules.
But different people can play the game with different rules. For example, Olympic hockey plays with a different sized rink from NHL hockey, has much harsher penalties for infractions like fighting, and a whole host of other technical changes. The rules of Olympic hockey are consistent, and the rules of NHL hockey are consistent. But the rules of both NHL and Olympic hockey, taken together, are inconsistent. You can't play both games at once.
Now suppose it came to be that, somehow, two rules in NHL hockey were inconsistent and therefore the game was unplayable. This would not imply that hockey as a whole is unplayableit may be that Olympic hockey would still be consistent. Nor would it imply that the rules of all games are inconsistent. The inconsistency in one set of rules has no bearing on a set of rules from a different game. And you could conceivably amend the inconsistent rules in the NHL to create a playable game... one that wouldn't, technically, be the same game as NHL hockey, but might have a lot of similar elements.
I think what you're trying to prove in your line of reasoning is that if one particular set of rules in mathematics is inconsistent, then all sets of possible rules in mathematics are inconsistent. I hope from my analogy above, it should be clear why this doesn't make sense... I mean, if someone were able to derive a clear contradiction in, say, ZFC, that would probably be quite an interesting and important result and would require a lot of headscratching among mathematicians to figure out how to fix the theory to remove the contradiction. Mathematics is a game, and if the rules lead to an inconsistency, then we would have to try a new set of rules.
The premises are the rules of the game: So many players on the ice, the size of the rink, how you are allowed to play the puck, etc. If these rules are consistent, then you can play the game in some sensible way. If the rules are inconsistentsay, there's a rule that says you can't touch the puck with your hands, and another rule that says, under the same conditions, you can touch the puck with your hands, then you arrive at a contradiction and nobody knows how to play anymore. This is no fun, so generally we try to make our games with consistent rules.
But different people can play the game with different rules. For example, Olympic hockey plays with a different sized rink from NHL hockey, has much harsher penalties for infractions like fighting, and a whole host of other technical changes. The rules of Olympic hockey are consistent, and the rules of NHL hockey are consistent. But the rules of both NHL and Olympic hockey, taken together, are inconsistent. You can't play both games at once.
Now suppose it came to be that, somehow, two rules in NHL hockey were inconsistent and therefore the game was unplayable. This would not imply that hockey as a whole is unplayableit may be that Olympic hockey would still be consistent. Nor would it imply that the rules of all games are inconsistent. The inconsistency in one set of rules has no bearing on a set of rules from a different game. And you could conceivably amend the inconsistent rules in the NHL to create a playable game... one that wouldn't, technically, be the same game as NHL hockey, but might have a lot of similar elements.
I think what you're trying to prove in your line of reasoning is that if one particular set of rules in mathematics is inconsistent, then all sets of possible rules in mathematics are inconsistent. I hope from my analogy above, it should be clear why this doesn't make sense... I mean, if someone were able to derive a clear contradiction in, say, ZFC, that would probably be quite an interesting and important result and would require a lot of headscratching among mathematicians to figure out how to fix the theory to remove the contradiction. Mathematics is a game, and if the rules lead to an inconsistency, then we would have to try a new set of rules.
Last edited by LaserGuy on Tue Jun 21, 2016 3:44 am UTC, edited 1 time in total.
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Re: Misunderstanding basic math concepts, help please?
They're not even that far  given that they think a set of inconsistent premises invalidates all of logic, a better analogy would be someone believing that if one hockey player cheats a single time, it means the rules of hockey are inconsistent everywhere.
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Re: Misunderstanding basic math concepts, help please?
Sorry for taking so long to respond  trying to make whatever point(s) I might have as clear and unambiguous as I can.
This post is mainly me trying to get to the same page as everyone else with regard assumptions/premises. The arguments I put forward are sincere  but not proof.
Falsifiability.
I've been assuming that Propositional Logic is, itself, a Logical System that follows the laws of Propositional Logic.
Whereas, Propositional Logic is a language and not falsifiable by logical mechanisms. As such, I'm attempting to disprove something that is defined to be unfalsifiable.
Silly me.
However, it looks to me like excluding languages (or, at least, Propositional Logic) from logical reasoning creates at least as many problems as it solves.
i. If Languages are a Logical System and the axioms of a language allow us to write (x & !x) then the axioms of that language are inconsistent with each other. But that would mean that nearly all languages are inconsistent and thus allow us to prove anything within those languages.
ii. So... Languages (such as Propositional Logic) are not themselves Logical Systems  they merely provide the framework within which we can describe actual Logical Systems.
iii. Seems reasonable.
iv. But, If we cannot disprove a language, nor can we prove anything using that language... We cannot prove that a word has a particular meaning (we cannot disprove a word has a particular meaning). A theorem that cannot be falsified is one that doesn't say anything.
v. Obviously we do think we can use language to say things. Since we are using a language (Propositional Logic) to describe individual Logic Systems, then we can prove things using language. But if we can prove individual Logical Systems, then some elements of language can be proven and disproven. But if Logical Reasoning applies to language (x & !x) disproves the axioms of whatever language allowed us to write (x & !x).
vi. ZFC set theory is sometimes used as a language to describe other axiomatic systems. If an inconsistency was found in ZFC Set Theory then we would consider the whole of ZFC Set Theory to be inconsistent. The same inconsistency within natural languages and semiformal languages (Propositional Logic) is discarded.
vii. No matter how intuitive and/or obvious any given natural language statement appears  if we specify that logical arguments (e.g. The Principle of Explosion) do not apply to natural languages  then it is impossible to prove (or disprove) a set of axioms that are stated in that natural language.
viii. So how do we start with a language in which we cannot prove or disprove anything; and prove or disprove anything? When does a statement stop belonging to a language (which cannot prove or disprove anything with respect to that statement) and start belonging to a logical system (which is expressed solely in terms of the given language)?
ix. Indeed, how do we start with something that is defined to function differently to Logical Systems (i.e. is immune to Logical Arguments) and build Logical Systems on top of that?
x. If we define language as being distinct from logical systems then we have specified that logical systems cannot describe all systems (they cannot describe languages). We have a real world mechanism (languages) that is specified not to behave in the same way as logical systems. Given that language is part of our universe, and Logical Systems cannot describe language  then Logical Systems cannot describe our universe.
...
I think that if we have a thing X that cannot be proven or disproven then, from a strictly logical perspective, X says nothing. Anything that depends on X to any degree also says nothing.
If Propositional Logic is itself a single Logical System  then every inconsistent system we create within Propositional Logic proves that the axioms of Propositional Logic are inconsistent with each other. This possibility is rejected  because it would mean throwing out all of Propositional Logic. So rather than accept the logical result of our reasoning, we assert that Propositional Logic is immune to our logical reasoning.
If Propositional Logic is immune to logical reasoning then we cannot prove or disprove any aspect of Propositional Logic.
...
I assume that most of you think that we can use a language that is defined to be unfalsifiable to build individual Logical Systems.
For example, we assert that a set of axioms is true. If those axioms lead to a contradiction then we know that the axioms that lead to that contradiction are inconsistent with each other.
So we do have a method of falsification. We don't have to start with absolute knowledge  we can pretend we have absolute knowledge and see where that gets us.
This appears to work pretty well. We can't prove that (e.g.) ZFC Set Theory is consistent  but we have a mechanism to potentially falsify axiomatic systems. The longer we go without falsifying a given system the more confidence we can have that the system is, indeed, consistent.
However, this is mathematics. There is a huge gulf between being really, really sure that a theorem is true and proving that a theorem is true.
...
I'm going to pause here for some feedback. Nobody has directly corrected my assumption that Propositional Logic is itself a logic system.
The idea that Propositional Logic is a language (or, at least, not itself a Logical System) seems to me to fit the majority of comments and to be consistent with other elements of mathematics that frequently regard Axiomatic Systems as being distinct from the language that describes Axiomatic Systems (ZFC used as a language for other axiomatic systems notwithstanding).
If I'm off base on this point then carrying on in this vein would be pointless.
...
@Twistar:
Thank you, again, for your detailed feedback.
There is one particular point you made that I'd like to strongly agree with and emphasise...
Showing that a system is inconsistent does not prove any single axiom to be wrong. It only shows that particular set of axioms is inconsistent with each other.
When we explicitly state a set of axioms there are an additional set of implicit axioms/assumptions embodied in the language that the explicit axioms are expressed in. That is, The rules of deduction that allow us to manipulate a given set of axioms are, themselves, assumptions/axioms.
When we find an inconsistent system, that inconsistency might be due to the explicit axioms or the rules of deduction (or a combination or the two).
Since a given set of Rules of Deduction typically apply to a number of axiomatic systems some of which appear to be consistent; it is generally assumed that the Rules of Deduction are consistent and any inconsistency is solely between the explicit axioms.
However, to the extent that the Rules of Deduction are part of a language that has been asserted unfalsifiable, it is impossible to prove that the Rules of Deduction themselves are consistent with each other.
@Cauchy & rmsgrey:
Your combined point regarding "existence" was extremely helpful. Working out how to clarify this one point really helped me to understand that my assumption of Propositional Logic itself being a logical system is not an assumption shared by anyone else. And that I needed to start to reframe my argument around the assumption that Propositional Logic is not a Logical System.
Worksheet Question 9:
As promised, albeit a post late.
1. (A & B) ⊃ C [prem.]
2. A & D [prem.]
3. B [prem.]
∴ B & C
4. A [2, Simp.]
5. A & B [3, 4, Conj.]
6. C [1, 5, M.P.]
7 ∴ B & C [3, 6 Conj.]
This post is mainly me trying to get to the same page as everyone else with regard assumptions/premises. The arguments I put forward are sincere  but not proof.
Falsifiability.
I've been assuming that Propositional Logic is, itself, a Logical System that follows the laws of Propositional Logic.
Whereas, Propositional Logic is a language and not falsifiable by logical mechanisms. As such, I'm attempting to disprove something that is defined to be unfalsifiable.
Silly me.
However, it looks to me like excluding languages (or, at least, Propositional Logic) from logical reasoning creates at least as many problems as it solves.
i. If Languages are a Logical System and the axioms of a language allow us to write (x & !x) then the axioms of that language are inconsistent with each other. But that would mean that nearly all languages are inconsistent and thus allow us to prove anything within those languages.
ii. So... Languages (such as Propositional Logic) are not themselves Logical Systems  they merely provide the framework within which we can describe actual Logical Systems.
iii. Seems reasonable.
iv. But, If we cannot disprove a language, nor can we prove anything using that language... We cannot prove that a word has a particular meaning (we cannot disprove a word has a particular meaning). A theorem that cannot be falsified is one that doesn't say anything.
v. Obviously we do think we can use language to say things. Since we are using a language (Propositional Logic) to describe individual Logic Systems, then we can prove things using language. But if we can prove individual Logical Systems, then some elements of language can be proven and disproven. But if Logical Reasoning applies to language (x & !x) disproves the axioms of whatever language allowed us to write (x & !x).
vi. ZFC set theory is sometimes used as a language to describe other axiomatic systems. If an inconsistency was found in ZFC Set Theory then we would consider the whole of ZFC Set Theory to be inconsistent. The same inconsistency within natural languages and semiformal languages (Propositional Logic) is discarded.
vii. No matter how intuitive and/or obvious any given natural language statement appears  if we specify that logical arguments (e.g. The Principle of Explosion) do not apply to natural languages  then it is impossible to prove (or disprove) a set of axioms that are stated in that natural language.
viii. So how do we start with a language in which we cannot prove or disprove anything; and prove or disprove anything? When does a statement stop belonging to a language (which cannot prove or disprove anything with respect to that statement) and start belonging to a logical system (which is expressed solely in terms of the given language)?
ix. Indeed, how do we start with something that is defined to function differently to Logical Systems (i.e. is immune to Logical Arguments) and build Logical Systems on top of that?
x. If we define language as being distinct from logical systems then we have specified that logical systems cannot describe all systems (they cannot describe languages). We have a real world mechanism (languages) that is specified not to behave in the same way as logical systems. Given that language is part of our universe, and Logical Systems cannot describe language  then Logical Systems cannot describe our universe.
...
I think that if we have a thing X that cannot be proven or disproven then, from a strictly logical perspective, X says nothing. Anything that depends on X to any degree also says nothing.
If Propositional Logic is itself a single Logical System  then every inconsistent system we create within Propositional Logic proves that the axioms of Propositional Logic are inconsistent with each other. This possibility is rejected  because it would mean throwing out all of Propositional Logic. So rather than accept the logical result of our reasoning, we assert that Propositional Logic is immune to our logical reasoning.
If Propositional Logic is immune to logical reasoning then we cannot prove or disprove any aspect of Propositional Logic.
...
I assume that most of you think that we can use a language that is defined to be unfalsifiable to build individual Logical Systems.
