Misunderstanding basic math concepts, help please?
Moderators: gmalivuk, Moderators General, Prelates
 gmalivuk
 GNU Terry Pratchett
 Posts: 25723
 Joined: Wed Feb 28, 2007 6:02 pm UTC
 Location: Here and There
 Contact:
Re: Misunderstanding basic math concepts, help please?
If you disagree with "x=x", you're not telling me something about 'x', you're telling me something about '=' (namely you're telling me that you use '=' differently from everyone else).
Re: Misunderstanding basic math concepts, help please?
You're the one claiming that everything ever is relative, Treatid. Are you saying you have an actual counterexample to the first law of thought, or is this more of your waving wildly in the air?
You seem to want to say
1. Axiomatic mathematics is an absolute system.
2. There are no absolute systems.
3. Axiomatic mathematics is... not a thing? I'm losing your argument here.
It's unclear what exactly "absolute" and "relative" mean here, and your stance conveniently seems to relieve you of the need to have hard definitions. I can see definitions of "absolute" for which 1 holds, and definitions for which 2 holds, but not any in which they both hold at the same time. Please enlighten us, and feel free to not add a pagelong rant.
You seem to want to say
1. Axiomatic mathematics is an absolute system.
2. There are no absolute systems.
3. Axiomatic mathematics is... not a thing? I'm losing your argument here.
It's unclear what exactly "absolute" and "relative" mean here, and your stance conveniently seems to relieve you of the need to have hard definitions. I can see definitions of "absolute" for which 1 holds, and definitions for which 2 holds, but not any in which they both hold at the same time. Please enlighten us, and feel free to not add a pagelong rant.
(∫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?
The axiom (for all x, x=x), says nothing about x, that is why we have the universal quantifier 'for all'.
That axiom is about a relation we choose to call equality(or label with =), and all it says is that equality(=) is reflexive. If you say 'but equality(=) isn't reflexive!', then you aren't using 'equality(=)' the same way we do, and are communicating badly.
That axiom is about a relation we choose to call equality(or label with =), and all it says is that equality(=) is reflexive. If you say 'but equality(=) isn't reflexive!', then you aren't using 'equality(=)' the same way we do, and are communicating badly.
Re: Misunderstanding basic math concepts, help please?
Cauchy wrote:It's unclear what exactly "absolute" and "relative" mean here, and your stance conveniently seems to relieve you of the need to have hard definitions.
Let me turn that around on you.
Mathematicians cannot define what absolute or relative means; they cannot define dimensions or integers; they can't define what axiomatic mathematics is; they can't define what true means. In short, mathematicians have never, ever defined anything  not even a little bit.
Whoops. Did I miss out "in an absolute sense"? I surely missed that they can't even define what 'define' means.
Yes, how convenient it is for me that mathematicians have not ever defined anything in an absolute sense.
If you have an actual argument  bring it. Meanwhile, you can shut me down by producing a single absolute definition. Perhaps you could try your hand at defining axiomatic mathematics?
But that is the problem, isn't it. You've conceded that mathematics does not define anything in an absolute sense... But you still think that mathematics does define things.
"Yeah  maybe we don't have absolute definitions  but we have good enough definitions. Technically they aren't absolute truths; but we still know what we mean when we say that something is true."
No. You don't know anything. I don't know anything either in this sense. My entire thesis at this point is that it is impossible to know anything. There is no fault in not being able to define 'true' within an axiomatic system. I'm not saying axiomatic mathematics is wrong because it can't formally define 'true' within a given axiomatic system. I'm saying axiomatic mathematics is wrong for trying to define anything. Mathematicians are wrong for pretending that they have defined anything, to any degree.
The one success of axiomatic mathematics is that it has demonstrated beyond reasonable doubt that the concept of defining something is null and void.
And your response?
"Yeah  well technically yes. But we totally know what axiomatic mathematics is. We totally know what the axiom of choice is. We totally know what dimensions are. We just can't quite write that knowledge down on paper."
There is no middle road here. Do you know something? No? Then you don't know anything.
Yes, of course we have a concept of knowledge. We have libraries full of books. I know when my mother's birthday is. I know how to switch on a computer, select a forum and type characters into the little window that appears.
And then I type the characters: "It is impossible to know anything" and your brain goes "squeak" and you get an intense urge to reach through the screen and throttle some sense into me.
But it is exactly this dichotomy that is the problem. We have a concept of knowing that appears rational and justifiable. You know what it means to say that you know what the theory of evolution is. You know what knowledge is. You know what it means to define something. These are perfectly normal words used in a productive manner all the time. It seems utterly perverse to tell an English speaking person that they don't know what the words they are using mean.
And then we have axiomatic mathematics. An attempt to properly formalise those meanings into a single consistent framework so that there is no messy informal doubt; so that all those arguments over the precise meaning of a given word can be settled in a formal environment with none of that tedious messing around with subjective interpretations.
And what happens?
You can't define true. You can't define consistent. You most definitely can't demonstrate consistency. You can't even define 'define'.
There is not a single word, phrase, group of symbols or anything else that you can assign a single definite meaning to.
And it isn't that you just haven't worked out how to do it yet. You have formal statements saying that there are fundamental limits on what axiomatic mathematics can do.
Whatever it is that axiomatic mathematics is trying to do (which you can't define), it isn't doing it. But because you can't define what it is that you aren't doing you are able persist in the fiction that you are doing something despite all the results telling you that you really, really are not defining anything in any sense.
Gargh! Again!
Right now you (as a group) are trying to tell me that it doesn't matter that axiomatic mathematics can't define anything and that it has always been a relativistic theory. You are arguing that it was never intended to define anything. Actually  I'm pretty sure you don't know what you are arguing  except you know you are arguing against whatever it is you think I'm arguing.
Yahweh! you are so full of crap.
You're like a kitten trying to cover up its shit on a wood floor.
Me: You can't define anything in an absolute sense.
You: Yeah. I suppose technically. But we were never trying to define things in an absolute sense. Yeah; that's it. We are defining things in a sense that is totally not absolute.
Me: Really? what other meaning/sense of definition are you using?
You: Umm... Well... we know it when we see it...
Me: But...? You can't actually define the sense of definition that you think you might be using, can you? You think you know what you are doing; but you can't quite put into words exactly what that might be. You're sure that axiomatic mathematics works in the way that you understand it; you're sure that you are being consistent in your behaviour  but you can't say what axiomatic mathematics is. You can't say what consistent is. You can't say what true is.
The first mantra of mathematics is: Define your terms.
And here you are trying to tell me that axiomatic mathematics has always been relative and it doesn't matter that you haven't defined anything. At all. Ever.
Lie to yourself all you like
You can tell me that I'm misunderstanding axiomatic mathematics and the Laws of Thought all the live day long. We could spend years arguing back and forth.
And the reason we could spend years arguing over exactly what is, and is not, the correct understanding of axiomatic mathematics and the Laws of Thought is that they aren't defined. At all. Not even a little bit.
This is the very problem that formal mathematics was intended to solve. If you don't define your terms then you can twist and turn and change your meaning part way through in order to argue any position you want to.
The trouble is that the result of this attempt to formalise definitions is to demonstrate that it is impossible to formalise definitions.
So here we are with you inventing new ways to interpret what axiomatic mathematics is and what it is trying to do.
And yes  you have a point  your interpretation is as valid as mine. Which is to say  neither of us can prove that axiomatic mathematics or the laws of thought have a single definite meaning exclusive of all other meanings.
The difference between you and me is that you are in denial. You think it is possible to know things. You think that there is a possibility of a fixed reference frame. You think you know what (x = x) means.
Oh  and incidentally
In order to determine if (x = y) you need to know what x and y are. You need to define x and y (and equality). You need to assume that x and y can have known values. You are assuming that x and y are absolutes.
If you are going to dick around reinterpreting fundamental assumptions of mathematics it might be a good idea to make sure that your reinterpretation is actually making the point you intended rather than... you know... the exact opposite point.
Whether you take the First Law as an argument about x or an argument about equality; it is clearly a statement of absolutes. That there is something that can be known. Whether that something is x, equality or identity is missing the point that it is a statement of certainty  of absolute knowledge  a thing is a thing
Don't be confused by the fact that the English Language works (to some degree)
You know what knowledge is.
Except that axiomatic mathematics has demonstrated that is is impossible to know anything in an absolute sense.
There is a mismatch here. A false assumption is being made.
Axiomatic mathematics has been crystal clear in telling us that it isn't possible to define anything.
Yet our personal experience of knowledge compels us to believe in the existence of definitions despite all the many ways in which formal analysis of axiomatic mathematics shouts, screams and begs us to understand that it isn't possible to define anything.
Thousands of years of Euclidean geometry: How could anyone say that Euclidean Geometry hasn't been defined? As mathematicians you've been working with Euclidean Geometry since you first learned to count (even if nobody named it the number line when you first learned 1 + 1).
You are sure that you've defined stuff. Even if you concede that the definition isn't absolute and universal  you just know that you have defined stuff.
So when I tell you that nothing has ever been defined to any degree  that it is impossible to define anything in any way then of course you are going to rebel. You feel right down to your bones that you know what Euclidean Geometry is.
In your mind, 'definition' and 'existence' are near synonyms. Denying the existence of definitions feels like denying the existence of the universe  or our own thoughts. You don't have to prove your own existence to yourself. You shouldn't have to justify the existence of definitions  they just are  it is what humans do  we define the world around us. It is what makes us humans. If we have never defined anything then (error... memory segment fault... does not compute... We have always been at war with East asia... axiomatic mathematics was always meant to be relativistic... non absolute definitions are definitely a thing (that I just can't quite define at this precise moment)... Fizz... Pop... Whirr...)
Specifically
The standard reaction here when conceding that there are no absolute definitions has been to immediately imply the existence of satisfactory nonabsolute definitions.
"Obvious really. Makes sense. We know we have English language definitions  so if we don't have absolute definitions then in the very worst case we have English language definitions. And axiomatic mathematics is more formal than English language so axiomatic mathematics definitions are automatically a step up from English language definitions  even if they don't reach all the way up to the lofty goal of absolute definitions."
Yes... That is all very plausible. Utter bollocks  but plausible bollocks.
You are getting away with sloppy thinking because you don't have any definitions to constrain yourself.
You can't even define 'definition'. What makes you think you are defining anything? No, really! Why do you think you've defined ZFC?
Yes, English works. Yes, English contains the word 'definition'.
And still I can sit here and confidently type: There has never been a definition of anything. Ever.
We can argue and fuss over what I mean by definition and what you mean by definition. Eventually we will agree that there is no definite, absolute, epistemically certain way to define definition.
"Oh  but there is totally this other meaning of definition."
Fine: tell me what it is so that I am certain that we agree on the definition.
"But that would require an absolute definition."
Yes. Quite. That is the point.
Reminder
I'm not being nihilistic. I'm not saying we should just give up on everything and live in caves.
I'm saying that everything we have achieved has been despite never defining everything. We don't need to define things. Stop trying. Stop pretending that anything is in any way a standin for an absolute definition (i.e. stop pretending that you have defined axiomatic mathematics, ZFC, dimensions, true, consistent, or anything else).
It isn't that the concepts just happen not to have been defined. It is that they are undefinable. It is not possible to define anything. To attempt to do so is a category error. There cannot be a fixed point. There cannot be a thing that only has one single meaning; one single definition.
Axiomatic mathematics cannot exist. No single definition can be unentangled from the infinitely many alternatives. There is no possible mechanism to choose one meaning over every other meaning.
