Misunderstanding basic math concepts, help please?
Moderators: gmalivuk, Moderators General, Prelates
Re: Misunderstanding basic math concepts, help please?
Fundamentally, Treatid, if you have different definitions and axioms than us, you aren't talking about axiomatic mathematics anymore, because that's *we're* talking about, and you're talking about something different. We disagree with your definitions and axioms in the sense that they don't do what you state, which is describe axiomatic mathematics. So *something* is contradictory, but your argument doesn't show that that thing is axiomatic mathematics if you're taking alternate definitions and axioms.
Separately, I used the term "axioms" here because you used it, but axioms are things inside the theories that axiomatic mathematics talks about. There aren't axioms *of* axiomatic mathematics; at the very least, we'd call whatever rules govern axiomatic mathematics something other than axioms so as to distinguish them from the axioms that axiomatic mathematics talks about. It's the same reason we talk about a family of sets instead of a set of sets, or a functional on functions instead of a function on functions. Families are sets and functionals are functions, but we use different terms to help distinguish the levels.
Separately, I used the term "axioms" here because you used it, but axioms are things inside the theories that axiomatic mathematics talks about. There aren't axioms *of* axiomatic mathematics; at the very least, we'd call whatever rules govern axiomatic mathematics something other than axioms so as to distinguish them from the axioms that axiomatic mathematics talks about. It's the same reason we talk about a family of sets instead of a set of sets, or a functional on functions instead of a function on functions. Families are sets and functionals are functions, but we use different terms to help distinguish the levels.
(∫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.
Re: Misunderstanding basic math concepts, help please?
Introduction (waffle  can skip)
gmaliveuk's warning that I'm close to another locked thread flustered me and I sort of skipped over the fact that rmsgrey, Cauchy and arbiteroftruth provided really good responses.
Rmsgrey provides a good high level overview. What he lays out is a good first pass approximation of my position. And at the same time is, I think, a fair representation of axiomatic mathematics and how we have arrived at the current state of the art.
Cauchy goes into detail about the differences between various concepts and how changing context changes pretty much everything including the language and rules. Again, I agree with a lot (not all) of what Cauchy says  to the point where I think that some of what he is describing supports my perspective more than opposing it.
arbiteroftruth backs up Cauchy on the distinctions between concepts, and in their most recent posts both point out that by changing axioms I am almost certainly changing what rules can be considered relevant to the new axioms. A point that I agree I was sliding over.
As such, the majority of responses to me have been relevant, constructive and deserve a detailed response from me. But length... and until we have agreed our assumptions there is a large chance we will misunderstand critical elements in one another's statements.
Wot I thunk
{not trying to be convincing  trying to describe where I'm coming from in the hopes it will provide extra context for understanding}
Rmsgrey summarises how difficult it has been for mathematics to get to a position where it could prove anything and that even this requires making some assumptions that we can't prove. But it seems to be working.
I have a problem with some of those assumptions, specifically:
1. That we can know* anything with certainty  even hypothetically.
2. That it is possible to create or destroy anything (this is closely linked with 1).
{definition of 'know' here is problematic. While I don't think we can have absolute knowledge in a sense that I think axiomatic mathematics implies; I do think that we can create a complete description of the universe without needing to make any assumptions.
I'm not saying "we can't describe anything perfectly so we might as well all go home". I'm saying "the assumptions behind axiomatic mathematics are leading us astray".}
Earlier in this thread a fair amount of time was spent on the difference between partial knowledge and full knowledge (or was it partial description...).
Newtonian Mechanics as a theory of the universe is wrong. At a local level (surface of this planet), it is mostly a very good approximation.
I think that something similar is the case when we try to define things. At a local level we have a high confidence that we have accurately described things. But such definitions are strictly local and cannot become 'absolute' definitions.
Example: Everyone knows what a human is. Anyone questioning the definition of human is silly. A few silly people argue over what stage a given human life starts, whether each sperm and egg is a human, or how many cells before a foetus becomes a human.
Or going back through our ancestry. Exactly which generation was it where protohumans stopped being proto and became humans.
Or if we upload a human consciousness into a computer, is that program running on a computer still human?
I'm in a bind
I think that fundamental assumptions of axiomatic mathematics are wrong. Unfortunately, pretty much the only mechanism of rigorous proof that we have is axiomatic mathematics.
At the same time, axiomatic mathematics cannot prove that its assumptions are 'true'. As compelling as the arguments each of you make are... they stop short of proving that axiomatic mathematics works in the way that is claimed.
This appears to be exactly where rmsgrey says we are. Axiomatic mathematics is necessarily based on unprovable assumptions  but it appears to work.
Impasse?
As much as this looks like an impasse  I think that axiomatic mathematics (possibly in the form of deductive reasoning) is sufficiently well designed that it is capable of showing itself to be flawed.
Specifically, if we can gather together every assumption and essential precursor to axiomatic mathematics in one place and make them all explicit we will see that there is no possible way for this group of rules to work in the way that is claimed.
The only slight hiccup in this plan of action is that we have to involve natural language at some point. Informal language, pretty much by definition, is not fully defined. If we could confidently extract every single assumption underlying natural language then we would have a fully formally defined language and mathematics wouldn't need to make any assumptions  all the details would already be explicit.
This is why we are in the situation of having to take at least of few assumptions on trust in the first place.
Do it all
While we might not be able to examine the specific rules of a specific natural language, we might be able to examine the essential rules that must apply for all languages. Hence my desire to use the extremely abstract concepts of symbols and rules as a starting point. I deliberately want to make general statements about everything and anything that can conceivably be thought of as 'symbols & rules'.
inb4 "but proving something with respect to every system doesn't necessarily apply to subsets of every system."
I'm confident that we will see one of two possible results. Either:
1. We see that there is no possible way to create the distinctions that you are telling me exist.
2. If those distinctions do exist, then we lose the essential requirements to describe those distinct systems.
Conclusion
There is nothing concrete in this post. I'm simply trying to communicate what my scepticism is and how I think it might be possible to demonstrate that going forward (and why I appear resistant to well reasoned arguments).
If there is some degree of acceptance that this might be a way forward, I'll have a more thorough go at specifying a start set of axioms. While I strongly disagree with Twistar that "simplicty" and "breadth" have any bearing on the validity of axioms  I have been assuming normal axiomatic assumptions  I need to make those explicit.
Edit: little bit of formatting.
gmaliveuk's warning that I'm close to another locked thread flustered me and I sort of skipped over the fact that rmsgrey, Cauchy and arbiteroftruth provided really good responses.
Rmsgrey provides a good high level overview. What he lays out is a good first pass approximation of my position. And at the same time is, I think, a fair representation of axiomatic mathematics and how we have arrived at the current state of the art.
Cauchy goes into detail about the differences between various concepts and how changing context changes pretty much everything including the language and rules. Again, I agree with a lot (not all) of what Cauchy says  to the point where I think that some of what he is describing supports my perspective more than opposing it.
arbiteroftruth backs up Cauchy on the distinctions between concepts, and in their most recent posts both point out that by changing axioms I am almost certainly changing what rules can be considered relevant to the new axioms. A point that I agree I was sliding over.
As such, the majority of responses to me have been relevant, constructive and deserve a detailed response from me. But length... and until we have agreed our assumptions there is a large chance we will misunderstand critical elements in one another's statements.
Wot I thunk
{not trying to be convincing  trying to describe where I'm coming from in the hopes it will provide extra context for understanding}
Rmsgrey summarises how difficult it has been for mathematics to get to a position where it could prove anything and that even this requires making some assumptions that we can't prove. But it seems to be working.
I have a problem with some of those assumptions, specifically:
1. That we can know* anything with certainty  even hypothetically.
2. That it is possible to create or destroy anything (this is closely linked with 1).
{definition of 'know' here is problematic. While I don't think we can have absolute knowledge in a sense that I think axiomatic mathematics implies; I do think that we can create a complete description of the universe without needing to make any assumptions.
I'm not saying "we can't describe anything perfectly so we might as well all go home". I'm saying "the assumptions behind axiomatic mathematics are leading us astray".}
Earlier in this thread a fair amount of time was spent on the difference between partial knowledge and full knowledge (or was it partial description...).
Newtonian Mechanics as a theory of the universe is wrong. At a local level (surface of this planet), it is mostly a very good approximation.
I think that something similar is the case when we try to define things. At a local level we have a high confidence that we have accurately described things. But such definitions are strictly local and cannot become 'absolute' definitions.
Example: Everyone knows what a human is. Anyone questioning the definition of human is silly. A few silly people argue over what stage a given human life starts, whether each sperm and egg is a human, or how many cells before a foetus becomes a human.
Or going back through our ancestry. Exactly which generation was it where protohumans stopped being proto and became humans.
Or if we upload a human consciousness into a computer, is that program running on a computer still human?
I'm in a bind
I think that fundamental assumptions of axiomatic mathematics are wrong. Unfortunately, pretty much the only mechanism of rigorous proof that we have is axiomatic mathematics.
At the same time, axiomatic mathematics cannot prove that its assumptions are 'true'. As compelling as the arguments each of you make are... they stop short of proving that axiomatic mathematics works in the way that is claimed.
This appears to be exactly where rmsgrey says we are. Axiomatic mathematics is necessarily based on unprovable assumptions  but it appears to work.
Impasse?
As much as this looks like an impasse  I think that axiomatic mathematics (possibly in the form of deductive reasoning) is sufficiently well designed that it is capable of showing itself to be flawed.
Specifically, if we can gather together every assumption and essential precursor to axiomatic mathematics in one place and make them all explicit we will see that there is no possible way for this group of rules to work in the way that is claimed.
The only slight hiccup in this plan of action is that we have to involve natural language at some point. Informal language, pretty much by definition, is not fully defined. If we could confidently extract every single assumption underlying natural language then we would have a fully formally defined language and mathematics wouldn't need to make any assumptions  all the details would already be explicit.
This is why we are in the situation of having to take at least of few assumptions on trust in the first place.
Do it all
While we might not be able to examine the specific rules of a specific natural language, we might be able to examine the essential rules that must apply for all languages. Hence my desire to use the extremely abstract concepts of symbols and rules as a starting point. I deliberately want to make general statements about everything and anything that can conceivably be thought of as 'symbols & rules'.
inb4 "but proving something with respect to every system doesn't necessarily apply to subsets of every system."
I'm confident that we will see one of two possible results. Either:
1. We see that there is no possible way to create the distinctions that you are telling me exist.
2. If those distinctions do exist, then we lose the essential requirements to describe those distinct systems.
Conclusion
There is nothing concrete in this post. I'm simply trying to communicate what my scepticism is and how I think it might be possible to demonstrate that going forward (and why I appear resistant to well reasoned arguments).
If there is some degree of acceptance that this might be a way forward, I'll have a more thorough go at specifying a start set of axioms. While I strongly disagree with Twistar that "simplicty" and "breadth" have any bearing on the validity of axioms  I have been assuming normal axiomatic assumptions  I need to make those explicit.
Edit: little bit of formatting.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:{definition of 'know' here is problematic. While I don't think we can have absolute knowledge in a sense that I think axiomatic mathematics implies; I do think that we can create a complete description of the universe without needing to make any assumptions.
At the same time, axiomatic mathematics cannot prove that its assumptions are 'true'. As compelling as the arguments each of you make are... they stop short of proving that axiomatic mathematics works in the way that is claimed.
Specifically, if we can gather together every assumption and essential precursor to axiomatic mathematics in one place and make them all explicit we will see that there is no possible way for this group of rules to work in the way that is claimed.
I've bolded the parts of the sentences where you are talking about some claim that someone is making about axiomatic mathematics.
Questions:
1) Who is making that claim and
2) What exactly is that claim?
In your mind there is some promise that someone has made about axiomatic mathematics about what it will deliver.
A) You are correct that axiomatic mathematics does not deliver on that promise.
B) You are wrong that anyone ever made that promise about axiomatic mathematics.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:[i]{definition of 'know' here is problematic. While I don't think we can have absolute knowledge in a sense that I think axiomatic mathematics implies; I do think that we can create a complete description of the universe without needing to make any assumptions.
How would you create a complete description of the universe without any assumptions? Wouldn't 'the universe exists' an assumption? If not how would you go about proving it exists?
Mathematics does not 'provide a complete description of the universe', whatever that means, I for one am not sure if it 'provides' any 'description of the universe' at all.
I see mathematics as either a tool used between likeminded people to convey human reasoning while relying on the parties understanding very basic concepts. That's why even computers can check mathematical proofs for errors. And why when you had a mistake in your solutions to one of the basic formal logic questions posted earlier, both me and Twistar both gave the same correction to your mistake.
Just my 2cents.
Been lurking here for a while.

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Re: Misunderstanding basic math concepts, help please?
Twistar wrote:Treatid wrote:{definition of 'know' here is problematic. While I don't think we can have absolute knowledge in a sense that I think axiomatic mathematics implies; I do think that we can create a complete description of the universe without needing to make any assumptions.At the same time, axiomatic mathematics cannot prove that its assumptions are 'true'. As compelling as the arguments each of you make are... they stop short of proving that axiomatic mathematics works in the way that is claimed.Specifically, if we can gather together every assumption and essential precursor to axiomatic mathematics in one place and make them all explicit we will see that there is no possible way for this group of rules to work in the way that is claimed.
I've bolded the parts of the sentences where you are talking about some claim that someone is making about axiomatic mathematics.
Questions:
1) Who is making that claim and
2) What exactly is that claim?
In your mind there is some promise that someone has made about axiomatic mathematics about what it will deliver.
A) You are correct that axiomatic mathematics does not deliver on that promise.
B) You are wrong that anyone ever made that promise about axiomatic mathematics.
This.
Treatid, you're correct that the underlying assumptions of mathematics cannot be proven true. You are correct that mathematics is constructed well enough that mathematics itself can prove the previous sentence. You are incorrect in your beliefs about what mathematics claims to be able to do. The results above do not violate the claims of what mathematics can do.
There was a period of time when mathematicians investigated this issue of whether mathematics could somehow prove its own consistency. They discovered, mathematically, that mathematics cannot prove its own consistency. That just means we're forever left with relying on fundamental assumptions rather than somehow bootstrapping rigor into existence out of nothing. It would've been nice if we could have done that, but we can't, so we stick with our fundamental assumptions as the best tool available and work with it.
You say your issue is that you don't think mathematics can do what (you think) it claims. If that's true, I only see two productive conclusions to draw from that.
1) Come up with some alternative to mathematics, such that this alternative can satisfy those claims that mathematics can't. Good luck with that.
2) "Reform" mathematics to accept its own limitations.
Option 1 is almost certainly impossible. Option 2 is essentially just "therefore mathematics shouldn't make these claims", and what we're telling you is that mathematics already doesn't make those claims.
Re: Misunderstanding basic math concepts, help please?