For example, we assert that a set of axioms is true. If those axioms lead to a contradiction then we know that the axioms that lead to that contradiction are inconsistent with each other.
So we do have a method of falsification. We don't have to start with absolute knowledge  we can pretend we have absolute knowledge and see where that gets us.
This appears to work pretty well. We can't prove that (e.g.) ZFC Set Theory is consistent  but we have a mechanism to potentially falsify axiomatic systems. The longer we go without falsifying a given system the more confidence we can have that the system is, indeed, consistent.
However, this is mathematics. There is a huge gulf between being really, really sure that a theorem is true and proving that a theorem is true.
...
I'm going to pause here for some feedback. Nobody has directly corrected my assumption that Propositional Logic is itself a logic system.
The idea that Propositional Logic is a language (or, at least, not itself a Logical System) seems to me to fit the majority of comments and to be consistent with other elements of mathematics that frequently regard Axiomatic Systems as being distinct from the language that describes Axiomatic Systems (ZFC used as a language for other axiomatic systems notwithstanding).
If I'm off base on this point then carrying on in this vein would be pointless.
...
@Twistar:
Thank you, again, for your detailed feedback.
There is one particular point you made that I'd like to strongly agree with and emphasise...
Showing that a system is inconsistent does not prove any single axiom to be wrong. It only shows that particular set of axioms is inconsistent with each other.
When we explicitly state a set of axioms there are an additional set of implicit axioms/assumptions embodied in the language that the explicit axioms are expressed in. That is, The rules of deduction that allow us to manipulate a given set of axioms are, themselves, assumptions/axioms.
When we find an inconsistent system, that inconsistency might be due to the explicit axioms or the rules of deduction (or a combination or the two).
Since a given set of Rules of Deduction typically apply to a number of axiomatic systems some of which appear to be consistent; it is generally assumed that the Rules of Deduction are consistent and any inconsistency is solely between the explicit axioms.
However, to the extent that the Rules of Deduction are part of a language that has been asserted unfalsifiable, it is impossible to prove that the Rules of Deduction themselves are consistent with each other.
@Cauchy & rmsgrey:
Your combined point regarding "existence" was extremely helpful. Working out how to clarify this one point really helped me to understand that my assumption of Propositional Logic itself being a logical system is not an assumption shared by anyone else. And that I needed to start to reframe my argument around the assumption that Propositional Logic is not a Logical System.
Worksheet Question 9:
As promised, albeit a post late.
1. (A & B) ⊃ C [prem.]
2. A & D [prem.]
3. B [prem.]
∴ B & C
4. A [2, Simp.]
5. A & B [3, 4, Conj.]
6. C [1, 5, M.P.]
7 ∴ B & C [3, 6 Conj.]
Re: Misunderstanding basic math concepts, help please?
When you talk about things like "disproving a language", you're using words in nonstandard ways that I don't understand. It's like talking about proving a horse or proving a sandwich  there's a category mismatch. A language is neither true nor false. Statements within that language may be true, false, undecidable, or nonsense, but the language as a whole is none of those things.
It's too late at night for me to criticise the rest of your post properly.
It's too late at night for me to criticise the rest of your post properly.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Sorry for taking so long to respond  trying to make whatever point(s) I might have as clear and unambiguous as I can.
This post is mainly me trying to get to the same page as everyone else with regard assumptions/premises. The arguments I put forward are sincere  but not proof.
Falsifiability.
I've been assuming that Propositional Logic is, itself, a Logical System that follows the laws of Propositional Logic.
Whereas, Propositional Logic is a language and not falsifiable by logical mechanisms. As such, I'm attempting to disprove something that is defined to be unfalsifiable.
Silly me.
However, it looks to me like excluding languages (or, at least, Propositional Logic) from logical reasoning creates at least as many problems as it solves.
i. If Languages are a Logical System and the axioms of a language allow us to write (x & !x) then the axioms of that language are inconsistent with each other. But that would mean that nearly all languages are inconsistent and thus allow us to prove anything within those languages.
Correct. If languages followed the rules of logic as you suggest, contradictions would be everywhere. However, this doesn't mean languages are without structure or rules, but that those rules are different from those we apply to deductive reasoning and logic.
ii. So... Languages (such as Propositional Logic) are not themselves Logical Systems  they merely provide the framework within which we can describe actual Logical Systems.
iii. Seems reasonable.
iv. But, If we cannot disprove a language, nor can we prove anything using that language... We cannot prove that a word has a particular meaning (we cannot disprove a word has a particular meaning). A theorem that cannot be falsified is one that doesn't say anything.
And here is where you go off the rails. You get that language can not be "disproven", but what makes you think we can't prove anything using a language? We can't prove anything using language alone, because language is not a Logical System. However, after adding the rules of deduction to our system, we actually can prove statements. You're making an assertion without basis.
Additionally, propositional logic is more than just a language, as has been discussed multiple times. It includes a language, but also rules of grammar and of inference.
v. Obviously we do think we can use language to say things. Since we are using a language (Propositional Logic) to describe individual Logic Systems, then we can prove things using language. But if we can prove individual Logical Systems, then some elements of language can be proven and disproven. But if Logical Reasoning applies to language (x & !x) disproves the axioms of whatever language allowed us to write (x & !x).
I've used this analogy before. My computer works. (Proof: you're reading what I wrote) I can type (X & !X). Therefore, it is not the case that the ability to write that statement causes everything to grind to a halt.
Your argument is not logically structured or sound. See if you can understand why these sentences do not follow from one another:
Language is useful (premise?)
Propositional logic is a language. (Premise maybe, which is false)
therefore, propositional logic is useful (true despite the previous statement)
If things can be proven, some sentences can be proven true or false. (Independent of the above, maybe another premise? But really a definition of proof)
If language is a logical system (not yet proven), being able to write contradictions invalidates that language. (False for Many reasons).
vi. ZFC set theory is sometimes used as a language to describe other axiomatic systems. If an inconsistency was found in ZFC Set Theory then we would consider the whole of ZFC Set Theory to be inconsistent. The same inconsistency within natural languages and semiformal languages (Propositional Logic) is discarded.
vii. No matter how intuitive and/or obvious any given natural language statement appears  if we specify that logical arguments (e.g. The Principle of Explosion) do not apply to natural languages  then it is impossible to prove (or disprove) a set of axioms that are stated in that natural language.
Just no. ZFC is a set of axioms. It is not a system of propositional logic, nor is it a language. You are correct that if one could demonstrate an inconsistency in ZFC, we could decide which axiom to remove, and thus which ones to keep, in order to recreate a consistent system.
Your claim that it is impossible to prove or disprove a statement because the language itself is not subject to the rules of inference in our system, well, demonstrates that you still don't understand the difference between language, rules of inference, and axioms, all of which come together to form a logical system. What follows is meaningless as a result, so I'll skip ahead a bit before commenting further.
There is an important distinction among provability, consistency, and truth. Statements can be true but not provable, sets of statements can be consistent but untrue, and some statements can be proven independent of a set of axioms, meaning they can't be proven true nor false in that axiomatic system.viii. So how do we start with a language in which we cannot prove or disprove anything; and prove or disprove anything? When does a statement stop belonging to a language (which cannot prove or disprove anything with respect to that statement) and start belonging to a logical system (which is expressed solely in terms of the given language)?
ix. Indeed, how do we start with something that is defined to function differently to Logical Systems (i.e. is immune to Logical Arguments) and build Logical Systems on top of that?
x. If we define language as being distinct from logical systems then we have specified that logical systems cannot describe all systems (they cannot describe languages). We have a real world mechanism (languages) that is specified not to behave in the same way as logical systems. Given that language is part of our universe, and Logical Systems cannot describe language  then Logical Systems cannot describe our universe.
...
I think that if we have a thing X that cannot be proven or disproven then, from a strictly logical perspective, X says nothing. Anything that depends on X to any degree also says nothing.
See above, re: language vs axioms vs rules of inference.If Propositional Logic is itself a single Logical System  then every inconsistent system we create within Propositional Logic proves that the axioms of Propositional Logic are inconsistent with each other. This possibility is rejected  because it would mean throwing out all of Propositional Logic. So rather than accept the logical result of our reasoning, we assert that Propositional Logic is immune to our logical reasoning.
If Propositional Logic is immune to logical reasoning then we cannot prove or disprove any aspect of Propositional Logic.
In the end, my analysis of the situation reveals two possible conclusions. One, you still don't grasp the difference between languages, axioms, rules of inference, and grammar, or the systems that include all 4, and that's tripping you up when trying to discuss one or more of them. Two, you've assumed that math is fucked from the start and are trying to prove yourself right, rather than starting from basic principles and seeing what that creates, so you're unwilling to consider that your initial assumption may be wrong. (It could be both)

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Re: Misunderstanding basic math concepts, help please?
Treatid wrote:i. If Languages are a Logical System and the axioms of a language allow us to write (x & !x) then the axioms of that language are inconsistent with each other. But that would mean that nearly all languages are inconsistent and thus allow us to prove anything within those languages.
Here it's important to remember that a logical Language has multiple sets of rules with different meanings. Grammar and syntax determine which formulas can be considered meaningful and which are meaningless. A meaningful sentence may be true, or it may be false, but the point is that it says something meaningful. "2+2=5" is false, but meaningful. A language that "allows us to write" something like (x & !x), in this sense, isn't doing anything controversial. We can manipulate (x & !x) as a meaningful sentence, albeit a false one, just like we can talk about "2+2=5" as a meaningful but false statement.
The other set of rules for a logical Language are the Rules of Inference. These allow us to prove certain sentences true, given some premises to start from. Some sentences, called tautologies, can be proven without using any premises, and are direct consequences of the Rules of Inference themselves. If a Language "allows us to write (x & !x)" in this sense, that is, if we can prove it without premises using the Rules of Inference, then there's a problem, because the Language has at that point proven a contradiction.
Nearly all Languages allow us to write (x & !x) in the first sense of being able to handle it as meaningful. Few if any useful Languages allow us to prove (x & !x) as a tautology of the language.
Notice I've been talking about proving (x & !x) without assuming any premises, and using only the properties of the Language itself. That's because once you assume some premises, you're operating within a Theory consisting of the Language together with the assumed premises. If you prove (x & !x) within some Theory, then that Theory is inconsistent and should probably be thrown out. But you can throw out the Theory by just throwing out one or more of the premises, and keep the Language. The Language does not by itself prove the contradiction. The Language proves something like "Theory 'T' implies a contradiction", which is a true and unproblematic statement.
Treatid wrote:iv. But, If we cannot disprove a language, nor can we prove anything using that language...
Where did you get "nor can we prove anything using that language"?
Treatid wrote:I think that if we have a thing X that cannot be proven or disproven then, from a strictly logical perspective, X says nothing. Anything that depends on X to any degree also says nothing.
This is a flawed way to look at things. This seems related to the same issue I started with: truth and meaning are separate things. "2+2=4" is meaningful and true. "2+2=5" is meaningful and false. "+2*=@" is meaningless and therefore is neither true nor false. Goodstein's Theorem is a meaningful and true statement about natural numbers, but which the axioms of Peano arithmetic are insufficient to prove.
You've got the relationship backwards. The inability of PA prove or disprove Goodstein's Theorem says something about the deficiency of PA, not about any deficiency in the meaningfulness of Goodstein's Theorem. It shows us that PA is not fully sufficient for describing the natural numbers.
Treatid wrote:If Propositional Logic is itself a single Logical System  then every inconsistent system we create within Propositional Logic proves that the axioms of Propositional Logic are inconsistent with each other.
If you are precise with what you mean by the terms "Logical System", 'create', and 'axiom', then you will either end up with something that no longer matches what mathematicians actually do, or you will end up with something that has no problems.
The only sense in which mathematicians "create a logical system within Propositional Logic" is to state some axioms to be assumed, and then Logic together with those axioms constitute a Theory that the mathematician has created. If that Theory is inconsistent, it must be thrown out. But again, this can be done by throwing out one or more of the axioms, and the Language can be kept. Logic by itself only proves something like "the theory is inconsistent". Logic doesn't actually prove the contradiction that the theory did, Logic by itself only proves that the Theory proves a contradiction.
Most of your post seems to be related to this one misunderstanding. So I want to draw attention to the relevant point here:
There is a difference between saying a sentence is meaningful, and saying a sentence is true.
Most or all of your misunderstandings are resolved by carefully distinguishing between which of those is being said at any given time.
Treatid wrote:1. (A & B) ⊃ C [prem.]
2. A & D [prem.]
3. B [prem.]
∴ B & C
4. A [2, Simp.]
5. A & B [3, 4, Conj.]
6. C [1, 5, M.P.]
7 ∴ B & C [3, 6 Conj.]
Looks good to me. 4/4
Re: Misunderstanding basic math concepts, help please?
