Axiomatic mathematics has no foundation

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arbiteroftruth
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Re: Axiomatic mathematics has no foundation

Postby arbiteroftruth » Tue Mar 18, 2014 10:49 pm UTC

Treatid wrote:
If you do still acknowledge the ability to state tautologies, do you or do you not recognize the possibility that a tautology might be of the form (something)-->(something else)?

Without axiomatic logic, the only thing that a statement proves is itself. Hence I can accept A --> A but not A --> B.


Would you accept (A&B)-->A? What about ((A-->B)&(B-->C))-->(A-->C)? If you object to the use of an ampersand, that second example can be written as (A-->B)-->((B-->C)-->(A-->C))

These are examples of statements that are tautologies but are not as trivial as A-->A. In the above examples, we don't have to assign any meaning whatsoever to A, B, and C in order to call the statement a tautology. If you don't accept non-trivial tautologies such as these because you don't think logical implication itself is sufficiently defined, then on what basis do you even accept the trivial case of A-->A?

But I'll assume for now that you're okay with non-trivial tautologies. If I were to express such an example as a mathematical theory, in "(A&B)-->A" I would call "A&B" the "axiom", and I would call "A" a "theorem" within the theory. This doesn't mean that "A&B" is considered true in any way, nor that it even means anything whatsoever. It neither has nor needs any definition. It's just the arbitrary string of characters that comprise the first part of a tautology, because tautologies that start with that particular string of characters are interesting to me. Mathematics is then ultimately the study of which strings of characters we can use as the rest of the tautology.

To put it in your language, when I take "A&B" as an "axiom" and "prove" "A" as a result, I am not defining "A&B" as meaning anything, nor am I accepting it as true. I am not defining the resulting "theorem" "A" as meaning anything either, nor am I accepting it as true. I'm only observing a relationship between the two, despite the fact that neither statement means anything by itself. Isn't that exactly what you're aiming for?

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Re: Axiomatic mathematics has no foundation

Postby gmalivuk » Tue Mar 18, 2014 11:58 pm UTC

Treatid wrote:@gmalivuk: Presumably you feel that “there are no objective truths” is, itself, an objective truth and therefore paradoxical and therefore disproves itself. So... does that mean you think there are objective truths? Or that there aren't objective truths?

There is no (technical) axiomatic mathematics... yet it was the attempt to construct axiomatic mathematics that showed axiomatic mathematics cannot exist.

Without being able to define axioms, we can't define proof either. We have no logic in a rigorous and absolutely defined sense.

I could have worded it better: “we cannot communicate objective truths”.
Okay, but in so doing you yourself claim to be communicating an objective truth.

And I'll also re-ask my other question (from a post which I admittedly deleted because I wanted my mic drop comment to come at the end of the last page rather than the top of this one): If it's impossible to have (or to communicate) objective truths, why on earth should any of the rest of us care about your subjective "truths"?
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Re: Axiomatic mathematics has no foundation

Postby Nicias » Wed Mar 19, 2014 12:42 am UTC

Schrollini wrote:
The XKCD Players
are proud to present

Pachyderm

being an Allegorie
in one itty-bitty little act

Dramatis Personae
Dit A. Ert, being the only sane man
Chorus, being Geek


A bare stage, conveying no meaning.
...
Exeunt.


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Re: Axiomatic mathematics has no foundation

Postby Twistar » Wed Mar 19, 2014 3:34 am UTC

Treatid wrote:The idea of axioms is that from the starting point, everything else follows in a consistent fashion. I'm completely happy with this concept. This is what I understand axioms to be.

The trouble is that the starting point cannot be defined in the first place. We cannot specify that set of rules that we are going to follow.

Axioms are stated in a language. But that language itself is not an axiomatic system. We can define ZFC in terms of Category Theory. But ultimately we rely on some natural language to define our first set of axioms. And that natural language isn't defined in the axiomatic sense.

The logic that is supposed to follow from a set of axioms is perfectly reasonable.

The trouble is that mathematicians forget to apply that logic in order to get the axioms in the first place: A statement only has meaning/significance/proof with respect to a set of axioms. So how do you state the axioms if the statements are not also part of an axiomatic system.

Where is the beginning?


What you are saying here is that it is impossible to have an language which is entirely formal. This is emphatically correct. What you are loathe to accept is the fact that formal languages have some informal aspects. See my thread on circularity in formal languages. However, one goal of formal languages is to minimize the amount informalities in formal languages to as few informal aspects as possible, and then the idea is that even though there are informal aspects, all mathematicians agree on what is meant by the informal features. In the language of first order logic (a language within which ZF can be written) one of the undefined concepts is the set itself. There is no formal definition for set. However, there is an informal definition of set. The informal definition relies on the fact that when I say "I have a set of symbols*" you still know what I mean by 'set' even though there is no rigorous definition. You know what I mean, everyone in this thread knows what I mean, every mathematician who has ever lived knows what I mean, and I would venture that nearly every single human being in the world knows what I mean by "set" even though I can't define it formally.
It is recognized that there are informal concepts in formal languages but mathematicians and logicians are o.k. with this because everyone agrees that everyone understands what we mean with the informal concepts.

Humans are not computers, and as such they aren't constrained to needing formal definitions to understand something. They can understand things in an informal way and all human thought relies on this and people are o.k. with it.

*symbol is also something which is not defined formally but we all know what I mean by symbol.

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Re: Axiomatic mathematics has no foundation

Postby ahammel » Wed Mar 19, 2014 4:13 am UTC

Twistar wrote: In the language of first order logic (a language within which ZF can be written) one of the undefined concepts is the set itself. There is no formal definition for set.

I thought ZF was the formal definition of 'set'. Perhaps I'm confused about what 'formal definition' means.
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Re: Axiomatic mathematics has no foundation

Postby Treatid » Wed Mar 19, 2014 3:15 pm UTC

Twistar wrote:What you are saying here is that it is impossible to have an language which is entirely formal. This is emphatically correct.

Thank you for taking the time to understand what I am saying.

What you are loathe to accept is the fact that formal languages have some informal aspects. See my thread on circularity in formal languages.

I have read that thread - I didn't jump into it because a) I felt you wanted an answer from the mainstream perspective of mathematics and b) I suspect some people prefer that my heresy be restricted to as few threads as possible.

However, one goal of formal languages is to minimize the amount informalities in formal languages to as few informal aspects as possible, and then the idea is that even though there are informal aspects, all mathematicians agree on what is meant by the informal features.

I agree that the attempt has been to minimise those informalities. What I don't agree with is that it is possible to agree on what is meant by those informal features.

i) Informal languages aren't magical. They are constrained by the same rules as everything else. By using informal languages as the root of mathematics you are merely hiding the problem of defining axioms - not resolving it.
ii) Agreement isn't equivalent to understanding. There was a time when everyone agreed that the sun went around the Earth. We could all agree that unicorns exist. That doesn't mean that slim rhinoceroses with magical horns and rainbow poop actually exist.
iii) Without being able to formally define the axioms - how do you know that we are agreeing on their meaning? How do you define that agreement when you can't define the meaning?
iv) A little bit pregnant is still pregnant. No matter how much we reduce the informal aspect... it is still sufficient to undermine the entirety of axiomatic mathematics. As much as the idea of a curve tending towards a limit leads us to feel that approaching the limit is sometimes nearly the same as reaching the limit... this is not one of those instances.

In the language of first order logic (a language within which ZF can be written) one of the undefined concepts is the set itself. There is no formal definition for set. However, there is an informal definition of set. The informal definition relies on the fact that when I say "I have a set of symbols*" you still know what I mean by 'set' even though there is no rigorous definition. You know what I mean, everyone in this thread knows what I mean, every mathematician who has ever lived knows what I mean, and I would venture that nearly every single human being in the world knows what I mean by "set" even though I can't define it formally.

I think you are a smidgen optimistic - but I agree with you that "set" is fine. A "set" as an abstract object is absolutely legitimate.

But Set Theory doesn't just have sets. It has The Empty Set... an object that assumes from the get go that we can distinguish one thing from another (zero from every other number). It also contains axioms that distinguish a self reference from every other type of reference (by excluding identity).

A better starting point is Category Theory.

It is recognized that there are informal concepts in formal languages but mathematicians and logicians are o.k. with this because everyone agrees that everyone understands what we mean with the informal concepts.

This, right here, is the problem.

There are a group of people who believe that a sky daddy watches their every movement, will judge their immortal souls when they die and take the 'good' ones to paradise for eternity. And this group of people agree that they understand what they are talking about.

Thinking that you understand, thinking that you agree with other people in that understanding, is not evidence, let alone proof, of anything.

I have no problem with doing what works. But mathematicians and Logicians shouldn't be OK with the idea that everyone just magically agrees and understands... the whole reason for trying to construct formal systems is that informal systems are notorious for not being consistently agreed upon and understood.

Humans are not computers, and as such they aren't constrained to needing formal definitions to understand something.

Again... the magical thinking. Humans are not something separate from the universe. We are governed by exactly the same rules as everything else. Whatever we can do - we can build a computer to do the same thing.

They can understand things in an informal way and all human thought relies on this and people are o.k. with it.

I, also, am okay with humans understanding things in the way that humans do. It is the only way for anything to be understood.

Mathematics, on the other hand, is trying to use a method of understanding that cannot work. Axiomatic theory is the wrong way to understand things.

Yes - I am being pedantic. Surely being pedantic is what mathematics is supposed to be.

In order to understand physics, we need to understand the terms being used. Even a term as simple as "empty" isn't straight forward. You know what I mean by an empty glass... except most empty glasses contain air. Hard Vacuum is the ultimate empty... except that we know that vacuum is a sea of virtual particles... that there is something through which photons and electrons (and gravitons?) travel. (Or that vacuum is curved... somehow).

Empty, zero, nothing... these feel intuitive enough. We all know what they mean. Until we try to actually find some nothing. Can you point to some nothingness? Can we create a volume that contains nothing?

On the number line, zero is just a point - almost indistinguishable from every other point. The number line zero isn't "empty" any more than any other number is "empty".

Category Theory does show that the problem is understood. Even the sets of set theory are a nod in the right direction... but to make sets useful you have to layer all those fixed point assumptions on top.

So... let us start with Category Theory. Using just abstract ideas of what objects and relationships are we know that we can construct everything else that is constructable.

A constructable object is the universe we inhabit.

The universe can be described by some sequence of relationship networks. There is no need to use terms we can't formally define.

The universe consists of a network of relationships and some (deterministic) process that operates on that network to change it.

...

@ahammel: Confusion about what 'formal definition' means is sensible. Ultimately, axioms themselves cannot be formally defined... which means we don't have formally defined axioms from which to formally define anything else.

However, ZF specifies Set Theory which includes sets and specifies their behaviour... but for the most part sets are abstract, undefined objects.

@gmalivuk: I don't know where you are coming from. Your questions imply that you think there might be some objective truth. I find it difficult to accept that this can genuinely be your position. The whole point and purpose of axioms is that a statement without context is meaningless. Even when we thought that axioms were a real thing, the truth of a statement was always judged with respect to a set of axioms.

So, you really must know that there is no objective truth. Which only leaves subjective truths.

In terms of "my subjective truths"... they aren't mine. Mathematics has demonstrated the points that I'm repeating in terms of there being no possible axioms. korona pointed out near the beginning of this thread that the limitations are already known. The wiki on foundational mathematics includes references to the various aspects of the foundational crises in mathematics.

arbiteroftruth wrote:Would you accept (A&B)-->A? What about ((A-->B)&(B-->C))-->(A-->C)? If you object to the use of an ampersand, that second example can be written as (A-->B)-->((B-->C)-->(A-->C))

These are examples of statements that are tautologies but are not as trivial as A-->A. In the above examples, we don't have to assign any meaning whatsoever to A, B, and C in order to call the statement a tautology. If you don't accept non-trivial tautologies such as these because you don't think logical implication itself is sufficiently defined, then on what basis do you even accept the trivial case of A-->A?

I accept A implies A on the same basis as I accept the existence of directed relationships. Or to put it another way: I think, therefore I am.

I don't think I can ever know what A is in an absolute (axiomatic) sense. I do think that part of A's purported existence includes A being related to itself. The identity relationship is significant when it comes to determining if two objects are distinguishable or not.

