## Axiomatic mathematics has no foundation

For the discussion of math. Duh.

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Treatid
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### Axiomatic mathematics has no foundation

A statement only has a specific meaning with respect to a set of axioms.

Axioms are statements.

It isn't possible to state a set of axioms without first creating an axiomatic system to define the axioms.

Which can't be done.

This is obvious.

And completely devastating to all of axiom based mathematics.

This isn't some minor tangential result.

Euclidean Geometry and set theory, for instance, have no basis.

The results achieved may say something about our biases – but they have no relevance to mathematics.

Revealing a (mistaken) assumption may seem trivial.

This assumption (that it is possible to state a set of axioms unambiguously) has underlain all of modern mathematics.

The magnitude of the implications will take a while to sink in... (much as I'd like to hammer home a few salient aspects...)

Physics will be happy to take many of the refugees from mathematics.

Mathematicians will need a different understanding to recover an aspect of the discipline.

Absolutist (axiomatic) mathematics is shown for the fallacy it always was...

Which leaves relativistic mathematics...

Robert'); DROP TABLE *;
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### Re: Axiomatic mathematics has no foundation

Treatid wrote:A statement only has a specific meaning with respect to a set of axioms.

Although my metamathematics is rather weak compared to some people here, I think you've confused axioms and languages. A language is a set of definitions of symbols and operators, whereas a set of axioms defines which statements are "true." (And, optionally, how true statements can be manipulated to produce more true statements.)
...And that is how we know the Earth to be banana-shaped.

Sizik
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### Re: Axiomatic mathematics has no foundation

Treatid wrote:Which leaves relativistic mathematics...

Does this actually mean anything, or are you just making up terms that don't exist?
gmalivuk wrote:
King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.
Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

cphite
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### Re: Axiomatic mathematics has no foundation

I knew all that math stuff was just nonsense! Now I'm actually glad I didn't pay attention in those classes...

Schrollini
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### Re: Axiomatic mathematics has no foundation

Sizik wrote:
Treatid wrote:Which leaves relativistic mathematics...

Does this actually mean anything, or are you just making up terms that don't exist?

It's better than their previous modus operandi, namely using existing terms in wholly novel ways.
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Treatid
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### Re: Axiomatic mathematics has no foundation

Robert'); DROP TABLE *; wrote:Although my metamathematics is rather weak compared to some people here, I think you've confused axioms and languages. A language is a set of definitions of symbols and operators, whereas a set of axioms defines which statements are "true." (And, optionally, how true statements can be manipulated to produce more true statements.)

True and False can only be determined with respect to meaning.

A bit string by itself has no meaning. Without context there is no way to determine how bit strings should interact.

Given a set of bit strings with specific meanings we can construct further bit strings with their own (related) meanings (the optional bit at the end of your post).

As such, axiomatic systems construct new meaning from existing meaning. A language is not a special case. A language is under the same constraints as any other set of statements in mathematics. In order for statements within a language to have a formally defined meaning it must be formally defined. Which means a language must be an axiomatic system. Which brings us to an infinite series of formally defined systems... or all axiomatic systems being undefined because there is no fixed point from which to build unambiguous meaning.

Axioms are generally taken to be this starting point. Obviously axioms need a specific language to be expressed.

So the language must be the source of the meaning...

Where, within language is the source of meaning?

The word "contains" (as in "set A contains set B") is defined in a dictionary. Using other words. Those other words can be looked up. These new words will be defined by yet more words... Until either a word is re-used - which is the very definition of tautology, or until we run out of words.

The only mechanism within mathematics that constructs meaning is within the scope of axioms. And this mechanism requires some meaning to have been already defined. Mathematics can extrapolate from existing meaning, but can't create meaning in the first place.

Okay - so there are hidden assumptions about what words mean... we just have to agree on a particular set of meanings...

But you can't do that either. The meaning is distinct from the symbols used. The symbol is just a label. There is no mechanism to access the meaning directly. There is no mechanism to convey the meaning directly. We can only convey the label. (The map is not the territory).
...

As humans, we do perceive meaning within various languages. But that meaning doesn't come from within the languages.