“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”
The malleability of words has been known, probably since mankind first grunted. "Was that a grunt yes? or a grunt no?"
Yes is it a pain when trying to communicate and the other person is misunderstanding your words. Tough. There aren't any shortcuts. There is no way to magically transfer your interpretation of a word to someone else. Symbols are just symbols. There is no special set of symbols that suddenly everyone will realise has a fixed definite meaning that can then be extrapolated to other symbols. They are just symbols.
Polite
I want to be polite. By being aggressive in my posts I get your backs up and it is that much harder for you to read anything I write in a sympathetic light. It makes you grasp at the flimsiest of counters just so you can stick it to me.
However, you are currently trying to tell me that axiomatic mathematics has always been a purely relativistic system, the Laws of Thought don't assert the assumption of absolute values and it was never part of axiomatic mathematics to define anything. (oh yes  and in the absence of being able to define axiomatic mathematics you still somehow know what axiomatic mathematics is)
So I'm trying another tack: criticise your ability to think rationally. Not a tactic that usually works on religious conviction, but maybe a couple of you take pride in your ability to think rationally and will notice that if you haven't defined anything, then you haven't defined axiomatic mathematics.
 gmalivuk
 GNU Terry Pratchett
 Posts: 25723
 Joined: Wed Feb 28, 2007 6:02 pm UTC
 Location: Here and There
 Contact:
Re: Misunderstanding basic math concepts, help please?
If you were impolite but logical, we wouldn't mind rudeness or even outright condescension. The bigger problem is that the one here who has most strongly demonstrated an inability to think rationally has been you yourself. You're unlikely to convince anyone else of anything about your own thinking, rational or otherwise, when all you continue to do here is refuse to use words the way everyone else is using them.
(In this case, you're evidently not using "definition" consistently with everyone else.)
If you think nothing can be defined at all, in a way that multiple agree on a single definition, then what are you even still doing here? You've decided real communication is impossible yet you're still apparently attempting to communicate with us.
(In this case, you're evidently not using "definition" consistently with everyone else.)
If you think nothing can be defined at all, in a way that multiple agree on a single definition, then what are you even still doing here? You've decided real communication is impossible yet you're still apparently attempting to communicate with us.
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5188
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Somerville, MA
 Contact:
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Polite
I want to be polite. By being aggressive in my posts I get your backs up and it is that much harder for you to read anything I write in a sympathetic light. It makes you grasp at the flimsiest of counters just so you can stick it to me.
I mean it's a better look on you than the Socratic faux humility but it is just an obstinate ignorance.
Mathematics is a language and language is a consensus game? Shit, the novel observations just keep coming.
Look, if you have a category, say, "absolute", such that nothing fits that category, you have a useless category and should jealously save all of your breath from such wastes.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
 gmalivuk
 GNU Terry Pratchett
 Posts: 25723
 Joined: Wed Feb 28, 2007 6:02 pm UTC
 Location: Here and There
 Contact:
Re: Misunderstanding basic math concepts, help please?
Also, you do not understand (what everyone else means by) the core concepts you're trying to discuss. That is hardly a "flimsy" counter.
Re: Misunderstanding basic math concepts, help please?
Look up definition by extension and definition by example. Both concepts of definition that don't require a full, formal description of the thing defined.
Re: Misunderstanding basic math concepts, help please?
Who the fuck cares whether mathematics can define something in an absolute sense? It's only you. You're the one putting words in other people's mouths, both in your posts explicitly and in axiomatic mathematics' mouth claiming that it wants to try to deal with absolutes. Everyone (except you) knows that there's ambiguity in math communication, we just deal with it the best we can because math is worth it. Your system of relative definitions where everything is only important based on its relationships is bullshit, it doesn't help anyone understand anything and will be a fruitless endeavor, as evidenced by the fact that you can't get it off the ground at all in this thread.
If your next post does not fit on my monitor screen I won't read it.
If your next post does not fit on my monitor screen I won't read it.
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Re: Misunderstanding basic math concepts, help please?
Treatid: Pretty sure I made this point pages ago, but all the whole of maths says is 'if x is true then y is also true'.
There's no 'absolutes' anywhere there. They are a figment of your imagination.
Maths doesn't even presuppose we all agree on what 'if' means or 'then' or anything else there. Ambiguity and confusion is certainly possible, as you yourself demonstrate.
Does that mean maths is worthless? Only if you believe all of life is worthless because we can't be sure of anything ever. Was the world created fully formed one second ago and will it be destroyed one second hence? Might be. Can't be sure.
You say you're not espousing a nihilist philosophy so, in that case, you should join the rest of us in not judging a discipline like maths on it's absolute provability but on its usefulness. Maths seems extraordinarily useful in predicting what will happen a day from now or a year from now.
There's no reason why it would have to be this way. Maybe in a different universe religion or magic would work better than logic. But that doesn't appear to be the case here, so we'll stick with what works up until it doesn't.
There's no 'absolutes' anywhere there. They are a figment of your imagination.
Maths doesn't even presuppose we all agree on what 'if' means or 'then' or anything else there. Ambiguity and confusion is certainly possible, as you yourself demonstrate.
Does that mean maths is worthless? Only if you believe all of life is worthless because we can't be sure of anything ever. Was the world created fully formed one second ago and will it be destroyed one second hence? Might be. Can't be sure.
You say you're not espousing a nihilist philosophy so, in that case, you should join the rest of us in not judging a discipline like maths on it's absolute provability but on its usefulness. Maths seems extraordinarily useful in predicting what will happen a day from now or a year from now.
There's no reason why it would have to be this way. Maybe in a different universe religion or magic would work better than logic. But that doesn't appear to be the case here, so we'll stick with what works up until it doesn't.
Re: Misunderstanding basic math concepts, help please?
elasto wrote:Treatid: Pretty sure I made this point pages ago, but all the whole of maths says is 'if x is true then y is also true'.
There's no 'absolutes' anywhere there. They are a figment of your imagination.
I note your use of quotes around 'absolute' to indicate that we haven't formally defined the term (which is perfectly reasonable  but I'm still going to turn it around on you).
In the absence of knowing what 'absolute' means, you are still, somehow, certain that the sentence you wrote does not imply absolutes.
Bloody good trick that. How do you do it? How do you arrive at certainty about something that hasn't been defined?
And there we have our first assumption of an absolute: you are certain there are no absolutes in that sentence. Would you go so far as to say you are absolutely certain there are no absolutes implied by that sentence?
I'm not playing word games with you. I'm pointing out a logical fallacy. You can't be certain about the nonexistence of absolutes. (you could be certain about the existence of absolutes  but you cannot be certain about their nonexistence).
If you claim to know anything with certainty then you are assuming and implying the existence of absolutes.
If you 'know' that mathematics doesn't assume absolutes then you are asserting an absolute. Catch 22 and instant contradiction.
A contradiction indicates that somewhere along the line you've made a set of incompatible (or impossible) assumptions.
...
elasto's post is a good one. He provided the hook for me to hang this particular argument on  but the argument is general  not specific to him.
If you claim to not be assuming absolutes, then you can't 'know' that I'm using a particular definition incorrectly; you can't 'know' that you are understanding a particular definition correctly.
Furthermore, as elasto rightly suggests, an axiomatic mathematics without absolute definitions is... nothing. The Laws of Thought don't allow for a middle ground; in the case of Excluded Middle  very explicitly and unsubtly so. 'partially true', 'sort of defined' and 'slightly epistemically certain', even if they were potentially meaningful terms; are explicitly excluded.
(x = y) != (x is probably much the same thing as y)
The fundamental assumption of axiomatic mathematics as stated by the Laws of Thought is that it is, in principle, possible to know something with certainty.
In fact, we know that we can't know anything with certainty. It isn't merely that we don't happen to have a lump of certainty but we might stumble upon some at some point. We will never be certain about anything.
You: "Yes  well... The bar is set too high. We need a more practical bar. And that's what we do with axiomatic mathematics  we just use a more feasible bar."
1. I agree wholeheartedly that the bar set by axiomatic mathematics isn't practical.
2. You can't just say you are lowering the bar. Lowering the bar is an action. Lowering the bar involves changing something such that the bar is lower.
3. The current height of the bar is established by the Laws of Thought. You can't have it both ways. You can't keep the thing that defines the height of the bar while changing the height of the bar. Attempting to do so is the contradiction that elasto illustrated so nicely.
So  your choice. Keep an impossibly high bar and never be able to define (or do) anything in accordance with that bar.
Or give up on the Laws of Thought and have a workable system (that works without the concept of certainty).
One or the other. Not both.
Right now, you recognise that you can't define things in an absolute sense. You recognise that you can't reach the strict bar set by axiomatic mathematics. You recognise that you are compromising by settling for a less strict sense of meaning within axiomatic mathematics (because it works).
What you aren't recognising is that your compromise is, in fact, a contradiction. If you are not following all the Laws of axiomatic mathematics, all the time, without exception then whatever you are doing is not axiomatic mathematics.
Fudging the results, even a little bit, even in the face of unreasonable (self inflicted) expectations, is bad science.
The result of the scientific experiment that is axiomatic mathematics is that we cannot hold to the assumptions (Laws of Thought) of axiomatic mathematics and get useful results.
@gmalivuk
So  in the middle of a conversation about the fact that it is impossible to define anything, you want to take me to task for not using your definitions  that you've just admitted you can't define? Cute.
Actually, to be fair, I have a reasonable idea of where you are coming from.
It is drummed into mathematicians that the greatest source of misunderstanding is improperly defined terms.
The knee jerk reaction (as you have handily demonstrated) is to question whether the initial definitions have been properly defined and understood.
However, given that we have never had an unambiguous definition and it is consequently impossible for anyone to say what their interpretation of anything is... Whatever mechanism is operating is not whatever you think it is.
You think that meaning is inherent in words. It isn't.
The purpose of communication is to convey patterns of relationships. Set Theory and Category Theory demonstrate that mathematicians already know this on some level.
The best way to convey patterns of relationships is with patterns of relationships. A sentence is a construction of relationships between words. The words are largely irrelevant. What matters is the pattern of relationships between the words.
You've been conditioned to assume that meaning is inherent in the words themselves to such an extent that even after admitting that you can't define any word, you are still demanding I use standard definitions. And you never questioned what you were doing while you were doing it.
The difference
You are hearing that we cannot have an absolute definition. That we cannot define something that is absolute.
What you should be hearing is that we cannot have absolutes. There cannot exist a thing such that it has a single interpretation/meaning.
It isn't that we can't define Euclidean Geometry in absolute terms. It is that there can be no thing that has the property of being definable.
So...
Despite the very best efforts of mathematicians  Sets have never been defined. Sets can't be defined.
Yet the pattern of relationships between sets does exist, can be described, and doesn't rely on us defining Sets. Indeed, the attempt to define Sets simply muddies the water when the only thing of any relevance is the pattern of relationships.
This is why you have the impression that axiomatic mathematics works. Describing a pattern of relationships works. Defining Sets doesn't
We don't need to define anything in order to describe a pattern of relationships. Axiomatic mathematics is 100% excess baggage. A formal definition of Sets doesn't add anything (we know because we know there has never been a formal definition of Sets).
And I can communicate with some degree of success because even while you think a word has a given meaning and I know that symbols don't have an inherent meaning  the patterns I'm weaving exist in the relationships between words and not, for the most part, within the words themselves.
You've seen the process of agreement in conversations and thought it was the chaff to be thrown away when it is the sole purpose. You've seen people arguing over meaning and sought to streamline the process by providing standard meanings  when there can be no standard meanings and the argument over meaning was always and entirely the only thing.