This is apparent in Set Theory. Most set Theories exclude sets containing themselves directly or indirectly. This is one of those "we just know" pieces of knowledge that creates a fixed point on which set theory is built. Excluding the identity relationship implies that it is possible to uniquely distinguish between an identity relationship and all other relationships.

But I'll assume for now that you're okay with non-trivial tautologies. If I were to express such an example as a mathematical theory, in "(A&B)-->A" I would call "A&B" the "axiom", and I would call "A" a "theorem" within the theory. This doesn't mean that "A&B" is considered true in any way, nor that it even means anything whatsoever. It neither has nor needs any definition. It's just the arbitrary string of characters that comprise the first part of a tautology, because tautologies that start with that particular string of characters are interesting to me. Mathematics is then ultimately the study of which strings of characters we can use as the rest of the tautology.

You are assuming more than you are stating.

Your last sentence implies that there is some constraint created by your initial string of characters. This is the assumption made by axiomatic mathematics.

It looks to me like you are trying to talk my language (appreciated) and express existing mathematics in those terms. However, I'm not struggling with the concept of axioms. I don't mind that an arbitrary declaration of axioms is arbitrary. Having declared an initial set of strings and a set of rules to be applied to those strings I'm completely on board with everything that happens afterwards - at least to the extent that the exercise is similar to programming a computer.

The problem lies in formally defining the axioms in the first place. Without a formal definition, axioms can mean anything (or nothing) you want them to. And it isn't possible to formally define a set of axioms.

To put it in your language, when I take "A&B" as an "axiom" and "prove" "A" as a result, I am not defining "A&B" as meaning anything, nor am I accepting it as true. I am not defining the resulting "theorem" "A" as meaning anything either, nor am I accepting it as true. I'm only observing a relationship between the two, despite the fact that neither statement means anything by itself. Isn't that exactly what you're aiming for?

It all falls down when you "prove" "A" as a result. Just choosing one result over another requires justification.

I accept your suggestion that neither the axioms or the result are themselves inherently meaningful. But there are infinitely many possible results... and choosing one of them over others has to be specified. It is this specification that cannot be justified by axiomatic logic.

@schrollini: :D

May I suggest a small alteration to your script: Instead of an Elephant we have a Unicorn:

Chorus: look, a Unicorn.

Pedant: No... that is a rhinoceros. It has four legs and a horn but that doesn't make it a Unicorn.

Chorus: Labels don't matter... we can call it a Unicorn if we like.

Pedant: Fair enough.

Chorus: Unicorns can fly... and poop rainbows... and their horns are magic. I bet if I ate ground unicorn horn it would give me a huge boner. Where's my chainsaw?

Pedant: Gah!

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Re: Axiomatic mathematics has no foundation

Postby arbiteroftruth » Wed Mar 19, 2014 3:53 pm UTC

Treatid, correct me if I'm wrong, but I think I've deciphered what you're getting at. Bear in mind that I'm going to suggest what I think is a better terminology for your point, so try not to get hung up on that.

I think your issue isn't about the ability to define axioms in the sense of ascribing any meaning to them, since we've gone round and round with the point that axioms don't have to actually mean anything but are just strings to be manipulated. It's also clear, from the very fact that you are attempting to use reason and to describe "networks of relationships", that you take no issue with the idea of defining the rules of some formal system, since the very act of describing "networks of relationships" sounds like an attempt at a formal system that follows some rules.

I think what you're really objecting to is the notion of arbitrary specification. I'll try to explain what I mean by that in terms of your preferred method of networks of relationships. Let's represent "objects" as Os and "directed relationships" as arrows. Consider the following network:

O->O
v
O

It seems that what you're objecting to is any attempt to talk about "the O on the right" as opposed to "the O on the bottom". You're contending that the only legitimate distinction is between "the O at the start of the arrows" and "the Os at the ends of the arrows", with no distinction allowed between the two Os at the ends of the arrows. Not even a distinction that's just an arbitrary labelling like:

O->A
v
B

Labelling them arbitrarily as A and B is something that you see as an illegitimate attempt to create a distinction where none exists. You contend that the very act of giving them different names assumes that there is some meaningful property distinguishing them, when there isn't. This same objection is why you have a problem with non-trivial tautologies like (A-->B)-->((B-->C)-->(A-->C)), because as long as A, B, and C are undefined, you think there is no justification for distinguishing them with three different names. It is this need for undefined-yet-somehow-distinct placeholders that you think renders axioms undefinable.

Is that a fair re-wording of your position?

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Re: Axiomatic mathematics has no foundation

Postby Schrollini » Wed Mar 19, 2014 5:27 pm UTC

From the director's extended cut of Pachyderm.

Dit A. Ert wrote:I can prove elephants don't exist.


Chorus wrote:Fine. Go ahead.


Dit A. Ert wrote:As everyone knows, elephants fly by flapping their ears. This is completely implausible. An elephant's ears have minimal structural support. There's no way it could properly maintain the correct cross-section to efficiently generate lift. Even if it could, the ears are clearly too small to generate sufficient lift. My calculations suggest barely a 2:1 glide ratio, and you can't really call that flying, now can you? In any event, the elephant's ears are located so far forward that the center of lift and center of gravity don't line up, and the elephant would pitch up and stall in no time at all!


Chorus wrote:Elephants can't fly!


Dit A. Ert wrote:My point exactly.


Chorus wrote:


Dit A. Ert wrote:


Chorus wrote:What we mean is, we never claimed that elephants can fly.


Dit A. Ert wrote:Now that I've proven that elephants don't exist, let's examine this strange non-flying creature you keep calling an elephant.


Chorus wrote:No. Wait. Why do you think elephants can fly?


Dit A. Ert wrote:Oh, everyone knows that. Here's a documentary that proves my point.
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Re: Axiomatic mathematics has no foundation

Postby Treatid » Wed Mar 19, 2014 6:08 pm UTC

arbiteroftruth wrote:Treatid, correct me if I'm wrong, but I think I've deciphered what you're getting at. Bear in mind that I'm going to suggest what I think is a better terminology for your point, so try not to get hung up on that.

I think your issue isn't about the ability to define axioms in the sense of ascribing any meaning to them, since we've gone round and round with the point that axioms don't have to actually mean anything but are just strings to be manipulated. It's also clear, from the very fact that you are attempting to use reason and to describe "networks of relationships", that you take no issue with the idea of defining the rules of some formal system, since the very act of describing "networks of relationships" sounds like an attempt at a formal system that follows some rules.

I think what you're really objecting to is the notion of arbitrary specification. I'll try to explain what I mean by that in terms of your preferred method of networks of relationships. Let's represent "objects" as Os and "directed relationships" as arrows. Consider the following network:

O->O
v
O

It seems that what you're objecting to is any attempt to talk about "the O on the right" as opposed to "the O on the bottom". You're contending that the only legitimate distinction is between "the O at the start of the arrows" and "the Os at the ends of the arrows", with no distinction allowed between the two Os at the ends of the arrows. Not even a distinction that's just an arbitrary labelling like:

O->A
v
B

Labelling them arbitrarily as A and B is something that you see as an illegitimate attempt to create a distinction where none exists. You contend that the very act of giving them different names assumes that there is some meaningful property distinguishing them, when there isn't. This same objection is why you have a problem with non-trivial tautologies like (A-->B)-->((B-->C)-->(A-->C)), because as long as A, B, and C are undefined, you think there is no justification for distinguishing them with three different names. It is this need for undefined-yet-somehow-distinct placeholders that you think renders axioms undefinable.

Is that a fair re-wording of your position?

Yes - this is absolutely the right ball-park.

The limits on axiomatic definitions tell us that the objects O, A and B and the relationships cannot be directly defined in any way. As such, we cannot directly distinguish these objects in any way. Any axiomatic declaration that directly or indirectly tries to make objects or relationships inherently distinguishable is attempting the impossible. This limitation holds for informal languages too. Informal languages are no more able to define objects or relationships directly than our formal languages.

With it being impossible to specify or observe any inherent properties in either objects or relationships... all we are left with is networks of relationships. Even with networks... we need distinguishing features (asymmetry) to be able to distinguish between different networks (or components of the same network).

Sets are a good abstract start - spoiled by the attempt to identify objects as being "the empty set" and excluding identity relationships.

Category Theory is a step better than sets. It doesn't try to identify objects in any way - they are almost entirely irrelevant. I've some quibbles over it still trying to save a bit of axiomatic theory (and trying to describe other axiomatic theories), but it's heart is in the right place and it is a good enough starting point.

Having established that defining objects, no matter how surreptitiously done, is impossible, we are left with networks of relationships to describe everything. Which includes the universe - physics.

@Schrollini: That documentary is the "Clay Institute Millennial Prizes". While anybody takes those seriously it means that mathematics as a whole is still claiming that Elephants can fly.

Category Theory is a strong argument that mathematics has taken on board some of the lessons. But until all the dots have been joined up - it is still a half-hearted acceptance. Clearly there are sufficient mathematicians who think that ZFC is sufficiently well defined that the question of whether P=NP is meaningful.

Nevertheless, We can move forward.

Physics.

We have now established that a network of relationships in which the objects and relationships cannot be specified is what we can communicate.

Therefore, physics is describable (can only be fully described) by a sequence of relationship networks. This is what the pedantry is about.

If you think that I'm arguing a point that mathematicians already accept... then you also accept that this is the only possible starting point for describing the fundamental structure of the universe. Is this the case?

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Re: Axiomatic mathematics has no foundation

Postby WibblyWobbly » Wed Mar 19, 2014 6:10 pm UTC

Schrollini wrote:From the director's extended cut of Pachyderm.

Dit A. Ert wrote:I can prove elephants don't exist.


Chorus wrote:Fine. Go ahead.


Dit A. Ert wrote:As everyone knows, elephants fly by flapping their ears. This is completely implausible. An elephant's ears have minimal structural support. There's no way it could properly maintain the correct cross-section to efficiently generate lift. Even if it could, the ears are clearly too small to generate sufficient lift. My calculations suggest barely a 2:1 glide ratio, and you can't really call that flying, now can you? In any event, the elephant's ears are located so far forward that the center of lift and center of gravity don't line up, and the elephant would pitch up and stall in no time at all!


Chorus wrote:Elephants can't fly!


Dit A. Ert wrote:My point exactly.


Chorus wrote:


Dit A. Ert wrote:


Chorus wrote:What we mean is, we never claimed that elephants can fly.


Dit A. Ert wrote:Now that I've proven that elephants don't exist, let's examine this strange non-flying creature you keep calling an elephant.


Chorus wrote:No. Wait. Why do you think elephants can fly?


Dit A. Ert wrote:Oh, everyone knows that. Here's a documentary that proves my point.

Will there be a version with director commentary? Something to explain a sort of Kafkaesque vibe where we've transcended the familiar notions of "math that actually works" into the nightmarish hellscape of "pseudo-pedantic strings of jargon hiding a lack of understanding that the things being defined to replace mathematics are themselves imprecise restatements of the ideas that made math work in the first place"? I'll buy that on Blu-ray. Especially if you get Nathan Fillion to narrate.

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Re: Axiomatic mathematics has no foundation

Postby Forest Goose » Wed Mar 19, 2014 6:59 pm UTC

It would make me ever so happy if you would explain what any of this has to do with Millennium Prize Problems (specifically P = NP). I've noticed that you don't complain about other results/problems - and given that you seem to accept computation and that P = NP is rather concretely grounded, I'm mighty confused why you keep bringing it up (and what exactly does it have to do with ZFC and axioms?).

Too, I'm not exactly sure how category theory gets a pass; actually, I'm not sure how anything gets a pass...I summarize your posts as:

"Nothing can have meaning"
"There is no objective truth"
...
"Here's what has meaning"
"Here's me stating my conclusions as objective facts"
...
"Turing machine, quantum physics, relativity"
...
"Only what I say goes".

My hang up is on how if there is no meaning, no objective truth, and this makes axioms impossible: why is anything else possible - including: why can you state and specify your "results" as if they were meaningful, well defined, absolute, and objective; if their very conclusion is nothing is any of those things?
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Re: Axiomatic mathematics has no foundation

Postby Dopefish » Wed Mar 19, 2014 7:06 pm UTC

I am somewhat uncomfortable with the use of "we" in statements like "We have now established/shown/proven..." when I haven't really seen anything that resembles you showing/proving anything despite the volume of words being used.