As a human being on Earth the symbols "up" and "down" are related to physical experiences. Many humans have similar experiences to which they can attach these symbols. This way a degree of communication is possible.

But you can't directly communicate your experience to me. You can only attach a set of experiences to symbols, pass the symbols, and hope that my understanding of those symbols is similar.

No matter how much it appears that the axioms of ZFC are unambiguously defined... there is no possible method within mathematics to unambiguously assign meaning to the the symbols describing the axioms.

Without an unambiguous set of axioms, there cannot be an unambiguous system.

Since all meaning comes from our physical experience within this universe:

i) any meaning perceived in mathematics derives from the universe around us. This is physics. Not mathematics.
ii) until we gain a perfect understanding of the universe, we cannot be sure that any perceived meaning is exactly as we interpret it. Anything based on that perception is uncertain at least to the degree we are uncertain of the precise mechanics of the universe.
...

The whole purpose of axiomatic mathematics is to create an unambiguous starting point from which to build.

To construct that unambiguous starting point requires not just definite axioms - but a definite language.

If the language is not subject to the same rigour as is applied to subsequent steps then the whole edifice is built on shaky foundations.

When examined, there is no formal definition of meanings within any language. The natural language definitions all appeal to experiences external to the mathematics/language. This applies just as much to the formal languages referenced in the article.

There is no definite (absolute) starting point upon which to build. Axiomatic mathematics knows this. It is the reason for axiomatic mathematics. But mathematicians neglected to notice that this constraint applies to the language used to describe axioms. The language does not have some special privileged access to a fixed point.

With no fixed point from which to build, there is no way to determine definite meaning for anything that follows.

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### Re: Axiomatic mathematics has no foundation

And yet, here you are, using a computer, sending millions of bits of encoded and encrypted data across the world, that is processed, routed, and translated automatically by machines, and it all works because the people who designed and built them used mathematics.

It works, bitches.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

Dopefish
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### Re: Axiomatic mathematics has no foundation

Aren't you trying to reason using logic, which itself boils down to some axioms?

If you want to argue that use of axioms is a fallacy, then you shouldn't use axioms in your argument. I'm also pretty sure that an axiom free argument would amount to you saying nothing, as there's assumptions/axioms deeply rooted in pretty much everything.

Treatid
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### Re: Axiomatic mathematics has no foundation

Dopefish wrote:Aren't you trying to reason using logic, which itself boils down to some axioms?

If you want to argue that use of axioms is a fallacy, then you shouldn't use axioms in your argument. I'm also pretty sure that an axiom free argument would amount to you saying nothing, as there's assumptions/axioms deeply rooted in pretty much everything.

You seem to be saying that the argument proves axiom based systems are invalid based on the logic of axiom systems - therefore the argument is invalid because it is based on axiomatic reasoning - therefore axiom systems are valid.

But if the axiom system hasn't collapsed then neither has the argument.

MartianInvader wrote:And yet, here you are, using a computer, sending millions of bits of encoded and encrypted data across the world, that is processed, routed, and translated automatically by machines, and it all works because the people who designed and built them used mathematics.

It works, bitches.

Physics works.

The meaning that we perceive in mathematics all comes from our direct experience of the universe we inhabit. As such, much of mathematics is relevant to the universe we inhabit. But it isn't really mathematics.... it is physics. The source of our natural languages is our experience within this universe. Everything built upon that biased view belongs to physics.

Plus... Euclidean Geometry worked for around 2,000 years despite the fifth postulate being omitted. But Euclidean Geometry without the fifth postulate is also non-Euclidean Geometry. Just because to works does not mean it is right.

Axiomatic mathematics isn't engineering. Engineering works to given tolerances. Axiomatic mathematics isn't supposed to be fuzzy.

...

Imagine a trans-dimensional alien arrived at earth and we tried to speak to them...

First thing is to establish a common language.

How do you do that?

You can't without there being some foundation upon which to build. (c.f.: Rosetta Stone).

Mathematics doesn't contain that foundation.