Cool. I agree with arbiteroftruth on your response to the worksheet question 4/4. Really glad to see we're on the same page regarding those exercises now. That is miles beyond where we were at the start of this thread.
Now, as others pointed out, you are using some terminology in a nonstandard way and it is leading to confusion. To alleviate this could you please define the following terms in your own words? All in the context of formal language. If we can't agree on the definitions of these words it isn't worth having any further discussion. How about you give your definitions and then in the next post I'll give my definitions.One more thing. I think it would help if in giving your definitions you gave specific examples from your response to worksheet question 9 since we all agree on the answer to that worksheet problem.
1) Well Formed Formula
2) Syntax
3) Rules of Deduction
4) Formal Language
5) Axiom
6) Theory
Now, as others pointed out, you are using some terminology in a nonstandard way and it is leading to confusion. To alleviate this could you please define the following terms in your own words? All in the context of formal language. If we can't agree on the definitions of these words it isn't worth having any further discussion. How about you give your definitions and then in the next post I'll give my definitions.One more thing. I think it would help if in giving your definitions you gave specific examples from your response to worksheet question 9 since we all agree on the answer to that worksheet problem.
1) Well Formed Formula
2) Syntax
3) Rules of Deduction
4) Formal Language
5) Axiom
6) Theory
Re: Misunderstanding basic math concepts, help please?
Definition of terms
Twistar's six definitions: They are all sets of symbols with a set of rules that apply to those symbols (or the result of having applied a set of rules to a set of symbols).
Set of symbols: Anything that isn't a set of rules. Also known as: any possible domain (and/or range) of a function.
Set of rules: Anything that can be applied to a set of symbols. Also known as: any possible function that takes a domain (and outputs a range).
Symbols and rules are here defined such that there is no other alternative. The premise here is that the only things that we can possibly have are rules or symbols. Any other words we might use are assumed to be a specific instance of symbols, rules or a combination of the two.
I'm going to refer to axiomatic mathematics. By this I mean any formal logical system that starts with premises/axioms/assumptions and applies rules to produce deductions.
I consider premises, axioms and assumptions to be synonyms for "known starting point". A known starting point will consist of a set of symbols and a set of rules that apply to those symbols.
I consider Propositional Logic, deductive logic and Axiomatic Mathematics to all be examples of axiomatic mathematics. Any argument I make referring to one of these should be taken as an argument with respect to all of these.
If at any point it is unclear what I mean by "object" or "thing" or "doohickey", I am deliberately trying to reference the largest possible set of "things" that can possibly be considered relevant in that context. Thus "symbol" is anything that can conceivably be thought of as a representation (writing on paper, sound waves, sequences of photons, ...). I'll try to keep this as clear as possible as I go  but in general I am trying to make broad, inclusive points that apply to the widest set of "systems" as opposed to a more narrow argument regarding a some specific system.
A Socratic Argument
The following is a quick summary of essential elements of this thread in terms of the above definitions (with only a smidgen of my bias showing)
A: A given system (language, theorem, set of well formed formulae, set of axioms) consists of a set of symbols and a set of rules that specify how we manipulate and interpret those symbols.
B: That seems reasonable. So long as we know the rules and how to apply them to the symbols I imagine we could construct lots of things. Indeed, it is difficult to think of anything that can't be described by an appropriate combination of symbols and rules.
A: Indeed. Here is Propositional Logic. And here is Axiomatic Mathematics. And this ZFC Set Theory is quite remarkable in how much we can construct from just a handful of axioms.
B: Wow. That is all very impressive. I very much want to understand the details of these things. I can see the symbols, but I'm a little hazy on the rules. Could you make the rules explicit for me please?
A: Certainly. {writes down the rules}.
B: Umm... You've written some symbols down. In order to interpret those symbols as a set of rules, I need a set of rules that tell me how to manipulate and interpret those symbols. Perhaps you could write down that set of rules?
A: Certainly. {writes down the rules for interpreting the rules that have been expressed as symbols}.
B: Umm... You've written some symbols down. In order to interpret those symbols as a set of rules, I need a set of rules that tell me how to manipulate and interpret those symbols. Perhaps you could write down that set of rules?
A: Certainly. {writes down the rules for interpreting the rules that have been expressed as symbols for interpreting the rules that have been expressed as symbols}.
B: Whoah. This leads to infinite recursion. Unless we start off with an agreed set of rules, there is no way to express a set of rules using just symbols  because we don't know how to interpret those symbols until we have an agreed set of rules.
A: Well... yes... technically that is correct. but it isn't a problem because we have informal languages that provide us with a bootstrap set of rules. Even better, we only need a few symbols and rules to get started. Granted, there may be a little fuzziness around some natural language definitions; but if that was seriously a problem I'm sure we would have realised that by now. C'mon  everybody else is onboard with the program. Stop rocking the boat and just accept that it all works.
B: Oh. Right. I didn't realise we had a universal language.
A: Well... no... natural languages aren't universal as such...
B: Oh. okay. Then we are all born with identical copies of this natural language?
A: Well... no... People learn natural languages. Probably some combination of imitation and trial and error within the local social group such that the group tends to associate similar rules to similar symbols.
B: Ah! I see. There are two different mechanisms. The mechanism of natural language allows us to write down a set of symbols and know what rules apply to those symbols. But natural language isn't sufficiently precise for some of our requirements. Whereas formal systems by themselves cannot specify any rules  but given an existing set of rules then we can construct and state any other rules that we like.
Starting Point > Logical Conclusion
p ⊃ q
Axiomatic mathematics (e.g. deductive logic) takes a set of premises and follows them to their logical conclusion through sequences of A ⊃ B ⊃ C ⊃ D ⊃ etc.
Axiomatic mathematics thus has two types of (presumably well formed) statements: Premises (axioms/assumptions) and deductions (statements that follow from premises according to a set of known rules).
As noted above, the distinction between symbols and rules is necessarily blurred. A set of symbols without rules doesn't tell us anything. A set of rules with nothing to apply them too is similarly unhelpful. In order to deduce anything from a set of premises we need both to state the premises (write down the appropriate symbols) and to know a set of rules that allows us to manipulate our initial premises to produce deductions.
Both premises and deductions consist of symbols & therulesthatapplytothosesymbols.
Without premises (both symbols and the rules that apply to them), axiomatic mathematics can't do anything. Axiomatic mathematics needs a starting point (axioms, assumptions, premises). (It is called 'axiomatic' mathematics for a reason).
Given that axiomatic mathematics does have a starting point, then other statements (symbols with associated rules) can be deduced from that starting point. These deductions may, in turn, be used for premises from which further deductions can be made.
What axiomatic mathematics cannot do is to create symbols & rules from nothing. Axiomatic mathematics can only construct deductions that follow directly from premises.
Fortunately we don't have to worry too much about how to create something from nothing. Natural language is something, and we can use that to specify the premises we need to get axiomatic mathematics going.
Note: as specified above  I'm using natural language in the widest possible sense. As such, a natural language is everything that is required to arrive at a set of premises (symbols and rules). If there is something that contributes in any degree to specifying a known set of symbols & their known rules; that something is part of natural language. Potentially this means that "natural language" and "the universe" are synonyms.
Given that natural language provides the first set of axioms for axiomatic systems, everything that can be deduced within axiomatic systems descends from natural language. Axiomatic mathematics cannot create new symbols or rules (nor remove them) except as a direct consequence of the initial natural language axioms.
Even millions of deductions later, the resulting symbols and rules are solely a consequence of those very first natural language axioms.
The only mechanism of creating symbols and rules from scratch is our initial natural language. Axiomatic Mathematics cannot invent rules.
If, while working with axiomatic mathematics, we perceive a difference between a later set of rules and an earlier set of rules; that change was implicit in the initial set of rules.
Axiomatic mathematics can only reveal what was implicit in the axioms.
There is no way to invent, delete or change rules except by the mechanism bequeathed to us by some previous step. And ultimately, the previous step is a natural language.
I'm going to be reiterating this point a lot more during the rest of this post. It is important.
Currently the defence of axiomatic mathematics that I'm seeing relies on magic rules that appear from nowhere along with magical differences between A and B. Given that B is entirely and completely specified by A (A ⊃ B), the idea that B can definitely be considered utterly separate and distinct from A appears (to me) to be ludicrous. Without A, B simply does not exist. B cannot be distinct from A.
The lines in the sand that axiomatic mathematics needs to draw in order to work... don't exist. They can't exist. For every A ⊃ B, B is, was and always will be only a consequence of A. Remove A and B vanishes. Separate B from A and B vanishes. B only exists as an implication of A.
Same point, different words
The following quotes are all from Gwydion. I found these particularly convenient to hang my points from. However, everyone (who responded) saw the need for me to clarify my definitions of terms. Both arbiteroftruth and Gwydion make similar points regarding the distinction between various concepts and how this allows mathematics to apply different rules to those concepts.
That each of you responded with similar requests and arguments gives me hope that we are actually communicating more successfully. At the same time, I think the arguments you are presenting are a fair reflection of mathematics as a whole.
I have a great deal of respect for your patience in having come this far; and for forcing me to be clearer (much clearer. much, much much clearer) with what I'm trying to convey.
Previously within this thread I would have refrained from including the following. I think the above makes my point fairly eloquently (I've thought the same before  and been wrong... we'll see how it pans out this time). Adding more generally just gives ammunition for skim readers to latch onto something, take it out of context and throw it back in my face.
On the other hand, being redundant, rephrasing and reiterating (ahem) provides an additional opportunity to convey my intent when I may not have been as clear as I hoped on a first attempt.
On the third hand; there is little more annoying than someone who is obviously wrong about everything being smug while spouting their supposed truths. And I do think I know what I'm talking about. And I do think I'm getting closer to expressing it in a form that people other than me can understand. (I'm absolutely open to the idea of you slapping me down and proving me wrong  I just think it is fairly unlikely  in the same way that you have considered it fairly unlikely that I have a relevant point).
So... if you have spotted a critical flaw in the above, I recommend saving yourself the anguish of watching a smug idiot double down on his idiocy and just get on with pointing out the flaw.
Forewarned is forearmed (perhaps a little belatedly).
"adding the rules of deduction to our system"
How? From where? Exactly what specifies those rules of deduction if they were not already part of the language?
What, exactly, are the rules for adding new rules to a system? What specifies how we specify the rules of addition (and presumably subtraction of old rules that no longer apply)? Are you proposing a new language that has the new rules? Are we adding two languages together when we add rules to a system?
Without the hyperbole: you have introduced a property that you haven't defined: adding new rules to an existing language.
I genuinely have no idea how you think that is supposed to work. What I'm hearing something like: "Q: How do you build a faster than light space ship? A: Build a faster than light engine and add crew quarters."
Please excuse me while I spend some time being as emphatic as possible. I don't see the following as a nitpick of a minor point. I think that this quote from Gwydion is a prime symptom of the problem I have with axiomatic mathematics in general.
i. Gwydion tells me that languages are distinct from deductive reasoning and logic; therefore different rules apply to languages than apply to deductive reasoning and logic. Arbiteroftruth tells me that, specifically, languages do not have premises (axioms/assumptions).
1. You are telling me that we start off with a set of symbols with a set of rules. Then suddenly a new set of rules appears and now this set of symbols with this new set of rules is definitely completely different to the old set of symbols and rules (even though we don't precisely know what the old set of symbols and rules were (if we did  it wouldn't be a natural/informal language  it would be a formal language)).
It seems to me that either the new set of rules are described by the original/first natural language and are therefore definitely part of the language and in no way distinct from the language... OR you are proposing another mechanism for specifying rules AND a mechanism for adding/subtracting this new mechanism to the existing language.
In the latter case  I am extraordinarily interested in learning what this new mechanism is for defining new rules, the rules for changing the rules of an existing system to make it a new system, and why these new mechanisms are not considered part of the original natural language.
ii. You all tell me that the Principle of Explosion does not apply to natural languages.
2. The first example on the Wikipedia page linked above is expressed entirely in natural language terms. It is right there on the page. The maniacs actually illustrate the Principle of Explosion using only elements of natural language. Insanity!
iii. Wikipedia isn't the last word in mathematics. Anybody can write a Wiki page  and errors/mistakes absolutely do happen. That page is wrong. The Principle of Explosion doesn't apply to natural languages.
3. Your opinion on the scope of the Principle of Explosion is irrelevant. The Principle of Explosion is an observation as much as it is an apriori rule. If we have a contradiction, then we can use that contradiction to prove any statement. There is no "only if mathematicians say it is okay" clause. There is no "only applies to formal systems" clause.
iv. Well... maybe... but that doesn't mean that the language is inconsistent. Inconsistency is only defined for formal systems.