You've got all the pieces already
I'm not holding out on you waiting for the right moment to reveal the secrets of the universe.
You've been trying to find meaning in the words. Whereas meaning resides in the gaps between words (poetic allusion to those candlestick illusions where the picture is in the negative space).
The idea that patterns of relationships are important isn't new. Set Theory and Category Theory are so, so close to understanding.
The only nudge you need now is to accept that you cannot define anything. The only way you can describe a given relationship is by reference to other relationships.
Granted, that last little step involves admitting to yourselves that Euclidean Geometry and ZFC Set Theory cannot possibly be things.
Even and despite admitting that you can't define anything, you've spent this entire thread trying to convince yourselves that you know what axiomatic mathematics is. You desperately want to believe that what you were taught and what you think you've been doing all your mathematical lives has some reality.
You are in good company. The entire field of mathematics for that last hundred years has had the information that it is impossible to define anything.
There is no need for another proof that a system cannot show itself consistent, or define true. I don't need to reinvent the foundational crises in mathematics.
The only problem  your only problem  is that you are reluctant to admit that you don't know what axiomatic mathematics is. All you have to do is embrace that ignorance. Take it seriously. Actually think about the consequences.
Re: Misunderstanding basic math concepts, help please?
I notice that I am confused.
The problem with the laws of thought is that they are absolutes, yes?
Any assertion that there are no absolutes is itself an absolute, yes?
It is asserted that axiomatic mathematics deals with absolutes when there are no absolutes, yes?
If you can handle all three of those ideas at once, I'd be very interested in hearing your explanation of what happens when an irresistible force is applied to an immovable object...
It may also be worth pointing out that there is one absolute  I exist.
If it helps, consider every single term in the English language to usually carry an unwritten rider along the lines of "assuming that an objective reality exists and is largely consistent with my purely subjective sensations, that my reasoning is generally reliable, and that my memory is a good approximation to events in an objective past"  including when used to set up axiomatic mathematics.
Aside from the bare fact of my existence, there is no sound basis for any pure absolute, but my memories are consistent with my having had experiences consistent with things happening the same way they would if the things treated as absolute truths in axiomatic mathematics were actually true.
In other words, as best I can tell, axiomatic mathematics comes closer to absolute truth than anything else short of the cogito, so, since the main alternative to accepting it is to abandon all faith in the reality of reality, I consider it to be correct to accept it in the absence of a meaningful and/or useful alternative.
Or, in plainer language: I know it's true.
The problem with the laws of thought is that they are absolutes, yes?
Any assertion that there are no absolutes is itself an absolute, yes?
It is asserted that axiomatic mathematics deals with absolutes when there are no absolutes, yes?
If you can handle all three of those ideas at once, I'd be very interested in hearing your explanation of what happens when an irresistible force is applied to an immovable object...
It may also be worth pointing out that there is one absolute  I exist.
If it helps, consider every single term in the English language to usually carry an unwritten rider along the lines of "assuming that an objective reality exists and is largely consistent with my purely subjective sensations, that my reasoning is generally reliable, and that my memory is a good approximation to events in an objective past"  including when used to set up axiomatic mathematics.
Aside from the bare fact of my existence, there is no sound basis for any pure absolute, but my memories are consistent with my having had experiences consistent with things happening the same way they would if the things treated as absolute truths in axiomatic mathematics were actually true.
In other words, as best I can tell, axiomatic mathematics comes closer to absolute truth than anything else short of the cogito, so, since the main alternative to accepting it is to abandon all faith in the reality of reality, I consider it to be correct to accept it in the absence of a meaningful and/or useful alternative.
Or, in plainer language: I know it's true.
Re: Misunderstanding basic math concepts, help please?
I don't want to speak for elasto, but I think his use of quotes indicates that he's using the term the way you've been using the term, without indicating his agreement that it is a reasonable way to use it. Or at least, that's what I would've done.Treatid wrote:elasto wrote:Treatid: Pretty sure I made this point pages ago, but all the whole of maths says is 'if x is true then y is also true'.
There's no 'absolutes' anywhere there. They are a figment of your imagination.
I note your use of quotes around 'absolute' to indicate that we haven't formally defined the term (which is perfectly reasonable  but I'm still going to turn it around on you).
OK, so that's a fair point  the statement "no statement is absolutely true" can not be true without creating a paradox. But isn't that your entire argument? Pretty much every one of your posts has told us that nothing is absolute, and by your own logic every one of your posts has been incorrect. Why do we need to pay attention to what you're saying, if it's all based on a false predicate?In the absence of knowing what 'absolute' means, you are still, somehow, certain that the sentence you wrote does not imply absolutes.
Bloody good trick that. How do you do it? How do you arrive at certainty about something that hasn't been defined?
And there we have our first assumption of an absolute: you are certain there are no absolutes in that sentence. Would you go so far as to say you are absolutely certain there are no absolutes implied by that sentence?
I'm not playing word games with you. I'm pointing out a logical fallacy. You can't be certain about the nonexistence of absolutes. (you could be certain about the existence of absolutes  but you cannot be certain about their nonexistence).
If you claim to know anything with certainty then you are assuming and implying the existence of absolutes.
If you 'know' that mathematics doesn't assume absolutes then you are asserting an absolute. Catch 22 and instant contradiction.
A contradiction indicates that somewhere along the line you've made a set of incompatible (or impossible) assumptions.
Claiming that nothing can be defined is a convenient way to get out of defining your terms, or of using the terms we're using in different ways leading to incorrect conclusions.If you claim to not be assuming absolutes, then you can't 'know' that I'm using a particular definition incorrectly; you can't 'know' that you are understanding a particular definition correctly.
Is it? I wouldn't necessarily agree. I think the fundamental assumption behind the Laws of Thought is that True and False are mutually exclusive of each other. Certainly there are logical systems in which that assumption doesn't hold, but it seems as reasonable a starting point as any.(x = y) != (x is probably much the same thing as y)
The fundamental assumption of axiomatic mathematics as stated by the Laws of Thought is that it is, in principle, possible to know something with certainty.
I fixed that last part for you  now it reads a little more like how the rest of us are thinking. In my reading, the "bar" you're asking for is one that you've conveniently stated is impossible to reach. The "bar" for the rest of us is lower  rather than asking for "absolute epistemic certainty", we're asking for "consistent with a stated list of assumptions and inference rules, which include the Laws of Thought among others."In fact, we know that we can't know anything with certainty. It isn't merely that we don't happen to have a lump of certainty but we might stumble upon some at some point. We will never be certain about anything.
You: "Yes  well... The bar is set too high. We need a more practical bar. And that's what we do with axiomatic mathematics  we just use a more feasible bar."
1. I agree wholeheartedly that the bar set byaxiomatic mathematicsme isn't practical.
2. You can't just say you are lowering the bar. Lowering the bar is an action. Lowering the bar involves changing something such that the bar is lower.
3. The current height of the bar is established bythe Laws of Thoughtme. You can't have it both ways. You can't keep the thing that defines the height of the bar while changing the height of the bar. Attempting to do so is the contradiction that elasto illustrated so nicely.
I'm not sure what contradiction you're seeing. If you take "axiomatic mathematics" as you see them, then yes there's a contradiction hidden in your claims about absolutes. If you take "axiomatic mathematics" which is literally just a phrase that means "a sort of math in which we assume some statements (axioms) true and then see what follows from them", then I don't see why anything about "Treatidian absolutes" is necessary.Right now, you recognise that you can't define things in an absolute sense. You recognise that you can't reach the strict bar set byaxiomatic mathematicsme. You recognise that you are compromising by settling for a less strict sense of meaning within axiomatic mathematics (because it works).
What you aren't recognising is that your compromise is, in fact, a contradiction. If you are not following all the Laws of axiomatic mathematics, all the time, without exception then whatever you are doing is not axiomatic mathematics.
The reason properly defined terms are so important in mathematics is simple, and your posts over the past several years underscore this point beautifully. If you want to talk about a problem, you need to express it in terms that other people can understand. So if you want to talk about "sets", but you're not using the definition of Set that the rest of the world agrees on, you don't get to use properties of Sets to talk about your sets, nor can you reason using theorems about Sets until you can demonstrate that your sets have the same properties and implications as the rest of our Sets.@gmalivuk
So  in the middle of a conversation about the fact that it is impossible to define anything, you want to take me to task for not using your definitions  that you've just admitted you can't define? Cute.
Actually, to be fair, I have a reasonable idea of where you are coming from.
It is drummed into mathematicians that the greatest source of misunderstanding is improperly defined terms.
The knee jerk reaction (as you have handily demonstrated) is to question whether the initial definitions have been properly defined and understood.
However, given that we have never had an unambiguous definition and it is consequently impossible for anyone to say what their interpretation of anything is... Whatever mechanism is operating is not whatever you think it is.
You think that meaning is inherent in words. It isn't.
The purpose of communication is to convey patterns of relationships. Set Theory and Category Theory demonstrate that mathematicians already know this on some level.
The best way to convey patterns of relationships is with patterns of relationships. A sentence is a construction of relationships between words. The words are largely irrelevant. What matters is the pattern of relationships between the words.
You've been conditioned to assume that meaning is inherent in the words themselves to such an extent that even after admitting that you can't define any word, you are still demanding I use standard definitions. And you never questioned what you were doing while you were doing it.
So if you want to talk about P=NP, you don't get to cancel out P from either side and say this is an easy problem provided that N=1. If you want to talk about quantum mechanics, you need to make sure that the system you're in is one to which QM applies, or you need to show how the system is analogous to another in which it applies, or something. If you want to talk about the Principle of Explosion, you need to show that you're defining it the same way as other people, or if you aren't then at least you need to justify why it applies in the situation you offer if a basic understanding of the subject doesn't immediately imply/demand that application.
OK fine, definitions don't exist  now what? This has been the missing piece of everything you've been saying for however many pages now: why should we care? If nothing is definable, but "I'm not being nihilistic. I'm not saying we should just give up on everything and live in caves." then what is the alternative? You've talked at a few points about relativistic mathematics, and it's convenient that you say nothing can be defined so it's not your fault that you can't tell us what that is or why we should be interested. You've created some nonexistent objects to try and demonstrate some points that ultimately fell flat. Now, you're saying that you don't actually have an alternative and all you're really looking for is that we all "take it seriously" and admit we're all clueless here.The difference
You are hearing that we cannot have an absolute definition. That we cannot define something that is absolute.
What you should be hearing is that we cannot have absolutes. There cannot exist a thing such that it has a single interpretation/meaning.
It isn't that we can't define Euclidean Geometry in absolute terms. It is that there can be no thing that has the property of being definable.
So what's with all the theorems and conjectures out there  if axiomatic mathematics is undefined nonsense, how come we all know what Fermat was trying to say with his last theorem? How do we know the Collatz Conjecture is a thing? We know it, because we've agreed to use common definitions, we've agreed to use terms in the same way as everyone else, and we've given examples and demonstrations that are consistent with these so that even if you don't fully understand the definitions and terms, you'll figure that out quickly.
I know how you're going to respond to that last paragraph already  "But how do you know? How can you agree on a definition when nothing can be defined?" If you're really stuck on that, then I'm not going to be able to help you. You've read everything I've wrote so far, and I'm going to assume you understood it, so even though I can't define any of the words with "absolute epistemic certainty" I believe we're speaking the same language. If we're not, then communication is impossible and I'll stop wasting both of our time.
 WibblyWobbly
 Can't Get No
 Posts: 506
 Joined: Fri Apr 05, 2013 1:03 pm UTC
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Despite the very best efforts of mathematicians  Sets have never been defined. Sets can't be defined.