This (Rosetta Stone) isn't something I'm inventing... really... linguistics has known that a language by itself is meaningless for at least a hundred years (c.f.: Rosetta Stone).

Without that foundation, everything else is meaningless. If the axioms aren't formally defined, then the system isn't formally defined. (Formal languages are not formally defined in this sense).
Last edited by Treatid on Thu Feb 06, 2014 10:27 pm UTC, edited 1 time in total.

Sizik
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### Re: Axiomatic mathematics has no foundation

There are two cases:

1. Axiom-based systems are invalid. Therefore, the argument, which relies on the axiom "It isn't possible to state a set of axioms without first creating an axiomatic system to define the axioms," is invalid.

2. Axiom-based systems are valid. Therefore, the argument is wrong.
gmalivuk wrote:
King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.
Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

Treatid
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### Re: Axiomatic mathematics has no foundation

Sizik wrote:There are two cases:

1. Axiom-based systems are invalid. Therefore, the argument, which relies on the axiom "It isn't possible to state a set of axioms without first creating an axiomatic system to define the axioms," is invalid.

2. Axiom-based systems are valid. Therefore, the argument is wrong.

There are loads more than just two cases:

3. The argument doesn't show that axiomatic mathematics is wrong - it shows that axiomatic mathematics is without foundation. Axiomatic mathematics continues to exist - but you can't unambiguously specify a set of axioms to use.

But really the only relevant point is whether or not you can unambiguously specify the language that specifies the axioms.

If you can define the meaning of a statement in mathematics such that there is zero ambiguity about the meaning then I am wrong.

All you need to do is define the definite meaning of a single statement without reference to anything outside formal mathematical systems. Should be easy, right?

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### Re: Axiomatic mathematics has no foundation

Treatid wrote:With no fixed point from which to build, there is no way to determine definite meaning for anything that follows.
Treatid wrote:Without that foundation, everything else is meaningless.

Maybe you need to define "meaning" and "everything else"/"anything that follows".

Because otherwise my response to "it isn't possible to state a set of axioms without first creating an axiomatic system to define the axioms" is: Who cares?

And that's not really a rhetorical question, I kind of want to know what sort of people are bothered by this existential "paradox" you've put forth. I'm an engineer who only deals with things I can describe in a drawing, so I obviously don't care.

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### Re: Axiomatic mathematics has no foundation

Adam H wrote:Because otherwise my response to "it isn't possible to state a set of axioms without first creating an axiomatic system to define the axioms" is: Who cares?

This is, of course, what the refutation really boils down to.

Of course math can't be shown to be some universal, perfect truth beyond what language can communicate, because language is all we have to communicate with. However, everyone who has spent any time with this stuff can agree on what is and isn't a valid logical progression, which makes it about as close as we can hope to achieve. Nobody actually cares that we are limited by language, we're far more concerned with what can can achieve by using these rules, where it's easy for everyone to agree on when they have or haven't been broken.

In other words: It works, bitches.

And hey, if you want to call all of math "physics" because it's based on analogies of physical objects and often has applications back in the real world, you can go right ahead, but you're not being revolutionary, you're just being a poor communicator.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

korona
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### Re: Axiomatic mathematics has no foundation

Mathematicians are fully aware of the "fallacy" you came up with.

The point is that
1) we cannot do any better than trying to build consistent axiom systems (no system can prove its own consistency; that is the second incompleteness theorem)
2) so far your axiom systems work pretty good.

If sometime in the future someone proves that our axioms are not consistent we will try to fix that and invent better axioms. And almost all theorems will continue to work with this new axiomatic foundation.
Last edited by korona on Thu Feb 06, 2014 11:31 pm UTC, edited 2 times in total.

brenok
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### Re: Axiomatic mathematics has no foundation

MartianInvader wrote:And hey, if you want to call all of math "physics" because it's based on analogies of physical objects and often has applications back in the real world, you can go right ahead, but you're not being revolutionary, you're just being a poor communicator.

And you know what we do with smug poor communicators...

Treatid
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### Re: Axiomatic mathematics has no foundation

korona wrote:Mathematicians are fully aware of the "fallacy" you came up with.