4. Red Herring. It doesn't matter whether you call it a paradox, an inconsistency or simply absurd. The point is that given (x & !x) we can prove (p & !p) for all p. Whatever allowed you to write (x & !x) also allows you to write (p & !p) for all p.
v. Yeah... I suppose. But that doesn't actually matter, because we don't have to use the whole language. Like we said above, we only actually need a few minor assumptions (axioms) and we can build pretty much everything from there.
5. No shit Sherlock. We know that we can prove absolutely anything using a language that contains a contradiction. We can take any formal inconsistent system and prove any statement (even the statements that aren't well formed). This is why we discard inconsistent systems  they don't do anything useful.
vi. No, no. You are misunderstanding. Yeah, maybe if you take the whole of a language, the whole language might, possibly, perhaps be considered inconsistent (if you really insist on applying formal language terms to an informal language), but at least some of the subsets of that formal language can be consistent (see ZFC (probably)).
6a. You tell me that if I start with (ZFC + !(the axiom of choice)) then I have an inconsistent system. And if we remove !(the axiom of choice) we get something completely different that is (probably) consistent.
6b. So... you are telling me that adding or removing symbols and or rules to/from a system fundamentally changes that system.
6c. So if we start with a known language... and then take some subset of that language  that subset is... what? We knew (to some degree) the rules of the whole language. But those rules don't apply to the subset. The subset is something else. But what else?
vii. You're an idiot. We've already told you that natural languages work by different rules than formal languages. Specifically, natural languages are magic and are capable of doing all the things that formal languages can't quite manage themselves. All you have to do is sweep any problems (like logic being unable to lift itself by its own bootstraps) into informal languages (which we can't quite specify formally at the moment  but are supersure do all the necessary things to make our formal systems work). But trust us, it isn't really a problem that informal languages need to do things that are impossible for formal languages in order for formal languages to work.
7. Good point. You are absolutely right that we can build a faster than light space ships if we just assume that we start with a faster than light engine. It helps even more if you forbid anyone from actually testing the faster than light engine because faster than light engines work by a different set of rules whereby actually switching it on isn't a valid test mechanism.
8. If we think that the Principle of Explosion applies wherever it can apply then "contradictions would be everywhere" (contradictions are everywhere  it is the consequences of those contradictions that are relevant).
If our starting point is something that can prove (p & !p) for all p, then we just might have a problem. Typically this situation proves that our axioms (assumptions/premises) are inconsistent with one another. Arbiterofthruth's claim that natural languages don't have any axioms notwithstanding (axioms = set of symbols & rules. Language doesn't have axioms? Seems like an absurd claim to me).
xiii. But that might prove that our assumptions behind axiomatic mathematics are inconsistent. Axiomatic mathematics is too important too throw out.
9. What is the point of inventing logic if you are going to disregard the results because you don't like them?
If a theorem is wrong it is wrong. You know this. A theorem over the natural numbers has to be true for all the natural numbers. The theorem could hold for the first million natural numbers, the first billion, trillion, ... every single natural number except one. If the theorem fails just once it is wrong. (maybe it can be twerked a bit to make a 'true' theorem, maybe it is a useful approximation in some circumstances  but as a theorem it is wrong).
Axiomatic mathematics depends entirely on natural language. Natural language is inconsistent if it allows us to write (x & !x).
ix. Oh  you are arguing that natural language doesn't work and that Gwydion's computer doesn't work. Obviously they do work  so you are wrong.
10. Don't be silly. Both Gwydion's computer and natural language work just fine. They just don't work the way mathematics must assume they work in order to make axiomatic mathematics work.
Maybe it is the assumptions of inconsistency that is the problem. Maybe it is the assumption that we can know something with certainty (see Descartes). maybe it is some subtle interaction of axioms that leads us astray.
Whatever the case, the Principle of Explosion simply tells us that if we have written down a contradiction (in any context) then the assumptions that allowed us to write that contradiction are inconsistent with one another. And when we have inconsistent assumptions we arrive at paradox, absurdity, "does not compute", proving everything and consequently proving nothing.
...
I agree that mathematics does attempt to distinguish between natural language and formal systems. It is absolutely essential that this be the case, otherwise we have just proven that large swathes of mathematics cannot work the way they are supposed to.
The trouble is that the only justification for that distinction that I can see is... if there isn't a distinction then large swathes of mathematics are based on inconsistent assumptions (and, even worse, that crackpot/idiot on the forums actually has a valid point).
It looks to me very much as if mathematics went to a lot of trouble to put together formal systems, found that their rigorous approach proved that the assumptions were inconsistent and stuck their fingers in their ears while loudly shouting "informal languages are fundamentally different therefore this result doesn't count".
What is the point of mathematics if you throw away the first result because it is inconvenient?
...
I am trivialising the issue somewhat. If Propositional Logic and Axiomatic Mathematics are shown to be fundamentally unworkable systems that leaves a massive vacuum at the centre of mathematics. Aside from the issue of how there can be so much Propositional Logic and Axiomatic Mathematics if neither works... we really would like a formal mechanism of describing the world around us. Throwing away what we have without a ready replacement is unattractive.
On the other hand, despite the impression that some people would like to give; having to start formal systems using informal languages is a known weakness  perhaps not to the extent of dismissing formal systems outright  but definitely a nagging hole that it would be nice to fill.
Because axiomatic mathematics is not completely satisfactory there has been considerable investment in alternative approaches (the nearby thread on the Liar's paradox links to type theory (which in turn references Intuitionistic type)).
Too long; Didn't read:
Axiomatic mathematics only survives by claiming that the Principle of Explosion only applies when we choose. If, instead, we apply the Principle of Explosion wherever we can apply it then axiomatic mathematics obviously falls apart.
Give me a reason why we should limit the scope of the Principle of Explosion that doesn't boil down to "well... otherwise axiomatic mathematics is wrong and we don't want it to be wrong".
Show me, don't tell me. Show me the difference between natural language and formal systems. Don't just tell me they are different ("because if they aren't we've been believing in something based on nothing more than faith").
Last Word
I have no illusions that this is the last word. I've thought I've made myself clear before and been very wrong.
Moreover there are other avenues to explore. We have chaos theory giving rise to emergent behaviours  maybe emergent rules provide a mechanism for distinction and new rules. Or maybe I've skipped an essential mechanism. Or my casting of everything into just symbols & rules is unreasonable.
At the very least, I hope that we have come far enough that you can see that there are genuine questions over the foundations of axiomatic mathematics that deserve serious consideration and shouldn't be dismissed just because the person raising the issues looks very similar to a typical internet crackpot.
P.S. Again  full credit to those who have (and hopefully are) helped me to refine what I'm trying to say. I don't know that I'm succeeding yet  but I do feel that I'm a lot closer then when I first leapt on these forums.
Twistar's six definitions: They are all sets of symbols with a set of rules that apply to those symbols (or the result of having applied a set of rules to a set of symbols).
Set of symbols: Anything that isn't a set of rules. Also known as: any possible domain (and/or range) of a function.
Set of rules: Anything that can be applied to a set of symbols. Also known as: any possible function that takes a domain (and outputs a range).
Symbols and rules are here defined such that there is no other alternative. The premise here is that the only things that we can possibly have are rules or symbols. Any other words we might use are assumed to be a specific instance of symbols, rules or a combination of the two.
I'm going to refer to axiomatic mathematics. By this I mean any formal logical system that starts with premises/axioms/assumptions and applies rules to produce deductions.
I consider premises, axioms and assumptions to be synonyms for "known starting point". A known starting point will consist of a set of symbols and a set of rules that apply to those symbols.
I consider Propositional Logic, deductive logic and Axiomatic Mathematics to all be examples of axiomatic mathematics. Any argument I make referring to one of these should be taken as an argument with respect to all of these.
If at any point it is unclear what I mean by "object" or "thing" or "doohickey", I am deliberately trying to reference the largest possible set of "things" that can possibly be considered relevant in that context. Thus "symbol" is anything that can conceivably be thought of as a representation (writing on paper, sound waves, sequences of photons, ...). I'll try to keep this as clear as possible as I go  but in general I am trying to make broad, inclusive points that apply to the widest set of "systems" as opposed to a more narrow argument regarding a some specific system.
A Socratic Argument
The following is a quick summary of essential elements of this thread in terms of the above definitions (with only a smidgen of my bias showing)
A: A given system (language, theorem, set of well formed formulae, set of axioms) consists of a set of symbols and a set of rules that specify how we manipulate and interpret those symbols.
B: That seems reasonable. So long as we know the rules and how to apply them to the symbols I imagine we could construct lots of things. Indeed, it is difficult to think of anything that can't be described by an appropriate combination of symbols and rules.
A: Indeed. Here is Propositional Logic. And here is Axiomatic Mathematics. And this ZFC Set Theory is quite remarkable in how much we can construct from just a handful of axioms.
B: Wow. That is all very impressive. I very much want to understand the details of these things. I can see the symbols, but I'm a little hazy on the rules. Could you make the rules explicit for me please?
A: Certainly. {writes down the rules}.
B: Umm... You've written some symbols down. In order to interpret those symbols as a set of rules, I need a set of rules that tell me how to manipulate and interpret those symbols. Perhaps you could write down that set of rules?
A: Certainly. {writes down the rules for interpreting the rules that have been expressed as symbols}.
B: Umm... You've written some symbols down. In order to interpret those symbols as a set of rules, I need a set of rules that tell me how to manipulate and interpret those symbols. Perhaps you could write down that set of rules?
A: Certainly. {writes down the rules for interpreting the rules that have been expressed as symbols for interpreting the rules that have been expressed as symbols}.
B: Whoah. This leads to infinite recursion. Unless we start off with an agreed set of rules, there is no way to express a set of rules using just symbols  because we don't know how to interpret those symbols until we have an agreed set of rules.
A: Well... yes... technically that is correct. but it isn't a problem because we have informal languages that provide us with a bootstrap set of rules. Even better, we only need a few symbols and rules to get started. Granted, there may be a little fuzziness around some natural language definitions; but if that was seriously a problem I'm sure we would have realised that by now. C'mon  everybody else is onboard with the program. Stop rocking the boat and just accept that it all works.
B: Oh. Right. I didn't realise we had a universal language.
A: Well... no... natural languages aren't universal as such...
B: Oh. okay. Then we are all born with identical copies of this natural language?
A: Well... no... People learn natural languages. Probably some combination of imitation and trial and error within the local social group such that the group tends to associate similar rules to similar symbols.
B: Ah! I see. There are two different mechanisms. The mechanism of natural language allows us to write down a set of symbols and know what rules apply to those symbols. But natural language isn't sufficiently precise for some of our requirements. Whereas formal systems by themselves cannot specify any rules  but given an existing set of rules then we can construct and state any other rules that we like.
Starting Point > Logical Conclusion
p ⊃ q
Axiomatic mathematics (e.g. deductive logic) takes a set of premises and follows them to their logical conclusion through sequences of A ⊃ B ⊃ C ⊃ D ⊃ etc.
Axiomatic mathematics thus has two types of (presumably well formed) statements: Premises (axioms/assumptions) and deductions (statements that follow from premises according to a set of known rules).
As noted above, the distinction between symbols and rules is necessarily blurred. A set of symbols without rules doesn't tell us anything. A set of rules with nothing to apply them too is similarly unhelpful. In order to deduce anything from a set of premises we need both to state the premises (write down the appropriate symbols) and to know a set of rules that allows us to manipulate our initial premises to produce deductions.
Both premises and deductions consist of symbols & therulesthatapplytothosesymbols.
Without premises (both symbols and the rules that apply to them), axiomatic mathematics can't do anything. Axiomatic mathematics needs a starting point (axioms, assumptions, premises). (It is called 'axiomatic' mathematics for a reason).
Given that axiomatic mathematics does have a starting point, then other statements (symbols with associated rules) can be deduced from that starting point. These deductions may, in turn, be used for premises from which further deductions can be made.
What axiomatic mathematics cannot do is to create symbols & rules from nothing. Axiomatic mathematics can only construct deductions that follow directly from premises.
Fortunately we don't have to worry too much about how to create something from nothing. Natural language is something, and we can use that to specify the premises we need to get axiomatic mathematics going.
Note: as specified above  I'm using natural language in the widest possible sense. As such, a natural language is everything that is required to arrive at a set of premises (symbols and rules). If there is something that contributes in any degree to specifying a known set of symbols & their known rules; that something is part of natural language. Potentially this means that "natural language" and "the universe" are synonyms.
Given that natural language provides the first set of axioms for axiomatic systems, everything that can be deduced within axiomatic systems descends from natural language. Axiomatic mathematics cannot create new symbols or rules (nor remove them) except as a direct consequence of the initial natural language axioms.
Even millions of deductions later, the resulting symbols and rules are solely a consequence of those very first natural language axioms.