I additionally don't think you should be making references to physics, and especially not quantum mechanics, as those topics are at best tangential to your point if not completely off-topic, and QM references frequently come across as "I don't understand this thing, and so no one understands this thing. QM is still real despite not being understood though, and so whatever I'm talking about is also real despite the lack of understanding.". (This is not to say you sound like that necessarily, but you should surely acknowledge there are many many cranks who make appeals to QM, so if you don't wish to be viewed as such you'd best avoid the topic if at all possible.)

I am still not convinced that there's not some self defeating logic being used here, since logic is part of mathematics, and so by saying something isn't logically valid (e.g. axiomatic mathematics), you're using that invalid system to reach that conclusion, and so your conclusion isn't valid. If the argument is that the logic still 'works' and it's just not for the reasons we might think, then that's a rather strong claim that logic just happens to actually work, but the rest of math doesn't necessarily. If the rest of math does work, than we're fine, and things like the clay prizes are merely for people finding something that 'works' for the claims involved rather than any deeper claims.

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Re: Axiomatic mathematics has no foundation

Postby lorb » Wed Mar 19, 2014 7:08 pm UTC

@Treatid: Have you yet familiarized yourself with the work of your precursors? I pointed out some earlier that adress a few of the arguments you make. Another one that already has been mentioned on this thread is Quine. I think you will like this quote of him when writing about set theory and it's foundations:
Willard van Orman Quine wrote:We find ourselves making deliberate choices and setting them forth unaccompanied by any attempt at justification other than in terms of their elegance and convenience,
(Source)
If you read further into his work you will find that he renders most of you objections to axiomatic maths moot, but arguing very similar to your own arguments.

@Everyone: There already is a profound body of work on many of the topics raised in this thread. Many different disciplines contributed to it and I think many assumptions made in this thread are not subjected to enough scrutiny when the topic at hand is very much ontological _and_ epistemological. For example: it is not universally agreed on that math is an abstract system of symbols and strings that have no a-priori relation to the empiric world. In fact some well known logicians and philosophers argue that such a distinction isn't even possible. (See here: http://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction)
imho there is a bunch of references missing in the overall discussion that are too easily dismissed at times:
- Kant holds that there are accessible objective truths (e.g. a priori truths, a possible foundation for axiomatic maths) (Link)
- Luhman argues that it is not only possible to define intersubjective meanings but without it we wouldn't even have a functioning society: by extension we can derive objective axioms from everyday communication (another possible foundation for axiomatic maths) (Link)
- Logical empiricism argues that theoretical terms derive meaning from correspondance to observation (Link)
And there is a wealth of "schools of thought" that stem from that centuries old debate that all have contributed many details and arguments.
It's just not true that mathematicians take the truth of their axioms simply for granted. Many logicians, mathematicians and philosophers whose area of work was touched by the topic at hand but many a thought into that and produced a lot of knowledge and viewpoints (sometimes conflicting, quine hates logical empricism) The rest just rely on what has been done before them: like everywhere in math ... you build on the work of others. No need to redo it from scratch all the time.

ps: a link to the Foundational Crisis has already been posted, but there is much more to it as anyone following the links above will find out

Sidenote:
Treatid wrote: Clearly there are sufficient mathematicians who think that ZFC is sufficiently well defined that the question of whether P=NP is meaningful.

Are you saying PNP is independent from ZFC?
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arbiteroftruth
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Re: Axiomatic mathematics has no foundation

Postby arbiteroftruth » Wed Mar 19, 2014 7:25 pm UTC

Okay Treatid, since I think I know what you're saying now, I'll try again to show how traditional mathematics can be constructed even within your restrictions.

We are allowed to talk about networks of relationships, and refer abstractly to objects that have these relationships. Since the relationships are directional, we can describe them as arrows and can distinguish between the property of being on the starting end of the arrow vs. the property of being on the destination end of the arrow. Let me know if you object to any of that so far.

Given all that, is there any reason we can't talk about the property of an object "not having any arrows pointing toward it"?

Separately, is there any reason we can't describe restrictions on the properties of networks that we're interested in studying?

If neither of those things are objectionable, then we can combine them. We can say that we're only interested in studying networks that have the property that it makes sense to talk about *the* object in the network with no arrows pointing toward it. The notion that it makes sense to talk about *the* (as opposed to *an*) object with that property is by no means universal to all possible networks. But if you've accepted the notion of restricting our studies to only certain networks, we can restrict our studies to those networks for which it does make sense to talk about *the* object with no incoming arrows.

We can further restrict our studies to networks in which all objects may be distinguished one way or another. In particular, we can focus on networks in which all objects may be distinguished purely in terms of the available paths to reach the object in question starting from the object with no incoming arrows. We can focus even more narrowly by requiring the network to always provide certain patterns of outgoing arrows.

In short, if you accept the notion of describing properties of networks, and you accept the notion of restricting our studies to networks that have certain properties, then the act of doing so is what the phrase "axiomatic system" is intended to convey, in terms of the language you've established.

So that leaves you four options that I can see.
1) Object to the act of describing properties of networks, in which case your entire attempt to use networks as a new form of mathematics breaks down.
2) Object to the act of choosing to restrict our studies to networks that have certain properties, in which case, why do you object?
3) Object to the specific restriction of looking at neworks in which objects can be distinguished, in which case, why do you object?
4) Accept that the act of specifying distinct objects is merely a restriction of which networks we're studying, rather than being a fundamentally broken approach to mathematics. In which case, axiomatic mathematics is vindicated.

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Re: Axiomatic mathematics has no foundation

Postby z4lis » Fri Mar 21, 2014 5:00 pm UTC

If you're OK with talking about the "O from which the arrows leave" and distinguishing objects based on their relationships to other objects in a network, then you should be OK with some primitive models of set theory. There's actually some "model" of a very weak set theory where sets are represented as trees, but you disregard any symmetries.

So the tree

O --> O
|
O

would represent the same tree as just O --> O, since you aren't allowed to talk about the "left/right" or "top/bottom" parts of the tree. Intuitively, each node represents the set containing its children. So in this model, the tree O represents the empty set. O --> O represents {empty set} and O --> O --> O represents {{empty set}}, while

O --> O --> O
|
O

represents {empty set, {empty set}}. Disregarding symmetries comes about from the fact that {empty set, empty set} should really be the same set as {empty set}.

PS I also need to point out lorb's post, above. Everything any of us have said in this thread has been debated, thought about, and written about for centuries by philosophers, scientists, mathematicians, writers,..., so please go find out what's out there on the topic!
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Re: Axiomatic mathematics has no foundation

Postby gmalivuk » Fri Mar 21, 2014 5:11 pm UTC

Yeah, if you accept the notion that there can be things and that there can be relationships between them, then you can pretty much build set theory. You have to define things a certain way, but defining things isn't the same as making baseless propositions. The empty set contains no other set because that's simply what I mean when I say "empty set". I'm not proclaiming that it corresponds to some real physical object or whatever. And then the axioms about the empty set and the things you can do with the "contains" relationship are further definitions, not declarations that some real thing exists in a concrete way that you can expect to touch and manipulate with physics or whatever.
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Re: Axiomatic mathematics has no foundation

Postby DR6 » Fri Mar 21, 2014 8:07 pm UTC

[quote="Treatid"]
Category Theory is a step better than sets. It doesn't try to identify objects in any way - they are almost entirely irrelevant. I've some quibbles over it still trying to save a bit of axiomatic theory (and trying to describe other axiomatic theories), but it's heart is in the right place and it is a good enough starting point.
/quote]


... but if category theory can be used to describe set theory, and category theory is a step better than sets, then set theory must be at least as well founded as category theory, since you can translate all statements about sets to category theory.

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Re: Axiomatic mathematics has no foundation

Postby Treatid » Sat Mar 22, 2014 2:05 am UTC

As much as the components of this have been around for a long time, there are some people working very hard to avoid putting the pieces of the puzzle together.

Let us start with an edge piece:

i) A statement without context is meaningless.
ii) We define context with axioms.
iii) Axioms are statements.

There is nothing complicated here. There isn't any wriggle room.

This is a closed loop. There is no entry or exit point for definitions in this closed loop.

The loop neither creates or destroys. It is just a loop. The very essence of tautology.

You don't need someone else to do your thinking for you on this one. The only place a closed loop will take you is back to your starting point.

...

But what if we could have a definite definition without starting with some axioms? Maybe natural languages have access to some mechanism that breaks this loop?

Okay... let us assume that you have a definite fixed point; something that is unambiguously definable. A starting point from which we can build.

The first thing you need to do is to communicate that fixed point to me so that I have the same starting point as you.

So... you communicate an unambiguous set of axioms... oh... I don't have the fixed point from which we can define a definite meaning for axioms. Until you have communicated that fixed point... I don't have a fixed point which allows you to communicate that fixed point.

Hmm... Perhaps you can show me the fixed point...

This fixed point... this starting point for axioms... it needs to not just be a fixed point - it needs to be a fixed point that anyone (or anything) looking at it will instantly recognise it as a fixed point and know all the properties of that fixed point.

Even if axiomatic mathematics had an initial fixed point - it would be useless unless everyone fully understood that fixed point without explanation or justification.

It isn't just that we don't have a starting point for axiomatic mathematics. Even if there was a starting point, it would require that everyone intuitively understood that starting point in every detail for it to be of any use.

Axiomatic mathematics requires omniscience as a starting point... and if you have omniscience you don't need axiomatic mathematics.

There are no axioms. Without axioms there is no axiomatic logic and there is no axiomatic proof.

Whatever existing mathematics is doing it has nothing to do with axioms, it has nothing to do with axiomatic logic and it has nothing to do with axiomatic proof.

...

The structure of axioms is fundamentally flawed. It isn't a system that would have been neat if only we had a way to build that initial set of axioms... it simply isn't a workable approach to knowledge.

Having said that... we had to learn that it wasn't a workable approach to knowledge. Mistakes are an essential part of learning, and attempting to create axiomatic systems has educated us about what is possible.

So, where does that leave Set Theories (as represented by ZFC)?

The astute will have noticed that having made axiomatic mathematics disappear in a flash of logic, we still have computers, bridges that don't fall down in the first gust of wind, economics, global positioning satellites and a bunch of other neat stuff.

As much as axiomatic mathematics is not the answer - there are still processes to be understood.

We can still reason, albeit we cannot formalise reasoning into a specific, defined, logic.

What we can't do is build anything in an axiomatic fashion. Anytime you are under the impression that you are building something axiomatically... you are simply wrong.

The claim has been made that the axioms of ZFC are immune to the critique of axioms in general because the axioms are abstract - sufficiently indefinite as to not get caught up in the issue of axioms not being defined.

As much as it may feel like no assumptions are being made... if you are managing to build anything at all in an axiomatic fashion it is because you are assuming the existence of a fixed point. Without an initial definition, axiomatic mathematics cannot do anything. So, obviously, if you feel like you are doing something, you either aren't using axiomatic logic, or you are pretending that you have a fixed point.

gmalivuk wrote:Yeah, if you accept the notion that there can be things and that there can be relationships between them, then you can pretty much build set theory.

"pretty much"...

Yeah... "pretty much", as in "not at all, in any way".

Sets themselves are not a problem. As a generally abstract object they are just a way of expressing relationships between objects.

But sets are not the same as set theory.

Remove 'the empty set' and the axiom of regularity (and any other axiom that precludes self reference (identity) for sets) and you are left with a really weak system. You cannot distinguish between sets. You cannot construct V. You cannot construct cardinality.

You have to define things a certain way, but defining things isn't the same as making baseless propositions.

???

Yes it is. That is exactly what it is.

As much as you may think a given definition is justified or consistent or unambiguous... we know that, in fact, we cannot define anything in the axiomatic sense.

Agreement is not the same as definition. Your perception that a thing is obvious doesn't imply that you actually know what that thing is.

The empty set contains no other set because that's simply what I mean when I say "empty set".

What is what you mean?

I'm not being perverse or arbitrary - although I am being pedantic.

In an absolute sense... you CANNOT know what a particular phrase means. As much as you feel otherwise... you do not know what "empty" means. Even if every other person on the planet appears to agree with you over the meaning of "empty"... you cannot distinguish between a single set with self reference (identity) and a ring of sets without pre-supposing that distinction.