Treatid, to go back to your penultimate post, I had read your entire reply but thought the others here did a better job of articulating my thoughts than I could have. I put together the first part of my post mostly as a joke, to inject a little levity into an otherwise somewhat tense discussion. I didn't mean it as an actual argument, though to be fair, your posts have on multiple occasions claimed that the ability for a language to construct contradictory sentences invalidates anything written in that language.
You never addressed the second part of my post, which was the fact that the principle of explosion is not a fundamental deductive rule or axiom on its own, but rather a consequence of some other fundamental rules. That's important because you're trying to "simplify all of math" to make your points, but in so doing you're taking away some of the tools that are necessary for further reasoning.
You believe that the fundamental assumptions of mathematics are "wrong"  this is not a helpful sentence. What are the assumptions that you don't believe? What does it mean for them to be wrong?
Are they "untrue" in the sense that they don't reflect how the real universe behaves? If so, that doesn't invalidate the deductions and work that is done within that system. It limits the applicability of those conclusions, but as you said before, Newtonian mechanics is applicable in certain situations and gives incredibly useful information in specific applications, despite not necessarily modeling things under extreme circumstances (such as near light speed).
Are they inconsistent with one another, such that it is not sensible to believe them simultaneously with one another? If so, please demonstrate the inconsistency. Mathematics as a system does provide the ability to prove its assumptions inconsistent  if you can show the inconsistency, then we can discuss which axioms or rules brought that inconsistent about and decide what "truth" remains once those are removed.
Would you prefer a mathematical that assumes one or more of those doesn't hold? That's great  you can do that! That's still mathematics! You can't apply your conclusions to other systems, but within the system you categorically define you can do all sorts of wonderful things.
I agree that what you're describing could be a "way forward" toward resolving some of the differences in this thread. I think some of the difficulty here has been that you are making unwritten assumptions about the way "mathematics" works, but that those assumptions don't necessarily produce the same results that you might expect when you change the system in which you're working. You've made appeals in the past to principles and theorems that may follow from the traditional ZFC axioms, but might not hold true in your simplified world of "everything is either a symbol or a rule".
It's especially important when defining simplified or alternative systems that you avoid using jargon  technical words that mean technical things in a specific system, even if that meaning doesn't apply outside that system  without defining it with respect to your new system. Let's take a sample from one of your earlier posts:
OK, so without trying particularly hard, I notice that you've defined the terms "set of symbols" and "set of rules". Your definitions are circular, in that they refer to one another, but that's sometimes unavoidable when starting with first principles and you did a nice job comparing to other things we might be able to reference. However, you don't define the word "set", which is troubling because a set is more than just "any collection" in mathematical terms. It's not gamebreaking, but you can't use any mathematical properties of sets that other mathematicians might use without asserting first that your "sets" have those same properties. You also assert that everything in the universe that you're exploring is either a symbol or rule, with no overlap, which is fine as long as you keep them separate. Functionals (functions that operate on functions) don't exist here, got it. But wait, "or a combination of the two"  what does it mean for something to be both? You've explicitly defined symbols as "not rules".
There may be other issues here, but that's a superficial approach to start with. Probably the most important issue here is that you don't provide any specific rules. Without those, your system is pretty boring and we can't do much with it. I'm looking forward to hearing you specify some of those rules for how we might interact with your theory, so that we can maybe find some common ground. As I said above, though, you need to specify your terms, rules, etc. much more clearly. They can still be simple and broad, but they can't be simplistic or poorlydefined. Also, please avoid using any math jargon unless you define it clearly *as it applies to your system of rules and symbols*, or you may end up with a system that is either too poorly constructed for it to be useful, or one that contains things that make it inconsistent or otherwise unusable.
You never addressed the second part of my post, which was the fact that the principle of explosion is not a fundamental deductive rule or axiom on its own, but rather a consequence of some other fundamental rules. That's important because you're trying to "simplify all of math" to make your points, but in so doing you're taking away some of the tools that are necessary for further reasoning.
treatid wrote:I think that fundamental assumptions of axiomatic mathematics are wrong. Unfortunately, pretty much the only mechanism of rigorous proof that we have is axiomatic mathematics.
At the same time, axiomatic mathematics cannot prove that its assumptions are 'true'. As compelling as the arguments each of you make are... they stop short of proving that axiomatic mathematics works in the way that is claimed.
This appears to be exactly where rmsgrey says we are. Axiomatic mathematics is necessarily based on unprovable assumptions  but it appears to work.
You believe that the fundamental assumptions of mathematics are "wrong"  this is not a helpful sentence. What are the assumptions that you don't believe? What does it mean for them to be wrong?
Are they "untrue" in the sense that they don't reflect how the real universe behaves? If so, that doesn't invalidate the deductions and work that is done within that system. It limits the applicability of those conclusions, but as you said before, Newtonian mechanics is applicable in certain situations and gives incredibly useful information in specific applications, despite not necessarily modeling things under extreme circumstances (such as near light speed).
Are they inconsistent with one another, such that it is not sensible to believe them simultaneously with one another? If so, please demonstrate the inconsistency. Mathematics as a system does provide the ability to prove its assumptions inconsistent  if you can show the inconsistency, then we can discuss which axioms or rules brought that inconsistent about and decide what "truth" remains once those are removed.
Would you prefer a mathematical that assumes one or more of those doesn't hold? That's great  you can do that! That's still mathematics! You can't apply your conclusions to other systems, but within the system you categorically define you can do all sorts of wonderful things.
If there is some degree of acceptance that this might be a way forward, I'll have a more thorough go at specifying a start set of axioms. While I strongly disagree with Twistar that "simplicty" and "breadth" have any bearing on the validity of axioms  I have been assuming normal axiomatic assumptions  I need to make those explicit.
I agree that what you're describing could be a "way forward" toward resolving some of the differences in this thread. I think some of the difficulty here has been that you are making unwritten assumptions about the way "mathematics" works, but that those assumptions don't necessarily produce the same results that you might expect when you change the system in which you're working. You've made appeals in the past to principles and theorems that may follow from the traditional ZFC axioms, but might not hold true in your simplified world of "everything is either a symbol or a rule".
It's especially important when defining simplified or alternative systems that you avoid using jargon  technical words that mean technical things in a specific system, even if that meaning doesn't apply outside that system  without defining it with respect to your new system. Let's take a sample from one of your earlier posts:
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.
OK, so without trying particularly hard, I notice that you've defined the terms "set of symbols" and "set of rules". Your definitions are circular, in that they refer to one another, but that's sometimes unavoidable when starting with first principles and you did a nice job comparing to other things we might be able to reference. However, you don't define the word "set", which is troubling because a set is more than just "any collection" in mathematical terms. It's not gamebreaking, but you can't use any mathematical properties of sets that other mathematicians might use without asserting first that your "sets" have those same properties. You also assert that everything in the universe that you're exploring is either a symbol or rule, with no overlap, which is fine as long as you keep them separate. Functionals (functions that operate on functions) don't exist here, got it. But wait, "or a combination of the two"  what does it mean for something to be both? You've explicitly defined symbols as "not rules".
There may be other issues here, but that's a superficial approach to start with. Probably the most important issue here is that you don't provide any specific rules. Without those, your system is pretty boring and we can't do much with it. I'm looking forward to hearing you specify some of those rules for how we might interact with your theory, so that we can maybe find some common ground. As I said above, though, you need to specify your terms, rules, etc. much more clearly. They can still be simple and broad, but they can't be simplistic or poorlydefined. Also, please avoid using any math jargon unless you define it clearly *as it applies to your system of rules and symbols*, or you may end up with a system that is either too poorly constructed for it to be useful, or one that contains things that make it inconsistent or otherwise unusable.
Re: Misunderstanding basic math concepts, help please?
I realize I didn't say this in my last post, but with regards to the Socratic argument part of your treatise, Treatid, you said: "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." I agree with most of that, but not about the rules specifying how we interpret those symbols. Take geometry, for instance. Euclid's first four axioms (that is, dropping the parallel postulate) can be interpreted in many ways. The three most famous are Euclidean geometry, geometry on the surface of a sphere, and hyperbolic geometry. These interpretations are not baked into the theory of axiomatic geometry that Euclid laid forth minus the parallel postulate, but it was a long time until someone realized that there were interpretations of the axioms other than Euclidean. Indeed, many mathematicians tried (and failed) to prove the parallel postulate from the other four axioms to cement that Euclidean geometry was the only interpretation. So it is very much not the case that the symbols and rules lay out an interpretation of those symbols and rules. That's up to the humans applying the theory to a specific example. And that's great, because it allows the package of a theory to potentially apply to many different areas that its original creator didn't (or did) anticipate. The theory for a group doesn't specify what the elements of the group mean, and so it allows you to take that package of deductions and apply it to anything that fits the meager qualifications necessary to be a group, be they numbers, rigid body rotations, or functions. The same deduction tells me that given any two numbers, there's a third I can add to the first to get to the second; that given any two positions of a rigid object, there's a rotation I can perform to get from the first to the second; and that given any two invertible functions, there's a third I can apply to the first to get to the second.
This is one example of something you seem to assume axiomatic mathematics says that it doesn't actually say.
This is one example of something you seem to assume axiomatic mathematics says that it doesn't actually say.
(∫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.
Re: Misunderstanding basic math concepts, help please?
Intro
When I first stumbled into these forums I had no subtly and it wasn't clear I'd even heard of 'nuance'. The responses I got were the ones I deserved.
Now Cauchy throws in the distinction between entirely prescriptive axiomatic systems and permissive ones in the full expectation that I'll appreciate that distinction and see how it relates to the discussion at hand. I'm sorry the road has been so frustrating, but you needed to teach me how to communicate (and still do).
As such I agree that I find myself in the position of taking some earlier comments (that I provoked) overly literally when the majority of recent comments have been far more nuanced.
Also, the pertinent point is made that it really doesn't matter even if I were to prove every aspect of axiomatic mathematics wrong  without a replacement it is the best we currently have  no matter how flawed, something is better than nothing. And axiomatic mathematics does appear to work to a significant degree. So what is the point of trying to tear down axiomatic mathematics?
So what next?
The obvious next step is to work out an alternative to axiomatic mathematics that overcomes the known flaws of axiomatic mathematics.
However, really smart people have tried really hard to do exactly this.
As much as I might have persuaded a few of you that I am slightly less naive regarding mathematics than first impressions suggested  I'm nowhere near convincing anybody that I know enough to even recognise an alternative to axiomatic mathematics let alone to have worked out (or stumbled upon) such a thing.
This is why my approach at this time has been to prove that the axioms/assumptions of axiomatic mathematics cannot be consistent with one another.
But, recent responses tell me that there is general agreement that (for example) needing to use natural languages to get axiomatic mathematics off the ground is an issue and that we are arguing over the degree rather than the existence of the issue.
And arguing over the degree doesn't move us forward. Unless there is an alternative  the argument is futile.
Moreover, I think that some people have stuck with this thread because they are interested in the idea of how precisely we can specify things when we get really, really pedantic about it. Indeed, I suspect the moderators have allowed me to go as far as I have because there is a known gap in axiomatic mathematics, and the responses I provoke are examples of good mathematical thinking even if my posts aren't.
Don't believe me
I'm going to try and describe the basis of an alternative to axiomatic mathematics. This is an absurd claim.
If your only motivation to continue reading is to laugh at the madman  that is fine.
Let's get right down to it
Definitions
Standard practice across mathematics is to start by defining your terms. In axiomatic mathematics  everything flows from the specified axioms. It is pretty important to know what your definitions/axioms are.
Except noone has yet worked out how to define anything in an absolute sense.
Typically we use words to define other words (c.f. dictionaries). Axiomatic mathematics might be thought of as a rigorous, formal dictionary... start with some defined terms (axioms) and build/define from there.
The trouble is that defining that first set of words in an absolute and incontrovertible manner is hard. Impossibly hard. Lots of people have put a lot of effort into establishing just one teeny tiny fixed point  not just with axiomatic mathematics. Physics and philosophy would love a definite, fixed foundation over which there was no possible argument.
If we only allow words to be defined by other words then we either run out of words, or we have created a loop and our definitions are all circular.
Absolute vs Relative
This is our first major clue. We cannot define anything in an absolute sense. We can define things relative to other things (ish  sort of  right now our definition of 'definition' is up in the air).
We (as a species) would very much like absolute answers. We really want to be able to say "this is true". Currently axiomatic mathematics tries to say "this is true with respect to the axioms". But try as we might we haven't found a way to pin down a set of axioms in such a way that they are definitely and only one absolute thing.
If we actually could define anything in absolute terms we would have very little need to A ⊃ B. We would just state what B is... Umm... by some mechanism other than using one group of words to describe another group of words.
First Hurdle
1. We cannot state or describe or define anything in absolute terms.
For the moment, the only evidence for this conclusion is that we haven't yet found a definitive way of specifying anything in absolute terms.
I expect this to be the single biggest choke point. My perception is that people really want there to be a definitive, God view from which absolutely true statements can be made.
Fortunately, General Relativity has given us a bit of practice at understanding that in some circumstances there isn't a single correct perspective. There are many perspectives that are related to each other  none of which is absolute (correct or more correct than any of the other perspectives).
And looking around mathematics as a whole, we can see many cases where the relationships between objects are more pertinent than the objects themselves. Indeed, Set Theory and, even more so, Category theory are so much about relationships rather than absolute definitions of axioms that I expect several people to dismiss me as just trying to reinvent those existing and well specified models.
In turn, I will suggest that those systems only work because of their relational parts. The axiomatic elements of those systems serve only to hold them back and prevent them being as useful as a purely relational system of mathematics.
But if we start with nothing...?
Whether we want an absolute statement or a relative statement, if we start with nothing  we still don't have anything to be relative to.
Or to put it another way: How on earth can we specify anything if we don't start with a definition of something?
Conclusion
I'm not trying to be coy. Pretty much everything is an emergent feature of a complex system. The trick is to specify the base complex system without being able to define anything.
Answering that properly is going to take some time and requires taking a number of things on faith  that will only be justified as a complete system. There is no starting point. A is going to be defined in terms of B. B is going to be defined in terms of A. And we are going to be spending a lot of time trying to work out why this isn't always and automatically a tautology.
Or maybe we need to spend more time proving that we cannot have an absolute, unequivocally true statement (whatever that actually means)?
Edit: added a word.
When I first stumbled into these forums I had no subtly and it wasn't clear I'd even heard of 'nuance'. The responses I got were the ones I deserved.
Now Cauchy throws in the distinction between entirely prescriptive axiomatic systems and permissive ones in the full expectation that I'll appreciate that distinction and see how it relates to the discussion at hand. I'm sorry the road has been so frustrating, but you needed to teach me how to communicate (and still do).
As such I agree that I find myself in the position of taking some earlier comments (that I provoked) overly literally when the majority of recent comments have been far more nuanced.