The only mechanism of creating symbols and rules from scratch is our initial natural language. Axiomatic Mathematics cannot invent rules.
If, while working with axiomatic mathematics, we perceive a difference between a later set of rules and an earlier set of rules; that change was implicit in the initial set of rules.
Axiomatic mathematics can only reveal what was implicit in the axioms.
There is no way to invent, delete or change rules except by the mechanism bequeathed to us by some previous step. And ultimately, the previous step is a natural language.
I'm going to be reiterating this point a lot more during the rest of this post. It is important.
Currently the defence of axiomatic mathematics that I'm seeing relies on magic rules that appear from nowhere along with magical differences between A and B. Given that B is entirely and completely specified by A (A ⊃ B), the idea that B can definitely be considered utterly separate and distinct from A appears (to me) to be ludicrous. Without A, B simply does not exist. B cannot be distinct from A.
The lines in the sand that axiomatic mathematics needs to draw in order to work... don't exist. They can't exist. For every A ⊃ B, B is, was and always will be only a consequence of A. Remove A and B vanishes. Separate B from A and B vanishes. B only exists as an implication of A.
Same point, different words
The following quotes are all from Gwydion. I found these particularly convenient to hang my points from. However, everyone (who responded) saw the need for me to clarify my definitions of terms. Both arbiteroftruth and Gwydion make similar points regarding the distinction between various concepts and how this allows mathematics to apply different rules to those concepts.
That each of you responded with similar requests and arguments gives me hope that we are actually communicating more successfully. At the same time, I think the arguments you are presenting are a fair reflection of mathematics as a whole.
I have a great deal of respect for your patience in having come this far; and for forcing me to be clearer (much clearer. much, much much clearer) with what I'm trying to convey.
Previously within this thread I would have refrained from including the following. I think the above makes my point fairly eloquently (I've thought the same before  and been wrong... we'll see how it pans out this time). Adding more generally just gives ammunition for skim readers to latch onto something, take it out of context and throw it back in my face.
On the other hand, being redundant, rephrasing and reiterating (ahem) provides an additional opportunity to convey my intent when I may not have been as clear as I hoped on a first attempt.
On the third hand; there is little more annoying than someone who is obviously wrong about everything being smug while spouting their supposed truths. And I do think I know what I'm talking about. And I do think I'm getting closer to expressing it in a form that people other than me can understand. (I'm absolutely open to the idea of you slapping me down and proving me wrong  I just think it is fairly unlikely  in the same way that you have considered it fairly unlikely that I have a relevant point).
So... if you have spotted a critical flaw in the above, I recommend saving yourself the anguish of watching a smug idiot double down on his idiocy and just get on with pointing out the flaw.
Forewarned is forearmed (perhaps a little belatedly).
Gwydion wrote:However, after adding the rules of deduction to our system, we actually can prove statements. You're making an assertion without basis.
"adding the rules of deduction to our system"
How? From where? Exactly what specifies those rules of deduction if they were not already part of the language?
What, exactly, are the rules for adding new rules to a system? What specifies how we specify the rules of addition (and presumably subtraction of old rules that no longer apply)? Are you proposing a new language that has the new rules? Are we adding two languages together when we add rules to a system?
Without the hyperbole: you have introduced a property that you haven't defined: adding new rules to an existing language.
I genuinely have no idea how you think that is supposed to work. What I'm hearing something like: "Q: How do you build a faster than light space ship? A: Build a faster than light engine and add crew quarters."
Gwydion wrote:Correct. If languages followed the rules of logic as you suggest, contradictions would be everywhere. However, this doesn't mean languages are without structure or rules, but that those rules are different from those we apply to deductive reasoning and logic.
Please excuse me while I spend some time being as emphatic as possible. I don't see the following as a nitpick of a minor point. I think that this quote from Gwydion is a prime symptom of the problem I have with axiomatic mathematics in general.
i. Gwydion tells me that languages are distinct from deductive reasoning and logic; therefore different rules apply to languages than apply to deductive reasoning and logic. Arbiteroftruth tells me that, specifically, languages do not have premises (axioms/assumptions).
1. You are telling me that we start off with a set of symbols with a set of rules. Then suddenly a new set of rules appears and now this set of symbols with this new set of rules is definitely completely different to the old set of symbols and rules (even though we don't precisely know what the old set of symbols and rules were (if we did  it wouldn't be a natural/informal language  it would be a formal language)).
It seems to me that either the new set of rules are described by the original/first natural language and are therefore definitely part of the language and in no way distinct from the language... OR you are proposing another mechanism for specifying rules AND a mechanism for adding/subtracting this new mechanism to the existing language.
In the latter case  I am extraordinarily interested in learning what this new mechanism is for defining new rules, the rules for changing the rules of an existing system to make it a new system, and why these new mechanisms are not considered part of the original natural language.
ii. You all tell me that the Principle of Explosion does not apply to natural languages.
2. The first example on the Wikipedia page linked above is expressed entirely in natural language terms. It is right there on the page. The maniacs actually illustrate the Principle of Explosion using only elements of natural language. Insanity!
iii. Wikipedia isn't the last word in mathematics. Anybody can write a Wiki page  and errors/mistakes absolutely do happen. That page is wrong. The Principle of Explosion doesn't apply to natural languages.
3. Your opinion on the scope of the Principle of Explosion is irrelevant. The Principle of Explosion is an observation as much as it is an apriori rule. If we have a contradiction, then we can use that contradiction to prove any statement. There is no "only if mathematicians say it is okay" clause. There is no "only applies to formal systems" clause.
iv. Well... maybe... but that doesn't mean that the language is inconsistent. Inconsistency is only defined for formal systems.
4. Red Herring. It doesn't matter whether you call it a paradox, an inconsistency or simply absurd. The point is that given (x & !x) we can prove (p & !p) for all p. Whatever allowed you to write (x & !x) also allows you to write (p & !p) for all p.
v. Yeah... I suppose. But that doesn't actually matter, because we don't have to use the whole language. Like we said above, we only actually need a few minor assumptions (axioms) and we can build pretty much everything from there.
5. No shit Sherlock. We know that we can prove absolutely anything using a language that contains a contradiction. We can take any formal inconsistent system and prove any statement (even the statements that aren't well formed). This is why we discard inconsistent systems  they don't do anything useful.
vi. No, no. You are misunderstanding. Yeah, maybe if you take the whole of a language, the whole language might, possibly, perhaps be considered inconsistent (if you really insist on applying formal language terms to an informal language), but at least some of the subsets of that formal language can be consistent (see ZFC (probably)).
6a. You tell me that if I start with (ZFC + !(the axiom of choice)) then I have an inconsistent system. And if we remove !(the axiom of choice) we get something completely different that is (probably) consistent.
6b. So... you are telling me that adding or removing symbols and or rules to/from a system fundamentally changes that system.
6c. So if we start with a known language... and then take some subset of that language  that subset is... what? We knew (to some degree) the rules of the whole language. But those rules don't apply to the subset. The subset is something else. But what else?
vii. You're an idiot. We've already told you that natural languages work by different rules than formal languages. Specifically, natural languages are magic and are capable of doing all the things that formal languages can't quite manage themselves. All you have to do is sweep any problems (like logic being unable to lift itself by its own bootstraps) into informal languages (which we can't quite specify formally at the moment  but are supersure do all the necessary things to make our formal systems work). But trust us, it isn't really a problem that informal languages need to do things that are impossible for formal languages in order for formal languages to work.
7. Good point. You are absolutely right that we can build a faster than light space ships if we just assume that we start with a faster than light engine. It helps even more if you forbid anyone from actually testing the faster than light engine because faster than light engines work by a different set of rules whereby actually switching it on isn't a valid test mechanism.
8. If we think that the Principle of Explosion applies wherever it can apply then "contradictions would be everywhere" (contradictions are everywhere  it is the consequences of those contradictions that are relevant).
If our starting point is something that can prove (p & !p) for all p, then we just might have a problem. Typically this situation proves that our axioms (assumptions/premises) are inconsistent with one another. Arbiterofthruth's claim that natural languages don't have any axioms notwithstanding (axioms = set of symbols & rules. Language doesn't have axioms? Seems like an absurd claim to me).
xiii. But that might prove that our assumptions behind axiomatic mathematics are inconsistent. Axiomatic mathematics is too important too throw out.
9. What is the point of inventing logic if you are going to disregard the results because you don't like them?
If a theorem is wrong it is wrong. You know this. A theorem over the natural numbers has to be true for all the natural numbers. The theorem could hold for the first million natural numbers, the first billion, trillion, ... every single natural number except one. If the theorem fails just once it is wrong. (maybe it can be twerked a bit to make a 'true' theorem, maybe it is a useful approximation in some circumstances  but as a theorem it is wrong).
Axiomatic mathematics depends entirely on natural language. Natural language is inconsistent if it allows us to write (x & !x).
ix. Oh  you are arguing that natural language doesn't work and that Gwydion's computer doesn't work. Obviously they do work  so you are wrong.
10. Don't be silly. Both Gwydion's computer and natural language work just fine. They just don't work the way mathematics must assume they work in order to make axiomatic mathematics work.
Maybe it is the assumptions of inconsistency that is the problem. Maybe it is the assumption that we can know something with certainty (see Descartes). maybe it is some subtle interaction of axioms that leads us astray.
Whatever the case, the Principle of Explosion simply tells us that if we have written down a contradiction (in any context) then the assumptions that allowed us to write that contradiction are inconsistent with one another. And when we have inconsistent assumptions we arrive at paradox, absurdity, "does not compute", proving everything and consequently proving nothing.
...
I agree that mathematics does attempt to distinguish between natural language and formal systems. It is absolutely essential that this be the case, otherwise we have just proven that large swathes of mathematics cannot work the way they are supposed to.
The trouble is that the only justification for that distinction that I can see is... if there isn't a distinction then large swathes of mathematics are based on inconsistent assumptions (and, even worse, that crackpot/idiot on the forums actually has a valid point).
It looks to me very much as if mathematics went to a lot of trouble to put together formal systems, found that their rigorous approach proved that the assumptions were inconsistent and stuck their fingers in their ears while loudly shouting "informal languages are fundamentally different therefore this result doesn't count".
What is the point of mathematics if you throw away the first result because it is inconvenient?
...
I am trivialising the issue somewhat. If Propositional Logic and Axiomatic Mathematics are shown to be fundamentally unworkable systems that leaves a massive vacuum at the centre of mathematics. Aside from the issue of how there can be so much Propositional Logic and Axiomatic Mathematics if neither works... we really would like a formal mechanism of describing the world around us. Throwing away what we have without a ready replacement is unattractive.
On the other hand, despite the impression that some people would like to give; having to start formal systems using informal languages is a known weakness  perhaps not to the extent of dismissing formal systems outright  but definitely a nagging hole that it would be nice to fill.
Because axiomatic mathematics is not completely satisfactory there has been considerable investment in alternative approaches (the nearby thread on the Liar's paradox links to type theory (which in turn references Intuitionistic type)).
Too long; Didn't read:
Axiomatic mathematics only survives by claiming that the Principle of Explosion only applies when we choose. If, instead, we apply the Principle of Explosion wherever we can apply it then axiomatic mathematics obviously falls apart.
Give me a reason why we should limit the scope of the Principle of Explosion that doesn't boil down to "well... otherwise axiomatic mathematics is wrong and we don't want it to be wrong".
Show me, don't tell me. Show me the difference between natural language and formal systems. Don't just tell me they are different ("because if they aren't we've been believing in something based on nothing more than faith").
Last Word
I have no illusions that this is the last word. I've thought I've made myself clear before and been very wrong.
Moreover there are other avenues to explore. We have chaos theory giving rise to emergent behaviours  maybe emergent rules provide a mechanism for distinction and new rules. Or maybe I've skipped an essential mechanism. Or my casting of everything into just symbols & rules is unreasonable.
At the very least, I hope that we have come far enough that you can see that there are genuine questions over the foundations of axiomatic mathematics that deserve serious consideration and shouldn't be dismissed just because the person raising the issues looks very similar to a typical internet crackpot.
P.S. Again  full credit to those who have (and hopefully are) helped me to refine what I'm trying to say. I don't know that I'm succeeding yet  but I do feel that I'm a lot closer then when I first leapt on these forums.
 gmalivuk
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Re: Misunderstanding basic math concepts, help please?
You *have* been shown the difference between languages and systems repeatedly, with at this point dozens of different examples and illustrations, since you started this thread.
I'll leave it unlocked in case anyone wants to try yet again, but I for one have lost all the hope I had when you first posted this, that maybe now you'd finally start listening to *and attempting to take to heart* the things everyone else has been telling you since you joined the forums.
I'll leave it unlocked in case anyone wants to try yet again, but I for one have lost all the hope I had when you first posted this, that maybe now you'd finally start listening to *and attempting to take to heart* the things everyone else has been telling you since you joined the forums.