Yet the pattern of relationships between sets does exist, can be described, and doesn't rely on us defining Sets. Indeed, the attempt to define Sets simply muddies the water when the only thing of any relevance is the pattern of relationships.
This is why you have the impression that axiomatic mathematics works. Describing a pattern of relationships works. Defining Sets doesn't
We don't need to define anything in order to describe a pattern of relationships.
We don't? What is a relationship? Are you talking about what happens when a mommy and a daddy love each other very much? Because that doesn't tell me anything about math, that only tells me where babies come from!
What is a pattern? Is it one of those things we use to help us sew together costumes for Halloween? Next year, I'm going to be a fighter pilot!
How does one describe a relationship between two ... objects, classes of objects, ideas of objects, or even abstract ideas ... without comparing properties? And how does one compare properties without describing said properties?
A challenge: since nothing is defined or definable, which I have to assume includes language, describe a pattern of relationships between sets without using words. Or pictures. (Since words have no inherent meaning  something I might agree with  and nothing is definable, we cannot define the meaning of pictures and thus cannot describe their purpose).
Re: Misunderstanding basic math concepts, help please?
Treatid: Definitions don't need to be absolute in some epistemological sense, they only need to be things we agree upon. This is as true of any maths you could come up with as traditional maths, as WibblyWobbly points out.
While you're right to say there's no cast iron proof that maths works  given that such a 'proof' would invariably end up circular  there's clearly a shitton of circumstantial evidence that it works for our physical world. For us to see your pages of waffle, billions of logical operations have to occur successfully  on your device, ours, and a whole host of devices in between.
Can you name one thing that your 'maths' can do that traditional maths can't? If not, your whole argument is nothing but nihilism despite all your protestations to the contrary.
While you're right to say there's no cast iron proof that maths works  given that such a 'proof' would invariably end up circular  there's clearly a shitton of circumstantial evidence that it works for our physical world. For us to see your pages of waffle, billions of logical operations have to occur successfully  on your device, ours, and a whole host of devices in between.
Can you name one thing that your 'maths' can do that traditional maths can't? If not, your whole argument is nothing but nihilism despite all your protestations to the contrary.
Re: Misunderstanding basic math concepts, help please?
Happy New Year!
Try to define distance
A strong subtext of what we are talking about is knowledge; what we can know to what degree of confidence.
While there are an unlimited number of things that axiomatic mathematics doesn't define  I think it is illuminating to concentrate on just one element: distance.
Geometry is largely about distance. The concepts of dimensions and Number Lines are (I would argue) essential aspects of distance.
These have all been explored in considerable detail within mathematics.
And yet noone has defined distance (in the same way that nothing else has been defined).
We all have a sense of distance. It isn't that the word is utterly meaningless to us.
But: "Why does distance exist?", "What creates distance?", "What, fundamentally, is distance?"; are as unanswered today as they were over two thousand years ago when Euclidean Geometry was first being laid down.
"But it works"
Yes... sort of. There is no question that we have equations that are practically useful (we can apply to the physical world).
But the only justification for those equations is that they work. We have no proofs; of anything.
When you say "Mathematics works"  you aren't entirely wrong. But if axiomatic mathematics cannot prove anything  then is it axiomatic mathematics that is working? Surely axiomatic mathematics without a single proof (of anything  at all) is not working? Whatever it is that is working is working despite axiomatic mathematics failing to prove anything  not because of it?
The caveat/rider that both rmsgrey and elasto have recently referred to is necessary. A strict reading of axiomatic mathematics tells us that if we can't prove something we can't claim to have any degree of knowledge with regard to that something. The Laws of thought tell us that we either know it or we don't (excluded middle  no 'partially true'  either true or false  nothing in between; noncontradiction  either true or false  not 'both true and false').
I wholeheartedly agree with that caveat/rider. We have to admit that the Laws of Thought don't work. If we try to stick to the Laws of Thought we find that we can't prove anything. We can't know anything. The result of trying to rigidly stick with the Laws of Thought is nihilistic. Which is why nobody here is actually committing to the Laws of Thought.
Choose one or the other:
1. Follow the Laws of Thought and find that rigid proof of anything is impossible  the inverse of which (excluded middle and noncontradiction) leaves us knowing nothing. Nihilism.
OR
2. Admit the the Laws of Thought lead to a dead end and that real life doesn't work in the sense explicitly stated by the Laws of Thought.
We are most of the way there
rmsgrey and elasto have explicitly stated that we don't follow the Laws of Thought in practice because that is impractical.
I completely agree with them on this point.
The bit we are currently stuck on is claiming that axiomatic mathematics is working even after admitting that you are disregarding the Laws of Thought. This is the contradiction that I'm criticising you for. You should either hold to the Laws of Thought or not.
Elasto (and I) are perfectly justified in claiming certainty with the caveat/rider provided. There is nothing contradictory about that. It only becomes contradictory when you try to combine the caveat/rider with the Laws of Thought.
Next
As above  admitting that mathematics requires the caveat/rider that rmsgrey suggests, and I'm pretty sure everyone here is already assuming exists, is a huge step.
You are smart enough to realise that the rider that rmsgrey suggests is incompatible with the Laws of Thought once it is pointed out to you. Again  you have all the pieces  you just need to put them together into one coherent whole.
Measurement and justification
A theory that cannot be tested is useless. The mathematical concept of proof (with respect to axiomatic mathematics) was intended to be a measure of the validity of an initial set of statements (and their consequences). It was a reasonable idea up until the point we found it was impossible to establish an unambiguous, known, set of axioms.
However, the idea of measuring/justifying concepts is sound.
Raging clue incoming...
How do we measure distance? (How do we quantify distance?)
Extra comments:
We cannot measure distance except by comparison to other distances (measurement) (It isn't possible to just know anything ex cathedra  as elasto points out everything we know is through circumstantial evidence).
Distance is a measure of how closely related things are (again  as elasto implies  our degree of certainty depends on the number (and quality) of relationships (circumstantial evidence).
There necessarily exists a minimum quantum of distance/relationship. (Among all the things we can't define: we can't define 'continuous'  we can only be aware of comparisons  we can't fundamentally know anything outside of a comparison (measurement).
As should be abundantly clear by now  included in the unlimited number of undefined things  fundamental particles are not defined. Yet I presume you have no problem understanding that humans (and their romantic relationships) are emergent properties of the underlying structure of the universe.
You are clinging to the idea that we must define things in order to understand them. Yet we've demonstrated that nothing has ever been defined. Everything you understand exists in the absence of any (absolute) definitions.
By rejecting the concept of definitions, I'm not changing anything. This is the default state  I'm just making it explicit. You are being incredulous over something that has already been established. I don't think you are alone in what you are expressing. Axiomatic mathematics was clearly built with the express purpose of defining things. Even where other people admit that axiomatic mathematics can't define anything, they are claiming that it works (at what?) in the very next breath.
Even while admitting that axiomatic mathematics has never defined anything  it is still a knee jerk response to see my rejection of definitions as a weakness. Even Gwydion has been trying to pass it off as 'convenient' for my argument rather than seeing it as a fundamental limit that applies to everyone equally.
You (general respondents) are saying the right things  and then spoiling it by not following through.
We all agree that there are no absolute definitions. Then you try and present my lack of definitions as a weakness while pretending it isn't an issue for axiomatic mathematics.
Try to define distance
This isn't a trick question.
We already know that there are limits on what we can definitively say about distance.
But we also know that we have reasonably successful equations that tell us how distance relates to velocity, time, acceleration, mass, force, rulers (measuring sticks), etc.
Those relationships are everything we know about distance. The same can be said for all those other qualities. The only things we know are what we can measure. And the only way to measure something is to compare it with something else.
This is a closed loop. Each thing is only known according to its relationships with other things.
But more than this  the only evidence for distance, velocity, time, acceleration, mass, force and metre long sticks are those relationships.
elasto is right on the money  all our evidence is circumstantial. All of it. Without exception.
As compelling as the concept of distance is  the only purpose it serves is to explain observed patterns of relationships.
So... try saying something meaningful about 'distance' that isn't a description of a relationship.
Try to define distance
A strong subtext of what we are talking about is knowledge; what we can know to what degree of confidence.
While there are an unlimited number of things that axiomatic mathematics doesn't define  I think it is illuminating to concentrate on just one element: distance.
Geometry is largely about distance. The concepts of dimensions and Number Lines are (I would argue) essential aspects of distance.
These have all been explored in considerable detail within mathematics.
And yet noone has defined distance (in the same way that nothing else has been defined).
We all have a sense of distance. It isn't that the word is utterly meaningless to us.
But: "Why does distance exist?", "What creates distance?", "What, fundamentally, is distance?"; are as unanswered today as they were over two thousand years ago when Euclidean Geometry was first being laid down.
"But it works"
Yes... sort of. There is no question that we have equations that are practically useful (we can apply to the physical world).
But the only justification for those equations is that they work. We have no proofs; of anything.
When you say "Mathematics works"  you aren't entirely wrong. But if axiomatic mathematics cannot prove anything  then is it axiomatic mathematics that is working? Surely axiomatic mathematics without a single proof (of anything  at all) is not working? Whatever it is that is working is working despite axiomatic mathematics failing to prove anything  not because of it?
The caveat/rider that both rmsgrey and elasto have recently referred to is necessary. A strict reading of axiomatic mathematics tells us that if we can't prove something we can't claim to have any degree of knowledge with regard to that something. The Laws of thought tell us that we either know it or we don't (excluded middle  no 'partially true'  either true or false  nothing in between; noncontradiction  either true or false  not 'both true and false').
I wholeheartedly agree with that caveat/rider. We have to admit that the Laws of Thought don't work. If we try to stick to the Laws of Thought we find that we can't prove anything. We can't know anything. The result of trying to rigidly stick with the Laws of Thought is nihilistic. Which is why nobody here is actually committing to the Laws of Thought.
Choose one or the other:
1. Follow the Laws of Thought and find that rigid proof of anything is impossible  the inverse of which (excluded middle and noncontradiction) leaves us knowing nothing. Nihilism.
OR
2. Admit the the Laws of Thought lead to a dead end and that real life doesn't work in the sense explicitly stated by the Laws of Thought.
We are most of the way there
rmsgrey and elasto have explicitly stated that we don't follow the Laws of Thought in practice because that is impractical.
I completely agree with them on this point.
The bit we are currently stuck on is claiming that axiomatic mathematics is working even after admitting that you are disregarding the Laws of Thought. This is the contradiction that I'm criticising you for. You should either hold to the Laws of Thought or not.
Elasto (and I) are perfectly justified in claiming certainty with the caveat/rider provided. There is nothing contradictory about that. It only becomes contradictory when you try to combine the caveat/rider with the Laws of Thought.
Next
As above  admitting that mathematics requires the caveat/rider that rmsgrey suggests, and I'm pretty sure everyone here is already assuming exists, is a huge step.
You are smart enough to realise that the rider that rmsgrey suggests is incompatible with the Laws of Thought once it is pointed out to you. Again  you have all the pieces  you just need to put them together into one coherent whole.
Measurement and justification
A theory that cannot be tested is useless. The mathematical concept of proof (with respect to axiomatic mathematics) was intended to be a measure of the validity of an initial set of statements (and their consequences). It was a reasonable idea up until the point we found it was impossible to establish an unambiguous, known, set of axioms.
However, the idea of measuring/justifying concepts is sound.
Raging clue incoming...
How do we measure distance? (How do we quantify distance?)