Also, the pertinent point is made that it really doesn't matter even if I were to prove every aspect of axiomatic mathematics wrong  without a replacement it is the best we currently have  no matter how flawed, something is better than nothing. And axiomatic mathematics does appear to work to a significant degree. So what is the point of trying to tear down axiomatic mathematics?
So what next?
The obvious next step is to work out an alternative to axiomatic mathematics that overcomes the known flaws of axiomatic mathematics.
However, really smart people have tried really hard to do exactly this.
As much as I might have persuaded a few of you that I am slightly less naive regarding mathematics than first impressions suggested  I'm nowhere near convincing anybody that I know enough to even recognise an alternative to axiomatic mathematics let alone to have worked out (or stumbled upon) such a thing.
This is why my approach at this time has been to prove that the axioms/assumptions of axiomatic mathematics cannot be consistent with one another.
But, recent responses tell me that there is general agreement that (for example) needing to use natural languages to get axiomatic mathematics off the ground is an issue and that we are arguing over the degree rather than the existence of the issue.
And arguing over the degree doesn't move us forward. Unless there is an alternative  the argument is futile.
Moreover, I think that some people have stuck with this thread because they are interested in the idea of how precisely we can specify things when we get really, really pedantic about it. Indeed, I suspect the moderators have allowed me to go as far as I have because there is a known gap in axiomatic mathematics, and the responses I provoke are examples of good mathematical thinking even if my posts aren't.
Don't believe me
I'm going to try and describe the basis of an alternative to axiomatic mathematics. This is an absurd claim.
If your only motivation to continue reading is to laugh at the madman  that is fine.
Let's get right down to it
Definitions
Standard practice across mathematics is to start by defining your terms. In axiomatic mathematics  everything flows from the specified axioms. It is pretty important to know what your definitions/axioms are.
Except noone has yet worked out how to define anything in an absolute sense.
Typically we use words to define other words (c.f. dictionaries). Axiomatic mathematics might be thought of as a rigorous, formal dictionary... start with some defined terms (axioms) and build/define from there.
The trouble is that defining that first set of words in an absolute and incontrovertible manner is hard. Impossibly hard. Lots of people have put a lot of effort into establishing just one teeny tiny fixed point  not just with axiomatic mathematics. Physics and philosophy would love a definite, fixed foundation over which there was no possible argument.
If we only allow words to be defined by other words then we either run out of words, or we have created a loop and our definitions are all circular.
Absolute vs Relative
This is our first major clue. We cannot define anything in an absolute sense. We can define things relative to other things (ish  sort of  right now our definition of 'definition' is up in the air).
We (as a species) would very much like absolute answers. We really want to be able to say "this is true". Currently axiomatic mathematics tries to say "this is true with respect to the axioms". But try as we might we haven't found a way to pin down a set of axioms in such a way that they are definitely and only one absolute thing.
If we actually could define anything in absolute terms we would have very little need to A ⊃ B. We would just state what B is... Umm... by some mechanism other than using one group of words to describe another group of words.
First Hurdle
1. We cannot state or describe or define anything in absolute terms.
For the moment, the only evidence for this conclusion is that we haven't yet found a definitive way of specifying anything in absolute terms.
I expect this to be the single biggest choke point. My perception is that people really want there to be a definitive, God view from which absolutely true statements can be made.
Fortunately, General Relativity has given us a bit of practice at understanding that in some circumstances there isn't a single correct perspective. There are many perspectives that are related to each other  none of which is absolute (correct or more correct than any of the other perspectives).
And looking around mathematics as a whole, we can see many cases where the relationships between objects are more pertinent than the objects themselves. Indeed, Set Theory and, even more so, Category theory are so much about relationships rather than absolute definitions of axioms that I expect several people to dismiss me as just trying to reinvent those existing and well specified models.
In turn, I will suggest that those systems only work because of their relational parts. The axiomatic elements of those systems serve only to hold them back and prevent them being as useful as a purely relational system of mathematics.
But if we start with nothing...?
Whether we want an absolute statement or a relative statement, if we start with nothing  we still don't have anything to be relative to.
Or to put it another way: How on earth can we specify anything if we don't start with a definition of something?
Conclusion
I'm not trying to be coy. Pretty much everything is an emergent feature of a complex system. The trick is to specify the base complex system without being able to define anything.
Answering that properly is going to take some time and requires taking a number of things on faith  that will only be justified as a complete system. There is no starting point. A is going to be defined in terms of B. B is going to be defined in terms of A. And we are going to be spending a lot of time trying to work out why this isn't always and automatically a tautology.
Or maybe we need to spend more time proving that we cannot have an absolute, unequivocally true statement (whatever that actually means)?
Edit: added a word.
Re: Misunderstanding basic math concepts, help please?
It's What the Tortoise Said to Achilles. I might grant you that the empty statement (which says nothing about anything) is vacuously true, but otherwise you're going to fall into this trap. You have to bootstrap somewhere if you want to get meaning, and bootstrapping is inherently this risky process with a chance of failure, or worse, a chance of different beings bootstrapping into different meanings. That's why people can talk past each other: you both think the other person is talking about what you're talking about, but they're not, even though they're using the same terms.
If you find a way of getting meaning without bootstrapping, let me know so that I can pick it apart and show you where you're bootstrapping. Until then, if you accept bootstrapping, then axiomatic mathematics is an attempt to push away as much ambiguity after the initial bootstrapping as possible, and it does a damn fine job.
As examples of where you might go wrong in finding meaning: if you write your final description of your new system in natural language, then you're relying on our having a shared understanding of the meaning of the words you used. If you write it in something not founded in natural language (itself?) then you're relying on my being able to figure out this thing's meaning from itself, which is just another form of bootstrapping. I don't see how else you could do it, but I bet you can surprise me.
If you find a way of getting meaning without bootstrapping, let me know so that I can pick it apart and show you where you're bootstrapping. Until then, if you accept bootstrapping, then axiomatic mathematics is an attempt to push away as much ambiguity after the initial bootstrapping as possible, and it does a damn fine job.
As examples of where you might go wrong in finding meaning: if you write your final description of your new system in natural language, then you're relying on our having a shared understanding of the meaning of the words you used. If you write it in something not founded in natural language (itself?) then you're relying on my being able to figure out this thing's meaning from itself, which is just another form of bootstrapping. I don't see how else you could do it, but I bet you can surprise me.
(∫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.
Re: Misunderstanding basic math concepts, help please?
First: I have read your entire post, and most of your previous posts in this thread and the earlier threads. I am going to raise some questions and thoughts I had while reading it. Please try to answer them as clearly as you can.
When have you proved "axiomatic mathematics" or aspects of it wrong(A post link and/or quote would be helpful)? Yes you have shown it is not "complete" in some sense, but you didn't show it was wrong.
This is an issue you seem to be having due to your perception of what mathematics should be, most other people here at this thread don't seem to be having an issue with it.
That is true, this is why we study mathematics to learn as best we can the language(s) it uses, so we can better understand what is being conveyed in an argument.
We didn't try, we use very simple terms so that someone with some level of education should be able to understand, and we know that this is what we do.
Also, "we as a species" don't have much consensus over nearly anything, with all the different religions, philosophical views, educational levels, opposing opinions, and so on...
We don't have any other means to communicate except by "words". The reason I used "words" is because I mean that in a broad sense of the word. Someone who is blind and deaf can't hear nor read words, and would have to communicate using his other senses, but in the end, we would communicate with that person by translating our spoken and written "words" into, say, tactile "words"(such as braille).
Who are you referring to by "people". From your argument it seems you are speaking about religious people, or people without much knowledge in mathematics and philosophy.
You are using a generalized idea of some "people" who are saying and thinking "things", yet you are not citing any specific sources. This leads me to believe you are citing your own perception of the people who are around you, who may not be a good representatives for people who have studies mathematics.
That is true, in mathematics we usually define objects by their relations to other objects. Such as the number 1 being the multiplicative identity, which just means that whatever number we multiply by 1, will result in that same number. Note that multiplication can also be seen as a type of relation.
What parts are exactly the "axiomatic elements"? Are these the axioms, which describe the relations? But then what are the "relational parts"?
In this case you seem to be adding a distinction that I can't seem to see.
To me, the trick is to communicate our thoughts and reasoning as best as we can, knowing that in the end, someone who's blind may not be able to read our formal proof, and someone who didn't learn the language we are using, will think it's ancient Egyptian hieroglyphs.
No one who has studied mathematics seriously should be under the impression that mathematics has some magical language that we can use to communicate perfectly with anyone, if it were we wouldn't really have to learn it. (If we had to learn it to be able to communicate with it, then we wouldn't be able to use it to communicate perfectly with anyone, because he'd have to learn it first to communicate with it).
That last paragraph is a bit convoluted, sorry for that.
My main point which has been iterated here for quite some time without your response:
You are under the impression that someone thinks certain stuff about mathematics and that he is of some authority over mathematics(whatever that means), without actually citing a source.
We say that we know of no one who thinks such things and that is also some authority over mathematics(whatever that means).
Treatid wrote:Also, the pertinent point is made that it really doesn't matter even if I were to prove every aspect of axiomatic mathematics wrong  without a replacement it is the best we currently have  no matter how flawed, something is better than nothing. And axiomatic mathematics does appear to work to a significant degree. So what is the point of trying to tear down axiomatic mathematics?
When have you proved "axiomatic mathematics" or aspects of it wrong(A post link and/or quote would be helpful)? Yes you have shown it is not "complete" in some sense, but you didn't show it was wrong.
Treatid wrote:This is why my approach at this time has been to prove that the axioms/assumptions of axiomatic mathematics cannot be consistent with one another.
But, recent responses tell me that there is general agreement that (for example) needing to use natural languages to get axiomatic mathematics off the ground is an issue and that we are arguing over the degree rather than the existence of the issue.
This is an issue you seem to be having due to your perception of what mathematics should be, most other people here at this thread don't seem to be having an issue with it.
Treatid wrote:Standard practice across mathematics is to start by defining your terms. In axiomatic mathematics  everything flows from the specified axioms. It is pretty important to know what your definitions/axioms are.
That is true, this is why we study mathematics to learn as best we can the language(s) it uses, so we can better understand what is being conveyed in an argument.
Treatid wrote:We (as a species) would very much like absolute answers. We really want to be able to say "this is true". Currently axiomatic mathematics tries to say "this is true with respect to the axioms". But try as we might we haven't found a way to pin down a set of axioms in such a way that they are definitely and only one absolute thing.
We didn't try, we use very simple terms so that someone with some level of education should be able to understand, and we know that this is what we do.
Also, "we as a species" don't have much consensus over nearly anything, with all the different religions, philosophical views, educational levels, opposing opinions, and so on...
Treatid wrote:If we actually could define anything in absolute terms we would have very little need to A ⊃ B. We would just state what B is... Umm... by some mechanism other than using one group of words to describe another group of words.
We don't have any other means to communicate except by "words". The reason I used "words" is because I mean that in a broad sense of the word. Someone who is blind and deaf can't hear nor read words, and would have to communicate using his other senses, but in the end, we would communicate with that person by translating our spoken and written "words" into, say, tactile "words"(such as braille).
Treatid wrote:1. We cannot state or describe or define anything in absolute terms.
For the moment, the only evidence for this conclusion is that we haven't yet found a definitive way of specifying anything in absolute terms.
I expect this to be the single biggest choke point. My perception is that people really want there to be a definitive, God view from which absolutely true statements can be made.
Who are you referring to by "people". From your argument it seems you are speaking about religious people, or people without much knowledge in mathematics and philosophy.
You are using a generalized idea of some "people" who are saying and thinking "things", yet you are not citing any specific sources. This leads me to believe you are citing your own perception of the people who are around you, who may not be a good representatives for people who have studies mathematics.
Treatid wrote:And looking around mathematics as a whole, we can see many cases where the relationships between objects are more pertinent than the objects themselves. Indeed, Set Theory and, even more so, Category theory are so much about relationships rather than absolute definitions of axioms that I expect several people to dismiss me as just trying to reinvent those existing and well specified models.
In turn, I will suggest that those systems only work because of their relational parts. The axiomatic elements of those systems serve only to hold them back and prevent them being as useful as a purely relational system of mathematics.
That is true, in mathematics we usually define objects by their relations to other objects. Such as the number 1 being the multiplicative identity, which just means that whatever number we multiply by 1, will result in that same number. Note that multiplication can also be seen as a type of relation.
What parts are exactly the "axiomatic elements"? Are these the axioms, which describe the relations? But then what are the "relational parts"?
In this case you seem to be adding a distinction that I can't seem to see.
Treatid wrote:I'm not trying to be coy. Pretty much everything is an emergent feature of a complex system. The trick is to specify the base complex system without being able to define anything.
Answering that properly is going to take some time and requires taking a number of things on faith  that will only be justified as a complete system. There is no starting point. A is going to be defined in terms of B. B is going to be defined in terms of A. And we are going to be spending a lot of time trying to work out why this isn't always and automatically a tautology.
Or maybe we need to spend more time proving that we cannot have an absolute, unequivocally true statement (whatever that actually means)?
To me, the trick is to communicate our thoughts and reasoning as best as we can, knowing that in the end, someone who's blind may not be able to read our formal proof, and someone who didn't learn the language we are using, will think it's ancient Egyptian hieroglyphs.
No one who has studied mathematics seriously should be under the impression that mathematics has some magical language that we can use to communicate perfectly with anyone, if it were we wouldn't really have to learn it. (If we had to learn it to be able to communicate with it, then we wouldn't be able to use it to communicate perfectly with anyone, because he'd have to learn it first to communicate with it).
That last paragraph is a bit convoluted, sorry for that.
My main point which has been iterated here for quite some time without your response:
You are under the impression that someone thinks certain stuff about mathematics and that he is of some authority over mathematics(whatever that means), without actually citing a source.
We say that we know of no one who thinks such things and that is also some authority over mathematics(whatever that means).
Re: Misunderstanding basic math concepts, help please?
One of the big ideas underlying formal mathematics is that you can start with entirely intuitive ideas and, by relating them to each other, and extending them and exploring their consequences you can reach the point where you can replace your original vague intuitive "definitions" with formal definitions that are based in the more rigorous structure you've built up, and do so without invalidating your deductions from the original version. So, for example, under ZF, what started originally as a vague concept of counting gets defined as a sequence of sets with a concept of a "next set", and, hey, if we define appropriate operations on those sets, they work just like the intuitive ideas of "addition" and "multiplication" and all the things we proved about arithmetic, and all the things we proved using arithmetic, still work.
We're essentially bootstrapping mathematics into rigour, and assuming it works because it seems to  it certainly produces useful results when we apply it to the real world.
We're essentially bootstrapping mathematics into rigour, and assuming it works because it seems to  it certainly produces useful results when we apply it to the real world.
Re: Misunderstanding basic math concepts, help please?