Re: Misunderstanding basic math concepts, help please?
A) Languages and systems are different types of things. A language is just a set of symbol strings that obey rules of syntax and grammar, so "The blue elephant sleeps sideways.", while nonsense, is a valid sentence in natural language, but "A an the some walrus fish banana joyfully purple all" is not a valid sentence in natural language. To distinguish between sentences like "The blue elephant sleeps sideways." and sentences like "My computer sits under my desk.", you need something richer than a "language"  you need some way of establishing the meaning of a sentence, some way of determining the existence (or otherwise) of referents, and some way of evaluating truth. In the case of natural language, the default is "by appeal to the universe"  or by doing things which have been observed to preserve agreement with the universe to sentences that have themselves been shown to agree with the universe. So because, from observation, elephants are grey, we can deduce that "the blue elephant" is a reference without a referent, but I can observe both my computer and my desk directly, and verify their relative positions.
A language is neither consistent nor inconsistent, in much the same way as justice is neither red nor blue (nor any other colour)  consistency is not a property languages have; it's a property systems have. If you're using the phrase "natural language" to refer to a system rather than a language, then you can't just write down a sentence that's part of the language and claim that it's part of the system without showing how it satisfies the additional requirements to be part of the system.
B) At one point, you extend your definition of "natural language" to be potentially synonymous with "the universe". If you're arguing that "natural language" under your definition is inconsistent, then you're arguing that the universe is inconsistent. If that's true, then we have bigger problems than whether or not axiomatic mathematics is consistent.
C) It's not news that we can prove almost nothing about anything  it took at least a couple of thousand years after the problem was raised for anyone to find anything we can prove, and noone's managed to come up with anything since  I can prove that I exist since the very fact I question my existence means I must exist in order to do the questioning. That tells me nothing about what I am, nor about the existence of other entities. To get beyond "cogito ergo sum", we need to make assumptions about the reality and persistence of the external universe, the validity of memory, etc. One of the fundamental assumptions we need to make in order to get anywhere is that it's possible to reason our way to valid conclusions  it's impossible to use reasoning to prove that our reasoning is valid because if our reasoning were invalid, our conclusions might or might not be true.
Ultimately, the best argument I'm aware of for our reasoning being okay is that it seems to work  assuming the internet exists, and functions as it appears to me that it does, that is evidence that the reasoning processes that led to it were sound. And the same goes for the myriad other examples of the fruits of human reason I encounter in my daily life. Yes, it's possible that tomorrow, I'll press the button to turn on my computer, and the monitor will turn into a giant blueberry, but my past experience leads me to discount that as a realistic possibility.
D) There may be a helpful parallel with computer programming languages  it's common for compilers for lowlevel programming languages to be written in that language rather than Assembler  the original version may have been compiled in some other language, but any errors or inefficiencies introduced by using that original compiler get fixed over multiple iterations of using more recent versions of the compiler to compile even more recent versions of the compiler's code. Provided the original compiler is good enough, you end up with a stable version of the compiler that has little or nothing to do with the original compiler. The point being that, provided the point you're iterating toward is stable, your starting point is largely irrelevant. Axiomatic mathematics is the result of a similar process  as the rules for axiomatic mathematics have been refined, they've been used to express the rules of axiomatic mathematics in more rigorous forms, with checks from reality that the resulting systems still describe the sort of everyday mathematics that describes things we encounter in the real world. Like everything else, axiomatic mathematics is metaphorically built on sand, but it's sand that's tightly packed, and has been shown from experience to be able to support some impressive skyscrapers...
A language is neither consistent nor inconsistent, in much the same way as justice is neither red nor blue (nor any other colour)  consistency is not a property languages have; it's a property systems have. If you're using the phrase "natural language" to refer to a system rather than a language, then you can't just write down a sentence that's part of the language and claim that it's part of the system without showing how it satisfies the additional requirements to be part of the system.
B) At one point, you extend your definition of "natural language" to be potentially synonymous with "the universe". If you're arguing that "natural language" under your definition is inconsistent, then you're arguing that the universe is inconsistent. If that's true, then we have bigger problems than whether or not axiomatic mathematics is consistent.
C) It's not news that we can prove almost nothing about anything  it took at least a couple of thousand years after the problem was raised for anyone to find anything we can prove, and noone's managed to come up with anything since  I can prove that I exist since the very fact I question my existence means I must exist in order to do the questioning. That tells me nothing about what I am, nor about the existence of other entities. To get beyond "cogito ergo sum", we need to make assumptions about the reality and persistence of the external universe, the validity of memory, etc. One of the fundamental assumptions we need to make in order to get anywhere is that it's possible to reason our way to valid conclusions  it's impossible to use reasoning to prove that our reasoning is valid because if our reasoning were invalid, our conclusions might or might not be true.
Ultimately, the best argument I'm aware of for our reasoning being okay is that it seems to work  assuming the internet exists, and functions as it appears to me that it does, that is evidence that the reasoning processes that led to it were sound. And the same goes for the myriad other examples of the fruits of human reason I encounter in my daily life. Yes, it's possible that tomorrow, I'll press the button to turn on my computer, and the monitor will turn into a giant blueberry, but my past experience leads me to discount that as a realistic possibility.
D) There may be a helpful parallel with computer programming languages  it's common for compilers for lowlevel programming languages to be written in that language rather than Assembler  the original version may have been compiled in some other language, but any errors or inefficiencies introduced by using that original compiler get fixed over multiple iterations of using more recent versions of the compiler to compile even more recent versions of the compiler's code. Provided the original compiler is good enough, you end up with a stable version of the compiler that has little or nothing to do with the original compiler. The point being that, provided the point you're iterating toward is stable, your starting point is largely irrelevant. Axiomatic mathematics is the result of a similar process  as the rules for axiomatic mathematics have been refined, they've been used to express the rules of axiomatic mathematics in more rigorous forms, with checks from reality that the resulting systems still describe the sort of everyday mathematics that describes things we encounter in the real world. Like everything else, axiomatic mathematics is metaphorically built on sand, but it's sand that's tightly packed, and has been shown from experience to be able to support some impressive skyscrapers...
Re: Misunderstanding basic math concepts, help please?
I'm not going to respond to Treatid's treatise in full. Heck, I'm not even going to read most of Treatid's treatise. His "Socratic argument" is rather insulting. But based on what I can glean from some parts I tried to read:
1. The difference between a language and a theory is the axioms. I don't understand what's so hard to understand about this. There's a difference between {symbols, wffs, deduction rules} and {symbols, wffs, deduction rules, axioms}. The difference is axioms, in case you didn't notice. This makes them different things. There are relationships between them (for instance, all the tautologies entailed by the deduction rules of a language will be true in any theory using that language) but you seem to be attaching a mystic significance to one being a subset of the other that everyone else here finds baffling.
2. There's a difference between being able to write down a wff and that wff being true. You have somehow conflated the two and it's again baffling. Just because a language can write (x & !x) does not make it true, because languages can write wffs that aren't tautologies and aren't true in some theories. This seems really obvious to me and I assume everyone else in the thread besides you, and I don't know why it's a stumbling block for you. I can write down the wffs of arithmetic 2+2=4 and 2+2=5, but that's not damning to arithmetic because one of these statements is true and the other is false.
3. The Principle of Explosion applies to languages insofar as that if a language has as tautologies X and !X for some wff X, and the deductive rule (or derived deductive rule) (X & !X) => P for any wffs X and P, then every statement will be a tautology. Not every language has these properties, and the vast majority of the ones studied by mathematicians do not take X and !X to both be tautologies for any wff X. As such, the Principle of Explosion can't get the claw grip it needs to tear the system apart. It's much more likely that the Principle of Explosion will get that grip when we introduce extra axioms by looking at a theory, compared to just looking at the tautologies of the language. I believe it's in that sense that people are saying that the Principle of Explosion does not apply to languages, though I can't speak for everyone.
4. Certainly if the Principle of Explosion applies to a theory it doesn't damn the language the theory was based on. And if a language suffers from the Principle of Explosion in the way I described in point 3, then axiomatic mathematics itself is not screwed.
5. If you don't understand the ways that axiomatic mathematics and natural languages are different, I'm not exactly sure what to tell you. Natural languages are this weird bootstrapping thing where you learn it by being exposed to it, making educated guesses, and then correcting when you find that your guesses don't match societal norms. Unless you have telepathy or are extremely skilled in the art of reading neurons, you can never be 100% sure that other people are understanding what you're saying in the same way that you understand it. Axiomatic mathematics is an effort to pull that confusion away from a swath of reasoning and contract it back to a starting point: if you understand the small amount of explanatory writing I do to set up a system, then all these things will follow for both of us. Furthermore, axiomatic mathematics attempts to write that explanatory writing in a minimally ambiguous way, setting up a language by specifying its symbols, wffs, and deduction rules at the beginning rather than expecting you to pull yourself into it by your own bootstraps.
6. I notice that you say "axioms = set of symbols & rules.". This is simply not true. A set of symbols and (deductive) rules is a language, if you pair those with the collection of wffs. The axioms for a theory are additional wffs that are taken as true. It's these kinds of basic misunderstandings that make us think your arguments are bullshit, when we can actually pierce the veil of what you're saying.
7. I'm trying to read through your "Starting Point > Logical Conclusion" argument, and I keep finding problems with it, mostly relating to how you use terms differently than everyone else. When you use words to mean different things than the people you're talking to, the conclusion is immediately suspect, because it's possible I've only erroneously agreed to it by accidentally substituting *my* definitions for words when I was confused, and your definitions when I wasn't.
7a. You say "Axiomatic mathematics (e.g. deductive logic) takes a set of premises and follows them to their logical conclusion through sequences of A ⊃ B ⊃ C ⊃ D ⊃ etc." What I read is "In a theory, we start with axioms (statements fiated to be true) and then apply rules of deduction to find more true statements." But I don't know if that's what you meant, because your term "premises" is vague. If you mean axioms, you should say that. If you mean something else, you should say that. When you say "axiomatic mathematics", if you mean "a theory within axiomatic mathematics", you should say that, and if you don't mean that then I don't think you're correct. When you say "follows them to their logical conclusion", if you mean "takes as true every wff that follows from a finite number of applications of deductive rules" then I'm with you, but otherwise, I don't know why you have a single chain when there could be multiple things that could be deduced from A. I was taking A to be a wff here, but if it's something else, then I think that using the "⊃" symbol is inappropriate since that's a symbol in one of the languages of axiomatic mathematics, not a symbol that's used on axiomatic mathematics itself.
And that's just the first sentence!
I hope this illustrates why it's important to use terms the same way as everyone else. Natural language (which you're using to communicate) relies on a common understanding of various things in order to function at peak efficiency. I'm not convinced that we have that common understanding, even after all that's gone on in the thread.
7b. You say "Axiomatic mathematics thus has two types of (presumably well formed) statements: Premises (axioms/assumptions) and deductions (statements that follow from premises according to a set of known rules)." This is just wrong. You've forgotten about the false statements, for instance. You're also blurring the levels of axiomatic mathematics. Within a given theory, the true statements can indeed be divided into the axioms and the rest, which will necessarily follow from the deductive rules of the language of the theory and from 0 or more axioms. (The tautologies in particular follow from no axioms, which is why they're true in every language.) But to talk about all the statements of axiomatic mathematics as falling into two types strikes me as asinine. Every wff of some language will be an axiom of some theory, and (almost) every wff will be a deduction but not an axiom of some other theory. So I don't know how you're dividing the statements up.
7c. You say "As noted above, the distinction between symbols and rules is necessarily blurred. A set of symbols without rules doesn't tell us anything. A set of rules with nothing to apply them too is similarly unhelpful." I don't see how the second and third sentences here entail the first. Functions and numbers need to be paired together to have the function work, but that doesn't blur the line between functions and numbers. Just because a set of symbols and a collection of deductive rules will be paired together in a theory doesn't mean that we can't tell one from the other. In fact, it will be rather easy in the outline of a language, because symbols will be in the "symbols" section and deductive rules will be in the "deductive rules" section.
7d. You say "Both premises and deductions consist of symbols & therulesthatapplytothosesymbols." This is wrong, unless you allow the collection of rules here to be empty. Axioms (what you call premises) and other true statements are wffs and thus consist of symbols. The means of deriving the truth of nonaxiom true statements is "rulesthatapplytothose symbols" (which you state as if there's only one collection of those rules ever, instead of a different collection for every language), but that doesn't mean that those rules are magically inside the statement. As a not that extreme example, what if there are two different derivations of a given true statement? Which rules should be "inside" that statement? All of them? Only the ones that show up in both derivations? Choose one derivation as canonical (how?) and then those rules are in the statement?