And then the axioms about the empty set and the things you can do with the "contains" relationship are further definitions, not declarations that some real thing exists in a concrete way that you can expect to touch and manipulate with physics or whatever.

This distinction you are trying to make here is irrelevant. The degree of abstractness has no impact on whether the reasoning is justified.


z4lis wrote:If you're OK with talking about the "O from which the arrows leave" and distinguishing objects based on their relationships to other objects in a network, then you should be OK with some primitive models of set theory.

The "O from which the arrows leave" may well also be the O to which the arrow is pointing.

Identity relationships really mess with set theory - which is why set theories tend to exclude them.

It means that we cannot, a priori, distinguish one end of the relationship from the other.

There's actually some "model" of a very weak set theory where sets are represented as trees, but you disregard any symmetries.

I'd hardly call ZFC a very weak set theory. V - the von Neumann hierarchy is exactly what you describe - trees in which symmetry is disregarded.

But in order to create the trees, ZFC needs both to exclude self reference in sets and to specify an arbitrarily identified starting point (the empty set).

Without those - in a genuinely weak set theory - it isn't possible to distinguish any given set from any other set. This is why set theory goes to so much trouble to axiomatically specify distinctions between sets (e.g. empty set is distinguishable from all other sets as an axiom rather than being derived). Without an imposed distinction... there is no way to distinguish sets in ZF or ZFC.

So the tree

O --> O
|
O

would represent the same tree as just O --> O, since you aren't allowed to talk about the "left/right" or "top/bottom" parts of the tree. Intuitively, each node represents the set containing its children. So in this model, the tree O represents the empty set. O --> O represents {empty set} and O --> O --> O represents {{empty set}}, while

O --> O --> O
|
O

represents {empty set, {empty set}}. Disregarding symmetries comes about from the fact that {empty set, empty set} should really be the same set as {empty set}.

This is the basic reasoning for disregarding symmetries that I was using... but allowing a set to have an identity relationship creates many more symmetries. Each of the trees you describe becomes indistinguishable. Specifically, a tree requires a definite start and end point. With identity relationships it becomes impossible to distinguish a start point from any other point.

PS I also need to point out lorb's post, above. Everything any of us have said in this thread has been debated, thought about, and written about for centuries by philosophers, scientists, mathematicians, writers,..., so please go find out what's out there on the topic!

You may have missed that lorb addressed the substantive portion of his post to "everyone".

It is those who are arguing for the status quo that do not understand how weak their position is.

arbiteroftruth wrote:We are allowed to talk about networks of relationships, and refer abstractly to objects that have these relationships. Since the relationships are directional, we can describe them as arrows and can distinguish between the property of being on the starting end of the arrow vs. the property of being on the destination end of the arrow. Let me know if you object to any of that so far.

Yes - objection.

ZFC specifically disallows identity relationships and arbitrarily distinguishes the empty set specifically so that a distinction can be made between the beginning and end of a relationship.

Without that assumed distinction - it is actually impossible to distinguish between two sets.

The axioms of ZFC aren't there for fun... they serve a purpose. A significant part of that purpose is to create a distinction between sets that doesn't arise by other means.

Category Theory recognises that there isn't a natural way to distinguish sets. Which is why it gives up on the effort entirely.


Given all that, is there any reason we can't talk about the property of an object "not having any arrows pointing toward it"?

Very much so.

This is why we have formal mathematics that pedantically tries to chase down all the details. As intuitive as the idea of distinguishing between objects based on the number of arrows pointing at them... it turns out that we can't create that distinction. If we assume that the distinction exists - then it exists... but if we don't start with that assumption then we can't build it from component parts. Things that we cannot build from component parts are assumptions of a fixed point - impossible assumptions (see also: axiom of choice).

Separately, is there any reason we can't describe restrictions on the properties of networks that we're interested in studying?

Umm...

We cannot describe anything in an absolute sense. We can and do describe all sorts of things subjectively. Whether a given restriction is well formed is not always trivial to determine.

If neither of those things are objectionable, then we can combine them. We can say that we're only interested in studying networks that have the property that it makes sense to talk about *the* object in the network with no arrows pointing toward it.

Ah. No - we can't.

While arrows are directional, they can also be circular. The from and to object might be the same object. Or they might not be. Unless we preclude the possibility of self reference, we cannot count the number of arrows in any degree.

We can further restrict our studies to networks in which all objects may be distinguished one way or another. In particular, we can focus on networks in which all objects may be distinguished purely in terms of the available paths to reach the object in question starting from the object with no incoming arrows. We can focus even more narrowly by requiring the network to always provide certain patterns of outgoing arrows.

This is true for something like ZFC... but ZFC achieves it by making impossible assumptions. Without those impossible assumptions all networks are similar in that we cannot definitely distinguish any set from any other set.

In short, if you accept the notion of describing properties of networks, and you accept the notion of restricting our studies to networks that have certain properties, then the act of doing so is what the phrase "axiomatic system" is intended to convey, in terms of the language you've established.

I agree that this is what people are claiming something like ZFC is. But the assumptions needed to build such systems are themselves violations of axiomatic mathematics.

So that leaves you four options that I can see.
1) Object to the act of describing properties of networks, in which case your entire attempt to use networks as a new form of mathematics breaks down.

The assumptions of axiomatic mathematics are really deeply implanted, aren't they?

You feel that if we can't describe a network in a certain way that there is nothing we can do. By excluding this option you are missing out on reality. Everything we know, understand and experience is contained within this option.

Axiomatic mathematics cannot understand this position... but frankly, axiomatic mathematics cannot understand anything.

Even within the fold of axiomatic mathematics... Category Theory completely disregards the distinction between sets. Objects are entirely irrelevant except to the degree that they provide notional anchors for relationships.

lorb wrote:@Treatid: Have you yet familiarized yourself with the work of your precursors? I pointed out some earlier that adress a few of the arguments you make. Another one that already has been mentioned on this thread is Quine. I think you will like this quote of him when writing about set theory and it's foundations:
Willard van Orman Quine wrote:We find ourselves making deliberate choices and setting them forth unaccompanied by any attempt at justification other than in terms of their elegance and convenience,
(Source)
If you read further into his work you will find that he renders most of you objections to axiomatic maths moot, but arguing very similar to your own arguments.

I wouldn't say I have exhaustively worked my way through all philosophy inside and outside of mathematics... I'm always eager to discover a new perspective - please do keep the references coming.

In response to another's comments... just because earlier philosophers/mathematicians haven't been able to definitively solve the puzzle is no reason to believe that we can't. We have the advantage of standing on their shoulders - and the view from up here is fantastic.

Sidenote:
Treatid wrote: Clearly there are sufficient mathematicians who think that ZFC is sufficiently well defined that the question of whether P=NP is meaningful.

Are you saying PNP is independent from ZFC?

I was specifically saying that there are no axiomatic systems. ZFC isn't (cannot) be axiomatically defined. In the absence of a base, anything supposedly constructed on that base is moot.

However, even within the conventionally accepted constraints of ZFC... P=NP is still independent. This has already been proven with respect to Relativising proofs. It has been shown that Relativising proofs cannot resolve the question. Generally this is taken as showing that some other sort of proof is needed. This is due to a mistaken interpretation of an aspect of Godel's incompleteness. The inability of Relativising proofs to resolve the question is taken as evidence that alternate proofs are required. However, all alternate proofs are necessarily external to the system being considered. We can choose any external system and prove whatever we like.

Specifically... Godel proposes statements that are "true but unprovable" within the system. However, there is no possible way to distinguish between "true but unprovable" statements and statements that are independent of the axioms.

Without any way to distinguish between the two classes of statement there is no justification for viewing them as two distinct objects. "True but unprovable" can be considered equal to "independent of the axioms" until some mechanism is proposed to distinguish the two cases.

Dopefish wrote:I am somewhat uncomfortable with the use of "we" in statements like "We have now established/shown/proven..." when I haven't really seen anything that resembles you showing/proving anything despite the volume of words being used.

It has been established that the lack of foundation for axioms is well covered. I don't need to prove anything because it has already been done. I'm leaning on existing, known mathematics. lorb provides a fresh link to the foundational crises which covers a wide variety of approaches to the same basic issue.

I additionally don't think you should be making references to physics, and especially not quantum mechanics, as those topics are at best tangential to your point if not completely off-topic, and QM references frequently come across as "I don't understand this thing, and so no one understands this thing. QM is still real despite not being understood though, and so whatever I'm talking about is also real despite the lack of understanding.". (This is not to say you sound like that necessarily, but you should surely acknowledge there are many many cranks who make appeals to QM, so if you don't wish to be viewed as such you'd best avoid the topic if at all possible.)

:) Ultimately my aim is to discuss physics.

In order to communicate I needed to understand the language that other people use... but when I tried to find a firm foundation in mathematics, I discovered instead that there is no foundation of any kind. The only thing that exists is a set of agreed assumptions. These assumptions are not suitable for discussing fundamental physics.

While I agree that there are many crackpots who diss existing physics... I'm already here dissing the entirety of axiomatic mathematics... And I'm only doing that so I can construct a common basis for discussing fundamental physics.

I am fully aware that Quantum Mechanics is an extremely successful theory in terms of making confirmed predictions right up to the limit of our ability to measure. However, I do feel that it is a triumph of mathematics over rationality. It makes predictions - but it doesn't provide an intuitive insight into the structure of things. By contrast, I find General Relativity to be a masterful piece of physics.

I am still not convinced that there's not some self defeating logic being used here, since logic is part of mathematics, and so by saying something isn't logically valid (e.g. axiomatic mathematics), you're using that invalid system to reach that conclusion, and so your conclusion isn't valid.

Mathematics already uses paradox as an indicator that a set of axioms aren't consistent with each other.

Axiomatic mathematic's own logic shows itself to contain a paradox. This is a pretty conventional way to demonstrate that a system isn't a valid system.

Granted, having disposed of axioms, the way forward without axiomatic logic may be a bit murky.

The loss of axiomatic logic is unfortunate... but in practice, nothing has been lost. We never had axioms. We never had axiomatic logic. Yet we have managed to achieved our current technology despite this handicap.

We have reasoning and progress even without axiomatic logic. We will never have the kind of definite knowledge that axiomatic mathematics tried to create. Love it or hate it... this is the situation we have.

The Relativistic Mathematics we can have is different in nature to the axiomatic approach to knowledge. However, as a system that actually works it has the potential to be far more effective than axiomatic mathematics.

If the argument is that the logic still 'works' and it's just not for the reasons we might think, then that's a rather strong claim that logic just happens to actually work, but the rest of math doesn't necessarily.

No - logic doesn't work. It never did work in the axiomatic sense. But reason does work. There are consistencies that we can work with.

Nothing has changed beyond us, potentially, becoming a bit more self-aware. Axiomatic mathematics hasn't stopped working... it never worked. The things we were doing when we thought we were doing axiomatic mathematics still exist and are still as practically useful as we found them previously. By seeing past axiomatic mathematics we have a chance to better understand those systems and processes. Better understanding will give us more conscious control.

As attractive as the idea of definite knowledge is... as useful as it seems to be able to be absolute about... anything... there isn't, and never was, objective truth. We never could unambiguously define something as true or provable. Yet we can still construct computers that (mostly) reliably reproduce a given behaviour millions of times over.

If the rest of math does work, than we're fine, and things like the clay prizes are merely for people finding something that 'works' for the claims involved rather than any deeper claims.

For varying degrees of 'works'.

Calculating the pay cheque at the end of the month is going to carry on much the same for the forseeable future.

One of the reasons we want to know whether P=NP is because it directly impacts cryptography. We need to know whether there is a shortcut for a given algorithms... whether a given encryption is secure. That question is intimately tied to the nature of mathematics. Anyone seriously involved in encryption cannot afford to continue to pretend that axiomatic mathematics exists.

Forest Goose wrote:My hang up is on how if there is no meaning, no objective truth, and this makes axioms impossible: why is anything else possible - including: why can you state and specify your "results" as if they were meaningful, well defined, absolute, and objective; if their very conclusion is nothing is any of those things?

This is a good question. It goes right to the heart of our experience as humans.

This is why it isn't sufficient to shrug off the foundational crises as some obscure technicality of little relevance.

We absolutely do have subjective meaning. We do perceive significance. Yet the axiomatic approach to knowledge cannot work.

There is something to be described and understood... and the assumed tool of axiomatic mathematics cannot gain the slightest traction.