Extra comments:
We cannot measure distance except by comparison to other distances (measurement) (It isn't possible to just know anything ex cathedra  as elasto points out everything we know is through circumstantial evidence).
Distance is a measure of how closely related things are (again  as elasto implies  our degree of certainty depends on the number (and quality) of relationships (circumstantial evidence).
There necessarily exists a minimum quantum of distance/relationship. (Among all the things we can't define: we can't define 'continuous'  we can only be aware of comparisons  we can't fundamentally know anything outside of a comparison (measurement).
WibblyWobbly wrote:We don't? What is a relationship? Are you talking about what happens when a mommy and a daddy love each other very much? Because that doesn't tell me anything about math, that only tells me where babies come from!
As should be abundantly clear by now  included in the unlimited number of undefined things  fundamental particles are not defined. Yet I presume you have no problem understanding that humans (and their romantic relationships) are emergent properties of the underlying structure of the universe.
You are clinging to the idea that we must define things in order to understand them. Yet we've demonstrated that nothing has ever been defined. Everything you understand exists in the absence of any (absolute) definitions.
By rejecting the concept of definitions, I'm not changing anything. This is the default state  I'm just making it explicit. You are being incredulous over something that has already been established. I don't think you are alone in what you are expressing. Axiomatic mathematics was clearly built with the express purpose of defining things. Even where other people admit that axiomatic mathematics can't define anything, they are claiming that it works (at what?) in the very next breath.
Even while admitting that axiomatic mathematics has never defined anything  it is still a knee jerk response to see my rejection of definitions as a weakness. Even Gwydion has been trying to pass it off as 'convenient' for my argument rather than seeing it as a fundamental limit that applies to everyone equally.
You (general respondents) are saying the right things  and then spoiling it by not following through.
We all agree that there are no absolute definitions. Then you try and present my lack of definitions as a weakness while pretending it isn't an issue for axiomatic mathematics.
Try to define distance
This isn't a trick question.
We already know that there are limits on what we can definitively say about distance.
But we also know that we have reasonably successful equations that tell us how distance relates to velocity, time, acceleration, mass, force, rulers (measuring sticks), etc.
Those relationships are everything we know about distance. The same can be said for all those other qualities. The only things we know are what we can measure. And the only way to measure something is to compare it with something else.
This is a closed loop. Each thing is only known according to its relationships with other things.
But more than this  the only evidence for distance, velocity, time, acceleration, mass, force and metre long sticks are those relationships.
elasto is right on the money  all our evidence is circumstantial. All of it. Without exception.
As compelling as the concept of distance is  the only purpose it serves is to explain observed patterns of relationships.
So... try saying something meaningful about 'distance' that isn't a description of a relationship.
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5188
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Somerville, MA
 Contact:
Re: Misunderstanding basic math concepts, help please?
d maps to R
d(x,y) ≥ 0
d(x,y) = 0 if and only if x = y
d(x,y) = d(y,x)
d(x,z) ≤ d(x,y) + d(y,z)
If it satisfies these properties, it is a distance. x and y are in M, and M is now a metric space.
It's not so hard.
d(x,y) ≥ 0
d(x,y) = 0 if and only if x = y
d(x,y) = d(y,x)
d(x,z) ≤ d(x,y) + d(y,z)
If it satisfies these properties, it is a distance. x and y are in M, and M is now a metric space.
It's not so hard.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
 gmalivuk
 GNU Terry Pratchett
 Posts: 25723
 Joined: Wed Feb 28, 2007 6:02 pm UTC
 Location: Here and There
 Contact:
Re: Misunderstanding basic math concepts, help please?
Treatid: You seem to keep mixing your incomplete understanding of mathematics with your incomplete understanding of physics.
Math is perfectly capable of defining both distance and continuity (in multiple ways in fact), even if we're not quite sure how exactly distance works at the smallest physical scales.
Math is perfectly capable of defining both distance and continuity (in multiple ways in fact), even if we're not quite sure how exactly distance works at the smallest physical scales.
Re: Misunderstanding basic math concepts, help please?
doogly wrote:d maps to R
d(x,y) ≥ 0
d(x,y) = 0 if and only if x = y
d(x,y) = d(y,x)
d(x,z) ≤ d(x,y) + d(y,z)
If it satisfies these properties, it is a distance. x and y are in M, and M is now a metric space.
It's not so hard.
1. You are putting the cart before the horse by using an assumption of distance (greater than/less than) as part of your definition of distance. If you start with arithmetic then of course you can demonstrate the number line. Until you have demonstrated an independent definition of equality, zero, addition, greater/less than; then all you are doing is writing down a tautology: Distance is a thing that has (ordered) separation.
I mean... yes... you're not wrong. But since separation is the quality of having distance....
gmalivuk wrote:Math is perfectly capable of defining both distance and continuity (in multiple ways in fact), even if we're not quite sure how exactly distance works at the smallest physical scales.
Really?
We know that mathematics hasn't defined anything in an absolute sense. So you mean something other than 'absolute definition'. What, exactly?
Perhaps you are referring to a dictionary definition where words are defined in terms of other words? As Doogly has done? So... you mean that mathematics has many tautologies for what 'distance' and 'continuous' are?
Fair enough. Words describing other words is the natural language definition of definition. And another word for a closed circle of words (or symbols) describing each other is 'tautology'.
So... if I'm reading you right  you are being smug about mathematics having a number of tautologies?
Cargo Cult
We agree that there are bits of mathematics that are practically useful: "It works".
The trouble is that you don't know why those bits work.
Axiomatic mathematics doesn't prove anything (A strict application of axiomatic mathematics cannot prove anything).
Using one group of symbols to describe other groups of symbols ultimately leads to running out of symbols or a closed loop where Symbol A defines Symbol B and Symbol B defines Symbol A  a tautology.
The standard 'definition' of a Metric Space that doogly provides says that a Metric Space is a thing that has the properties of a Metric Space.
"Ah  but we can, at least distinguish between Metric Spaces and nonMetric Spaces"
No. You can't.
The operators (functions) 'd(a, b)', '+', '=', '≥', '≤' and the identity zero are just symbols. The meaning of those symbols are specified by other symbols.
When we discover that we don't have any absolute definitions  what we have discovered is that we cannot pin a definite meaning to a given symbol or group of symbols.
Under the standard 'definition' of a Metric Space provided by doogly  a thing is a Metric Space if there is any conceivable set of meanings for the involved symbols that fits the pattern. Alternatively, nothing is a metric space because we cannot specify the meaning of a given set of symbols.
"But we do have known meanings for given symbols"
No. You don't. Sort of.
A Cargo Cultist could build a paper airplane by bowing towards the east, muttering prayers to the gods, folding the paper just so and voila  a paper airplane that flies. This cultist could teach his methods to the next generation who will be amazed to discover they can make flying paper. But without understanding  none of the new cultists will realise that most of what they are being taught is redundant to the process of making paper fly.
In case I'm being too subtle:
'Definition' is the 'prayer to allah' part of this analogy. We've never had absolute definitions. Symbols defining other symbols can only lead to tautology.
Everything ever achieved has been with our very best definitions being tautologies.
So where does meaning come from?
We do use words in a (subjectively) meaningful way. We draw patterns of relationships using words.
Addition on the number line specifies a particular pattern of relationship between numbers. Multiplication on the number line is another pattern of relationships. And so on for all possible functions on the Number Line. The Number Line is then the sum of these patterns of relationships.
While a Metric Space will typically be specified in terms of the behaviour of the functions over that domain... the behaviour of those functions over that domain will create a pattern of relationships between the elements of that domain. It is entirely within the purview of existing mathematics to start with a given topology and fit functions and domain to that topology rather than to construct a topology from the functions and domain.
From this perspective, mathematics is already all about describing patterns of relationships.
"So what are you making a fuss about if that is what we are already doing?"
Because we can describe a pattern. We cannot define a function.
Everyone here knows what addition on the Number Line is. You just can't define it.
You'll recognise it when you see it  but as hard as axiomatic mathematics has tried  it isn't possible to define addition in absolute terms. And any nonabsolute definition of addition boils down to a tautology.
This is very much Cogito Ergo Sum. We recognise the pattern of our own existence even while we cannot prove our existence in absolute terms.
Likewise, we recognise the patterns of arithmetic even while we cannot meaningfully define arithmetic in an unambiguous way.
Humans are good at recognising patterns. We can reproduce a pattern and spot departures from a pattern.
We still can't define a pattern. Like our own existence, a pattern is itself. The most we can do is to compare how similar two patterns are (this is a lot... like, everything a lot).
Re: Misunderstanding basic math concepts, help please?
So far you have been rambling on and on about how we can't describe anything, so what makes the 'relationship of patterns of words' that you suggest not be included in the quantifier 'anything'? You have yet to show a single concrete example of how you would do things differently.
Could you show what your methodology would consider a proof(and make it a simple one, it shouldn't be more than a paragraph, seeing as if even simple proofs in your methodology require a page long thesis, 3 analogies and a nudge to physics, it would make it far less useful than first order logic that we've shown you a few pages ago how we use it to communicate, understand and correct each other, all while being concise)
As for "we can't describe addition for numbers", I present you to Presburger Arithmetic
Could you show what your methodology would consider a proof(and make it a simple one, it shouldn't be more than a paragraph, seeing as if even simple proofs in your methodology require a page long thesis, 3 analogies and a nudge to physics, it would make it far less useful than first order logic that we've shown you a few pages ago how we use it to communicate, understand and correct each other, all while being concise)
As for "we can't describe addition for numbers", I present you to Presburger Arithmetic
 gmalivuk
 GNU Terry Pratchett
 Posts: 25723
 Joined: Wed Feb 28, 2007 6:02 pm UTC
 Location: Here and There
 Contact:
Re: Misunderstanding basic math concepts, help please?
Yes, things like = and < are symbols, but they are symbols that work in a particular way that can be precisely described using firstorder logic. It takes a bit more work to get to the real numbers and hence to measures of distance, but that can be done in a systematic, stepbystep way, too.
A set with a distance function (which must have the properties doogly described because otherwise it's not a distance function) is a metric space. All the other properties of a metric space can be derived from that, but we don't really put those properties in knowingly at the beginning.
A set with a distance function (which must have the properties doogly described because otherwise it's not a distance function) is a metric space. All the other properties of a metric space can be derived from that, but we don't really put those properties in knowingly at the beginning.
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5188
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Somerville, MA
 Contact:
Re: Misunderstanding basic math concepts, help please?
1.
You use the word "absolute" a lot. What does this word mean to you?
2.
I think this is a very good analogy! The question is whether "definition" is the unnecessary prayer part, or the actually important folding up bits.
The enlightened westerner can demonstrate that a paper airplane still flies by doing the folding and eliminating the praying, and then the silly islander must acknowledge how silly the praying was. (Egads this is an embarrassing example, but I'm running with it.)
So, can you accomplish anything without definitions? Because the people who operate in the shared reality of mathematics can do things like go to the moon, split the atom and build robots that do jumping jacks. Using definitions of distance. It very much is a shared reality that it would be better if you could participate in.
You use the word "absolute" a lot. What does this word mean to you?
2.
Treatid wrote:'Definition' is the 'prayer to allah' part of this analogy.
I think this is a very good analogy! The question is whether "definition" is the unnecessary prayer part, or the actually important folding up bits.
The enlightened westerner can demonstrate that a paper airplane still flies by doing the folding and eliminating the praying, and then the silly islander must acknowledge how silly the praying was. (Egads this is an embarrassing example, but I'm running with it.)