I am pleased to hear it. However, it appears that it wasn't just me who didn't get the memo.

The point is that
1) we cannot do any better than trying to build consistent axiom systems (no system can prove its own consistency; that is the second incompleteness theorem)

Obviously throwing out axiomatic mathematics isn't an attractive proposition. A huge amount of work has been invested in it.

And if our biases are predicated on physics, then much of that mathematics should still be a reasonable approximation for useful work in the universe. There is unquestionably a degree to which it works.

But...

i) Mathematics based on a fundamental flaw is not what I understood mathematics to be. Engineering, physics and economics are fine with a system that works. I was under the impression that pure mathematics had higher ideals. I feel that mathematics shouldn't be afraid of hard answers. Why add the fifth postulate to Euclidean Geometry? Everyone knew what they meant without it... But then we wouldn't have non-Euclidean Geometry.
ii) This isn't a minor detail. If the crutch of physics is removed - there is no way to specify an axiomatic system. "Yeah, but what ya gonna do?", just... seems inadequate.
iii) We absolutely can do better. Or at least - less flawed. The Relativistic Mathematics I referenced helped me to become aware of a lack of foundation for axiomatic maths.

2) so far your axiom systems work pretty good.

Newtonian mechanics works pretty good too, most of the time.

But most people don't claim that Newtonian mechanics is correct. It is a not bad approximation. It is easy to use. But it isn't correct.

If sometime in the future someone proves that our axioms are not consistent we will try to fix that and invent better axioms. And almost all theorems will continue to work with this new axiomatic foundation.

You say "Mathematicians are fully aware of the "fallacy" you came up with."... but it isn't clear to me that you actually understand the implications.

There is absolutely no way to specify a set of axioms within mathematics. We can (and do) borrow meaning from physics.... but there is no way to specify that meaning. Mathematicians can only hope that they have roughly the same assumptions with regard to a set of axioms. Talking about "finding inconsistency" and "making better axioms" seems to completely miss the fact that it is absolutely impossible to define a set of axioms without reference to something external.

Having said that... I'm sure you do understand the implications. I'm just surprised that you seem to regard it as a minor detail. It seems to me to be a very big and significant deal.

While I sort of understand mathematician's decision not to throw out so much invested effort... I also find it baffling for mathematicians to turn a blind eye to such a fundamental flaw in the axiomatic model.

...

Do you have any references to where this issue is discussed? I'd like some background on this particular decision and how mathematicians feel it impacts their field.

Dopefish
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### Re: Axiomatic mathematics has no foundation

I'm pretty sure physics is basically just math with extra assumptions about the nature of the universe and measurement errors, rather than the other way around, so if you're dissatisfied with the state of math, physics is apt to be far worse off.

Sizik
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### Re: Axiomatic mathematics has no foundation

gmalivuk wrote:
King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.
Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

Treatid
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### Re: Axiomatic mathematics has no foundation

Dopefish wrote:I'm pretty sure physics is basically just math with extra assumptions about the nature of the universe and measurement errors, rather than the other way around, so if you're dissatisfied with the state of math, physics is apt to be far worse off.

I think my issue is/was expectation. I don't expect physics to aim for lofty ideals. Physics is only relevant to the degree it can predict the universe. No matter how elegant/beautiful/compelling something is in physics - the ultimate arbiter is whether it predicts the universe accurately. "It works" trumps all other considerations in physics.

I did expect mathematics to go with "right" rather than "convenient".

On the other hand, it would have been ridiculous for this hole to have been missed by the entire field of mathematics. I expected that mathematics would put such an issue front and center... so I allowed myself to believe that since it wasn't front and center, it had been missed. It is going to take me a while to digest that mathematics is is no more noble than engineering. It is like discovering that your parents can lie...

elasto
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### Re: Axiomatic mathematics has no foundation

All of mathematics is of the form 'if A is true then B is also true'. I am rather incredulous that you weren't aware of this given how much you think and write about the topic.

We often choose which A's we assume true based on the real world. But mathematicians are playful creatures and sometimes choose weird and wonderful A's - some of which then turn out to have surprising real-world applications.