Demki wrote:Also, "we as a species" don't have much consensus over nearly anything, with all the different religions, philosophical views, educational levels, opposing opinions, and so on...
Which is exactly why  in mathematics  we don't take anything for granted.
Instead, we start from a set of axioms and effectively say "do you accept all these to be true? If so, then here is what we can deduce". There's no "absolute truth" in mathematics. Only truths which are dependent on the axioms you choose to accept.
Ofcourse, for this entire endeavor to have any meaning, mathematicians usually strive to make their axioms as uncontroversial as possible. Things like "we can add 1 to any number" or "a union of two sets is also a set" (both of these are actual axioms  in PA and ZF respectively). We can't PROVE either of these things, which is exactly what the word "axiom" means. But we can say "assuming that these things are true, we can also prove that X is true" where "X" is something which isn't as selfevident (like  say  Fermat's Last Theorem) as the original axioms.
That's all mathematics really is. It's a way to codify basic common sense into a more rigorous language. It helps us not to get lost with all our different assumptions, and also help us guarantee that we aren't contradicting ourselves.
Now, if a set of axioms seem really really obvious, one might be tempted to regard them (and anything proven by them) as an absolute truth. Not many people would doubt that the numbers go on forever or that we can "stick together" two sets of objects to create a new set. So while things that we deduce from such axioms are not exactly "guranteed" to be true, they are as close to an absolute truth as mere mortals can get.
Re: Misunderstanding basic math concepts, help please?
I did not intend to claim that I had proven axiomatic mathematics to be 'wrong'. I had intended to convey that the subsequent clause was a hypothetical by my use of "even if".
Having said that, various comments including those be Demki, rmsgrey and the nearby thread on the Liar's Paradox prompted a thought that I'll sketch at the end of this post.
As I tried to note in my last post, I feel it is more important to show a potential alternative to axiomatic mathematics than to criticise axiomatic mathematics for failing to be 'perfect' when nothing else does better.
Not axiomatic mathematics
The structure for presenting a new theorem/model in most of mathematics is generally to present an initial set of axioms/assumptions/definitions/premises and then proceed from there by some form of deduction.
I want to try and show that there is a productive alternative to this approach.
This approach is:
A. largely unfamiliar (as opposed to axioms + deduction)
B. Communicated by me  (we've all felt that frustration)
C. Implausible. (really... what are the chances it is actually constructive rather than the delusions of a madman?)
And to top it all off:
D. Explicitly rejects the idea that individual elements can be considered in isolation. Each element is specified in relation to the other elements. No element is a fixed point.
On the plus side:
i. There are necessarily very few core elements.
What is it
A set/group of relationships that change in a deterministic manner.
That's it?
Yep. Now all we need is a rigorous understanding of those words when I've stated that it is impossible to define any of those words in an absolute sense.
Relationships
The word 'relationship' here is a label. It really is impossible to define 'relationship' in an absolute sense. But we do have a sense of what 'relationships' are in general and that sense is useful. A good starting place is to think of a relationship as the most generic thing possible  something like the relationship of a set containing another set  but even more abstract.
Change
Not loose coins in your pocket.
Functions and Rules... The process or mechanism of change is a substantial part of mathematics.
While it is convenient to talk about change as distinct from the subject of change (Functions AND domains, Symbols AND rules); without something to change, change can't happen. Just as without rules/functions, symbols just sit there doing and meaning nothing.
Here, 'change' is a fundamental property of 'relationships'. Or relationships are a fundamental property of change. Or the terms aren't quite right to convey what is meant  but are used as a first pass crutch because we can't jump from here to there in one leap.
Deterministic
No gaps. No reliance on picking one route through infinitely many worlds. No God.
Fully self contained. No requirement for anything external to bootstrap or otherwise justify any step of a specific model within the system.
Useful background
The importance of relationships (over the objects being related) is already widely recognised in mathematics. While I don't think the following go far enough, they are very much large steps in the right direction:
Rough roadmap of where I think I'm going
Not included  pretty much everything we name/describe/define at our local level is an emergent feature of a complex system. Complex systems are well described in many places. While it isn't automatically obvious that such things as Dimensions are necessarily an emergent property  the idea of emergent properties is well enough understood that it is plausible that Dimensions are an emergent property (and exactly this has been seriously proposed already within mathematics/physics).
End post bit
Proof and disproof are opposite sides of the same coin. By applying the same symmetry to the subject of proof and disproof we find an equivalence:
(prove A = B) is equivalent to (disprove A != B)
(disprove A = B) is equivalent to (prove A != B)
For all nontrivial axiomatic systems A it is shown to be impossible to (prove A (is consistent)). However, for all axiomatic systems A there exists an axiomatic systems !A such that (prove A (is consistent)) is equivalent to (disprove !A (is consistent)) or (prove !A is inconsistent) and vice versa.
But if it is impossible to (prove A (is consistent)), then it must be impossible to (disprove !A (is consistent)). Likewise, if it is impossible to (prove !A (is consistent)) then it must be impossible to (disprove A (is consistent)).
Yet there appears to be many examples of nontrivial axiomatic systems that have been disproven or proven inconsistent.
Having said that, various comments including those be Demki, rmsgrey and the nearby thread on the Liar's Paradox prompted a thought that I'll sketch at the end of this post.
As I tried to note in my last post, I feel it is more important to show a potential alternative to axiomatic mathematics than to criticise axiomatic mathematics for failing to be 'perfect' when nothing else does better.
Not axiomatic mathematics
The structure for presenting a new theorem/model in most of mathematics is generally to present an initial set of axioms/assumptions/definitions/premises and then proceed from there by some form of deduction.
I want to try and show that there is a productive alternative to this approach.
This approach is:
A. largely unfamiliar (as opposed to axioms + deduction)
B. Communicated by me  (we've all felt that frustration)
C. Implausible. (really... what are the chances it is actually constructive rather than the delusions of a madman?)
And to top it all off:
D. Explicitly rejects the idea that individual elements can be considered in isolation. Each element is specified in relation to the other elements. No element is a fixed point.
On the plus side:
i. There are necessarily very few core elements.
What is it
A set/group of relationships that change in a deterministic manner.
That's it?
Yep. Now all we need is a rigorous understanding of those words when I've stated that it is impossible to define any of those words in an absolute sense.
Relationships
The word 'relationship' here is a label. It really is impossible to define 'relationship' in an absolute sense. But we do have a sense of what 'relationships' are in general and that sense is useful. A good starting place is to think of a relationship as the most generic thing possible  something like the relationship of a set containing another set  but even more abstract.
Change
Not loose coins in your pocket.
Functions and Rules... The process or mechanism of change is a substantial part of mathematics.
While it is convenient to talk about change as distinct from the subject of change (Functions AND domains, Symbols AND rules); without something to change, change can't happen. Just as without rules/functions, symbols just sit there doing and meaning nothing.
Here, 'change' is a fundamental property of 'relationships'. Or relationships are a fundamental property of change. Or the terms aren't quite right to convey what is meant  but are used as a first pass crutch because we can't jump from here to there in one leap.
Deterministic
No gaps. No reliance on picking one route through infinitely many worlds. No God.
Fully self contained. No requirement for anything external to bootstrap or otherwise justify any step of a specific model within the system.
Useful background
The importance of relationships (over the objects being related) is already widely recognised in mathematics. While I don't think the following go far enough, they are very much large steps in the right direction:
 Graph Theory (vis. The Travelling Salesman problem  not equations plotted on axis)
 Category Theory (the best, closest formal approach to a purely relational mathematical theory so far)
 Set Theory (a good first pass at a relational theory  has deficits that prompted the creation of Category Theory)
 Chaos/Complexity Theory (the orbits of chaotic systems are directly relevant  It is useful to identify individual systems by their orbits (irrespective of an internal mechanism)
Rough roadmap of where I think I'm going
 Observation (in order to observe, the observer must be part of the system of the observed. Related: description and emulation.)
 Absolute (what can 'absolute' mean given the above.)
 Difference (An element of definitions is distinguishing A from B. I'll argue that the distinction between A and B is more important than A or B.)
 Relationships (in this instance  the importance of a relationship is to specify notidentity  A is not identical to B but has some connection.)
 Impossible (once we have ruled out the impossible  whatever is left... Without absolute definitions  there aren't many ways to specify first principles.)
 Change (change implies a discernible difference between before and after. See difference above.)
 Existence (I think, therefore I am. In order to exist, we need change & difference & relationships)
 First principle (until we have created 'difference'  we cannot distinguish between things. If we cannot distinguish between change, difference and relationships but require these to exist, then change, difference and relationships are indistinguishable... and are our first principle even while we cannot define exactly what that first principle is.)
 Orbits (Orbits of complex systems  one way of visualising change.)
 Relationships that change (an internal representation of change.)
 Enumerating all possible systems under this schema (Everything that is describable (and why we cannot describe anything else))
 The simplest conceivable describable systems.
Not included  pretty much everything we name/describe/define at our local level is an emergent feature of a complex system. Complex systems are well described in many places. While it isn't automatically obvious that such things as Dimensions are necessarily an emergent property  the idea of emergent properties is well enough understood that it is plausible that Dimensions are an emergent property (and exactly this has been seriously proposed already within mathematics/physics).
End post bit
Proof and disproof are opposite sides of the same coin. By applying the same symmetry to the subject of proof and disproof we find an equivalence:
(prove A = B) is equivalent to (disprove A != B)
(disprove A = B) is equivalent to (prove A != B)
For all nontrivial axiomatic systems A it is shown to be impossible to (prove A (is consistent)). However, for all axiomatic systems A there exists an axiomatic systems !A such that (prove A (is consistent)) is equivalent to (disprove !A (is consistent)) or (prove !A is inconsistent) and vice versa.
But if it is impossible to (prove A (is consistent)), then it must be impossible to (disprove !A (is consistent)). Likewise, if it is impossible to (prove !A (is consistent)) then it must be impossible to (disprove A (is consistent)).
Yet there appears to be many examples of nontrivial axiomatic systems that have been disproven or proven inconsistent.
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Re: Misunderstanding basic math concepts, help please?
What is !A, for an axiomatic system? Is it the negation of all the axioms of A? What about the rules of inference?
If A is a set of statements and !A is the set of the negations of all the statements in A, then I don't see any simple way that the consistency of either set is related to the consistency of the other. They could both be inconsistent (if A contains obviously inconsistent statements like x=y and x!=y, then !A contains x!=y and x=y), or they could presumably both be consistent.
It seems like you're still mixing levels of analysis. "x=y is false" is equivalent to "x!=y is true", but that doesn't mean "A is consistent" is equivalent to "!A is inconsistent", nor does it mean "S is provable in A" is equivalent either to "!S is provable in !A" or "!S is disprovable in A" or "S is disprovable in !A".
Simplistically:
Consistency is a quality of a set of statements given rules of inference, of which an axiomatic system is a specific example.
Provability is a quality of a statement, given a set of statements and rules of inference.
Truth is a (weaker) quality of a statement, given a model which may include a set of statements and rules of inference.
Yet you keep saying things about consistency, provability, and truth value as though they all follow the same rules and are basically interchangeable.
If A is a set of statements and !A is the set of the negations of all the statements in A, then I don't see any simple way that the consistency of either set is related to the consistency of the other. They could both be inconsistent (if A contains obviously inconsistent statements like x=y and x!=y, then !A contains x!=y and x=y), or they could presumably both be consistent.
It seems like you're still mixing levels of analysis. "x=y is false" is equivalent to "x!=y is true", but that doesn't mean "A is consistent" is equivalent to "!A is inconsistent", nor does it mean "S is provable in A" is equivalent either to "!S is provable in !A" or "!S is disprovable in A" or "S is disprovable in !A".
Simplistically:
Consistency is a quality of a set of statements given rules of inference, of which an axiomatic system is a specific example.
Provability is a quality of a statement, given a set of statements and rules of inference.
Truth is a (weaker) quality of a statement, given a model which may include a set of statements and rules of inference.
Yet you keep saying things about consistency, provability, and truth value as though they all follow the same rules and are basically interchangeable.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Relationships
The word 'relationship' here is a label. It really is impossible to define 'relationship' in an absolute sense. But we do have a sense of what 'relationships' are in general and that sense is useful. A good starting place is to think of a relationship as the most generic thing possible  something like the relationship of a set containing another set  but even more abstract.
Are you kidding me? You're going through all of these threads over all of these years talking about how there's no "basis for axiomatic mathematics" no "starting point" it is "not defined" and then you are going to let yourself get away presenting something as vague as the paragraph I've quoted as the cornerstone of your theory? Let me quote you again just so you can think about what you've said:
"It really is impossible to define 'relationship' in an absolute sense. But we do have a sense of what 'relationships' are in general and that sense is useful."
This is exactly what we say about the justification for formal languages and you remain unconvinced.
I'm going to repeat my last post which you never responded to:
Twistar wrote:Treatid wrote:{definition of 'know' here is problematic. While I don't think we can have absolute knowledge in a sense that I think axiomatic mathematics implies; I do think that we can create a complete description of the universe without needing to make any assumptions.At the same time, axiomatic mathematics cannot prove that its assumptions are 'true'. As compelling as the arguments each of you make are... they stop short of proving that axiomatic mathematics works in the way that is claimed.Specifically, if we can gather together every assumption and essential precursor to axiomatic mathematics in one place and make them all explicit we will see that there is no possible way for this group of rules to work in the way that is claimed.
I've bolded the parts of the sentences where you are talking about some claim that someone is making about axiomatic mathematics.
Questions:
1) Who is making that claim and
2) What exactly is that claim?
In your mind there is some promise that someone has made about axiomatic mathematics about what it will deliver.
A) You are correct that axiomatic mathematics does not deliver on that promise.
B) You are wrong that anyone ever made that promise about axiomatic mathematics.
Any system you try to create which fulfills this mystical promise that you have not yet explicitly stated will fail to fulfill that promise for the same reasons that axiomatic fails to fulfill that promise and for the same reasons that your proposed replacement also fails to fulfill that promise.
edit: made it less angry.
Re: Misunderstanding basic math concepts, help please?
I'm an idiot.
Regarding the End post bit:
(A != B) != !(A = B)
I should not have referred to !A. I should have referred to the complement of A: A'.
That was a stupid mistake on my part.
gmalivuk's point regarding exactly how one goes about constructing a complement of an axiomatic system A such that (prove A is consistent) is equivalent to (disprove A' is consistent) is still very much relevant.
If we take it as a given that for all A there exists A' then the possible paradox stands.
So we probably don't want to take it as a given that there exists A' for all A.
I take the point regarding levels of analysis. But I don't think I am mixing that up (you know  other than confusing 'not' for 'complement')
If we can disprove an axiomatic system (i.e. prove an axiomatic system inconsistent) then it is appropriate to consider proof with respect to axiomatic systems. As such my argument is intending to illustrate the inconsistency of being able to disprove an axiomatic system (is consistent) but not be able to prove an axiomatic system (is consistent) when both proof and disproof are proofs expressed in different ways.