Approaching understanding without the use of axiomatic mathematics is going to take some learning. The first step of the process is to understand that the existing assumptions of axiomatic systems really are entirely the wrong approach. Fortunately for your sanity - you don't have to take my word on this point. The foundational crises contains a wealth of approaches to showing this. All you have to do is understand that the foundational crises is actually something important that is worthy of attention...

DR6 wrote:... but if category theory can be used to describe set theory, and category theory is a step better than sets, then set theory must be at least as well founded as category theory, since you can translate all statements about sets to category theory.

:) I was expecting someone to have picked up on this sooner.

First... basic Category Theory needs additional axioms in order to describe conventional set theories.

But mainly... a Universal Turing Machine can emulate any other Turing Machine. Both Category Theory and ZFC are attempts at Universal Turing Machine Equivalents. They are designed to be able to describe every other describable system.

In general, all systems can be described by a network of relationships. In this regard - Category Theory can describe that pattern of relationships (set theory too).

However, a pattern of relationships is not the same thing as an axiomatic system. To illustrate... there are infinitely many axioms for Peano arithmetic. Peano Arithmetic cannot be expressed as a definite network of relationships... even an infinite network. There are many systems that are axiomatically defined that cannot be expressed by a definite network of relationships. A prime reason is that the relevant systems are constructed using an impossible axiom (or several). As well formed as these system might appear they cannot be expressed as a network of relationships. To fix this - the same impossible axioms can be inserted into Category Theory.

So... Category Theory, and to a lesser extent, ZFC are Universal Turing Machine Equivalents that can emulate all describable systems. But not all axiomatic systems are describable. The feature that makes those systems not describable can be added to Category Theory (and tend to be built into ZFC already).

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Re: Axiomatic mathematics has no foundation

Postby arbiteroftruth » Sat Mar 22, 2014 2:55 am UTC

Treatid wrote:
gmalivuk wrote:Yeah, if you accept the notion that there can be things and that there can be relationships between them, then you can pretty much build set theory.

"pretty much"...

Yeah... "pretty much", as in "not at all, in any way".


No, "pretty much" as in "in every way but name" because you're using a different starting point.

You have to define things a certain way, but defining things isn't the same as making baseless propositions.

???

Yes it is. That is exactly what it is.

As much as you may think a given definition is justified or consistent or unambiguous... we know that, in fact, we cannot define anything in the axiomatic sense.


Where did he say anything about "in the axiomatic sense"? Is there a sense in which you think things can be defined? If not, adding the phrase "in the axiomatic sense" is pointless. If so, try applying his reasoning to that notion of "define".

The empty set contains no other set because that's simply what I mean when I say "empty set".

What is what you mean?

I'm not being perverse or arbitrary - although I am being pedantic.

In an absolute sense... you CANNOT know what a particular phrase means.


You mean, like, for example, the phrase "network of relationships"? Your approach to mathematics is not immune to the philosophical obstacles of language.

There's actually some "model" of a very weak set theory where sets are represented as trees, but you disregard any symmetries.

I'd hardly call ZFC a very weak set theory. V - the von Neumann hierarchy is exactly what you describe - trees in which symmetry is disregarded.

But in order to create the trees, ZFC needs both to exclude self reference in sets and to specify an arbitrarily identified starting point (the empty set).


And you haven't given any reason we aren't allowed to talk about "a network of relationships that just so happens to not have any self-reference", and then say to ourselves "hey, that's interesting. I wonder what else I can reason about such a network".

arbiteroftruth wrote:We are allowed to talk about networks of relationships, and refer abstractly to objects that have these relationships. Since the relationships are directional, we can describe them as arrows and can distinguish between the property of being on the starting end of the arrow vs. the property of being on the destination end of the arrow. Let me know if you object to any of that so far.

Yes - objection.

ZFC specifically disallows identity relationships and arbitrarily distinguishes the empty set specifically so that a distinction can be made between the beginning and end of a relationship.

Without that assumed distinction - it is actually impossible to distinguish between two sets.

The axioms of ZFC aren't there for fun... they serve a purpose. A significant part of that purpose is to create a distinction between sets that doesn't arise by other means.

Category Theory recognises that there isn't a natural way to distinguish sets. Which is why it gives up on the effort entirely.


Kindly point out where the fuck you think I said anything about ZFC, identity relationships, or Category Theory anywhere in the paragraph you're supposedly objecting to.

Given all that, is there any reason we can't talk about the property of an object "not having any arrows pointing toward it"?

Very much so.

This is why we have formal mathematics that pedantically tries to chase down all the details. As intuitive as the idea of distinguishing between objects based on the number of arrows pointing at them... it turns out that we can't create that distinction. If we assume that the distinction exists - then it exists... but if we don't start with that assumption then we can't build it from component parts. Things that we cannot build from component parts are assumptions of a fixed point - impossible assumptions (see also: axiom of choice).


Okay, let me put it this way. If you think we should study networks of relationships and the patterns they have, give an example of a type of pattern you think we *can* reason about. If you can't even have the notion of "there are no arrows pointing here", how do you expect to engage in any sort of reasoning about any patterns whatsoever?

Separately, is there any reason we can't describe restrictions on the properties of networks that we're interested in studying?

Umm...

We cannot describe anything in an absolute sense. We can and do describe all sorts of things subjectively.


Kindly point out where the fuck you think I said anything about describing things specifically "in an absolute sense" as opposed to in a subjective sense that you would approve of.

If the argument is that the logic still 'works' and it's just not for the reasons we might think, then that's a rather strong claim that logic just happens to actually work, but the rest of math doesn't necessarily.

No - logic doesn't work. It never did work in the axiomatic sense. But reason does work.


Logic is literally nothing more than an attempt to communicate reason. If reason is valid, then the use of logic within my own mind is valid, because within my own mind they are one and the same. And to the extent that your objection is that communication is impossible, nothing you attempt will ever be immune from that.

However, a pattern of relationships is not the same thing as an axiomatic system. To illustrate... there are infinitely many axioms for Peano arithmetic. Peano Arithmetic cannot be expressed as a definite network of relationships... even an infinite network.


Give me a clear explanation of what we're actually allowed to talk about regarding networks of relationships, and I can virtually guarantee that either your notion of networks of relationships will be too trivial to be interesting as a form of mathematics, or I will be able to describe Peano arithmetic in terms you approve of (or you'll move the goalposts by suddenly insisting that I haven't played by your rules even though I have). With the caveat that "induction" might be difficult to construct.

This is the big problem here. You keep talking about how everything can be described as networks of relationships, but as soon as anyone else tries to describe something about a network of relationships, you object that they're trying to define things "absolutely", as opposed to whatever it is you think you're doing differently. Unless you give us a clear explanation of the rules of the game for talking about networks, all you're doing is communicating badly and acting smug about it.

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Re: Axiomatic mathematics has no foundation

Postby gmalivuk » Sat Mar 22, 2014 3:31 am UTC

Treatid wrote:
gmalivuk wrote:Yeah, if you accept the notion that there can be things and that there can be relationships between them, then you can pretty much build set theory.

"pretty much"...

Yeah... "pretty much", as in "not at all, in any way".

Sets themselves are not a problem. As a generally abstract object they are just a way of expressing relationships between objects.

But sets are not the same as set theory.
Right, but it's set theory that you can build from things and relations between things. The "pretty much" was facetious: you really can get the same entire structure.
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Re: Axiomatic mathematics has no foundation

Postby doogly » Sat Mar 22, 2014 3:59 am UTC

Also, if you think GR is intuitive but QM is not, this just means you do not understand Hilbert spaces. And are probably overestimating the degree to which you understand differential geometry anyway.
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Re: Axiomatic mathematics has no foundation

Postby el matematico » Sat Mar 22, 2014 4:00 am UTC

Treatid wrote:As much as the components of this have been around for a long time, there are some people working very hard to avoid putting the pieces of the puzzle together.

Let us start with an edge piece:

i) A statement without context is meaningless.
ii) We define context with axioms.
iii) Axioms are statements.

There is nothing complicated here. There isn't any wriggle room.

This is a closed loop. There is no entry or exit point for definitions in this closed loop.

The loop neither creates or destroys. It is just a loop. The very essence of tautology.

You don't need someone else to do your thinking for you on this one. The only place a closed loop will take you is back to your starting point.

Axioms and mathematics in general don't need any intrinsic "meaning" and they don't need to. You are the one who keeps bringing the meaning of zero or the empty set. Math is meaningless, and only because we as humans can give meaning to stuff is that we can use math to model meaningful stuff.
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Re: Axiomatic mathematics has no foundation

Postby doogly » Sat Mar 22, 2014 4:07 am UTC

Also:
Treatid wrote:you CANNOT know what a particular phrase means. As much as you feel otherwise... you do not know what "empty" means. Even if every other person on the planet appears to agree with you over the meaning of "empty"...

You should rethink what "meaning" means. Sets aren't magic. It's language, and if we can communicate, then the meaning is legit. cf the semiotics links before.
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Re: Axiomatic mathematics has no foundation

Postby Gwydion » Sat Mar 22, 2014 5:28 am UTC

Treatid, are you confusing the math-definition of axiomatic (using a set of simple definitions and constructions to build elegant systems) with the English-definition of axiomatic (unquestionably true and self-evident)? It would make me a sad panda if four pages of argument boiled down to this.

If that's not the case, then please define axiom and axiomatic mathematics. Also, please don't tell me that my request is invalid or can't be done - this is not a philosophical point but rather a plain-English request for clarification. If you're using these words to mean something other than what everyone else is saying, then we'll just go around in circles forever.

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Re: Axiomatic mathematics has no foundation

Postby Dopefish » Sat Mar 22, 2014 6:23 am UTC

Treatid wrote:
Dopefish wrote:I am somewhat uncomfortable with the use of "we" in statements like "We have now established/shown/proven..." when I haven't really seen anything that resembles you showing/proving anything despite the volume of words being used.

It has been established that the lack of foundation for axioms is well covered. I don't need to prove anything because it has already been done. I'm leaning on existing, known mathematics. lorb provides a fresh link to the foundational crises which covers a wide variety of approaches to the same basic issue.


That's fine, people have shown some things and there are indeed deeper issues here (which I'm inclined to think of as mostly philosophical, but of course that's debatable). But, I don't think you've shown anything new here or really added any meaningful insights, so it's less "We have shown..." and more "Mathematicians/philosophers elsewhere have shown...", as I typically think of 'we' in this context as you (and to a lesser degree, the other posters in this thread), rather than the more general collective.


I also don't think it's honest to make a thread about something with the aim to talk about something unrelated. If you want to talk about physics, there's a science forum for that. There's even a fictional science sub-forum if you have a flavour of physics that people wouldn't recognise as physics without being told. If you need to prove something mathematical first before you think your physics can even begin to be presented, then you can do try to do so here first and then make a science thread, but trying to do both at the same time in here seem like it'd only serve to muddy already murky waters as people attempt to figure out what you're getting at.

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Re: Axiomatic mathematics has no foundation

Postby Forest Goose » Sat Mar 22, 2014 2:52 pm UTC

Rather than argue the spew of nonsense you posted, I'll just leave this here (from your other thread, about P=NP) to show you're disingenuous.

You said in this thread:

I was specifically saying that there are no axiomatic systems. ZFC isn't (cannot) be axiomatically defined. In the absence of a base, anything supposedly constructed on that base is moot.

However, even within the conventionally accepted constraints of ZFC... P=NP is still independent. This has already been proven with respect to Relativising proofs. It has been shown that Relativising proofs cannot resolve the question. Generally this is taken as showing that some other sort of proof is needed. This is due to a mistaken interpretation of an aspect of Godel's incompleteness. The inability of Relativising proofs to resolve the question is taken as evidence that alternate proofs are required. However, all alternate proofs are necessarily external to the system being considered. We can choose any external system and prove whatever we like.

Specifically... Godel proposes statements that are "true but unprovable" within the system. However, there is no possible way to distinguish between "true but unprovable" statements and statements that are independent of the axioms.

Without any way to distinguish between the two classes of statement there is no justification for viewing them as two distinct objects. "True but unprovable" can be considered equal to "independent of the axioms" until some mechanism is proposed to distinguish the two cases.


But, from your http://forums.xkcd.com/viewtopic.php?f=17&t=107069&p=3505102#p3505102, you'll notice what I already said and linked a paper for.