You can do maths without regard for real-world physics, but you can't do physics without maths, so maths is definitely the more fundamental discipline. Arguably philosophy is more fundamental again - though the division between maths and philosophy is somewhat blurry.

Treatid
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### Re: Axiomatic mathematics has no foundation

elasto wrote:All of mathematics is of the form 'if A is true then B is also true'. I am rather incredulous that you weren't aware of this given how much you think and write about the topic.

I am and was fully aware of axiomatic theory and how it proceeds.

However, while it is technically possible to assign "true" and "false" to any arbitrary bit string... a bit string by itself tells us nothing about other bit strings.

We must attach "meaning" or "intention" to a bit string in order to work with it (e.g. "equals", "greater than", "contains", "addition").

It is this attachment of "meaning" to symbols that cannot be done unambiguously.

We often choose which A's we assume true based on the real world. But mathematicians are playful creatures and sometimes choose weird and wonderful A's - some of which then turn out to have surprising real-world applications.

Actually... A's are always based on the real world. There is no other source for them. There is no mechanism within mathematics to create A's from scratch. (although given A's, mathematics can propagate them via axiomatic systems)

You can do maths without regard for real-world physics,

No. You can't. (axiomatic) Mathematics requires some foundation to work from. And no such foundation exists within mathematics. Mathematics must always rely on an outside reference for its foundation... and the only external reference we have available is the universe.

but you can't do physics without maths, so maths is definitely the more fundamental discipline.

That is what I once believed. But it isn't true. ish...

It is true that modern mathematics provides the underpinnings for modern physics as a discipline. But mathematics is intimately tied to the physics of this universe. So - mathematics as a discipline can be considered more fundamental than physics as a discipline. But the physics of the universe provides the foundation for modern mathematics as a discipline.

Arguably philosophy is more fundamental again - though the division between maths and philosophy is somewhat blurry.

I recommend reading the link to Foundations of Mathematics that Sizik provided a couple of messages ago. It has a brief section on the relationship between mathematics and philosophy (and "Foundational crises").

korona
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### Re: Axiomatic mathematics has no foundation

Treatid wrote:No. You can't. (axiomatic) Mathematics requires some foundation to work from. And no such foundation exists within mathematics. Mathematics must always rely on an outside reference for its foundation... and the only external reference we have available is the universe.

What is the physical importance of the existence of arbitrary products in the category of sets? That's an axiom we use constantly but its only justification is that our theorems become more beautiful. Sometimes we just choose our axioms so that we get more general theorems.

Schrollini
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### Re: Axiomatic mathematics has no foundation

Treatid wrote:We must attach "meaning" or "intention" to a bit string in order to work with it (e.g. "equals", "greater than", "contains", "addition").

I reject this assertion. We can work with a set of symbols and rules for manipulating them without understanding them. Computers do it all the time. Computers are commonly used to generate and check proofs, but I don't think anyone will claim that the computer understands the meaning of the symbols it's manipulating.

Now it's true that we do attach meaning to symbols when we're working on them; this is what makes math useful. But the definition of an axiomatic system is independent of and not reliant upon this meaning.

It's also true that we choose which axioms to use based on the meaning we assign to them and (generally) our real-world intuition. So what?
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### Re: Axiomatic mathematics has no foundation

Treatid wrote:I think my issue is/was expectation. I don't expect physics to aim for lofty ideals. Physics is only relevant to the degree it can predict the universe. No matter how elegant/beautiful/compelling something is in physics - the ultimate arbiter is whether it predicts the universe accurately. "It works" trumps all other considerations in physics.