@Twistar
I note your anger  but I'm not quite sure what is prompting it.
Your summary of what I'm trying to do seems pretty good to me.
Yes  I face exactly the same limitations as axiomatic mathematics currently faces with regard to creating an unequivocally, fully specified, nopossibleroomforanydegreeofdoubt, definite starting point.
The difference, I think, is that I fully embrace that limitation. I think that by embracing that limitation and following it to its logical conclusion we can arrive at a system of description that works better than axiomatic mathematics.
Whereas, I view axiomatic mathematics as trying to minimise that limitation. I see axiomatic mathematics try to suggest that our natural language understanding of particular words may not be perfect  but is good enough to justify these various conclusions.
And I do agree that our local use of words reflects our local experience of the universe such that it is productive to rely on our understanding of those words for local results. What I don't agree with is that those words have any significance outside our local experience.
There is a point where assuming that the local geometry is flat no longer works. Likewise, I think there is a point where being reasonably sure we know what words and symbols mean is insufficient to support being certain that what our mathematics says is exactly and precisely what we think it says with no possible alternative.
You may be right that I'm imputing more to axiomatic mathematics than is claimed. However:
1. The discussion over whether mathematics is discovered or invented suggests that some people believe that elements of mathematics are fundamentally true in some sense and not just 'a handy rule of thumb that appears to work  at least locally'.
2. I think we can have a mathematics without any doubt or uncertainty (although still without absolute definitions). Where we don't have to rely on informal definitions of axioms to kickstart our models.
I appreciate your scepticism that I can succeed. But I don't understand your anger at me trying.
Edit: bouncing around on what !(A = B) is or isn't.
Regarding the End post bit:
(A != B) != !(A = B)
I should not have referred to !A. I should have referred to the complement of A: A'.
That was a stupid mistake on my part.
gmalivuk's point regarding exactly how one goes about constructing a complement of an axiomatic system A such that (prove A is consistent) is equivalent to (disprove A' is consistent) is still very much relevant.
If we take it as a given that for all A there exists A' then the possible paradox stands.
So we probably don't want to take it as a given that there exists A' for all A.
I take the point regarding levels of analysis. But I don't think I am mixing that up (you know  other than confusing 'not' for 'complement')
If we can disprove an axiomatic system (i.e. prove an axiomatic system inconsistent) then it is appropriate to consider proof with respect to axiomatic systems. As such my argument is intending to illustrate the inconsistency of being able to disprove an axiomatic system (is consistent) but not be able to prove an axiomatic system (is consistent) when both proof and disproof are proofs expressed in different ways.
@Twistar
I note your anger  but I'm not quite sure what is prompting it.
Your summary of what I'm trying to do seems pretty good to me.
Yes  I face exactly the same limitations as axiomatic mathematics currently faces with regard to creating an unequivocally, fully specified, nopossibleroomforanydegreeofdoubt, definite starting point.
The difference, I think, is that I fully embrace that limitation. I think that by embracing that limitation and following it to its logical conclusion we can arrive at a system of description that works better than axiomatic mathematics.
Whereas, I view axiomatic mathematics as trying to minimise that limitation. I see axiomatic mathematics try to suggest that our natural language understanding of particular words may not be perfect  but is good enough to justify these various conclusions.
And I do agree that our local use of words reflects our local experience of the universe such that it is productive to rely on our understanding of those words for local results. What I don't agree with is that those words have any significance outside our local experience.
There is a point where assuming that the local geometry is flat no longer works. Likewise, I think there is a point where being reasonably sure we know what words and symbols mean is insufficient to support being certain that what our mathematics says is exactly and precisely what we think it says with no possible alternative.
You may be right that I'm imputing more to axiomatic mathematics than is claimed. However:
1. The discussion over whether mathematics is discovered or invented suggests that some people believe that elements of mathematics are fundamentally true in some sense and not just 'a handy rule of thumb that appears to work  at least locally'.
2. I think we can have a mathematics without any doubt or uncertainty (although still without absolute definitions). Where we don't have to rely on informal definitions of axioms to kickstart our models.
I appreciate your scepticism that I can succeed. But I don't understand your anger at me trying.
Edit: bouncing around on what !(A = B) is or isn't.
 gmalivuk
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Re: Misunderstanding basic math concepts, help please?
You're not trying, not really, and that's why people are frustrated with you. You've been making the same mistakes for four years, despite numerous people trying to explain things to you every single time you start yet another thread about the same damn thing.Treatid wrote:I don't understand your anger at me trying.
Re: Misunderstanding basic math concepts, help please?
What is the complement of an axiomatic system? Does it have the same symbols? The same wellformed formulae? The same rules of inference? The same axioms? You use the term like you expect us to know what it means, but no one else talks about the complement of a theory. Te complement of a set, I know. Of a theory, not at all. So you have to define this so that we know what you mean. This isn't me being pedantic; I actually do not know what the complement of a system could possibly mean.
You talk about disproving an axiomatic system, but no one else uses that term to mean demonstrating the system is inconsistent. In fact, no one else uses the term with regards to systems at all. To disprove a statement is to prove that the statement is false. False and inconsistent are not the same thing. I feel like you're trying to use the term "disprove" so that you can port over logic about proofs to axiomatic systems, which is exactly the kind of mixing of levels that gmalivuk was talking about. Either way, your point about proving systems consistent and inconsistent misses the mark. Godel's second incompleteness theorem only talks about systems proving themselves consistent. It says nothing about proving other systems are inconsistent. That's why ZF can prove that ZFC is consistent.
I understand Twistar's anger quite well. You spent most of this thread trash talking axiomatic mathematics due to its reliance on an informal language, and now you turn around and say that its problem is that it doesn't rely enough of informal language, and that the solution is to embrace informality at full strength. It comes across as a 180 from your previous position, as though you don't care why axiomatic mathematics gets replaced so long as it does.
Regarding your new theory, I don't understand what a relationship is. Can you provide a clearer definition? Because if not, the whole thing seems like a nonstarter.
You talk about disproving an axiomatic system, but no one else uses that term to mean demonstrating the system is inconsistent. In fact, no one else uses the term with regards to systems at all. To disprove a statement is to prove that the statement is false. False and inconsistent are not the same thing. I feel like you're trying to use the term "disprove" so that you can port over logic about proofs to axiomatic systems, which is exactly the kind of mixing of levels that gmalivuk was talking about. Either way, your point about proving systems consistent and inconsistent misses the mark. Godel's second incompleteness theorem only talks about systems proving themselves consistent. It says nothing about proving other systems are inconsistent. That's why ZF can prove that ZFC is consistent.
I understand Twistar's anger quite well. You spent most of this thread trash talking axiomatic mathematics due to its reliance on an informal language, and now you turn around and say that its problem is that it doesn't rely enough of informal language, and that the solution is to embrace informality at full strength. It comes across as a 180 from your previous position, as though you don't care why axiomatic mathematics gets replaced so long as it does.
Regarding your new theory, I don't understand what a relationship is. Can you provide a clearer definition? Because if not, the whole thing seems like a nonstarter.
(∫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.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Yes  I face exactly the same limitations as axiomatic mathematics currently faces with regard to creating an unequivocally, fully specified, nopossibleroomforanydegreeofdoubt, definite starting point. The difference, I think, is that I fully embrace that limitation.
It is exactly the other way around.
We all embrace that limitation. Among other things, we embrace those limitations because people like Godel have already proven that there's no way to circumnavigate them.
On the other hand, it seems that you  and not us  is the one looking for a loophole around these limitations. You've said in that big post of yours:
Treatid wrote:No Gaps... No Doubt... No Uncertainty...
Which is a requirement that cannot be fulfilled. It is fundamentally impossible.
Besides, your socalled "alternative to axiomatic mathematics" will also need to include some assumptions (that is: axioms) if you want it to get anywhere. The only difference is that since you're writing your framework in the sloppy language called english, you don't even notice how many assumptions you're making.
You may be right that I'm imputing more to axiomatic mathematics than is claimed. However:
The discussion over whether mathematics is discovered or invented suggests that some people believe that elements of mathematics are fundamentally true in some sense and not just 'a handy rule of thumb that appears to work  at least locally'.
They are fundamentally true, but only in the sense that once you've defined the objects you want to talk about in a certain way, certain conclusions must follow.
Is the statement "there does not exist a largest prime number" fundamentally true?
Yes... But only once we've decided what we mean  exactly  by the words "number" and "larger" and "prime". This is hardly a limitation or a cause for doubt, since without these definitions the above statement is rendered meaningless. There's no point in asking whether a statement about X is true, until we've decided what we mean by "X" in the first place.
And the only place where the "we do this because it works" ruleofthumb comes in, is when we decide which concepts interest us. We are interested in the counting numbers (as created  for example  by the axioms of PA) rather than some other alternative "number" concept because:
1. It jives with our intuitive notion of counting.
2. It is immensly useful in the physical world.
3. It forms the basis for very rich and deep mathematical theories.
These 3 reasons can be regarded as at least somewhat subjective. But even if they are, this doesn't change the fact that once we've chosen the concepts we want to talk about and defined them, we can say things about them which are definitely absolutely true.
I really don't understand why you have so much trouble getting this simple and basic point. Unless Gmalivuk is right and you're simply not trying.
As such my argument is intending to illustrate the inconsistency of being able to disprove an axiomatic system (is consistent) but not be able to prove an axiomatic system (is consistent) when both proof and disproof are proofs expressed in different ways.
There's no contradiction here. The fact that proving inconsistency is easier than proving the opposite is actually very easy to understand:
Proving the inconsistency of an axiomatic system is easy (in theory): You just need to find a single contradicition in the system. Find a single line of reasoning that leads to (say) 0=1, and you're done.
Proving the opposite is much trickier. To prove that system is consistent you need to ensure that ALL proofs in that system do not end in a contradiction. That's infinitely many proofs, so the only hope of proving consistency is by some really clever argument that covers them all.
So no, there's no symmetry.
By the way, what you call "the complement of an axiomatic system" cannot be an axiomatic system by itself, unless one of the two systems is trivial (that is: contains no axioms at all). This is very easy to prove:
1.If A is consistent (and nonempty) then:
(a) If X is a contradictory statement, then X is not in A.
(b) Therefore, X is in A'. (by your definition of "complement")
(c) Therefore, if A' was an axiomatic system, it would be an inconsistent one.
(d) Therefore, if A' was an axiomatic system, you could prove ANYTHING with it.
(e) In particular, any theorem of A would be provable in A'.
(f) Which contradicts our assumption that A' is the complement of A.
(g) Therefore if A' is the complement of A, it cannot be an axiomatic system.
QED
And if A itself is inconsistent, the proof is even shorter:
2. If A is inconsistent, then:
(a) Anything can be proven in A.
(b) Therefore, nothing can be proven in A' (by your definition of complement).
(c) In particular, A' cannot have any axioms.
QED
In other words:
The only way for your A/A' symmetry to hold is if one of those systems is inconsistent and the other is trivial (has no axioms). In any other case, if A is an axiomatic system, A' wouldn't be one.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:@Twistar
I note your anger  but I'm not quite sure what is prompting it.
Your summary of what I'm trying to do seems pretty good to me.
Yes  I face exactly the same limitations as axiomatic mathematics currently faces with regard to creating an unequivocally, fully specified, nopossibleroomforanydegreeofdoubt, definite starting point.
The difference, I think, is that I fully embrace that limitation. I think that by embracing that limitation and following it to its logical conclusion we can arrive at a system of description that works better than axiomatic mathematics.
As PsiCubed said, everyone in this thread except you fully embraces this limitation of axiomatic mathematics. YOU are the one who thinks OTHER PEOPLE don't embrace this limitation yet you haven't produced example of anyone arguing that such a limitation doesn't exist. There certainly aren't any examples in this thread.
You may be right that I'm imputing more to axiomatic mathematics than is claimed. However:
1. The discussion over whether mathematics is discovered or invented suggests that some people believe that elements of mathematics are fundamentally true in some sense and not just 'a handy rule of thumb that appears to work  at least locally'.
2. I think we can have a mathematics without any doubt or uncertainty (although still without absolute definitions). Where we don't have to rely on informal definitions of axioms to kickstart our models.
1. Ok, this is your first time citing an example of someone thinking "elements of mathematics are fundamentally true in some sense". No one in this thread thinks that. Modern mathematics founded in the axioms of set theory was invented, not discovered. The rules of set theory were invented. Period. No debate. If someone wants to debate against this point it is because they have some misunderstandings similar to yours regarding the structure of formal langauges.
2. There's going to be informal definitions somewhere and I'm not going to let you get away with a single damn informal anything after how hard you've railed against set theory. Show me a theory with nothing informal and I will be supremely impressed. It would change my world view. Show me a theory that relies on some informal definitions (even if they're different than those of set theory) and I'll tell you your theory has the same issues that set theory does and I'll yawn.
You're railing against a Platonist strawman that you've created who thinks that mathematical objects are somehow fundamentally true in some sense and we are using axiomatic set theory to discover those truths. The person who you've made up who thinks this isn't real and their beliefs aren't represented by anyone here. Maybe you can send us links of articles you find where people seem to be arguing this point and then we can tell you what we think of it, whether we agree or disagree.
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:1. Ok, this is your first time citing an example of someone thinking "elements of mathematics are fundamentally true in some sense". No one in this thread thinks that. Modern mathematics founded in the axioms of set theory was invented, not discovered. The rules of set theory were invented. Period. No debate. If someone wants to debate against this point it is because they have some misunderstandings similar to yours regarding the structure of formal langauges.
To be fair, most mathematicians do adhere to Platonism to a certain degree. You won't find many of them who'll claim that (say) Fermat's Last Theorem isn't fundamentally true  at least in some sense.
And while more complex theories (like  say  ZF) are definitely "invented", this doesn't change the fact that once we set up the rules, there's a complete "world" for us to discover. Theorems are proved, and they are either valid or not. This is a process of discovering external truths, and not inventing things as we see fit.
At least that's the way I see it. And many professional mathematicians agree with this view. Others, ofcourse, disagree. It is a matter of personal choice, which stems from where you decide to draw the line between "absolute truths" and "assumptions we should doubt". Some people will say "doubt everything", which is fine. Others would draw the line at various places, which is also fine. This a philosophical issue and not a factual/mathematical one, so it is perfectly okay for educated people to disagree on it.
At any rate, Treatid is right when he says that (at least some) mathematicians take this platonist view seriously. I'm all for calling the guy out when he spouts nonesense, but in this case he is actually correct.