So, can you accomplish anything without definitions? Because the people who operate in the shared reality of mathematics can do things like go to the moon, split the atom and build robots that do jumping jacks. Using definitions of distance. It very much is a shared reality that it would be better if you could participate in.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: Misunderstanding basic math concepts, help please?
Ho boy!
We have Demki insisting that I formally prove that it is impossible to formally prove (or define) anything.
We have gmalivuk confidently asserting the existence of definite definitions (without providing any support for his assertions) despite acknowledging the existence of the Foundational Crises in Mathematics (which clearly states that there is a foundational element of mathematics that is categorically Terra Incognita).
Even doogly feels I'm rejecting the status quo despite the status quo being that there has never been a definition or proof of anything  ever.
No absolute definitions
Most people have acknowledged (or at least allowed) that we do not, in fact, have any epistemically certain, absolute, unambiguous, definitions.
The most extreme and direct statement of this is the solipsism resulting from Cogito Ergo Sum. I.E. We can't prove that our sensory input isn't being tampered with, therefore we can't know anything with certainty.
I'm pretty sure everyone is on the same page with regard to our lack of this level of certainty.
Obviously we can't have complete and utter certainty  but equally obviously it is ridiculous to reject all possibility of sentience. Our existence demonstrates... something.
It is the next step I think is causing various knickers to be knotted:
"We don't have absolute definitions  but we do have nonabsolute definitions"
No. No we don't.
Let us look at the problem in more abstract terms
We have two sets.
1. Set A consists of all the possible symbols we can express.
2. Set B consists of all possible meanings we could associate with symbols from set A.
Every possible label and symbol we can write down is a member of Set A. So: 'definition', 'description', 'proof', '+', 'meaning', 'knowledge' and all the etcs...
Every possible meaning (and/or definition, and/or concept) are members of Set B.
For instance, all the possible mappings between Set B and Set A are members of Set B. Also, all the possible ways we could order Set A or Set B are members of Set B.
i. Using only members of Set A, specify a known member of set B.
I'll wait....
I got bored
This is obviously an impossible "by your bootstraps" problem. There is no way to specify a mapping between B and A (remember  the mapping is a member of B) using only members of A. To even begin we need a preexisting, known mapping between A and B... But then how do we represent that known mapping? How do we communicate a known mapping to someone who doesn't start with that known mapping?
This problem is utterly intractable
If we have something that we perceive as working  then it cannot be equivalent to this problem.
Given that we perceive some aspects of mathematics working... then mathematics cannot be represented as a Set of Symbols plus a Set of Concepts (or any equivalent thereof).
To answer doogly's first point:
This is what I perceive 'absolute' to be; what the Laws of Thought are expressing; and the intention of axiomatic mathematics:
To specify a mapping (or mappings) between concepts and symbols.
And, as expressed here  it is clear that there is no mechanism available that allows us to express such a mapping.
I think this is where we are
Your comments strongly suggests that each of you believes that there is a mapping between a set of concepts and a set of symbols  even if that mapping isn't 'absolute' (whatever that means).
But where the elements needed to specify the mapping are elements of the sets we want to map to each other  it can't work. Even if you started with a known mapping  you can't uniquely express that mapping using members of Set A.
Don't take my word for it. Try it for yourself. This is the root of the Foundational Crises in mathematics. You will not be able to find any deterministic mechanism of mapping to symbols that provides a constructive result. Not even partially.
It isn't possible to map an idea to a symbol.
Doogly's point 2.
The lack of definitions isn't something I've invented. The Foundational Crises started long before I was born. It has never been possible to specify a particular mapping between A and B.
All those things that doogly describes, have been accomplished without a single definition. They have been achieved despite the Foundational Crises.
Just think how much more could be accomplished if we correctly understood what we were doing.
We have Demki insisting that I formally prove that it is impossible to formally prove (or define) anything.
We have gmalivuk confidently asserting the existence of definite definitions (without providing any support for his assertions) despite acknowledging the existence of the Foundational Crises in Mathematics (which clearly states that there is a foundational element of mathematics that is categorically Terra Incognita).
Even doogly feels I'm rejecting the status quo despite the status quo being that there has never been a definition or proof of anything  ever.
No absolute definitions
Most people have acknowledged (or at least allowed) that we do not, in fact, have any epistemically certain, absolute, unambiguous, definitions.
The most extreme and direct statement of this is the solipsism resulting from Cogito Ergo Sum. I.E. We can't prove that our sensory input isn't being tampered with, therefore we can't know anything with certainty.
I'm pretty sure everyone is on the same page with regard to our lack of this level of certainty.
Obviously we can't have complete and utter certainty  but equally obviously it is ridiculous to reject all possibility of sentience. Our existence demonstrates... something.
It is the next step I think is causing various knickers to be knotted:
"We don't have absolute definitions  but we do have nonabsolute definitions"
No. No we don't.
Let us look at the problem in more abstract terms
We have two sets.
1. Set A consists of all the possible symbols we can express.
2. Set B consists of all possible meanings we could associate with symbols from set A.
Every possible label and symbol we can write down is a member of Set A. So: 'definition', 'description', 'proof', '+', 'meaning', 'knowledge' and all the etcs...
Every possible meaning (and/or definition, and/or concept) are members of Set B.
For instance, all the possible mappings between Set B and Set A are members of Set B. Also, all the possible ways we could order Set A or Set B are members of Set B.
i. Using only members of Set A, specify a known member of set B.
I'll wait....
I got bored
This is obviously an impossible "by your bootstraps" problem. There is no way to specify a mapping between B and A (remember  the mapping is a member of B) using only members of A. To even begin we need a preexisting, known mapping between A and B... But then how do we represent that known mapping? How do we communicate a known mapping to someone who doesn't start with that known mapping?
This problem is utterly intractable
If we have something that we perceive as working  then it cannot be equivalent to this problem.
Given that we perceive some aspects of mathematics working... then mathematics cannot be represented as a Set of Symbols plus a Set of Concepts (or any equivalent thereof).
To answer doogly's first point:
This is what I perceive 'absolute' to be; what the Laws of Thought are expressing; and the intention of axiomatic mathematics:
To specify a mapping (or mappings) between concepts and symbols.
And, as expressed here  it is clear that there is no mechanism available that allows us to express such a mapping.
I think this is where we are
Your comments strongly suggests that each of you believes that there is a mapping between a set of concepts and a set of symbols  even if that mapping isn't 'absolute' (whatever that means).
But where the elements needed to specify the mapping are elements of the sets we want to map to each other  it can't work. Even if you started with a known mapping  you can't uniquely express that mapping using members of Set A.
Don't take my word for it. Try it for yourself. This is the root of the Foundational Crises in mathematics. You will not be able to find any deterministic mechanism of mapping to symbols that provides a constructive result. Not even partially.
It isn't possible to map an idea to a symbol.
Doogly's point 2.
The lack of definitions isn't something I've invented. The Foundational Crises started long before I was born. It has never been possible to specify a particular mapping between A and B.
All those things that doogly describes, have been accomplished without a single definition. They have been achieved despite the Foundational Crises.
Just think how much more could be accomplished if we correctly understood what we were doing.
Re: Misunderstanding basic math concepts, help please?
There seems to be something inconsistent about using symbols to attempt to convince people that you can't use symbols to represent ideas.
It's reasonable enough to observe that we don't understand how language starts, but to conclude from that that there can be no meaningful communication is like concluding that the universe doesn't exist because we don't understand how the Big Bang came about. The reality is that language exists and communicates sufficiently well that two groups of people can agree to bore a 30 mile tunnel from both ends and meet successfully in the middle (well, actually slightly closer to one end than the other, but close enough) with less than half a meter error in alignment at first contact. Any argument that it's impossible to establish accurate definitions for symbols and concepts contradicts everyday experience.
What we can say is that we don't understand where meaning comes from originally  how we get from nothing to even the simplest protolanguage  but once we accept the original miracle of some symbol acquiring sense, we can sketch out programs for bootstrapping into more rigorous symbology or into richer language. You can use nonrigorous natural language to describe formal, rigorous languages  in a model where everything is black or white, it doesn't matter that in the real world, all you have are shades of grey, and that you can't sharply divide white from grey from black.
Put it another way: you expect us to understand what you mean by the phrase "absolute definitions"  there is some process whereby that symbol (or set of symbols) has gained enough meaning that you expect us to understand what you mean when you say that absolute definitions don't exist. That means that that process you implicitly accept as having happened in this instance provides some level of definition. If the phrase were completely undefined, you wouldn't be using it as though it had a definite meaning. By making an argument, you are accepting that the words you are using have (nonabsolute) definitions  that there is something that the symbols you assemble mean. If there were no definitions of any sort then anflkjewbuv rkjlgbelwabvknsn oheoiurn eurkwngalj kjeaehhakljrtal;!3n45.,se989 wouldn't you agree?
It's reasonable enough to observe that we don't understand how language starts, but to conclude from that that there can be no meaningful communication is like concluding that the universe doesn't exist because we don't understand how the Big Bang came about. The reality is that language exists and communicates sufficiently well that two groups of people can agree to bore a 30 mile tunnel from both ends and meet successfully in the middle (well, actually slightly closer to one end than the other, but close enough) with less than half a meter error in alignment at first contact. Any argument that it's impossible to establish accurate definitions for symbols and concepts contradicts everyday experience.
What we can say is that we don't understand where meaning comes from originally  how we get from nothing to even the simplest protolanguage  but once we accept the original miracle of some symbol acquiring sense, we can sketch out programs for bootstrapping into more rigorous symbology or into richer language. You can use nonrigorous natural language to describe formal, rigorous languages  in a model where everything is black or white, it doesn't matter that in the real world, all you have are shades of grey, and that you can't sharply divide white from grey from black.
Put it another way: you expect us to understand what you mean by the phrase "absolute definitions"  there is some process whereby that symbol (or set of symbols) has gained enough meaning that you expect us to understand what you mean when you say that absolute definitions don't exist. That means that that process you implicitly accept as having happened in this instance provides some level of definition. If the phrase were completely undefined, you wouldn't be using it as though it had a definite meaning. By making an argument, you are accepting that the words you are using have (nonabsolute) definitions  that there is something that the symbols you assemble mean. If there were no definitions of any sort then anflkjewbuv rkjlgbelwabvknsn oheoiurn eurkwngalj kjeaehhakljrtal;!3n45.,se989 wouldn't you agree?
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Ho boy!
We have Demki insisting that I formally prove that it is impossible to formally prove (or define) anything.
And this here shows you didn't read my post.
I didn't ask you to prove that you can't prove anything.
I asked you to use whatever methodology you propose to replace axiomatic mathematics to prove/show some result, however small it may be.
This is because so far you've been thrashing axiomatic mathematics, rambling on with anecdotes and winks to physics without directly replying to our questions and arguments, without giving us anything that may actually replace/improve mathematics.
If that's not bad criticism, I don't know what is.
 WibblyWobbly
 Can't Get No
 Posts: 506
 Joined: Fri Apr 05, 2013 1:03 pm UTC
Re: Misunderstanding basic math concepts, help please?
rmsgrey wrote:There seems to be something inconsistent about using symbols to attempt to convince people that you can't use symbols to represent ideas.
It's reasonable enough to observe that we don't understand how language starts, but to conclude from that that there can be no meaningful communication is like concluding that the universe doesn't exist because we don't understand how the Big Bang came about. The reality is that language exists and communicates sufficiently well that two groups of people can agree to bore a 30 mile tunnel from both ends and meet successfully in the middle (well, actually slightly closer to one end than the other, but close enough) with less than half a meter error in alignment at first contact. Any argument that it's impossible to establish accurate definitions for symbols and concepts contradicts everyday experience.