Forest Goose wrote:I don't understand your proof, you need to provide background and definitions, this is said everytime you post a theory.

Why isn't this also a proof that A = B is independent of ZFC? It looks you are just using P and NP as names without invoking any properties.

That P=NP doesn't relativize is exactly as strong as being independent of RCT; see http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf and http://www.cs.sfu.ca/~kabanets/papers/act-full.pdf

Is your idea connected to something related to Baker, Gill, and Solovay, I don't see the connection in the words you typed, would you clarify?

To be perfectly honest, you'd probably be better served if you quit saying "I solved super difficult problem X, here's a proof" followed by 4-5 garbled paragraphs and, instead, just said "I have a neat idea, here's a sketch of what I was thinking" followed by as clear and concise a representation of your idea you can give. When I see someone saying "I proved X" and X is hard, I assume that they are either 14, following up with a proof that could be published, or are a crank.
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Re: Axiomatic mathematics has no foundation

Postby Gwydion » Sat Mar 22, 2014 4:03 pm UTC

ForestGoose, you reminded me of another point I wanted to make earlier.
Specifically... Godel proposes statements that are "true but unprovable" within the system. However, there is no possible way to distinguish between "true but unprovable" statements and statements that are independent of the axioms.

Without any way to distinguish between the two classes of statement there is no justification for viewing them as two distinct objects. "True but unprovable" can be considered equal to "independent of the axioms" until some mechanism is proposed to distinguish the two cases.
Treatid, without any way of distinguishing between two empty sets, what is your justification for viewing them as two distinct objects? By what mechanism do you propose we distinguish them?

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Re: Axiomatic mathematics has no foundation

Postby z4lis » Sat Mar 22, 2014 5:23 pm UTC

What do you mean by the quoted statement below? I'm not trying to be argumentative, I just want a clear explanation of what you mean. If you have any books or papers that touch on the topic, that'd be awesome.

Identity relationships really mess with set theory - which is why set theories tend to exclude them.


Now for the more argumentative part. You seem to be missing the point of mathematics. What you're doing here is like to going to a forum about the Lord of the Rings series and telling everyone that statements like "Frodo was a hobbit." are incorrect on the grounds that the books are works of fiction, and therefore Frodo and hobbits don't exist and so anything said about them is incorrect. Of course they are. Everybody on that forum is well aware of the fact that the books are works of fiction, but that's not what people are discussing. The assumed context is that "you've read the Lord of the Rings and are assuming the things written there are more or less true, for this discussion".

"Mathematics" as a whole doesn't need a fixed set of axioms. We were doing mathematics and reasoning about plane geometry and elementary number theory way before "sets" ever existed as a concept. People since then have attempted to come up with a fixed set of axioms for mathematics, but we've discovered that it doesn't work out so well. It's a purely human activity, and the context behind mathematics assumes you have the same internal model of numbers, logic, sets, and the like as the writer does. The fact that it's perhaps incomprehensible, inconsistent nonsense to some being who knows only Truth is irrelevant, because many, many humans, who do not need absolute truths to function, do understand it (or at least think they do), and find it useful and interesting.

That said, it is a miracle that both you and I and Einstein and Newton and Archimedes have internal models mutually consistent enough that we can all write down, grasp, understand, and reason about shapes and numbers, whatever they may or may not be. I do think about what's going on there sometimes. You seem to think the answer is... nothing's going on?
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Re: Axiomatic mathematics has no foundation

Postby lightvector » Sat Mar 22, 2014 7:52 pm UTC

Treatid, you're still not understanding. Everyone already agrees with the main point of your thesis (although not all the supporting details). The apparent disagreement is because you are using words like "axiom" to mean something different than what everyone else means when they say "axiom".

If we use the word "axiom" to mean an Objective Truth, with a universal inherent meaning independent of any interpretation, then there are no axioms and no way to define an axiom, and as you said, axiomatic mathematics is impossible. Pretty much everyone agrees with you on this.

If we use the word "axiom" to mean one of the many arbitrary rules that we've made up about how various imaginary systems could behave, a rule whose consequences would be interesting to study, then inventing axioms and doing mathematics with them is quite possible. That is, it is easily possible to make up some rules (ex: the rules of tic-tac-toe), communicate them to someone else well enough so that they can almost always understand and play by the same rules (ex: "We take turns marking empty boxes in this 3x3 grid for ourselves. If you mark three in a row, column, or diagonal before the other person does, then you win, if the board fills up without either player winning then the game is a draw"), and then investigate the consequences of these rules (ex: "If I play in the corner first and you reply anywhere other than the center, you're going to lose, but if you reply in the center, it's a draw unless someone messes up after that."). Tic-tac-toe is a bit simpler than the things people usually study in math, but it's a legitimate example.

When people other than you use the word "axiom", they mean the second thing, rather than the first. When they refer to "axiomatic mathematics" they mean the thing I described above, the study of the consequences of different bunches of rules we've made up that we find interesting or useful in practice for modeling the way certain things behave in real life. They do not mean the pursuit of Objective Knowledge from a fixed base of assumptions that are inherently Meaningful and True, which as you have argued, is indeed not possible.

Do you understand?

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Re: Axiomatic mathematics has no foundation

Postby snowyowl » Sun Mar 23, 2014 5:46 pm UTC

z4lis wrote:That said, it is a miracle that both you and I and Einstein and Newton and Archimedes have internal models mutually consistent enough that we can all write down, grasp, understand, and reason about shapes and numbers, whatever they may or may not be. I do think about what's going on there sometimes. You seem to think the answer is... nothing's going on?

I think it's even more miraculous that the universe seems to agree with us. See The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Gravity is proportional to the inverse-square of distance, which is an approximation to a certain tensor equation. Complex numbers are central to quantum mechanics, despite the fact that complex numbers were invented centuries earlier and nobody at the time could have known they'd be an accurate description. The universe seems to run on fundamental rules that are mathematical in nature.

(It's not obvious that a universe has to be mathematical. Fiction runs on laws of narrative and drama, not mathematics.)
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Re: Axiomatic mathematics has no foundation

Postby doogly » Sun Mar 23, 2014 5:56 pm UTC

I think it is pretty obvious that the rules of the universe are mathematical, because there really aren't any alternatives to math. If you tried to write something down besides math, you could very quickly just turn that thing into math.
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Re: Axiomatic mathematics has no foundation

Postby gmalivuk » Sun Mar 23, 2014 6:14 pm UTC

Yeah, mathematics at its most basic is the study of consistent systems, so if the universe is in any sense a consistent system, math will be what you use to describe it.
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Re: Axiomatic mathematics has no foundation

Postby snowyowl » Mon Mar 24, 2014 11:59 am UTC

gmalivuk wrote:Yeah, mathematics at its most basic is the study of consistent systems, so if the universe is in any sense a consistent system, math will be what you use to describe it.

Fair enough. It's still not completely obvious that the universe has to be a consistent system. Although I can't think of any interesting counter-examples right now.
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Re: Axiomatic mathematics has no foundation

Postby gmalivuk » Mon Mar 24, 2014 7:15 pm UTC

It doesn't "have to" be, but if it isn't, then science is impossible from the get-go. So we start off with the "axiom" that the universe is in fact consistent and see where that gets us.
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Re: Axiomatic mathematics has no foundation

Postby lorb » Mon Mar 24, 2014 7:50 pm UTC

For all we know ZFC could be inconsistent but we still do a lot of math that is working very nicely.
In the same way science works, even if the universe is incosistent (See for example problem of induction and critical rationalism)
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Re: Axiomatic mathematics has no foundation

Postby snowyowl » Mon Mar 24, 2014 9:16 pm UTC

gmalivuk wrote:It doesn't "have to" be, but if it isn't, then science is impossible from the get-go. So we start off with the "axiom" that the universe is in fact consistent and see where that gets us.

Fair enough. It's hardly unexpected that the study of consistent systems is useful in analysing one specific consistent system. It is convenient though.
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Re: Axiomatic mathematics has no foundation

Postby Treatid » Tue Mar 25, 2014 4:12 am UTC

Parable: You buy a brand new car from your local dealership and they deliver it to your home. You get in it to take it for its first drive and find it won't start. After a little investigation you find it doesn't have an engine. Somewhat perturbed, you ring the dealer to inquire why your car has no engine.

After some umming and ahhing the dealer finally admits that he knows the car doesn't have an engine. "But it does have seats, wheels and a sun-roof. I don't see what the problem is."

You take a moment to agree that the car does indeed have quite a few of the features associated with cars... except for the "motorised" bit. The car doesn't go anywhere.

"Sure it goes places", the dealer says." Just hook it up to another car and it'll go where ever you want it to.

...

Mathematics is practiced by humans. Humans tend not to be identical. Many human traits are distributed according to a normal distribution (bell curve). Allowing for some skewing, the degree to which any given topic of knowledge is understood by a given group of people can be expressed as a normal distribution. Some people don't have the first clue what is going on; some people have a deep grasp of the intricacies of the matter; and most people fall somewhere between these two extremes.

Category Theory demonstrates to me that some people have a significant (if incomplete) grasp of the impossibility of defining axioms. On the other hand, some of the posts in this thread convince me that this degree of understanding is not universal.

For all that some people would like me to believe that I am being redundant in arguing a point that is already understood... I don't see evidence that anyone here actually understands the significance. Telling me that you understand 'something' is somewhat lass persausive than showing me that you understand something.

Normally this forum is very good at demonstrating an answer... at justifying an answer... rather than simply asserting an answer.

Which is directly relevant to the argument at hand. The reasoning after a set of axioms is well defined. Whereas the axioms themselves are, by definition, just asserted.

inb4: I have no issue with arbitrariness itself. Some comments seem to suggest that my objection is over how justified I feel a set of axioms are... that I feel there is some distinction between 'real' and mathematics... that I think there should be a preferred set of axioms... that, in short, I can't handle abstractions.

What I actually think is that it is impossible to state, define, infer or agree on any axiom at all.

Some people seem to have intellectually accepted this point - they seem to agree that this is true. And then they defend ZFC. This leaves me confused.

Without axioms there are no starting points. There is no ZFC.

Planet Earth is a significant lump. As humans infecting her surface, the distinction between a truly fixed reference point and Earth is not all that obvious. But having discovered that there is a difference, that difference is crucial.

Let us go back to 'the empty set'.

The empty set isn't a problem because I have a grudge against 'empty'. Axioms don't have to be plausible or sensible or necessarily possible. Propose that the moon is made of special cheese, and the unicorns living there poop rainbows after eating the cheese and I'll go along with the argument.

The problem is that you can't state an axiom. As much as you think you know what an empty set is... as much as it seems like other people are agreeing with you... YOU CANNOT STATE AN AXIOM.

We can see the result of this in this thread. I tell you that we cannot know what an empty set is and your counter-argument is...? "It's obvious"? This is a mathematics forum. "it's obvious" doesn't apply to anything.

Axioms are outside the formal mechanisms of definition. You have no tools with which to demonstrate or exclude a definition. All you can do is use other words... which are also not formally defined... until you end up back where you started.

Of course words are not useless. We can communicate. Language works.

But the mechanisms of communication have nothing to do with axiomatic mathematics. There is no aspect of axioms that is relevant to anything we actually do. Whatever ZFC might be, it isn't an axiomatic system and there is nothing that can be constructed on it. Any perception that other 'axiomatic' systems are constructed from ZFC is an illusion.

I can see that this idea is vaguely understood. Category Theory is a product of that understanding and is clearly trying to address the issue (just as ZFC was prior to that). As a discipline, mathematics is aware that there is an issue. It is the individual mathematicians that seem determined not to address the issue.

From my perspective, the responses I've been getting are defenses of axiomatic mathematics and that there is no reason not to carry on doing things the way they have always been done. That existing mathematics is useful is a reasonable argument for not instantly discarding existing approaches. But it isn't a reason to pretend that existing mathematics has no flaws. More importantly it isn't a reason to avoid looking at alternatives.

Which leaves me in the position where I can see that there is some degree of understanding that axioms don't exist. At the same time, I see a great deal of effort put into defending and justifying axiomatic approaches (e.g. ZFC).