It doesn't help his argument, though. What Treatid is missing, is that everyone (including himself) must assume SOME things, if we want to have a meaningful conversation about anything. Ditching the methods of axiomatic mathematics does not free yourself of the need for making assumptions. It just makes your assumptions sloppier and more prone to selfcontradiction (like what he did when assumed the existence of "the complementary axiomatic system" of A).
To summarize:
Educated people may have different views regarding how "absolute" the truths of mathematics are. But no educated person will ever suggest that pulling assumptions out of hat is a better way than writing them as axioms in a rigorous and orderedly fashion.
Re: Misunderstanding basic math concepts, help please?
PsiCubed wrote:Twistar wrote:1. Ok, this is your first time citing an example of someone thinking "elements of mathematics are fundamentally true in some sense". No one in this thread thinks that. Modern mathematics founded in the axioms of set theory was invented, not discovered. The rules of set theory were invented. Period. No debate. If someone wants to debate against this point it is because they have some misunderstandings similar to yours regarding the structure of formal langauges.
To be fair, most mathematicians do adhere to Platonism to a certain degree. You won't find many of them who'll claim that (say) Fermat's Last Theorem isn't fundamentally true  at least in some sense.
But there's the rub. If you say Fermat's last theorem is fundamentally true then all of Treatid's arguments against axiomatic mathematics hold water against you. So you hedge it by saying "at least in some sense". When push comes to shove this is just a confusing and misleading way of sweeping under the rug the reality that we can never have 100% certain knowledge or understanding of "fundamental truths". In my opinion it's better just admit outright the fact that math doesn't provide absolute knowledge and then go from there. Formal languages can do a fine job of this if thought about/presented in the right way.
And while more complex theories (like  say  ZF) are definitely "invented", this doesn't change the fact that once we set up the rules, there's a complete "world" for us to discover. Theorems are proved, and they are either valid or not. This is a process of discovering external truths, and not inventing things as we see fit.
Yes but those truths are all relative to whatever you set up at the outset. This discussion in this thread, at it's heart, isn't about what happens after the rules are set up (which I agree might be a discovery thing) it is about what is happening while the rules are being set up. If you try to say the axioms are discovered, or you're not clear about the fact that the axioms are invented, then you're not communicating clearly and I'm reminded of comic 169.
At least that's the way I see it. And many professional mathematicians agree with this view. Others, ofcourse, disagree. It is a matter of personal choice, which stems from where you decide to draw the line between "absolute truths" and "assumptions we should doubt". Some people will say "doubt everything", which is fine. Others would draw the line at various places, which is also fine. This a philosophical issue and not a factual/mathematical one, so it is perfectly okay for educated people to disagree on it.
At any rate, Treatid is right when he says that (at least some) mathematicians take this platonist view seriously. I'm all for calling the guy out when he spouts nonesense, but in this case he is actually correct.
Well, seen as an argument against those mathematicians who take the platonist view seriously, most of what Treatid is saying makes a lot of sense and constitutes a legitimate attack against that point of view.
Anyways, clearly I'm antiPlatonist when it comes to axiomatic mathematics. A large part of the reason for that is because the philosophy does radiate out the idea that math somehow finds and communicates absolute truths. However, for all of the reasons Treatid and everyone else in this thread have pointed out, that is actually NOT the case. All that happens is you get the potential to confuse people like Treatid.
Maybe a question to everyone in the thread Agree or Disagree (no hedging, no explanation, one word answer which has to be Agree or Disagree) to the following statement:
"Mathematics does not discover or communicate any absolute truths"
Twistar: Agree
edit: and by absolute I mean absolute, no bullshit, not relative to anything. The strongest sense of the word. The obvious sense of the word. I'm trying to be clear, not confusing.
Re: Misunderstanding basic math concepts, help please?
For Twistar's question: Agree.
Other than that, all I had to say was already pointed out, and in my previous post, and it seems that treatid decided to ignore the questions I raised in that post.
Other than that, all I had to say was already pointed out, and in my previous post, and it seems that treatid decided to ignore the questions I raised in that post.

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Re: Misunderstanding basic math concepts, help please?
Disagree.
You're asking a philosophical question that hinges on the distinction, if there is any, between truth and knowledge. For all the reasons we've explored here, human confidence in mathematical truth can never be absolute. At some point you get back to What the Tortoise Said to Achilles and you're forced to take basic logical concepts for granted. If you define "absolute truth" to mean "absolute epistemic certainty", then mathematics never deals in "absolute truth". But if you take truth as a primitive concept in its own right, which I do, then something can be "absolute truth" even if we can never be absolutely certain of it. I place logical proofs in that category.
You're asking a philosophical question that hinges on the distinction, if there is any, between truth and knowledge. For all the reasons we've explored here, human confidence in mathematical truth can never be absolute. At some point you get back to What the Tortoise Said to Achilles and you're forced to take basic logical concepts for granted. If you define "absolute truth" to mean "absolute epistemic certainty", then mathematics never deals in "absolute truth". But if you take truth as a primitive concept in its own right, which I do, then something can be "absolute truth" even if we can never be absolutely certain of it. I place logical proofs in that category.
Re: Misunderstanding basic math concepts, help please?
arbiteroftruth wrote:Disagree.
You're asking a philosophical question that hinges on the distinction, if there is any, between truth and knowledge. For all the reasons we've explored here, human confidence in mathematical truth can never be absolute. At some point you get back to What the Tortoise Said to Achilles and you're forced to take basic logical concepts for granted. If you define "absolute truth" to mean "absolute epistemic certainty", then mathematics never deals in "absolute truth". But if you take truth as a primitive concept in its own right, which I do, then something can be "absolute truth" even if we can never be absolutely certain of it. I place logical proofs in that category.
You lost me. I'm definitely in the first camp of defining "absolute truth" to mean "absolute epistemic certainty". Your next sentence didn't make sense to me, can you explain more?
But if you take truth as a primitive concept in its own right, which I do, then something can be "absolute truth" even if we can never be absolutely certain of it.
I understand the possibility that something can be absolute truth but that we can't be certain of it. Basically something can be true but humans just can't know with certainty that it's true. However, there's yet a further step needed to answer my original question. Let's say things can be absolutely true (but humans can't know it), the question remains if mathematics gets at that absolute truth.
Said another way, it's possible that
1) there is such a thing as absolute truth
2) Human's can't know absolute truth with absolute certainty
3) axiomatic mathematics also doesn't get at the essence of absolute truth
In any case, are you opposed to the idea of defining "absolute truth" to mean "absolute epistemic certainty" for purposes of this discussion?

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Re: Misunderstanding basic math concepts, help please?
Twistar wrote:
I understand the possibility that something can be absolute truth but that we can't be certain of it. Basically something can be true but humans just can't know with certainty that it's true. However, there's yet a further step needed to answer my original question. Let's say things can be absolutely true (but humans can't know it), the question remains if mathematics gets at that absolute truth.
Said another way, it's possible that
1) there is such a thing as absolute truth
2) Human's can't know absolute truth with absolute certainty
3) axiomatic mathematics also doesn't get at the essence of absolute truth
Sure, that's possible in the epistemic sense. But with "disagree" I'm asserting that that possibility is not actually the case. That's not a proof of course, but you simply asked what we thought on the matter.
In any case, are you opposed to the idea of defining "absolute truth" to mean "absolute epistemic certainty" for purposes of this discussion?
For the purposes of discussion, I'm fine with that. And by that definition, I agree that math never accesses "absolute truth".
Re: Misunderstanding basic math concepts, help please?
I believe that the one provable truth is the cogito. I also believe that there is an external reality, populated with people generically like myself (though different in detail) and that mathematics provides highly accurate descriptions of aspects of that external reality.
Whether that's "agree" or "disagree", I leave to people who understand the question to answer.
Whether that's "agree" or "disagree", I leave to people who understand the question to answer.
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:But there's the rub. If you say Fermat's last theorem is fundamentally true then all of Treatid's arguments against axiomatic mathematics hold water against you.
Not really.
Because Treatid's arguments can be used against ANYTHING. Including, by the way, his very own ideas.
Sure, philosophically speaking, there's always place for doubt. Maybe our entire understanding of logic is faulty. Maybe our most basic concepts of numbers and sets, even though they seem obvious, are inconsistent. Or maybe we are all living in a giant simulation, and what we think as "logic" and "pure thought" is nothing more than an elaborate prank pulled on us by the guy in charge of the everything.
But since we have absolutely no way of testing these ideas or putting them to use, they are of zero practical value.
And none of this changes the fact that we are far more sure of things like Fermat's last theorem, then we are sure of nonmathematical claims we think are true. You want to say that we aren't 100% certain? Fine. But we are as certain of the validity of such truths as mere mortals can ever be certain of anything.
Twistar wrote:So you hedge it by saying "at least in some sense". When push comes to shove this is just a confusing and misleading way of sweeping under the rug the reality that we can never have 100% certain knowledge or understanding of "fundamental truths".
Not at all.
First of all, the phrase "in some sense" is a direct quote from Treatid, rather than some apologetic phrase I've come up with.
Secondly, his use of this term is quite apt. Mathematical truths can only be regarded as true within their respective axiomatic systems. The absolute truths in mathematics take the general form of "Theorem X follows from axioms X,Y,Z". They do not take the form of "Theorem X is absolutely true on its own". Such statements are not even untrue, because without a set of axioms and definitions, "Theorem X" is just a meaningless set of symbols.
Acknowledging this fact isn't sweeping things under the rug. It is a simple statement regarding the way mathematics really work.
In my opinion it's better just admit outright the fact that math doesn't provide absolute knowledge and then go from there. Formal languages can do a fine job of this if thought about/presented in the right way.
It is better to stick to the truth.
And the truth is that:
(1) the entire debate of platonism vs "math is just a tool that works" is completely irelevant to the validity of Treatid's claims (or lack of).
(2) When Treatid spoke of people who believe that "mathematical results are, in a sense, fundamentally and absolutely true", he wasn't creating a strawman. There are many mathematicians who are platonists at heart, and denying such an obvious fact isn't going to help you get your points across.
Yes but those truths are all relative to whatever you set up at the outset. This discussion in this thread, at it's heart, isn't about what happens after the rules are set up (which I agree might be a discovery thing) it is about what is happening while the rules are being set up. If you try to say the axioms are discovered, or you're not clear about the fact that the axioms are invented, then you're not communicating clearly and I'm reminded of comic 169.
It isn't that simple.
Yes, the axioms are invented. But usually axioms are chosen because they represent something we are familiar with in the real world.
Take numbers, for example. Peano invented his axioms in order to create a structure which would conform with the properties we usually associate with the counting numbers. Things like "every number has a successor" or the axiom of induction. While the axioms of PA were technically invented by man, they were also constructed in a very specific way to represent what we intuitively know as "the world of natural numbers".
So what  exactly  is the difference between saying "Theorem X is true in the context of PA" and saying "Theorem X is an absolutely true statement about natural numbers"? Since PA was constructed in a way that ordinary numbers are a model for it, these two statements say exactly the same thing!
Well, seen as an argument against those mathematicians who take the platonist view seriously, most of what Treatid is saying makes a lot of sense and constitutes a legitimate attack against that point of view.
If so, then what is exactly your problem with him?
The simple fact is that many mathematicians are platonists. So if you're regarding his statements as a justified attack on the platonist point of view, why do you debate him instead of joining him?
(Ofcourse, as you probably expect, I strongly disagree with your assessment)
Maybe a question to everyone in the thread Agree or Disagree (no hedging, no explanation, one word answer which has to be Agree or Disagree) to the following statement:
"Mathematics does not discover or communicate any absolute truths"
Since you're forcing me to answer with a single word: Disagree.
But any single word answer would be misleading, as I've already pointed out.
Forcing people to give one word answers to questions which are genuinely complicated, does not count as "being clear". It is exactly these kinds of blackandwhite extreme views which confuse people like Treatid.
At any rate, looking at the answers of the people who responded to your poll we can see that:
1. The actual answers split about 50:50
2. Pretty much nobody was willing to leave his answer uncommented.
Which just proves that your simplistic approach of "yes or no" is inadequate.
Re: Misunderstanding basic math concepts, help please?
Before responding to specifics I'll just say that I think we agree on more things that it is sounding like at the moment. I'm just taking a hard stance against the notion that mathematics can access "absolute truths" because I think holding on to such a notion is a great cause for confusion and this whole thread is devoted to stamping out certain confusions. Confusions which are extremely close to the idea that mathematics can access "absolute truths".
I agree with all of this. The last bit is important. I emphatically DO want to say that we aren't 100% certain, however, I still also want to emphasize the fact that mathematical knowledge does give us more certainty about things than do other ways of knowing.
I obviously agree with you that mathematical truths only make sense relative to whatever (human) defined system we are working in at the time. We differ in our definitions of "fundamental truth". You seem to imply that the statement "Theorem X follows from axioms X,Y,Z" can be absolutely true under the right conditions. I, however, would not ascribe the title "fundamentally true" to such a statement. In my eyes, it is a much more mundane observation about one of the emergent features of whatever system we are working in. My statement would be something like "Theorem X follows from axioms X,Y,Z relative to system S" or something of the sort. That the statement has to be referenced to whatever specific system we are working in removes it as a candidate for "fundamental truth".
I'll just be clear that I realize I'm setting an absurdly (impossibly) high bar for "fundamental truth" by how I'm defining it (complete epistemic certainty) and I'm not surprised that nothing can live up to it. Basically, my thinking is that when many people talk about "fundamental" or "absolute" truth they are using the same bar I am, so I go with that definition. Furthermore, I think that there is better language that can be used to describe statements which don't rise to the bar I am setting. See the previous paragraph.
(1) A few things here. I think we might have some confusion.
a) the debate regarding Platonism is clearly relevant to this thread given how much we are worried about the absolute vs. relative truth of axioms in this thread. It is also clearly Platonistic ideas that have driven Treatid to try to rewrite mathematics.
b) Not sure which claims either of us are talking about. A lot of Treatid's claims would be valid against someone trying to argue a Platonistic interpretation of mathematics. However, many of his claims are misguided, confused, and badly communicated still.
(2) You're right. I was too quick to dismiss platonists. Basically I harshly dismiss that philosophy precisely because it can lead to confusions like those in this thread so I think I was seeking to put it down harshly to try to stamp out the confusion. But you're right that it's better to recognize the fact that actually a lot of mathematicians are platonists at heart and go from there. I guess a conclusion that I draw at this point is that someone who is a Platonist is going to have a very different conversation with Treatid trying to get him to realize that what he's proposing isn't better than mathematics as we have it now than the conversation I would want to have with Treatid. Basically I can't defend Platonism to Treatid because I think it is the root of his confusion.