What we can say is that we don't understand where meaning comes from originally  how we get from nothing to even the simplest protolanguage  but once we accept the original miracle of some symbol acquiring sense, we can sketch out programs for bootstrapping into more rigorous symbology or into richer language. You can use nonrigorous natural language to describe formal, rigorous languages  in a model where everything is black or white, it doesn't matter that in the real world, all you have are shades of grey, and that you can't sharply divide white from grey from black.
Put it another way: you expect us to understand what you mean by the phrase "absolute definitions"  there is some process whereby that symbol (or set of symbols) has gained enough meaning that you expect us to understand what you mean when you say that absolute definitions don't exist. That means that that process you implicitly accept as having happened in this instance provides some level of definition. If the phrase were completely undefined, you wouldn't be using it as though it had a definite meaning. By making an argument, you are accepting that the words you are using have (nonabsolute) definitions  that there is something that the symbols you assemble mean. If there were no definitions of any sort then anflkjewbuv rkjlgbelwabvknsn oheoiurn eurkwngalj kjeaehhakljrtal;!3n45.,se989 wouldn't you agree?
So, if I'm reading you right (and let me know if I'm not), what you're saying is that there is a fundamental disconnect, but it's not in mathematics, it's in humanity itself. Extracting meaning from a representation may well go beyond even the first protolanguages; hell, can you imagine the first attempts to use a drawing (or a charade) to represent a structure (or an action) that isn't physically present (or being performed) at that specific moment in time? Using a drawing of a herd animal to represent an actual nearby herd of animals? The very first time something like this happens, though (the first time someone understands "representation"), doesn't that satisfy Treatid's mapping between sets A and B, i.e., doesn't everything else become possible? If not, Treatid, please give us an explanation of why this isn't the case.
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5188
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Somerville, MA
 Contact:
Re: Misunderstanding basic math concepts, help please?
And treatid's A and B are not sets. I mean, his argument doesn't depend on their being sets, but look, this word has a definition, and the rest of us know it. Somehow this works for us. Maybe it's magic?
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
 WibblyWobbly
 Can't Get No
 Posts: 506
 Joined: Fri Apr 05, 2013 1:03 pm UTC
Re: Misunderstanding basic math concepts, help please?
doogly wrote:And treatid's A and B are not sets. I mean, his argument doesn't depend on their being sets, but look, this word has a definition, and the rest of us know it. Somehow this works for us. Maybe it's magic?
But it's not an absolute definition, so it's not actually a definition, and it may actually be a squirrel. We just don't know.
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5188
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Somerville, MA
 Contact:
Re: Misunderstanding basic math concepts, help please?
Or hmm, maybe there's a more charitable interpretation?
Definitions of things do evolve as we think about them. Let's say someone wanted to define a manifold as
M1) A topological space which is locally Euclidean.
This is really what we would all like a manifold to be. In my heart of hearts, I feel like this is a manifold. And there are certain results I would like to prove about a manifold.
But, there are some sneaky counterexamples you could build with this definition! It's a little too loose! Now we say
M2) A topological space which is second countable, Hausdorff, and locally Euclidean.
It corresponds to our basic notion plus "no whammies." It is more precise.
Is this definition absolutely what it means to be a manifold? Well, for now yes. Maybe we'll revise the definition of manifold. I mean, practically speaking, we won't, because this has been around long enough that when we refine things further, we will add qualifiers like "Smooth Manifold" rather than redefining manifold to be "smooth."
( Unless I always, always work with smooth manifolds. Then I probably stick the smooth assumption in there and you will have to back up your discussion a bit if you were really talking about a C^3 manifold. You have now been exposed as an analyst. Or likewise, what a physicist calls a "tensor" is what a matho would call a "tensor field." We almost always do this thing and it is fine that it's not absolute, we can work it out. If your first semester of grad school you take differential topology in the math department and GR in the physics department, the whole thing may take a month, but hopefully not two.)
So the definition of "manifold" is in a sense absolute. But it can be understood and articulated in a precise way when it is articulated, and we can explore the bounds of what is meant by that definition.
Which is why set originally defined in such a way that treatid's above A and B would be sets, but now the definition of set is such that they cannot. If the definition of set were absolute across time this wouldn't be a possible thing to have happen with the word. But it means a thing right now, so use the word the right way. If you look at the current definition, there's not ambiguity.
Definitions of things do evolve as we think about them. Let's say someone wanted to define a manifold as
M1) A topological space which is locally Euclidean.
This is really what we would all like a manifold to be. In my heart of hearts, I feel like this is a manifold. And there are certain results I would like to prove about a manifold.
But, there are some sneaky counterexamples you could build with this definition! It's a little too loose! Now we say
M2) A topological space which is second countable, Hausdorff, and locally Euclidean.
It corresponds to our basic notion plus "no whammies." It is more precise.
Is this definition absolutely what it means to be a manifold? Well, for now yes. Maybe we'll revise the definition of manifold. I mean, practically speaking, we won't, because this has been around long enough that when we refine things further, we will add qualifiers like "Smooth Manifold" rather than redefining manifold to be "smooth."
( Unless I always, always work with smooth manifolds. Then I probably stick the smooth assumption in there and you will have to back up your discussion a bit if you were really talking about a C^3 manifold. You have now been exposed as an analyst. Or likewise, what a physicist calls a "tensor" is what a matho would call a "tensor field." We almost always do this thing and it is fine that it's not absolute, we can work it out. If your first semester of grad school you take differential topology in the math department and GR in the physics department, the whole thing may take a month, but hopefully not two.)
So the definition of "manifold" is in a sense absolute. But it can be understood and articulated in a precise way when it is articulated, and we can explore the bounds of what is meant by that definition.
Which is why set originally defined in such a way that treatid's above A and B would be sets, but now the definition of set is such that they cannot. If the definition of set were absolute across time this wouldn't be a possible thing to have happen with the word. But it means a thing right now, so use the word the right way. If you look at the current definition, there's not ambiguity.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
 WibblyWobbly
 Can't Get No
 Posts: 506
 Joined: Fri Apr 05, 2013 1:03 pm UTC
Re: Misunderstanding basic math concepts, help please?
I still think it's a squirrel.
I suppose I'm still stuck at the question of "why is an absolute definition required in order to be used?" and perhaps a little at the question "what is an absolute definition?", which is where I most recently entered the debate, asking about the concept of "absolute" and whether a relationship between concepts and symbols (should it exist) is far deeper than mathematics. Treatid seems to think no such relationship exists. I am befuddled as to why. Treatid seems to think such a relationship must itself be a concept, but I would think it outside concepts and symbols; we can create a concept of it, but it is just another representation. and a representation of a thing is not the thing. But he's also incorrectly using language, such as "sets", "mapping", etc., and insofar as I used his terminology, that's my mistake. I'm aware of the proper uses of such terms, and should be more careful. Hopefully, that makes at least a little sense.
Unless you weren't replying to me, in which case I am now concerned that it may be a ferret.
I suppose I'm still stuck at the question of "why is an absolute definition required in order to be used?" and perhaps a little at the question "what is an absolute definition?", which is where I most recently entered the debate, asking about the concept of "absolute" and whether a relationship between concepts and symbols (should it exist) is far deeper than mathematics. Treatid seems to think no such relationship exists. I am befuddled as to why. Treatid seems to think such a relationship must itself be a concept, but I would think it outside concepts and symbols; we can create a concept of it, but it is just another representation. and a representation of a thing is not the thing. But he's also incorrectly using language, such as "sets", "mapping", etc., and insofar as I used his terminology, that's my mistake. I'm aware of the proper uses of such terms, and should be more careful. Hopefully, that makes at least a little sense.
Unless you weren't replying to me, in which case I am now concerned that it may be a ferret.
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5188
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Somerville, MA
 Contact:
Re: Misunderstanding basic math concepts, help please?
Oh yeah of course "absolute" is not a meaningful statement.
This is like how my flavor of atheism gets the fancy name "noncognitivism."
This is like how my flavor of atheism gets the fancy name "noncognitivism."
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: Misunderstanding basic math concepts, help please?
To summarize:
Treatid: Maths can't prove itself to be true! It uses symbols that are not absolute!
Us: So? It works well enough to put a man on the moon
Treatid: I have something better!
Us: Do tell
Treatid: {babbles with symbols}
Us: Doesn't that contradict what you said earlier?
Treatid: <crickets>
Treatid: Maths can't prove itself to be true! It uses symbols that are not absolute!
Us: So? It works well enough to put a man on the moon
Treatid: I have something better!
Us: Do tell
Treatid: {babbles with symbols}
Us: Doesn't that contradict what you said earlier?
Treatid: <crickets>
Re: Misunderstanding basic math concepts, help please?
Hiding simplicity behind words
Last post I posited two sets. One set of symbols and A.N.Other set. I then asserted that there was no way to specify a single known mapping between these sets using only symbols.
None so blind as those who will not see
In doogly's case, it can be argued that he is recasting my argument into a form he is more comfortable with.
Other responses appear to me to be nothing more than deflection.
Since you are all reluctant to admit that you can't see a way out of the mapping problem...
I'm going to assume that you all have enough experience and sense to have some comprehension of the mapping problem I outlined. And further  that if you had a trivial solution to the problem you would be rubbing my face in it right now.
Since I think that thinking about this mapping problem will illustrate many of my points: I'll just drop a mention of Russell's Paradox here.
The flip side of definitions
In order to define something you need something to define. A thing.
In the mapping problem I described we have two distinct sets (Symbols and a set of nonsymbols). Implied within the structure of a set is that each element of a set is distinct.
The explicit second and third Laws of thought: Either that or the other  nothing in between and not both. Plus the Law of Identity.
From sets to integers mathematics explicitly assumes that it is possible to draw a hard line between this and that. That time is distinct from distance, that temperature is distinct from velocity, that each integer is distinct from every other integer. (Hmmm... maybe I should have given more thought to those first two examples...)
Categorisation
It is really useful to be able to categorise things. For example, we could categorise humans into racist and nonracists. We can then round up all the racists and shoot them.
Mathematics simply takes categorisation to its logical conclusion: while 'racist' might be a somewhat subjective term, we can extrapolate the idea of categorisation and we arrive at some sets which are totally and utterly distinct with no middle ground (excluded middle) and where a given element is definitely in one set or the other  but not both (noncontradiction).
Of course, if it was just a matter of extrapolation we could just go faster and faster until we match, then exceed the speed of light in vacuum.
So... is it reasonable to extrapolate simple categorisation to an extreme?
At the extreme: Can we have two unrelated things?
Create two unrelated ideas.
Create one unrelated idea.
What would an unrelated idea even be?
What steps would you have to take in order to make a pair of ideas less related?
Saying A is unrelated to B actually reinforces a relationship between A and B. Indeed, it isn't clear what 'unrelated' could possibly mean given that it isn't possible to give an (Laws of Thought) example of completely unrelated. Lessrelated  that's plausible. UnRelated? Any thought you have is related to every thought I have by virtue of us existing within the same universe.
This isn't word play  everything that exists within and as a consequence of this universe is related to every other thing that exists within and as a consequence of this universe.
In terms of categorisation, we can definitely group things. Up to a limit. But when you want to say that this thing is definitely distinct from that thing you've gone over the limit.
Principle of Explosion & Euclidean and nonEuclidean Geometry
There has been a long standing argument over my determination to apply the Principle of Explosion to the widest possible scope. We have also argued over Euclidean Geometry and nonEuclidean Geometry being mutually inconsistent with each other.