What I am being told, and what I can see do not match up. I can see mathematicians giving serious consideration to whether P=NP despite it being impossible to define any axiomatic problem. I can see people in this thread thinking they know what 'empty' means with respect to the empty set. I see people who are aware of some slight technical issue... but who are sure this doesn't really apply to axiomatic mathematics... After all... "axiom" is just a label... "if there is a problem with axioms... well we aren't really trying to do axioms... no... it is something else entirely that just happens to look the same".

Coupled with the fact that despite my frequent references to Category Theory, nobody has agreed that the only thing we can communicate are relationships... I am seriously skeptical of claims of (fully) understanding that we cannot define axioms.

This is why I keep repeating points that in theory have been agreed upon (plus it helps get new-comers to the thread up to speed).

If it was just a matter of agreeing that there is an issue... fine - we've agreed there is an issue. Being pointed at the foundational crises is enough to show that, at some level, someone knows there is an issue.

However, there is a solution to the issue. We know there is no absolute knowledge. Which leaves subjective knowledge and networks of relationships. If we perceive anything other than networks of relationships then the perception involves emergent properties of systems. For perception (and subjective knowledge) we need an observer... better have them as an explicit part of the system. A shiny new way to understand what we understand (that actually works) and all the feedback I get is defending axioms which are known to be broken.

lorb wrote:For all we know ZFC could be inconsistent but we still do a lot of math that is working very nicely.
In the same way science works, even if the universe is incosistent (See for example problem of induction and critical rationalism)

Everyone should read the links lorb provides... they are short passages in non-technical language, but interesting in their own right and relevant here (to the point where I am shown that there is genuine understanding rather than being told).

Consistent and inconsistent are properties of axiomatic systems. Without axioms, the question of whether a system is consistent or not is moot.

We find counting to be useful. We don't need to justify our ability to count through an axiomatic system. I am very much in the Popper camp on this.

z4lis wrote:What do you mean by the quoted statement below? I'm not trying to be argumentative, I just want a clear explanation of what you mean. If you have any books or papers that touch on the topic, that'd be awesome.

Identity relationships really mess with set theory - which is why set theories tend to exclude them.

There are two parts to this:

i) identifying when two sets are equivalent.
ii) having definite start and end points.

Two sets are the same set when they contain the same sets. Since all instances of the empty set contain the same (zero) sets... all instances are considered the same object.

Suppose we have five sets in a ring such that A contains B contains C contains D contains E contains A. Each set also includes a self reference so set A contains {A, B}.

Examining each set in turn we can see that each set is indistinguishable from the other sets (excluding our labeling). Specifically, the set contained in each set is indistinguishable. If all empty sets are the same set - then by the same rules, each of these sets is the same set. Which means that set A actually contains {A, A}.

While it might seem like a small issue... we now have a set with two self references. What is the correct cardinality of this set?

If a set is permitted self references, then it is difficult to count the number of self references. The empty set with identity relationship goes from being the empty set to a set with an indeterminate number of self references. You might consider that it creates a series of identity sets with increasing cardinality... but they all only contain itself - even if there are many instances of itself. The worst case scenario is that a set containing only itself may have any cardinality.

Let us suppose now we have a set that contains what was the empty set but is now the identity set. How do we distinguish this new set from the identity set? We can't. We have a set that contains itself and the identity set. Which is exactly what the identity set is... a set which contains an indeterminate number of instances of the identity set.

Any set that contains only the identity set is indistinguishable from the identity set.

Which leaves us with the challenge of creating a set which isn't the identity set, so that another set can be distinguishable from the identity set. Another chicken and egg problem. Until we have a set that can be distinguished from the identity set - we can't create a set that is distinguishable from the identity set.

The fact that it's perhaps incomprehensible, inconsistent nonsense to some being who knows only Truth is irrelevant, because many, many humans, who do not need absolute truths to function, do understand it (or at least think they do), and find it useful and interesting.

This is the "it works" defense. Which is a very reasonable argument... up to the point where it doesn't work.

That said, it is a miracle that both you and I and Einstein and Newton and Archimedes have internal models mutually consistent enough that we can all write down, grasp, understand, and reason about shapes and numbers, whatever they may or may not be. I do think about what's going on there sometimes. You seem to think the answer is... nothing's going on?

Ooh - another instance that confounds the argument I gave up above. You are showing me that you understand.

I think that nothing 'axiomatic' is going on. There is definitely stuff happening. What frustrates me is that the explanation for that stuff is obviously incorrect - but there appears to be little interest in pursuing a less-incorrect explanation.

I agree that it is amazing that so many people can come to agreement without an absolute reference frame upon which to construct that agreement. So amazing that there is no way it is just luck or coincidence. There is a specific mechanism that allows us to communicate at a high degree of abstraction despite a lack of any absolutes to anchor that communication. Axioms and the associated logic are not that mechanism... but despite trying to use a model of knowledge that cannot work, we still manage to communicate. What could we achieve if we fully understood the actual mechanism?

Gwydion wrote:Treatid, without any way of distinguishing between two empty sets, what is your justification for viewing them as two distinct objects? By what mechanism do you propose we distinguish them?

I don't. If two things are indistinguishable then they are indistinguishable. I think that sets cannot be distinguished. Trying to make them distinguishable is folly... Which is a large part of the reason why Category Theory doesn't try to distinguish between categories (sets) or (for the most part) relationships.

Dopefish wrote:I also don't think it's honest to make a thread about something with the aim to talk about something unrelated.

My aim when starting this thread was to talk about mathematics. I also have a longer term aim of talking about physics.

There's even a fictional science sub-forum if you have a flavour of physics that people wouldn't recognise as physics without being told. If you need to prove something mathematical first before you think your physics can even begin to be presented, then you can do try to do so here first and then make a science thread, but trying to do both at the same time in here seem like it'd only serve to muddy already murky waters as people attempt to figure out what you're getting at.

I see your argument. I'll counter with the suggestion that an illustration of why and where the argument is particularly relevant helps to provide a context. Our intuitive understanding of Elephants make it difficult to realise that we cannot define Elephants. Whereas Quantum Mechanical objects are sufficiently far from everyday human experience that it is easier to see that natural language and intuition are insufficient anchors for axiomatic mathematics.

Gwydion wrote:Treatid, are you confusing the math-definition of axiomatic (using a set of simple definitions and constructions to build elegant systems) with the English-definition of axiomatic (unquestionably true and self-evident)? It would make me a sad panda if four pages of argument boiled down to this.

Nope - no confusion on that score.

If that's not the case, then please define axiom and axiomatic mathematics.

A set of axioms are an arbitrary starting point. They specify a set of initial conditions and rules. Anything that can be reached by applying the rules from the initial conditions is considered provable (true) with respect to the axioms. Godel's definition of a system includes the language in which the axioms are specified. As such, systems may also include statements that are disprovable (false) with respect to the axioms or undefined with respect to the axioms.

A set of axioms may also be consistent: all statements are definitively true or false. Inconsistent axioms are generally regarded as poor system in that given one inconsistency, arbitrarily many other inconsistencies may be created. (inconsistent axioms are regarded as containing a contradiction within the axioms).

A set of axioms may be complete: If all the possible statements in the language describing the system are either true or false with respect to the axioms then the system is complete.

Oh yes... And it is completely impossible to specify a set of axioms.

lightvector wrote:If we use the word "axiom" to mean an Objective Truth, with a universal inherent meaning independent of any interpretation, then there are no axioms and no way to define an axiom, and as you said, axiomatic mathematics is impossible. Pretty much everyone agrees with you on this.

If we use the word "axiom" to mean one of the many arbitrary rules that we've made up about how various imaginary systems could behave, a rule whose consequences would be interesting to study, then inventing axioms and doing mathematics with them is quite possible.

You see a distinction where I don't think there is one. Those various arbitrary rules are either definite arbitrary rules - in which case they are objective truths - or they are undefined arbitrary rules - in which case they aren't rules.

Objective Truth isn't some esoteric god level knowledge... it is anything that you define in an absolute sense. And there isn't any alternative to defining things in an absolute sense when it comes to axioms. You have either specified a definite set of axioms in an unambiguous way... or you haven't specified anything at all. There is no middle ground. And you can't specify a set of axioms in an unambiguous way

That is, it is easily possible to make up some rules (ex: the rules of tic-tac-toe), communicate them to someone else well enough so that they can almost always understand and play by the same rules (ex: "We take turns marking empty boxes in this 3x3 grid for ourselves. If you mark three in a row, column, or diagonal before the other person does, then you win, if the board fills up without either player winning then the game is a draw"), and then investigate the consequences of these rules (ex: "If I play in the corner first and you reply anywhere other than the center, you're going to lose, but if you reply in the center, it's a draw unless someone messes up after that."). Tic-tac-toe is a bit simpler than the things people usually study in math, but it's a legitimate example.

"almost always"

Two humans communicating common interests are not the same as defining a set of axioms. I am obviously not arguing that humans cannot communicate. That would be silly.

Yes - what mathematicians actually do has a lot in common with specifying the rules of a game. It has nothing to do with axioms - in either of the senses that you have described axioms.

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Re: Axiomatic mathematics has no foundation

Postby gmalivuk » Tue Mar 25, 2014 5:25 am UTC

elephant_stop.jpg
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Re: elephantine matematics has no foundation

Postby gmalivuk » Sat Mar 29, 2014 8:27 pm UTC

Quoting the first and last post with wordfilters, for posterity.
Treatid wrote:A statement only has a specific meaning wid respect to a set ah elephants.

elephants be statements.



It isn't possible to state a set ah elephants widdout first creating an elephantine system to define de elephants.

Which kyan't be done.

dis be obvious.

an' completely devastating to all ah elephant based matematics.

dis isn't some minor tangential result.

Euclidean Geometry an' set theory, for instance, have no basis.

de results achieved may say someting about our biases – but dey have no relevance to matematics.

Revealing a (mistaken) assumption may seem trivial.

dis assumption (dat it be possible to state a set ah elephants unambiguously) has underlain all ah modern matematics.

de magnitude ah de implications will tek a while to sink in... (many as I'd like to hammer home a few salient aspects...)

Physics will be happy a tek many ah de refugees from matematics.

Mathematicians will need a different understanding to recover an aspect ah de discipline.

Absolutist (elephantine) matematics be shown for de fallacy it always was...

Which leaves relativistic matematics...
Treatid wrote:Parable: You buy a brand new cyar from you local dealership an' dey deliver it to you home. You get in it a tek it for its first drive an' find it won't start. After a few investigation you find it nuh have an engine. Somewhat perturbed, you ring de dealer to inquire why you cyar has no engine.

After some umming an' ahhing de dealer finally admits dat he knows de cyar nuh have an engine. "But it does have seats, wheels an' a sun-roof. me nuh see wa de problem be."

You tek a moment to agree dat de cyar does indeed have quite a few ah de features associated wid cars... except for de "motorised" bit. de cyar nuh go anywhere.

"Sure it goes places", de dealer says." Just hook it up to anudder cyar an' it'll go weh ever you wan it to.

...

matematics be practiced by humans. Humans tend not a be identical. Many human traits be distributed according to a normal distribution (bell curve). Allowing for some skewing, de degree to which any given topic ah knowledge be understood by a given group ah people kyan be expressed as a normal distribution. Some people nuh have de first clue wa be a go pon; some people have a deep grasp ah de intricacies ah de matter; an' most people fall somewhere between dese two extremes.

Category Theory demonstrates to me dat some people have a significant (if incomplete) grasp ah de impossibility ah defining elephants. pon de udder hand, some ah de posts in dis tread convince me dat dis degree ah understanding be not universal.

For all dat some people would like me to believe dat I be a be redundant in arguing a point dat be already understood... me nuh see evidence dat anyone here actually understands de significance. a tell me dat you understand 'someting' be somewhat lass persausive dan showing me dat you understand someting.

Normally dis forum be very good at demonstrating an answer... at justifying an answer... radder dan simply asserting an answer.

Which be directly relevant to de argument at hand. de reasoning after a set ah elephants be well defined. Whereas de elephants themselves be, by definition, just asserted.

inb4: I have no issue wid arbitrariness itself. Some comments seem to suggest dat me objection be over how justified I feel a set ah elephants be... dat I feel deh be some distinction between 'real' an' matematics... dat I tink deh should be a preferred set ah elephants... dat, in short, I kyan't handle abstractions.

wa I actually tink be dat it be impossible to state, define, infer or agree pon any elephant at all.