But your objection to mathematics applies just as equally here. What does "predicts the universe" mean? How can we know what a prediction actually is? How can we know it actually came true, or did not come true? How can we know what the values of measurements really were? At some point, we have to just buckle down and say, "Look, we can all agree that we understand what we mean when we talk about 10 grams, 1 centimeter, 3 seconds, etc., and we'll base our studies on this mutual understanding and go from there." So it goes with math, and pretty much any other discipline for that matter.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

Cleverbeans
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### Re: Axiomatic mathematics has no foundation

Treatid wrote:Actually... A's are always based on the real world. There is no other source for them. There is no mechanism within mathematics to create A's from scratch. (although given A's, mathematics can propagate them via axiomatic systems)

This is not true. Non-Euclidean geometry being an easy first example. At first everyone agreed that the interior angles of a triangles sum to 180 degrees, but then a few enterprising mathematicians found that you could assume they either had greater than or less than 180 degrees and be entirely consistent. At first no one thought this would ever bear real world applications but eventually we discovered using empirical methods that we were wrong to assume that classic geometry was correct and were forced to review our assumptions.

I consider science to be the empirical verification of which axioms best describe the real world, but the axioms are still there. Theoretical science is the development of certain axioms in an effort to find a verifiable claim which we can empirically test. I think you'll find that the more you study epistemology the murkier knowledge gets and you'll be forced to find peace with this to function in the world.

For example we know that we can't state every assumption we make, and we know that some concepts must be undefined. You can always play the "define it" game where you insist on each word being defined without referencing the other words you're defining. Start with something like "line", which might be described as the " the unique collection of points joining two points which minimizes distance" and now you have to define points, collection, joining, two, minimize and distance all without referring to lines. You can do this ad nauseam.

This has led me to conclude that mathematics is essentially socially constructed. We *agree* to certain definitions which are collectively viewed to be useful, and choose to ignore some definitions and axiomatic systems which aren't. If a mathematician wants to explore a new system of axioms they have to convince their peers there is something of interest in it, and demonstrate that these axioms lead to provable claims that are somehow distinct or equivalent to existing systems. You may have noticed you have a hard time getting non-standard terminology accepted by the community because of this. This can be a little unsettling for people with a high school or undergraduate understanding of the subject because they've come to believe that mathematics represents our best attempt at universal truth or a language that we could use to talk to aliens. the fact is it's very human endeavor with all the trappings of other less objective subjects.
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Treatid
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### Re: Axiomatic mathematics has no foundation

korona wrote:What is the physical importance of the existence of arbitrary products in the category of sets? That's an axiom we use constantly but its only justification is that our theorems become more beautiful. Sometimes we just choose our axioms so that we get more general theorems.

It looks to me like you are arguing that Set Theory, in which the only objects are sets, is independent of physical references? Perhaps you are also arguing that ZFC is rigorously defined, has no physical references and can be used to construct most other axiomatic systems? The lack of foundation that you perceive is that there is no absolute reason to choose ZFC over some alternate systems? But having chosen ZFC, everything that follows is unambiguous?

I have no problem with any of the above. My argument is not that there is no way to prefer one set of axioms over another.

My argument is that it is not possible to unambiguously specify any set of axioms in the first place. [Edit: I've been lumping "meaning" in with the definition of axioms. It is possible to define a set of rules (symbol manipulations) unambiguously. It isn't possible to assign meaning to those rules unambiguously. This is a fairly important distinction that I now realise I haven't been making clear. Sorry for this mistake. I can now see that mathematicians tend to put weight onto the symbol manipulation over specific meaning for given symbols. As such - at least some of the following is arguing against a strawman.]

ZFC cannot be defined unambiguously. The words and symbols used to describe the axioms of ZFC are not defined within mathematics. Their meaning derives (possibly indirectly) from our experience in the physical world. And that experience is subjective, cannot be communicated directly, is partial, and in no way provides an unambiguous platform on which to build unambiguous systems.

As far as I can tell this isn't the "foundational crises" in mathematics.

When defining terms ("meaning") we do so by reference to other terms. But those other terms are defined by reference to yet further terms.

If any set of terms can be constructed in an unambiguous manner then we can construct further terms from those initial ones (axiomatic mathematics).

But if those initial terms are in any way ambiguous, then anything we construct from them will also be ambiguous (cross-referencing may help to reduce ambiguity...).

If we have no starting point (trans-dimensional alien), there is nothing to build upon. No way to create communication ("meaning").

We do have a starting point... us and the physical universe we inhabit. But we don't have a full and complete understanding of that starting point. Any "meaning" that we derive is incomplete, partial, ambiguous. Anything we build on that is also ambiguous.