So, ok, I'll admit my response here is going to be weak and might sound a little weird but here goes: First, I think there is a difference between "Theorem X is true in the context of PA" and "Theorem X is an absolutely true statement about natural numbers". It's not clear how you're using "natural numbers" in the second sentence. Something like "humans were using natural numbers before the peano axioms, those are what I mean by 'natural numbers'". So, what happens next in my mind is I boil this down and say well, the natural numbers are somehow intrinsic to humans, and how we think about the world. So I would be down with the statement "Theorem X is a true statement relative to how humans naturally think about the world" or something like that. The point is the phrase "absolute truth" will always have a more privileged place than to be able to be applied to any particular theorem.
I'll also note that the ideas in this paragraph stem from my personal idea that the axioms humans lay out for mathematics is reflective of their own psychology as opposed to something intrinsic about the world.
Hmmm, I can say it in a few words to make a strong point or I could try to use more words to soften the point but I don't think I will.
I think Treatid is a platonist at heart, however, he sees the flaws with platonism as they are manifested in the foundations of formal language. As such, he is trying to come up with a new system to get around these perceived flaws to recover something resembling the absolute certainty which he seeks. I won't join him because I think very strongly that the idea of absolute certainty is at best a fools quest and worst an impediment to furthering understanding.
I'll say again that I don't think all of his statements are justified attacks against the platonist point of view. Many of his ideas are ill informed. However, one of the main thrusts of his argument is a valid attack against platonism which I agree with. I guess this is a big reason why I'm sympathetic to him.
Well I'm surprised people are saying they disagree, so from where I'm sitting I draw two conclusions.
1) I understand Treatid's confusion a little more.
2) I see that the rest of us in the thread probably aren't arguing a unified front with Treatid. It seems to me that I can have a particular conversation with Treatid based on my belief that mathematics cannot access absolute truth. Someone who believes that mathematics can access absolute truth is going to have to have a very different conversation with him. Maybe we should think about how to approach that in this thread.
It's fine that people leave it commented I guess. I just wanted to make sure people did give a definitive answer one way or the other. I'm happy with the responses people have given! They're interesting! Like I said I'm surprised the answer is split, so that makes me optimistic that maybe we'll learn something.
PsiCubed wrote:Twistar wrote:But there's the rub. If you say Fermat's last theorem is fundamentally true then all of Treatid's arguments against axiomatic mathematics hold water against you.
Not really.
Because Treatid's arguments can be used against ANYTHING. Including, by the way, his very own ideas.
Sure, philosophically speaking, there's always place for doubt. Maybe our entire understanding of logic is faulty. Maybe our most basic concepts of numbers and sets, even though they seem obvious, are inconsistent. Or maybe we are all living in a giant simulation, and what we think as "logic" and "pure thought" is nothing more than an elaborate prank pulled on us by the guy in charge of the everything.
But since we have absolutely no way of testing these ideas or putting them to use, they are of zero practical value.
And none of this changes the fact that we are far more sure of things like Fermat's last theorem, then we are sure of nonmathematical claims we think are true. You want to say that we aren't 100% certain? Fine. But we are as certain of the validity of such truths as mere mortals can ever be certain of anything.
I agree with all of this. The last bit is important. I emphatically DO want to say that we aren't 100% certain, however, I still also want to emphasize the fact that mathematical knowledge does give us more certainty about things than do other ways of knowing.
Twistar wrote:So you hedge it by saying "at least in some sense". When push comes to shove this is just a confusing and misleading way of sweeping under the rug the reality that we can never have 100% certain knowledge or understanding of "fundamental truths".
Not at all.
First of all, the phrase "in some sense" is a direct quote from Treatid, rather than some apologetic phrase I've come up with.
Secondly, his use of this term is quite apt. Mathematical truths can only be regarded as true within their respective axiomatic systems. The absolute truths in mathematics take the general form of "Theorem X follows from axioms X,Y,Z". They do not take the form of "Theorem X is absolutely true on its own". Such statements are not even untrue, because without a set of axioms and definitions, "Theorem X" is just a meaningless set of symbols.
Acknowledging this fact isn't sweeping things under the rug. It is a simple statement regarding the way mathematics really work.
I obviously agree with you that mathematical truths only make sense relative to whatever (human) defined system we are working in at the time. We differ in our definitions of "fundamental truth". You seem to imply that the statement "Theorem X follows from axioms X,Y,Z" can be absolutely true under the right conditions. I, however, would not ascribe the title "fundamentally true" to such a statement. In my eyes, it is a much more mundane observation about one of the emergent features of whatever system we are working in. My statement would be something like "Theorem X follows from axioms X,Y,Z relative to system S" or something of the sort. That the statement has to be referenced to whatever specific system we are working in removes it as a candidate for "fundamental truth".
I'll just be clear that I realize I'm setting an absurdly (impossibly) high bar for "fundamental truth" by how I'm defining it (complete epistemic certainty) and I'm not surprised that nothing can live up to it. Basically, my thinking is that when many people talk about "fundamental" or "absolute" truth they are using the same bar I am, so I go with that definition. Furthermore, I think that there is better language that can be used to describe statements which don't rise to the bar I am setting. See the previous paragraph.
In my opinion it's better just admit outright the fact that math doesn't provide absolute knowledge and then go from there. Formal languages can do a fine job of this if thought about/presented in the right way.
It is better to stick to the truth.
And the truth is that:
(1) the entire debate of platonism vs "math is just a tool that works" is completely irelevant to the validity of Treatid's claims (or lack of).
(2) When Treatid spoke of people who believe that "mathematical results are, in a sense, fundamentally and absolutely true", he wasn't creating a strawman. There are many mathematicians who are platonists at heart, and denying such an obvious fact isn't going to help you get your points across.
(1) A few things here. I think we might have some confusion.
a) the debate regarding Platonism is clearly relevant to this thread given how much we are worried about the absolute vs. relative truth of axioms in this thread. It is also clearly Platonistic ideas that have driven Treatid to try to rewrite mathematics.
b) Not sure which claims either of us are talking about. A lot of Treatid's claims would be valid against someone trying to argue a Platonistic interpretation of mathematics. However, many of his claims are misguided, confused, and badly communicated still.
(2) You're right. I was too quick to dismiss platonists. Basically I harshly dismiss that philosophy precisely because it can lead to confusions like those in this thread so I think I was seeking to put it down harshly to try to stamp out the confusion. But you're right that it's better to recognize the fact that actually a lot of mathematicians are platonists at heart and go from there. I guess a conclusion that I draw at this point is that someone who is a Platonist is going to have a very different conversation with Treatid trying to get him to realize that what he's proposing isn't better than mathematics as we have it now than the conversation I would want to have with Treatid. Basically I can't defend Platonism to Treatid because I think it is the root of his confusion.
Yes but those truths are all relative to whatever you set up at the outset. This discussion in this thread, at it's heart, isn't about what happens after the rules are set up (which I agree might be a discovery thing) it is about what is happening while the rules are being set up. If you try to say the axioms are discovered, or you're not clear about the fact that the axioms are invented, then you're not communicating clearly and I'm reminded of comic 169.
It isn't that simple.
Yes, the axioms are invented. But usually axioms are chosen because they represent something we are familiar with in the real world.
Take numbers, for example. Peano invented his axioms in order to create a structure which would conform with the properties we usually associate with the counting numbers. Things like "every number has a successor" or the axiom of induction. While the axioms of PA were technically invented by man, they were also constructed in a very specific way to represent what we intuitively know as "the world of natural numbers".
So what  exactly  is the difference between saying "Theorem X is true in the context of PA" and saying "Theorem X is an absolutely true statement about natural numbers"? Since PA was constructed in a way that ordinary numbers are a model for it, these two statements say exactly the same thing!
So, ok, I'll admit my response here is going to be weak and might sound a little weird but here goes: First, I think there is a difference between "Theorem X is true in the context of PA" and "Theorem X is an absolutely true statement about natural numbers". It's not clear how you're using "natural numbers" in the second sentence. Something like "humans were using natural numbers before the peano axioms, those are what I mean by 'natural numbers'". So, what happens next in my mind is I boil this down and say well, the natural numbers are somehow intrinsic to humans, and how we think about the world. So I would be down with the statement "Theorem X is a true statement relative to how humans naturally think about the world" or something like that. The point is the phrase "absolute truth" will always have a more privileged place than to be able to be applied to any particular theorem.
I'll also note that the ideas in this paragraph stem from my personal idea that the axioms humans lay out for mathematics is reflective of their own psychology as opposed to something intrinsic about the world.
Well, seen as an argument against those mathematicians who take the platonist view seriously, most of what Treatid is saying makes a lot of sense and constitutes a legitimate attack against that point of view.
If so, then what is exactly your problem with him?
The simple fact is that many mathematicians are platonists. So if you're regarding his statements as a justified attack on the platonist point of view, why do you debate him instead of joining him?
(Ofcourse, as you probably expect, I strongly disagree with your assessment)
Hmmm, I can say it in a few words to make a strong point or I could try to use more words to soften the point but I don't think I will.
I think Treatid is a platonist at heart, however, he sees the flaws with platonism as they are manifested in the foundations of formal language. As such, he is trying to come up with a new system to get around these perceived flaws to recover something resembling the absolute certainty which he seeks. I won't join him because I think very strongly that the idea of absolute certainty is at best a fools quest and worst an impediment to furthering understanding.
I'll say again that I don't think all of his statements are justified attacks against the platonist point of view. Many of his ideas are ill informed. However, one of the main thrusts of his argument is a valid attack against platonism which I agree with. I guess this is a big reason why I'm sympathetic to him.
Maybe a question to everyone in the thread Agree or Disagree (no hedging, no explanation, one word answer which has to be Agree or Disagree) to the following statement:
"Mathematics does not discover or communicate any absolute truths"
Since you're forcing me to answer with a single word: Disagree.
But any single word answer would be misleading, as I've already pointed out.
Forcing people to give one word answers to questions which are genuinely complicated, does not count as "being clear". It is exactly these kinds of blackandwhite extreme views which confuse people like Treatid.
Well I'm surprised people are saying they disagree, so from where I'm sitting I draw two conclusions.
1) I understand Treatid's confusion a little more.
2) I see that the rest of us in the thread probably aren't arguing a unified front with Treatid. It seems to me that I can have a particular conversation with Treatid based on my belief that mathematics cannot access absolute truth. Someone who believes that mathematics can access absolute truth is going to have to have a very different conversation with him. Maybe we should think about how to approach that in this thread.
At any rate, looking at the answers of the people who responded to your poll we can see that:
1. The actual answers split about 50:50
2. Pretty much nobody was willing to leave his answer uncommented.
Which just proves that your simplistic approach of "yes or no" is inadequate.
It's fine that people leave it commented I guess. I just wanted to make sure people did give a definitive answer one way or the other. I'm happy with the responses people have given! They're interesting! Like I said I'm surprised the answer is split, so that makes me optimistic that maybe we'll learn something.
 Cleverbeans
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 Joined: Wed Mar 26, 2008 1:16 pm UTC
Re: Misunderstanding basic math concepts, help please?
I recall having a similar crisis of confidence as I began to understand exactly how limited mathematical models are in practice. I've come to the following conclusions:
1) Mathematical knowledge is the most certain and enduring form of knowledge we have.
2) Mathematics only vaguely resembles reality and we have to simplify our assumptions about the world to make it fit the data.
3) Most laymen are not aware that math is vague and imperfect which leads to unjustified faith in the math.
It's the third that leads to most conflicts about axiomatic methods. Some high school teacher says math is the universal language and since math skills are rewarded socially they're encouraged to revere the subject. Nothing could be further from the truth however. It's just another kind of language that follows highly simplified rules so we can reach logical conclusions but only a very small part of human knowledge can be described this way. Your objection that none of these axioms are true is of course fully correct. That's why they're called assumptions not truths.
Treatid do you think you're capable of changing your mind on this subject or are you just looking for more flaws in the established system? I'm starting to think you're being disingenuous at this point.
1) Mathematical knowledge is the most certain and enduring form of knowledge we have.
2) Mathematics only vaguely resembles reality and we have to simplify our assumptions about the world to make it fit the data.
3) Most laymen are not aware that math is vague and imperfect which leads to unjustified faith in the math.
It's the third that leads to most conflicts about axiomatic methods. Some high school teacher says math is the universal language and since math skills are rewarded socially they're encouraged to revere the subject. Nothing could be further from the truth however. It's just another kind of language that follows highly simplified rules so we can reach logical conclusions but only a very small part of human knowledge can be described this way. Your objection that none of these axioms are true is of course fully correct. That's why they're called assumptions not truths.
Treatid do you think you're capable of changing your mind on this subject or are you just looking for more flaws in the established system? I'm starting to think you're being disingenuous at this point.
"Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration."  Abraham Lincoln
Re: Misunderstanding basic math concepts, help please?
Cleverbeans wrote:I recall having a similar crisis of confidence as I began to understand exactly how limited mathematical models are in practice. I've come to the following conclusions:
1) Mathematical knowledge is the most certain and enduring form of knowledge we have.
2) Mathematics only vaguely resembles reality and we have to simplify our assumptions about the world to make it fit the data.
3) Most laymen are not aware that math is vague and imperfect which leads to unjustified faith in the math.
It's the third that leads to most conflicts about axiomatic methods. Some high school teacher says math is the universal language and since math skills are rewarded socially they're encouraged to revere the subject. Nothing could be further from the truth however. It's just another kind of language that follows highly simplified rules so we can reach logical conclusions but only a very small part of human knowledge can be described this way. Your objection that none of these axioms are true is of course fully correct. That's why they're called assumptions not truths.
Treatid do you think you're capable of changing your mind on this subject or are you just looking for more flaws in the established system? I'm starting to think you're being disingenuous at this point.
Nice. Yeah this is what I'm trying to get at with the whole "cause of Treatid's confusion" deal. Thanks for saying it a lot more clearly and leaving out all the philosophy junk!
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:Before responding to specifics I'll just say that I think we agree on more things that it is sounding like at the moment.
You're probably right.
Which is kinda the point I was trying to make: This philosophical divide between the platonists and the nonplatonists doesn't really matter, as long as we agree on the way mathematics actually work (and we do).
Twistar wrote:I obviously agree with you that mathematical truths only make sense relative to whatever (human) defined system we are working in at the time. We differ in our definitions of "fundamental truth". You seem to imply that the statement "Theorem X follows from axioms X,Y,Z" can be absolutely true under the right conditions. I, however, would not ascribe the title "fundamentally true" to such a statement. In my eyes, it is a much more mundane observation about one of the emergent features of whatever system we are working in. My statement would be something like "Theorem X follows from axioms X,Y,Z relative to system S" or something of the sort. That the statement has to be referenced to whatever specific system we are working in removes it as a candidate for "fundamental truth".
See, here I don't agree.
I don't see anything "mundane" in the fact that countless of complex theorems can be proven from any small set of axioms, let alone from a small set of seemingly obvious axioms. It is nothing short of mindboggling that this can be done.