So that's 4 "paragraphs" out of 18 for the section, and I've found a problem or ambiguity or both in each one. The next one looks just as bad, but I'm tired of writing at this point.(Hint: you've conflated axiomatic mathematics with a system inside axiomatic mathematics yet again.)
So no, we're not sticking our fingers in our ears. Your claims have very basic problems and they have for as long as I've seen you post them. You continue to take down not axiomatic mathematics but instead your warped view of what axiomatic mathematics is. It's becoming tiresome, and I'll reiterate what everyone else has told you quite a few times: educate yourself on what axiomatic mathematics is. Maybe take a course in firstorder logic at your local community college. Because otherwise you're going to keep making arguments that betray your lack of knowledge about axiomatic mathematics, and people are going to keep calling you on it.
1. The difference between a language and a theory is the axioms. I don't understand what's so hard to understand about this. There's a difference between {symbols, wffs, deduction rules} and {symbols, wffs, deduction rules, axioms}. The difference is axioms, in case you didn't notice. This makes them different things. There are relationships between them (for instance, all the tautologies entailed by the deduction rules of a language will be true in any theory using that language) but you seem to be attaching a mystic significance to one being a subset of the other that everyone else here finds baffling.
2. There's a difference between being able to write down a wff and that wff being true. You have somehow conflated the two and it's again baffling. Just because a language can write (x & !x) does not make it true, because languages can write wffs that aren't tautologies and aren't true in some theories. This seems really obvious to me and I assume everyone else in the thread besides you, and I don't know why it's a stumbling block for you. I can write down the wffs of arithmetic 2+2=4 and 2+2=5, but that's not damning to arithmetic because one of these statements is true and the other is false.
3. The Principle of Explosion applies to languages insofar as that if a language has as tautologies X and !X for some wff X, and the deductive rule (or derived deductive rule) (X & !X) => P for any wffs X and P, then every statement will be a tautology. Not every language has these properties, and the vast majority of the ones studied by mathematicians do not take X and !X to both be tautologies for any wff X. As such, the Principle of Explosion can't get the claw grip it needs to tear the system apart. It's much more likely that the Principle of Explosion will get that grip when we introduce extra axioms by looking at a theory, compared to just looking at the tautologies of the language. I believe it's in that sense that people are saying that the Principle of Explosion does not apply to languages, though I can't speak for everyone.
4. Certainly if the Principle of Explosion applies to a theory it doesn't damn the language the theory was based on. And if a language suffers from the Principle of Explosion in the way I described in point 3, then axiomatic mathematics itself is not screwed.
5. If you don't understand the ways that axiomatic mathematics and natural languages are different, I'm not exactly sure what to tell you. Natural languages are this weird bootstrapping thing where you learn it by being exposed to it, making educated guesses, and then correcting when you find that your guesses don't match societal norms. Unless you have telepathy or are extremely skilled in the art of reading neurons, you can never be 100% sure that other people are understanding what you're saying in the same way that you understand it. Axiomatic mathematics is an effort to pull that confusion away from a swath of reasoning and contract it back to a starting point: if you understand the small amount of explanatory writing I do to set up a system, then all these things will follow for both of us. Furthermore, axiomatic mathematics attempts to write that explanatory writing in a minimally ambiguous way, setting up a language by specifying its symbols, wffs, and deduction rules at the beginning rather than expecting you to pull yourself into it by your own bootstraps.
6. I notice that you say "axioms = set of symbols & rules.". This is simply not true. A set of symbols and (deductive) rules is a language, if you pair those with the collection of wffs. The axioms for a theory are additional wffs that are taken as true. It's these kinds of basic misunderstandings that make us think your arguments are bullshit, when we can actually pierce the veil of what you're saying.
7. I'm trying to read through your "Starting Point > Logical Conclusion" argument, and I keep finding problems with it, mostly relating to how you use terms differently than everyone else. When you use words to mean different things than the people you're talking to, the conclusion is immediately suspect, because it's possible I've only erroneously agreed to it by accidentally substituting *my* definitions for words when I was confused, and your definitions when I wasn't.
7a. You say "Axiomatic mathematics (e.g. deductive logic) takes a set of premises and follows them to their logical conclusion through sequences of A ⊃ B ⊃ C ⊃ D ⊃ etc." What I read is "In a theory, we start with axioms (statements fiated to be true) and then apply rules of deduction to find more true statements." But I don't know if that's what you meant, because your term "premises" is vague. If you mean axioms, you should say that. If you mean something else, you should say that. When you say "axiomatic mathematics", if you mean "a theory within axiomatic mathematics", you should say that, and if you don't mean that then I don't think you're correct. When you say "follows them to their logical conclusion", if you mean "takes as true every wff that follows from a finite number of applications of deductive rules" then I'm with you, but otherwise, I don't know why you have a single chain when there could be multiple things that could be deduced from A. I was taking A to be a wff here, but if it's something else, then I think that using the "⊃" symbol is inappropriate since that's a symbol in one of the languages of axiomatic mathematics, not a symbol that's used on axiomatic mathematics itself.
And that's just the first sentence!
I hope this illustrates why it's important to use terms the same way as everyone else. Natural language (which you're using to communicate) relies on a common understanding of various things in order to function at peak efficiency. I'm not convinced that we have that common understanding, even after all that's gone on in the thread.
7b. You say "Axiomatic mathematics thus has two types of (presumably well formed) statements: Premises (axioms/assumptions) and deductions (statements that follow from premises according to a set of known rules)." This is just wrong. You've forgotten about the false statements, for instance. You're also blurring the levels of axiomatic mathematics. Within a given theory, the true statements can indeed be divided into the axioms and the rest, which will necessarily follow from the deductive rules of the language of the theory and from 0 or more axioms. (The tautologies in particular follow from no axioms, which is why they're true in every language.) But to talk about all the statements of axiomatic mathematics as falling into two types strikes me as asinine. Every wff of some language will be an axiom of some theory, and (almost) every wff will be a deduction but not an axiom of some other theory. So I don't know how you're dividing the statements up.
7c. You say "As noted above, the distinction between symbols and rules is necessarily blurred. A set of symbols without rules doesn't tell us anything. A set of rules with nothing to apply them too is similarly unhelpful." I don't see how the second and third sentences here entail the first. Functions and numbers need to be paired together to have the function work, but that doesn't blur the line between functions and numbers. Just because a set of symbols and a collection of deductive rules will be paired together in a theory doesn't mean that we can't tell one from the other. In fact, it will be rather easy in the outline of a language, because symbols will be in the "symbols" section and deductive rules will be in the "deductive rules" section.
7d. You say "Both premises and deductions consist of symbols & therulesthatapplytothosesymbols." This is wrong, unless you allow the collection of rules here to be empty. Axioms (what you call premises) and other true statements are wffs and thus consist of symbols. The means of deriving the truth of nonaxiom true statements is "rulesthatapplytothose symbols" (which you state as if there's only one collection of those rules ever, instead of a different collection for every language), but that doesn't mean that those rules are magically inside the statement. As a not that extreme example, what if there are two different derivations of a given true statement? Which rules should be "inside" that statement? All of them? Only the ones that show up in both derivations? Choose one derivation as canonical (how?) and then those rules are in the statement?
So that's 4 "paragraphs" out of 18 for the section, and I've found a problem or ambiguity or both in each one. The next one looks just as bad, but I'm tired of writing at this point.(Hint: you've conflated axiomatic mathematics with a system inside axiomatic mathematics yet again.)
So no, we're not sticking our fingers in our ears. Your claims have very basic problems and they have for as long as I've seen you post them. You continue to take down not axiomatic mathematics but instead your warped view of what axiomatic mathematics is. It's becoming tiresome, and I'll reiterate what everyone else has told you quite a few times: educate yourself on what axiomatic mathematics is. Maybe take a course in firstorder logic at your local community college. Because otherwise you're going to keep making arguments that betray your lack of knowledge about axiomatic mathematics, and people are going to keep calling you on it.
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.

 Posts: 449
 Joined: Wed Sep 21, 2011 3:44 am UTC
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:A Socratic Argument
The following is a quick summary of essential elements of this thread in terms of the above definitions (with only a smidgen of my bias showing)
A: A given system (language, theorem, set of well formed formulae, set of axioms) consists of a set of symbols and a set of rules that specify how we manipulate and interpret those symbols.
B: That seems reasonable. So long as we know the rules and how to apply them to the symbols I imagine we could construct lots of things. Indeed, it is difficult to think of anything that can't be described by an appropriate combination of symbols and rules.
A: Indeed. Here is Propositional Logic. And here is Axiomatic Mathematics. And this ZFC Set Theory is quite remarkable in how much we can construct from just a handful of axioms.
B: Wow. That is all very impressive. I very much want to understand the details of these things. I can see the symbols, but I'm a little hazy on the rules. Could you make the rules explicit for me please?
A: Certainly. {writes down the rules}.
B: Umm... You've written some symbols down. In order to interpret those symbols as a set of rules, I need a set of rules that tell me how to manipulate and interpret those symbols. Perhaps you could write down that set of rules?
A: Certainly. {writes down the rules for interpreting the rules that have been expressed as symbols}.
B: Umm... You've written some symbols down. In order to interpret those symbols as a set of rules, I need a set of rules that tell me how to manipulate and interpret those symbols. Perhaps you could write down that set of rules?
A: Certainly. {writes down the rules for interpreting the rules that have been expressed as symbols for interpreting the rules that have been expressed as symbols}.
B: Whoah. This leads to infinite recursion. Unless we start off with an agreed set of rules, there is no way to express a set of rules using just symbols  because we don't know how to interpret those symbols until we have an agreed set of rules.
A: Well... yes... technically that is correct. but it isn't a problem because we have informal languages that provide us with a bootstrap set of rules. Even better, we only need a few symbols and rules to get started. Granted, there may be a little fuzziness around some natural language definitions; but if that was seriously a problem I'm sure we would have realised that by now. C'mon  everybody else is onboard with the program. Stop rocking the boat and just accept that it all works.
B: Oh. Right. I didn't realise we had a universal language.
A: Well... no... natural languages aren't universal as such...
B: Oh. okay. Then we are all born with identical copies of this natural language?
A: Well... no... People learn natural languages. Probably some combination of imitation and trial and error within the local social group such that the group tends to associate similar rules to similar symbols.
B: Ah! I see. There are two different mechanisms. The mechanism of natural language allows us to write down a set of symbols and know what rules apply to those symbols. But natural language isn't sufficiently precise for some of our requirements. Whereas formal systems by themselves cannot specify any rules  but given an existing set of rules then we can construct and state any other rules that we like.
You're doing okay up until "Oh. Right. I didn't realise we had a universal language." The actual responses moving forward from there would be:
A: We don't have a universal language, but then again, we don't need one.
B: But then these mathematical symbols and rules aren't a universal language either!
A: So?
B: So, aren't they supposed to be?
A: No. Where did you get that idea?
B: I thought math was supposed to provide perfect, totally unambiguous descriptions. How can that be possible if we still have to deal with natural language barriers?
A: It isn't, and math was never supposed to provide perfect, totally unambiguous descriptions. Math is a tool for maximizing clarity of communication. It's still rooted in whatever natural language you and your audience happen to share, but we take that natural language, use it to describe some very simple rules so that things are as clear as natural language can possibly make them, and then go from there.
B: But then math isn't a universal language!
A: I thought we covered this...
Treatid wrote:Axiomatic mathematics (e.g. deductive logic) takes a set of premises and follows them to their logical conclusion through sequences of A ⊃ B ⊃ C ⊃ D ⊃ etc.
No, deductive logic describes how to take a set of premises and follow them to their logical conclusions. The act of actually taking a set of premises is the act of choosing some particular theory to work with.
Treatid wrote:Given that natural language provides the first set of axioms for axiomatic systems, everything that can be deduced within axiomatic systems descends from natural language. Axiomatic mathematics cannot create new symbols or rules (nor remove them) except as a direct consequence of the initial natural language axioms.
Even millions of deductions later, the resulting symbols and rules are solely a consequence of those very first natural language axioms.
The rules of deduction can ultimately be derived from natural language/natural human intuitions about reasoning. And what a given axiom is understood to mean, if anything, can be traced back to natural language. But the act of taking an axiom as true is a human action that occurs at a particular time and place. It is not derived from natural language: it is a decision that a human makes. When he then expresses that decision to others, it's written in some formal language which is understood by reference back to natural language. But that doesn't make the axiom itself a product of natural language. Just like seeing a tree and saying "that's a tree" doesn't make the tree a product of natural language. The tree is an object in its own right, and natural language is just being used to say something about it. The same is true of the axioms.
This is the same distinction we've been hammering for some time now. There is a difference between saying that a sentence is meaningful, and saying that a sentence is true. The former is ultimately traced back to natural language. The latter is determined by separate means. In the case of a tree, it's determined by physical reality. In the case of a mathematical axiom, it's determined by a human decision to define a theory.