Some people seem a have intellectually accepted dis point - dey seem to agree dat dis be true. an' den dey defend ZFC. dis leaves me confused.

widdout elephants deh be no starting points. deh be no ZFC.

Planet Earth be a significant lump. As humans infecting she surface, de distinction between a truly fixed reference point an' Earth be not all dat obvious. But a have discovered dat deh be a difference, dat difference be crucial.

Let ahbe go back to 'de empty set'.

de empty set isn't a problem because I have a grudge against 'empty'. elephants nuh have a be plausible or sensible or necessarily possible. Propose dat de moon be made ah special cheese, an' de unicorns living deh poop rainbows after eating de cheese an' I'll go along wid de argument.

de problem be dat you kyan't state an elephant. As many as you tink you know wa an empty set be... as many as it seems like udder people be agreeing wid you... YOU CANNOT STATE AN elephant.

We kyan see de result ah dis in dis tread. I tell you dat we cannot know wa an empty set be an' you counter-argument be...? "It's obvious"? dis be a matematics forum. "it's obvious" nuh apply to anyting.

elephants be outside de formal mechanisms ah definition. You have no tools wid which to demonstrate or exclude a definition. All you kyan do be use udder words... which be also not formally defined... till you end up back weh you started.

ah course words be not useless. We kyan communicate. Language works.

But de mechanisms ah communication have nuhting a do wid elephantine matematics. deh be no aspect ah elephants dat be relevant to anyting we actually do. Whatever ZFC might be, it isn't an elephantine system an' deh be nuhting dat kyan be constructed pon it. Any perception dat udder 'elephantine' systems be constructed from ZFC be an illusion.

I kyan see dat dis idea be vaguely understood. Category Theory be a product ah dat understanding an' be clearly a try to address de issue (just as ZFC was prior to dat). As a discipline, matematics be aware dat deh be an issue. It be de individual mathematicians dat seem determined not to address de issue.

From me perspective, de responses I've been a get be defenses ah elephantine matematics an' dat deh be no reason not to carry pon a do tings de way dey have always been done. dat existing matematics be useful be a reasonable argument for not instantly discarding existing approaches. But it isn't a reason to pretend dat existing matematics has no flaws. More importantly it isn't a reason to avoid a look at alternatives.

Which leaves me in de position weh I kyan see dat deh be some degree ah understanding dat elephants nuh exist. At de same time, I see a great deal ah effort put into defending an' justifying elephantine approaches (e.g. ZFC).

wa I be a be told, an' wa I kyan see do not match up. I kyan see mathematicians a give serious consideration to whedder P=NP despite it a be impossible to define any elephantine problem. I kyan see people in dis tread a tink dey know wa 'empty' means wid respect to de empty set. I see people who be aware ah some slight technical issue... but who be sure dis nuh really apply to elephantine matematics... After all... "elephant" be just a label... "if deh be a problem wid elephants... well we aren't really a try a do elephants... no... it be someting else entirely dat just happens a look de same".

Coupled wid de fact dat despite me frequent references to Category Theory, nobody has agreed dat de only ting we kyan communicate be relationships... I be seriously skeptical ah claims ah (fully) understanding dat we cannot define elephants.

dis be why I keep repeating points dat in theory have been agreed pon (plus it helps get new-comers to de tread up to speed).

If it was just a matter ah agreeing dat deh be an issue... fine - we've agreed deh be an issue. a be pointed at de foundational crises be enough to show dat, at some level, someone knows deh be an issue.

However, deh be a solution to de issue. We know deh be no absolute knowledge. Which leaves subjective knowledge an' networks ah relationships. If we perceive anyting udder dan networks ah relationships den de perception involves emergent properties ah systems. For perception (an' subjective knowledge) we need an observer... better have dem as an explicit part ah de system. A shiny new way to understand wa we understand (dat actually works) an' all de feedback I get be defending elephants which be known a be broken.

lorb wrote:For all we know ZFC could be inconsistent but we still do a lot ah mats dat be a wuk very nicely.
In de same way science works, even if de universe be incosistent (See for example problem ah induction an' critical rationalism)

Everyone should read de links lorb provides... dey be short passages in non-technical language, but interesting in dey own right an' relevant here (to de point weh I be shown dat deh be genuine understanding radder dan a be told).

Consistent an' inconsistent be properties ah elephantine systems. widdout elephants, de question ah whedder a system be consistent or not be moot.

We find counting a be useful. We nuh need to justify our ability to count tru an elephantine system. I be very many in de Popper camp pon dis.

z4lis wrote:wa do you mean by de quoted statement below? I'm not a try a be argumentative, I just wan a clear explanation ah wa you mean. If you have any books or papers dat touch pon de topic, dat'd be awesome.

Identity relationships really mess wid set theory - which be why set theories tend to exclude dem.

deh be two parts to dis:

i) identifying when two sets be equivalent.
ii) a have definite start an' end points.

Two sets be de same set when dey contain de same sets. Since all instances ah de empty set contain de same (zero) sets... all instances be considered de same object.

Suppose we have five sets in a ring such dat A contains B contains C contains D contains E contains A. Each set also includes a self reference so set A contains {A, B}.

Examining each set in turn we kyan see dat each set be indistinguishable from de udder sets (excluding our labeling). Specifically, de set contained in each set be indistinguishable. If all empty sets be de same set - den by de same rules, each ah dese sets be de same set. Which means dat set A actually contains {A, A}.

While it might seem like a squeasey issue... we now have a set wid two self references. wa be de correct cardinality ah dis set?

If a set be permitted self references, den it be difficult to count de number ah self references. de empty set wid identity relationship goes from a be de empty set to a set wid an indeterminate number ah self references. You might consider dat it creates a series ah identity sets wid increasing cardinality... but dey all only contain itself - even if deh be many instances ah itself. de worst case scenario be dat a set containing only itself may have any cardinality.

Let ahbe suppose now we have a set dat contains wa was de empty set but be now de identity set. How do we distinguish dis new set from de identity set? We kyan't. We have a set dat contains itself an' de identity set. Which be exactly wa de identity set be... a set which contains an indeterminate number ah instances ah de identity set.

Any set dat contains only de identity set be indistinguishable from de identity set.

Which leaves ahbe wid de challenge ah creating a set which isn't de identity set, so dat anudder set kyan be distinguishable from de identity set. anudder chicken an' egg problem. till we have a set dat kyan be distinguished from de identity set - we kyan't create a set dat be distinguishable from de identity set.

de fact dat it's perhaps incomprehensible, inconsistent nonsense to some a be who knows only Truth be irrelevant, because many, many humans, who do not need absolute truths to function, do understand it (or at fewerest tink dey do), an' find it useful an' interesting.

dis be de "it works" defense. Which be a very reasonable argument... up to de point weh it nuh wuk.

dat said, it be a miracle dat bote you an' I an' Einstein an' Newton an' Archimedes have internal models mutually consistent enough dat we kyan all write down, grasp, understand, an' reason about shapes an' numbers, whatever dey may or may not be. I do tink about wa's a go pon deh sometimes. You seem a tink de answer be... nuhting's a go pon?

Ooh - anudder instance dat confounds de argument I gave up above. You be showing me dat you understand.

I tink dat nuhting 'elephantine' be a go pon. deh be definitely stuff happening. wa frustrates me be dat de explanation for dat stuff be obviously incorrect - but deh appears a be few interest in pursuing a less-incorrect explanation.

I agree dat it be amazing dat so many people kyan come to agreement widdout an absolute reference frame pon which to construct dat agreement. So amazing dat deh be no way it be just luck or coincidence. deh be a specific mechanism dat allows ahbe to communicate at a high degree ah abstraction despite a lack ah any absolutes to anchor dat communication. elephants an' de associated logic be not dat mechanism... but despite a try a use a model ah knowledge dat cannot wuk, we still manage to communicate. wa could we achieve if we fully understood de actual mechanism?

Gwydion wrote:Treatid, widdout any way ah distinguishing between two empty sets, wa be you justification for viewing dem as two distinct objects? By wa mechanism do you propose we distinguish dem?

me nuh. If two tings be indistinguishable den dey be indistinguishable. I tink dat sets cannot be distinguished. a try to mek dem distinguishable be folly... Which be a large part ah de reason why Category Theory nuh try to distinguish between categories (sets) or (for de most part) relationships.

Dopefish wrote:I also nuh tink it's honest to mek a tread about someting wid de aim to talk about someting unrelated.

me aim when starting dis tread was to talk about matematics. I also have a longer term aim ah talking about physics.

deh's even a fictional science sub-forum if you have a flavour ah physics dat people wouldn't recognise as physics widdout a be told. If you need to prove someting mathematical first before you tink you physics kyan even begin a be presented, den you kyan do try a do so here first an' den mek a science tread, but a try a do bote at de same time in here seem like it'd only serve to muddy already murky waters as people attempt to figure out wa you're a get at.

I see you argument. I'll counter wid de suggestion dat an illustration ah why an' weh de argument be particularly relevant helps to provide a context. Our intuitive understanding ah Elephants mek it difficult to realise dat we cannot define Elephants. Whereas Quantum Mechanical objects be sufficiently far from everyday human experience dat it be easier a see dat natural language an' intuition be insufficient anchors for elephantine matematics.

Gwydion wrote:Treatid, you be confusing de math-definition ah elephantine (a use a set ah simple definitions an' constructions to build elegant systems) wid de English-definition ah elephantine (unquestionably true an' self-evident)? It would mek me a sad panda if four pages ah argument boiled down to dis.

Nope - no confusion pon dat score.

If dat's not de case, den please define elephant an' elephantine matematics.

A set ah elephants be an arbitrary starting point. dey specify a set ah initial conditions an' rules. anyting dat kyan be reached by applying de rules from de initial conditions be considered provable (true) wid respect to de elephants. Godel's definition ah a system includes de language in which de elephants be specified. As such, systems may also include statements dat be disprovable (false) wid respect to de elephants or undefined wid respect to de elephants.

A set ah elephants may also be consistent: all statements be definitively true or false. Inconsistent elephants be generally regarded as poor system in dat given one inconsistency, arbitrarily many udder inconsistencies may be created. (inconsistent elephants be regarded as containing a contradiction widdin de elephants).

A set ah elephants may be complete: If all de possible statements in de language describing de system be either true or false wid respect to de elephants den de system be complete.

Oh yes... an' it be completely impossible to specify a set ah elephants.

lightvector wrote:If we use de word "elephant" to mean an Objective Truth, wid a universal inherent meaning independent ah any interpretation, den deh be no elephants an' no way to define an elephant, an' as you said, elephantine matematics be impossible. Pretty many everyone agrees wid you pon dis.

If we use de word "elephant" to mean one ah de many arbitrary rules dat we've made up about how various imaginary systems could behave, a rule whose consequences would be interesting to study, den inventing elephants an' a do matematics wid dem be quite possible.

You see a distinction weh me nuh tink deh be one. dose various arbitrary rules be either definite arbitrary rules - in which case dey be objective truths - or dey be undefined arbitrary rules - in which case dey aren't rules.

Objective Truth isn't some esoteric god level knowledge... it be anyting dat you define in an absolute sense. an' deh isn't any alternative to defining tings in an absolute sense when it comes to elephants. You have either specified a definite set ah elephants in an unambiguous way... or you haven't specified anyting at all. deh be no middle ground. an' you kyan't specify a set ah elephants in an unambiguous way

dat be, it be easily possible to mek up some rules (ex: de rules ah tic-tac-toe), communicate dem to someone else well enough so dat dey kyan almost always understand an' play by de same rules (ex: "We tek turns marking empty boxes in dis 3x3 grid for ourselves. If you mark tree in a row, column, or diagonal before de udder person does, den you win, if de board fills up widdout either player winning den de game be a draw"), an' den investigate de consequences ah dese rules (ex: "If I play in de corner first an' you reply anywhere udder dan de center, you're a go to lose, but if you reply in de center, it's a draw unless someone messes up after dat."). Tic-tac-toe be a bit simpler dan de tings people usually study in mats, but it's a legitimate example.

"almost always"

Two humans communicating common interests be not de same as defining a set ah elephants. I be obviously not arguing dat humans cannot communicate. dat would be silly.

Yes - wa mathematicians actually do has a lot in common wid specifying de rules ah a game. It has nuhting a do wid elephants - in either ah de senses dat you have described elephants.
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