Besides, the suffix of "from axioms X,Y,Z relative to system S" is not something that is artificially limiting the scope of Theorem X. It is simply telling us what Theorem X is talking about (plus refering to the basic rules of deduction). So I'm sorry, but I don't understand why you think this makes the statement any less "absolute".
(1) A few things here. I think we might have some confusion.
a) the debate regarding Platonism is clearly relevant to this thread given how much we are worried about the absolute vs. relative truth of axioms in this thread. It is also clearly Platonistic ideas that have driven Treatid to try to rewrite mathematics.
It would be more accurate to say that what driven Treatid to try and rewrite mathematics is a very extreme (and misguided) version of Platonism which no serious mathematician believes in.
So attacking platonism as whole isn't going help his confusion. On the contrary: It serves to deepen the confusion, because it lumps up together Treatid's own misguided ideas with something completely different
(2) You're right. I was too quick to dismiss platonists. Basically I harshly dismiss that philosophy precisely because it can lead to confusions like those in this thread so I think I was seeking to put it down harshly to try to stamp out the confusion
The problem is that this very same excuse can be used to dismiss anything.
For example, one could argue that your own belief about "lack of any absolute truths" could lead to some kind of postmodernism where anything goes. Or that it could lead people to conclude that mathematics is useless.
So, should people dismiss your view "because it can lead to confusion"? I'm sure you wouldn't appreciate it if someone did that.
As to this specific thread:
The questions that Treatid originally raised here are complicated. If one isn't prepared to think of the finer points when discussing the foundations and limitations of mathematics, he will be confused. And if others try to oversimplify the issue to the point of losing the complexities of the original discussion, this serves to perpetuate said confusion.
1) I understand Treatid's confusion a little more.
2) I see that the rest of us in the thread probably aren't arguing a unified front with Treatid. It seems to me that I can have a particular conversation with Treatid based on my belief that mathematics cannot access absolute truth. Someone who believes that mathematics can access absolute truth is going to have to have a very different conversation with him. Maybe we should think about how to approach that in this thread.
Well, I think we all agree that meeting Treatid's standard of "absolute truth" is impossible to achieve. He's setting the bar about as high as you do, which is a completely impossible standard for any human endeavor.
And I don't really think that being a platonist, I should have a problem with that. I fully agree that we humans can never be 100% certain of anything, and don't see any contradiction between this and voicing the opinion that abstract ideas (such as numbers) have some absolute existence "out there".
Re: Misunderstanding basic math concepts, help please?
Ok, Platonism can be more nuanced than I give it credit for. Apologies. I still think it's confusing and misleading and don't like it.
I guess the moral of this sub discussion we've had is that you're using a different definition for absolute truth than I am and I suspect Treatid is. I'm pretty sure that if we define "absolute truth" as "complete epistemic certainty" you would agree that math does not access "absolute truth" defined in this sense.
If we define "absolute truth" in the way that I think you prefer to define it then math CAN access "absolute truth".
Does that sum up the disagreement/confusion?
PsiCubed wrote:Well, I think we all agree that meeting Treatid's standard of "absolute truth" is impossible to achieve. He's setting the bar about as high as you do, which is a completely impossible standard for any human endeavor.
And I don't really think that being a platonist, I should have a problem with that. I fully agree that we humans can never be 100% certain of anything, and don't see any contradiction between this and voicing the opinion that abstract ideas (such as numbers) have some absolute existence "out there".
I guess the moral of this sub discussion we've had is that you're using a different definition for absolute truth than I am and I suspect Treatid is. I'm pretty sure that if we define "absolute truth" as "complete epistemic certainty" you would agree that math does not access "absolute truth" defined in this sense.
If we define "absolute truth" in the way that I think you prefer to define it then math CAN access "absolute truth".
Does that sum up the disagreement/confusion?
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:I guess the moral of this sub discussion we've had is that you're using a different definition for absolute truth than I am and I suspect Treatid is. I'm pretty sure that if we define "absolute truth" as "complete epistemic certainty" you would agree that math does not access "absolute truth" defined in this sense.
Probably. I'm not sure what you mean by "complete epistemic certainty", though.
If by that you mean something that we  on the human end of things  are 100% certain of, then I agree.
But in my view, the existence of an absolute truth should not depend on our ability to know that it is absolute. To me, the requirement that "humans must be 100% certain that it is an absolute truth" is adding a needless anthropocentric bias which has nothing to do with objective facts. So I can happily say that things like "2+2=4" are objective absolute truths, while still acknowledging that we'll never be 100% certain of that.
On the other hand, I can also see where you're coming from, too.
It's interesting. To you this additional requirement of human certainty is "setting the bar higher". To me, it is the exact opposite. And in a way, we are both right. Perhaps the bigger moral here is that the concept of "a purely absolute truth" is logically untenable. Funnily enough, any discussion of the "absolute" requires us to first answer the question "absolute by what standards?", and there's no "catchall" golden standard that will be completely satisfactory.
(the only exception to this is Descartes' cogito, which was already mentioned a few posts ago. Since the cogito is already an anthropocentric statement, our two standards do not contradict one another in this specific case)
Re: Misunderstanding basic math concepts, help please?
Cleverbeans wrote:3) Most laymen are not aware that math is vague and imperfect which leads to unjustified faith in the math.
It's the third that leads to most conflicts about axiomatic methods. Some high school teacher says math is the universal language and since math skills are rewarded socially they're encouraged to revere the subject.
Huh?
In what way is math "vague"? Imperfect  I agree. Nothing is perfect. But vague?
Also, the laymen revere math? Math skills are socially rewarded? That is a very strange land of which you speak. Never seen such a place in my life, unfortunately.
Perhaps you meant to say that the laymen believe that the mathematicians themselves think math is perfect? Because in a way, that's the basis for what Treatid is arguing here: "you guys think math is perfect. I'm the only person who accepts the fact that it isn't".
Re: Misunderstanding basic math concepts, help please?
PsiCubed wrote:Twistar wrote:I guess the moral of this sub discussion we've had is that you're using a different definition for absolute truth than I am and I suspect Treatid is. I'm pretty sure that if we define "absolute truth" as "complete epistemic certainty" you would agree that math does not access "absolute truth" defined in this sense.
Probably. I'm not sure what you mean by "complete epistemic certainty", though.
If by that you mean something that we  on the human end of things  are 100% certain of, then I agree.
Yes that is what I mean. In the end "absolute truth" is two words and I define them to mean this particular thing. I take this meaning because I think it is the meaning most people intuitively associate with those words when they hear them.
But in my view, the existence of an absolute truth should not depend on our ability to know that it is absolute. To me, the requirement that "humans must be 100% certain that it is an absolute truth" is adding a needless anthropocentric bias which has nothing to do with objective facts. So I can happily say that things like "2+2=4" are objective absolute truths, while still acknowledging that we'll never be 100% certain of that.
On the other hand, I can also see where you're coming from, too.
It's interesting. To you this additional requirement of human certainty is "setting the bar higher". To me, it is the exact opposite. And in a way, we are both right. Perhaps the bigger moral here is that the concept of "a purely absolute truth" is logically untenable. Funnily enough, any discussion of the "absolute" requires us to first answer the question "absolute by what standards?", and there's no "catchall" golden standard that will be completely satisfactory.
absolute by the highest standards thinkable is what I'm going for.
PsiCubed wrote:Cleverbeans wrote:
3) Most laymen are not aware that math is vague and imperfect which leads to unjustified faith in the math.
It's the third that leads to most conflicts about axiomatic methods. Some high school teacher says math is the universal language and since math skills are rewarded socially they're encouraged to revere the subject.
Huh?
In what way is math "vague"? Imperfect  I agree. Nothing is perfect. But vague?
Also, the laymen revere math? Math skills are socially rewarded? That is a very strange land of which you speak. Never seen such a place in my life, unfortunately.
Perhaps you meant to say that the laymen believe that the mathematicians themselves think math is perfect? Because in a way, that's the basis for what Treatid is arguing here: "you guys think math is perfect. I'm the only person who accepts the fact that it isn't".
Math is vague in the sense that we have to at some point rely on informal language to define mathematical objects and ideas such as "set" and "is an element of" and a few other things. It is probably less vague than any other language humans have come up with so far, but there is still a degree of vagueness. This is the central point of the entire thread.
Maybe revere is a strange word to describe how laymen view math but I think they do see it as some complicated thing that grants access to higher knowledge. Maybe it grants access to absolute truth defined in the sense above! Who knows? They haven't looked into it that closely (by definition of layman).
Math skills are socially rewarded in the sense that STEM jobs usually bring in larger salaries than many other professions might.
It's obvious to me how someone who doesn't study math in depth could get the idea that math accesses absolute truths (in the sense I have defined the term). Maybe it's obvious to me because I was such a person at one time and I experienced those thought patterns.
Re: Misunderstanding basic math concepts, help please?
If you discard the first Graham's Number digits of Pi, then look at the next digit, then it's unknown, and possibly even unknowable, whether that digit is 7 or not, but most people would be happy to regard the proposition that it is 7 as being either true or false, in a fairly absolute sense  generally people are happy drawing a distinction between facts, beliefs, and knowledge, with truth being an attribute of facts; certainty, of belief.
There's a fairly common intuition, tied in with the belief in a persistent objective universe, that truths are true regardless of whether any human has, or ever will, discovered them. When I shuffle a pack of cards, the common intuition is that whether or not the bottom card is the ace of spades is settled when I stop shuffling, not when I reveal the card, despite the latter being the moment when my uncertainty about the matter is resolved.
Quantum mechanics, of course, adopts a different view, but the quantum mechanical model of the universe has never been claimed to be an intuitive one...
There's a fairly common intuition, tied in with the belief in a persistent objective universe, that truths are true regardless of whether any human has, or ever will, discovered them. When I shuffle a pack of cards, the common intuition is that whether or not the bottom card is the ace of spades is settled when I stop shuffling, not when I reveal the card, despite the latter being the moment when my uncertainty about the matter is resolved.
Quantum mechanics, of course, adopts a different view, but the quantum mechanical model of the universe has never been claimed to be an intuitive one...
Re: Misunderstanding basic math concepts, help please?
rmsgrey wrote:If you discard the first Graham's Number digits of Pi, then look at the next digit, then it's unknown, and possibly even unknowable, whether that digit is 7 or not, but most people would be happy to regard the proposition that it is 7 as being either true or false, in a fairly absolute sense  generally people are happy drawing a distinction between facts, beliefs, and knowledge, with truth being an attribute of facts; certainty, of belief.
This question is knowable. We have algorithms which can compute Pi to arbitrary precision in a finite number of computational steps. We might need to wait orders and orders of magnitude of time longer than the current age of the universe to get our answer but that's fine. The question is decidable. Other questions such as the continuum hypothesis are undecidable under certain sets of axioms. With these sorts of questions it really doesn't make sense to ascribe truth values to particular statements. Take the following example:
1) A [Prem]
2) A>C [Prem]
is B true or false in this system? It is neither. It hasn't been defined and can't be derived. In mathematics we can't work with something that hasn't been defined. To say that B still has a truth value independent of how we've defined it might be one possible way to explain the situation, but there are more clear ways to explain the situation. Namely, reserving the assignment of truth values to decidable statements and leaving undecidable statements in their own category.
There's a fairly common intuition, tied in with the belief in a persistent objective universe, that truths are true regardless of whether any human has, or ever will, discovered them. When I shuffle a pack of cards, the common intuition is that whether or not the bottom card is the ace of spades is settled when I stop shuffling, not when I reveal the card, despite the latter being the moment when my uncertainty about the matter is resolved.
Sure, that can be what you mean by "absolute truth". I think that jives with the definition I've given for "absolute truth". I guess we can argue over whether absolute truths exist or not but that's maybe not the point. The point (which we all agree on) is that humans and humans using math cannot access absolute truth.
Quantum mechanics, of course, adopts a different view, but the quantum mechanical model of the universe has never been claimed to be an intuitive one...
Things are no different in quantum mechanics. Quantum mechanics makes predictions about the state of the world which can be tested. Just like in classical mechanics, our tests/experiments do not give us absolute epistemic certainty about the state of the world. We should try really really hard to avoid confusing this discussion by bringing quantum mechanics in....
edit: clarification
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Re: Misunderstanding basic math concepts, help please?
The (g_{64}+1)th digit if pi is knowable in principle, because we know the (finite sequence of) steps we'd need to take to find out even if the universe won't last long enough to take all those steps. However, we know there must be true but unprovable statements in any powerful consistent system, and I can't think of what "knowable" would mean for a mathematical statement if not provable.
For example, all real numbers are exactly one of rational, irrational but algebraic, or transendental, yet there are many (welldefined) numbers whose set membership is unknown. It's conceivable to me that for at least a few such numbers, which set they belong to may in fact be unknowable. I don't think you'll find much agreement with the claim that "x is rational" or "x is transcendental" ever lacks a truth value when x is a real number.
For example, all real numbers are exactly one of rational, irrational but algebraic, or transendental, yet there are many (welldefined) numbers whose set membership is unknown. It's conceivable to me that for at least a few such numbers, which set they belong to may in fact be unknowable. I don't think you'll find much agreement with the claim that "x is rational" or "x is transcendental" ever lacks a truth value when x is a real number.
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:Yes that is what I mean. In the end "absolute truth" is two words and I define them to mean this particular thing. I take this meaning because I think it is the meaning most people intuitively associate with those words when they hear them.
Most people intuitively expect to have it both ways:
On the one hand, people expect absolute truths to exist objectively and independently of human thought. On the other hand, people expect absolute truths to give them 100% certainty.
That's exactly the crux of the problem.
BTW I think that this is one of the reasons (not the only one, ofcourse) that revelationbased religions are so popular. If we assume, on a leap of faith, that the words in some Holy Book are absolute truths, we can have it both ways (or at least delude ourselves that we can).
Math is vague in the sense that we have to at some point rely on informal language to define mathematical objects and ideas such as "set" and "is an element of" and a few other things.
That's a vagueness in our intuitive thought process, rather than a problem with math itself.
An axiomatic system is nothing more than a "game" of manipulating symbols which has a clear, precise set of rules. You could write a computer program that checks whether a proof in (say) ZF is valid. Indeed, this is exactly the idea at the center of Godel's incompleteness theorems: His entire work is based on converting axiomatic systems into a "manipulating symbols game".
gmalivuk wrote:For example, all real numbers are exactly one of rational, irrational but algebraic, or transendental, yet there are many (welldefined) numbers whose set membership is unknown. It's conceivable to me that for at least a few such numbers, which set they belong to may in fact be unknowable. I don't think you'll find much agreement with the claim that "x is rational" or "x is transcendental" ever lacks a truth value when x is a real number.
That's an excellent example. Especially since there are numerical definitions which can be proven to be uncomputable.
For example, you can convert the answer to the infamous halting problem into a single real number between 0 and 1. This number would be, by definition, uncomputable. Yet it most certainly is a specific, welldefined number.
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