Treatid wrote:3. Your opinion on the scope of the Principle of Explosion is irrelevant. The Principle of Explosion is an observation as much as it is an apriori rule. If we have a contradiction, then we can use that contradiction to prove any statement. There is no "only if mathematicians say it is okay" clause. There is no "only applies to formal systems" clause.
There is, when you take care to understand what it means to "have a contradiction". Being able to physically write down one sentence, and physically write down a contrary sentence, does not constitute "having a contradiction" in the sense mean by the PoE. "Having a contradiction" means taking some sentence to be true, and also taking a contrary sentence to be true, within a shared context.
This is the same distinction we've been hammering for some time now. There is a difference between saying that a sentence is meaningful, and saying that a sentence is true. Language, whether natural or formal, deals primarily with meaning. You can express two contradictory sentences and understand what they both mean, and this does not constitute "having a contradiction" in the sense the PoE is talking about. A theory takes some sentences to be true and then uses the rules of deduction to prove that other sentences must also be true within the same theory. If those deductions lead to contradictory sentences being simultaneously true within the same theory, that constitutes "having a contradiction" in the sense that PoE is talking about. And as expected, you'll be able to derive whatever sentence you want within the same theory.
So that means it's not a good idea to use that theory. Cool. We can still use the same language though, because the language by itself doesn't "have a contradiction" in the relevant sense.
When we've said that "the principle of explosion doesn't apply to languages", that's meant as a shorthand for the distinction I made above. As Cauchy said, if you could derive a contradiction as a tautology of a language, the principle of explosion would still apply, and the language would be inconsistent. But that's an entirely separate matter from simply being able to write down two contradictory sentences. Writing them down, and taking them as true, are two different things. Language only "contains contradictions" in the former sense of being able to write them down and understand them as meaningful. But the pinciple of explosion deals with "having contradictions" in the latter sense of taking them as true. Again, there is a difference between saying that a sentence is meaningful, and saying that a sentence is true.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:4. Red Herring. It doesn't matter whether you call it a paradox, an inconsistency or simply absurd. The point is that given (x & !x) we can prove (p & !p) for all p. Whatever allowed you to write (x & !x) also allows you to write (p & !p) for all p.
Treatid wrote:(x & !x)
Guys, Treatid just wrote (x & !x), and that's a contradiction! That means everything he said is inconsistent! Oh wait, so did I! What does this mean? *thread explodes*
_______
Wikipedia article on the principle of explosion wrote:The principle is not a universal rule; rather it exists as a consequence of a choice of which logic to use.
We should limit the scope of the principle of explosion because it has a limited scope. Trying to apply things where they don't belong leads to nonsensical conclusions, like my computer exploding as I'm typing this. As has been stated, the principle of explosion is not a standalone fact, but rather one that derives from other concepts including the Law of Excluded Middle (a wff X is either true or false, not neither nor both) and our rules of deduction. It's not the axiom of explosion, after all. If you remove the Law of Excluded Middle, then the principle of explosion ceases to hold. If you change the rules of deduction, you might also make it impossible to prove a sentence P even from a contradiction.
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:1) Well Formed Formula
2) Syntax
3) Rules of Deduction
4) Formal Language
5) Axiom
6) Theory
Treatid wrote:Definition of terms
Twistar's six definitions: [...]
It would be rather important that you define each of the six separately because when people use the words "Syntax" and "Rules of Deduction" they mean them to refer to different things. If you/we can't distinguish between all of them and define what the difference is, there is no discussion to be had.
Please be gracious in judging my english. (I am not a native speaker/writer.)
http://decodedarfur.org/
http://decodedarfur.org/
Re: Misunderstanding basic math concepts, help please?
@gmaliveuk: thank you for your forebearance.
1. I posted a lot and I understand the reluctance to read all of it, especially when you vehemently disagree with me.
However, some of the points you are making I directly addressed. I may have been wrong  but you aren't responding to the argument I've actually made. (Gwydion: I saw your point  I responded to it in some depth. I don't mind that you didn't want to thoroughly read everything I wrote. I do mind that you are constructing a strawman argument based on your guess of what I might have written).
2. A great deal of what is being said I agree with. Unfortunately, a lot of what is being said assumes a different set of definitions/axioms/assumptions than those I stated at the beginning of my post.
3. I can't respond to every point that has been raised without producing another massive screed that nobody is going to read (at least not in a single post).
Definition of terms
This is obviously critical. And I have the very strong feeling that we are talking past each other.
What it looks to me is happening is that I'm looking at the similarities between concepts and others are looking at the differences. A Toyota is different to a Volvo  but they are both cars.
You are telling me that there are differences between A & B. I'm telling you that there are similarities between A & B.
You are not wrong in seeing differences between A & B. I'm not wrong in seeing similarities in A & B.
The importance of the degree of similarity or difference between A & B varies according to the context. A replacement part for a car needs us to specify the make, model and potentially year of car. A driving licence only distinguishes based on engine capacity and weight. You don't have to get a new driving licence when you swap your Nissan for a Ford (Pedantic: Ford also make lorries so...)
My premises/definitions
Anything we can write down (communicate in some way) is a set of symbols.
Any manipulation or interpretation of a set of symbols is done through a set of rules.
For some B there exists A such that A ⊃ B
(plus the other points made under definitions in my previous post)
Done
These are my axioms and my general definition of the various terms. I absolutely accept that there are more detailed definitions based on the differences between sets of symbols and the differences between sets of rules. I don't see your definitions as being exclusive of my definitions.
Some of the responses come very close to saying that my axioms (definitions) are wrong. I'm hoping I'm misreading. Specifically, many of the counterarguments seem to be based on things other than my stated axioms/definitions. If I dismiss your axioms and substitute my own, I'm going to have trouble even understanding what you have written let alone making a coherent counter argument.
It is important that we have a common basis. This point has rightfully been made to me.
So, before we take another step... I have stated some axioms as the basis for further discussion (this post and last post). Do you understand the definitions and accept that they are axioms? Do you have some other reason for rejecting these axioms?
Edit: removed a sentence.
1. I posted a lot and I understand the reluctance to read all of it, especially when you vehemently disagree with me.
However, some of the points you are making I directly addressed. I may have been wrong  but you aren't responding to the argument I've actually made. (Gwydion: I saw your point  I responded to it in some depth. I don't mind that you didn't want to thoroughly read everything I wrote. I do mind that you are constructing a strawman argument based on your guess of what I might have written).
2. A great deal of what is being said I agree with. Unfortunately, a lot of what is being said assumes a different set of definitions/axioms/assumptions than those I stated at the beginning of my post.
3. I can't respond to every point that has been raised without producing another massive screed that nobody is going to read (at least not in a single post).
Definition of terms
This is obviously critical. And I have the very strong feeling that we are talking past each other.
What it looks to me is happening is that I'm looking at the similarities between concepts and others are looking at the differences. A Toyota is different to a Volvo  but they are both cars.
You are telling me that there are differences between A & B. I'm telling you that there are similarities between A & B.
You are not wrong in seeing differences between A & B. I'm not wrong in seeing similarities in A & B.
The importance of the degree of similarity or difference between A & B varies according to the context. A replacement part for a car needs us to specify the make, model and potentially year of car. A driving licence only distinguishes based on engine capacity and weight. You don't have to get a new driving licence when you swap your Nissan for a Ford (Pedantic: Ford also make lorries so...)
My premises/definitions
Anything we can write down (communicate in some way) is a set of symbols.
Any manipulation or interpretation of a set of symbols is done through a set of rules.
For some B there exists A such that A ⊃ B
(plus the other points made under definitions in my previous post)
Done
These are my axioms and my general definition of the various terms. I absolutely accept that there are more detailed definitions based on the differences between sets of symbols and the differences between sets of rules. I don't see your definitions as being exclusive of my definitions.
Some of the responses come very close to saying that my axioms (definitions) are wrong. I'm hoping I'm misreading. Specifically, many of the counterarguments seem to be based on things other than my stated axioms/definitions. If I dismiss your axioms and substitute my own, I'm going to have trouble even understanding what you have written let alone making a coherent counter argument.
It is important that we have a common basis. This point has rightfully been made to me.
So, before we take another step... I have stated some axioms as the basis for further discussion (this post and last post). Do you understand the definitions and accept that they are axioms? Do you have some other reason for rejecting these axioms?
Edit: removed a sentence.
Last edited by Treatid on Wed Jul 27, 2016 6:17 am UTC, edited 1 time in total.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:So, before we take another step... I have stated some axioms as the basis for further discussion (this post and last post). Do you understand the definitions and accept that they are axioms? Do you have some other reason for rejecting these axioms?
I reject your definitions and axioms. They are too simplistic and broad and not well enough defined. I asked for explicit definitions of 6 things and you gave a few vague sentences attempting to brush over all of them in one broad stroke. The nuances that you miss in this are important if not critical for the discussion.
But I think we have a different issue here than just not having clear enough definitions. I think we are struggling with what it means to "define" something mathematically. At first glance (as you pointed out in your previous post) it seems like we are just adding rules willynilly and just getting rid of the ones that cause problems because they cause problems. When students are first starting into proofbased mathematics this is a common confusion. I direct you to this stackexchange thread: Why do we not have to prove definitions?
I think the thing that is bothering you is that logic is "madeup". People (such as us in this thread) are just telling your rules and its not clear where those rules are coming from and what the rules are for making new rules.
Before I go on, one clarification. If I change the rules then it means we are working in a new thing. A couple examples:
1) Say I have a language in which the following is one of the rules of syntax:
'S1: if "p" is a wellformed formula and "q" is a well formed formula then "p&q" is a wellformed formula'
I can now say: '''let's get rid of this rule of syntax and see what happens'''. However, there is a subtext which isn't stated in the triplequoted statement. The subtext is that in '''getting rid of''' the rules of syntax I am actually creating a whole new LANGUAGE. The original language with the rules is L1 and the new language is L2 so we are now working in a totally different language. To fully define the language I would need to rewrite out the definition of the alphabet, all of the other rules of syntax and all of the rules of deduction etc. The two languages might have almost exactly the same rules in every other way but since they differ by this syntax rule they are totally different languages.
2) Another example: say we're doing a proof where the premises are:
"
1) A V B [Prem]
2) B ⊃ C [Prem]
3) ~C [Prem]
"
We could go ahead and in a few lines prove "A". But I can say '''Let's get rid of premise 2 and see what happens. The subtext here is that by removing that premise I am actually now working on a new problem or a new theory. There was the original theory T1, which had 3 premises, but now there is a new theory T2 which has different premises. However, there is more context dependent subtext which says that both of these theories are expressed in the same formal language. However, we could ALSO consider the same theory expressed in a DIFFERENT formal language*
In any case the point I'm trying to make here is that you can make up rules and change them however you like. The two things you can't do are
1) be unclear about what your trying to do or say.
2) break the rules you've already stated.
But anyways, I don't have a lot of time now so I have to wrap it up here. I think the point I'm trying to get across is the following argument:
a) Yes, the rules of math are just made up.
b) no, this is not a surprise to any of us and
c) no, this doesn't invalidate any of mathematics.
But yeah, maybe you can read up some on your own but I'll reiterate that I think the confusion now boils down to what mathematicians actually mean when we talk about defining something vs. the more common language meaning of defining something.
*Note, in this post I'm using double quotes for statements made in the formal language, single quotes for statements about the formal langauge made in the metalanguage (may include statements in the formal language which will appear in nested double quotes) and triple quotes for statements purely in the metalanguage.
**Assuming the syntax rules are the same so that wffs in one language L1 are wffs in the other language L2. If the rules of syntax are different then the axioms of the theory might be nonsense in the second language.

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Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Some of the responses come very close to saying that my axioms (definitions) are wrong. I'm hoping I'm misreading. Specifically, many of the counterarguments seem to be based on things other than my stated axioms/definitions.
There is nothing apriori 'wrong' with defining things as you have. But the problem comes in when you use those definitions in the same post where you're analyzing something in standard terminology, like a statement of the principle of explosion. If you take a standard statement of the PoE, but interpret it based on your own definitions rather than the standard ones, then you're guilty of the fallacy of equivocation, and you're not really discussing the PoE. Rather, you're discussing an imitation of the PoE in which the terms mean different things than they're intended to, and in this case the resulting statement is false. This is because the PoE depends crucially on the finer distinctions that you're glossing over in your definitions. (in particular, the relevant distinction is between claiming that a sentence is meaningful vs. claiming that a sentence is true)
That doesn't mean there's nothing to be gained by tentatively moving forward with your proposed definitions. But it does limit what we can meaningfully discuss without requiring finer distinctions than your definitions allow.
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