In both cases interlocutors have argued that there is a definite difference such that the Principle of Explosion cannot be considered with regard to natural languages and inconsistent axiomatic systems cannot be considered alongside each other.
But we can't specify an absolute distinction any more than we can specify an absolute definition. And without a hard distinction between inconsistent axiom sets... everything in axiomatic mathematics is inconsistent.
"You are doing nothing but bashing axiomatic mathematics"
In this post I have pointed out the fundamental interconnectedness of all things.
"Whoa  hippy dippy"
No  just a simple (lower case t) truth. The Laws of Thought are a simple statement that it is possible to have a clear distinction.
A simple thought experiment shows that we cannot have unrelated ideas  let alone anything else.
I'm guessing that you want to say that I'm using too vague and general a meaning for 'related' or 'unrelated'. "Yes  obviously everything is related if you go to that extreme".
I'm stating my position, "everything is related", as a direct contradiction of the Laws of Thought. If you do, indeed, see that it is obvious that everything (within this universe) is related  then we have taken a giant step forward. If you don't think it is obvious that everything (in this universe) is related then provide some reasoning and we'll engage in some productive dialogue.
Bonus Content:
Synonyms for Symbol: ... Pattern ...
Last post I posited two sets. One set of symbols and A.N.Other set. I then asserted that there was no way to specify a single known mapping between these sets using only symbols.
None so blind as those who will not see
In doogly's case, it can be argued that he is recasting my argument into a form he is more comfortable with.
Other responses appear to me to be nothing more than deflection.
Since you are all reluctant to admit that you can't see a way out of the mapping problem...
I'm going to assume that you all have enough experience and sense to have some comprehension of the mapping problem I outlined. And further  that if you had a trivial solution to the problem you would be rubbing my face in it right now.
Since I think that thinking about this mapping problem will illustrate many of my points: I'll just drop a mention of Russell's Paradox here.
The flip side of definitions
In order to define something you need something to define. A thing.
In the mapping problem I described we have two distinct sets (Symbols and a set of nonsymbols). Implied within the structure of a set is that each element of a set is distinct.
The explicit second and third Laws of thought: Either that or the other  nothing in between and not both. Plus the Law of Identity.
From sets to integers mathematics explicitly assumes that it is possible to draw a hard line between this and that. That time is distinct from distance, that temperature is distinct from velocity, that each integer is distinct from every other integer. (Hmmm... maybe I should have given more thought to those first two examples...)
Categorisation
It is really useful to be able to categorise things. For example, we could categorise humans into racist and nonracists. We can then round up all the racists and shoot them.
Mathematics simply takes categorisation to its logical conclusion: while 'racist' might be a somewhat subjective term, we can extrapolate the idea of categorisation and we arrive at some sets which are totally and utterly distinct with no middle ground (excluded middle) and where a given element is definitely in one set or the other  but not both (noncontradiction).
Of course, if it was just a matter of extrapolation we could just go faster and faster until we match, then exceed the speed of light in vacuum.
So... is it reasonable to extrapolate simple categorisation to an extreme?
At the extreme: Can we have two unrelated things?
Create two unrelated ideas.
Create one unrelated idea.
What would an unrelated idea even be?
What steps would you have to take in order to make a pair of ideas less related?
Saying A is unrelated to B actually reinforces a relationship between A and B. Indeed, it isn't clear what 'unrelated' could possibly mean given that it isn't possible to give an (Laws of Thought) example of completely unrelated. Lessrelated  that's plausible. UnRelated? Any thought you have is related to every thought I have by virtue of us existing within the same universe.
This isn't word play  everything that exists within and as a consequence of this universe is related to every other thing that exists within and as a consequence of this universe.
In terms of categorisation, we can definitely group things. Up to a limit. But when you want to say that this thing is definitely distinct from that thing you've gone over the limit.
Principle of Explosion & Euclidean and nonEuclidean Geometry
There has been a long standing argument over my determination to apply the Principle of Explosion to the widest possible scope. We have also argued over Euclidean Geometry and nonEuclidean Geometry being mutually inconsistent with each other.
In both cases interlocutors have argued that there is a definite difference such that the Principle of Explosion cannot be considered with regard to natural languages and inconsistent axiomatic systems cannot be considered alongside each other.
But we can't specify an absolute distinction any more than we can specify an absolute definition. And without a hard distinction between inconsistent axiom sets... everything in axiomatic mathematics is inconsistent.
"You are doing nothing but bashing axiomatic mathematics"
In this post I have pointed out the fundamental interconnectedness of all things.
"Whoa  hippy dippy"
No  just a simple (lower case t) truth. The Laws of Thought are a simple statement that it is possible to have a clear distinction.
A simple thought experiment shows that we cannot have unrelated ideas  let alone anything else.
I'm guessing that you want to say that I'm using too vague and general a meaning for 'related' or 'unrelated'. "Yes  obviously everything is related if you go to that extreme".
I'm stating my position, "everything is related", as a direct contradiction of the Laws of Thought. If you do, indeed, see that it is obvious that everything (within this universe) is related  then we have taken a giant step forward. If you don't think it is obvious that everything (in this universe) is related then provide some reasoning and we'll engage in some productive dialogue.
Bonus Content:
Synonyms for Symbol: ... Pattern ...
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5188
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Somerville, MA
 Contact:
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:In doogly's case, it can be argued that he is recasting my argument into a form he is more comfortable with.
Maybe how I set things up most recently did come closer to this.
But if that is the case, then we still get to wildly different conclusions.
You seem to saying that if language is interconnected and a purely social convention... things can mean anything and you can apply a Principle of Explosion to Euclidean Geometry if you want.
I am saying that if language is interconnected and a purely social convention... we need to pay close attention to each other and refine our sense of definitions until we reach a functional state of mutual understanding. If you join with the community that has defined things like "set" in a such a way that your example sets are not sets, then we can do something useful. *Especially* if you want to then use results that are based on definitions meaning a certain thing. Like you want to use a property of sets as mathematical set theory gives them, but not the definition it gives. No bueno! Likewise with the principle of explosion. It's just not doing the work you want and you can't make it do the work you want.
The definitions are not brought down from JHVH on stone tablets, they are conventions we all agree to. If you think something is ambiguous and needs some clarifications somewhere, that is maybe 99% likely to reveal a gap in your education which we can help with, or maybe 1% likely to reveal that a definition was ill posed and is what needs to be clarified.
And the way this process works is that the definitions get *stricter* as they are scrutinized. Look at an integral. We have lots of definitions for integrals  Reimann, Steltjes, Lebesgue... why does this come about? What's the progression been?
If something seems ambiguous to the people trying to speak the language, we have to get *less* ambiguous.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
 WibblyWobbly
 Can't Get No
 Posts: 506
 Joined: Fri Apr 05, 2013 1:03 pm UTC
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Other responses appear to me to be nothing more than deflection.
Why are other responses deflections? Why do you get to decide what is and isn't a deflection? And why isn't your decision that other responses are deflections (presented without evidence) not itself simply a deflection and an unwillingness to deal with questions you don't want to try to answer? This is also the hallmark of a bad argument and a badfaith debater  you shouldn't be picking and choosing whose questions are "important enough" for you to answer. You haven't illustrated anything of your own principles that replaces that which you've set out to destroy. It's simply not enough to say "this is wrong". Fix it.
Treatid wrote:In the mapping problem I described we have two distinct sets (Symbols and a set of nonsymbols). Implied within the structure of a set is that each element of a set is distinct.
The explicit second and third Laws of thought: Either that or the other  nothing in between and not both. Plus the Law of Identity.
No, you didn't posit a set of symbols and a set of nonsymbols. You posited a set of symbols and a set of "meanings". Setting aside the question of whether those are actual sets, the Law of Excluded Middle doesn't apply to those, because one is not the negation of the other. Now you're changing the problem. The middle you were trying to exclude contains objects (not symbols representing objects, but the objects themselves, which have no intrinsic meanings) and actions. More than that, though, why is it necessary to do this mapping? Why must we be able to go from only symbols to meanings at all? It's absolutely not what axiomatic mathematics is doing, so why have you created this false dilemma? It's not at all obvious to this "lesser mind", but you should bloody well be interested in making it so before you claim any sort of victory.
Stop being a dick and start actually getting to work.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Since you are all reluctant to admit that you can't see a way out of the mapping problem...
I'm going to assume that you all have enough experience and sense to have some comprehension of the mapping problem I outlined. And further  that if you had a trivial solution to the problem you would be rubbing my face in it right now.
I've not seen a way into the "mapping problem" yet.
If you want people to rebut (or accept) your arguments, you're going to need to present them in a way that lets people see that there's a case there to answer...

 Posts: 359
 Joined: Wed Sep 21, 2011 3:44 am UTC
Re: Misunderstanding basic math concepts, help please?
rmsgrey wrote:Treatid wrote:Since you are all reluctant to admit that you can't see a way out of the mapping problem...
I'm going to assume that you all have enough experience and sense to have some comprehension of the mapping problem I outlined. And further  that if you had a trivial solution to the problem you would be rubbing my face in it right now.
I've not seen a way into the "mapping problem" yet.
If you want people to rebut (or accept) your arguments, you're going to need to present them in a way that lets people see that there's a case there to answer...
He's just giving a slightly fancier description of the same philosophical obstacle we've acknowledged for quite some time, in this very thread and in many of his past threads. Namely, that you can't teach someone how a language works using only that language. You can't learn English just by reading a dictionary written solely in English. You can't teach firstorder logic using only propositions expressed in firstorder logic. Etc. Treatid sees this as a fundamental flaw in the entire concept of mathematics, because he still sees mathematics as attempting to be completely unambiguous in the strongest possible sense of the word, and how can anything be unambiguous when the entire concept of conveying ideas to each other is necessarily built on just assuming that we all mean the same thing when we use a given word?
Of course, the answer to his objection is the same as we've said time and time again. Math isn't trying to be unambiguous in the ridiculouslystrong sense that he thinks it is. Math takes it for granted that we're not going to worry about the philosophical obstacles with the concept of communication. It takes it for granted that we all have a basic grasp of whatever human language we happen to be communicating in. And then, within the context of that human language, it starts 'defining' things as 'unambiguously' as possible.
 MartianInvader
 Posts: 769
 Joined: Sat Oct 27, 2007 5:51 pm UTC
Re: Misunderstanding basic math concepts, help please?
arbiteroftruth wrote:He's just giving a slightly fancier description of the same philosophical obstacle we've acknowledged for quite some time, in this very thread and in many of his past threads. Namely, that you can't teach someone how a language works using only that language.
Even this seems false to me. I'm pretty sure I didn't learn English by having people explain it to me in some other language.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!
 gmalivuk
 GNU Terry Pratchett
 Posts: 25723
 Joined: Wed Feb 28, 2007 6:02 pm UTC
 Location: Here and There
 Contact:
Re: Misunderstanding basic math concepts, help please?
No, but you still made extensive use of things that weren't English, such as the physical world around you.
 Xanthir
 My HERO!!!
 Posts: 5198
 Joined: Tue Feb 20, 2007 12:49 am UTC
 Location: The Googleplex
 Contact:
Re: Misunderstanding basic math concepts, help please?
Yeah, the "other language" used was a combination of instinctual shared understanding and various nonverbal methods of communication. You didn't learn your first English *in* English, tho once you started you began bootstrapping yourself (accompanied by the previous methods).
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))
Who is online
Users browsing this forum: No registered users and 9 guests