Without a fixed, definite starting point there is no way to create a fixed, definite set of axioms. We cannot create ZFC in an unambiguous fashion.

That some of the systems that have been created work is a defense, to a degree. But to pretend that ZFC is a definite, unambiguously defined system is going to far. It isn't a question of finding inconsistencies within ZFC... it is that ZFC cannot be properly defined in the first place.

Hilary Putnam has certainly addressed this point... so it is absolutely known to mathematics (e.g.):
Twin Earth Thought Experiment

From the referenced wiki page:
In practice, most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system.

So, you are certainly correct that this isn't news to mathematics. And "it works, bitch" is a fair summary of the approach mathematics takes.

But at the same, time, anything based on axiomatic mathematics (no matter how indirectly) is nothing more than a statement of our biases. And no doubt there is value in understanding what our biases are....

However, I had the idea that there was a mathematics that existed independent of humans. That there was a realm of concepts not tied to our physical existence that could be explored. It turns out that modern mathematics doesn't even attempt to address anything that isn't intimately tied to our biases. I find this disappointing.

So, I'm going to make my own mathematics (with blackjack and hookers) which isn't hobbled by the impossibility of defining a fixed point upon which to build everything else.

MartianInvader wrote:But your objection to mathematics applies just as equally here. What does "predicts the universe" mean? How can we know what a prediction actually is? How can we know it actually came true, or did not come true? How can we know what the values of measurements really were? At some point, we have to just buckle down and say, "Look, we can all agree that we understand what we mean when we talk about 10 grams, 1 centimeter, 3 seconds, etc., and we'll base our studies on this mutual understanding and go from there." So it goes with math, and pretty much any other discipline for that matter.

I agree that physics is no better founded than mathematics. My point was simply that physics doesn't claim to be well founded, whereas I was under the illusion that mathematics had loftier ideals. I even thought that mathematics might strive to reveal "universal truths". As it turns out, mathematics is just about finding better ways to count money.

There's nothing wrong with that, per se. It is my illusions that are at fault for regarding mathematics as something it isn't.

And your last point is extremely relevant. It isn't obvious that there is an alternative. This is a hard limitation on how knowledge can be expressed and communicated. Lacking any alternative, we work with what we have no matter how flawed. Doing nothing is not an option.

However, I think it is possible to construct something that doesn't depend on a fixed starting point that axiomatic mathematics requires.

Schrollini wrote:
Treatid wrote:We must attach "meaning" or "intention" to a bit string in order to work with it (e.g. "equals", "greater than", "contains", "addition").

I reject this assertion. We can work with a set of symbols and rules for manipulating them without understanding them. Computers do it all the time. Computers are commonly used to generate and check proofs, but I don't think anyone will claim that the computer understands the meaning of the symbols it's manipulating.

We constructed the computer (and the program it runs). That construction is where we assigned meaning.

The computer is just manipulating symbols. You are right - the computer does not perceive any meaning in those symbols. It simply does what it does according to its structure.

But by the same token, a computer does not check proofs, we may interpret the output symbols as a proof.

The symbols never hold any meaning themselves. The symbols are just labels that we attach to meaning. A computer manipulates symbols. It does not manipulate meaning.

Meaning is abstract. We have no way of directly expressing or communicating meaning. We can communicate symbols.

Now it's true that we do attach meaning to symbols when we're working on them; this is what makes math useful. But the definition of an axiomatic system is independent of and not reliant upon this meaning.

Ah... we could express a set of axioms as a computer program. The hardware and software are known deterministic systems. By expressing the axioms within this deterministic system, we are unambiguously specifying a set of axioms for a system. There is no argument over what the system will do (run it and see). There is no sense in which this system is ambiguous.

Except for my last sentence, I agree.

But the "attaching meaning to symbols to make math useful" bit is a fairly essential part of the process. And it is this part that cannot be done unambiguously.

It's also true that we choose which axioms to use based on the meaning we assign to them and (generally) our real-world intuition. So what?

So - we cannot communicate the meaning we have attached to a symbol. If the meaning you attach to a symbol is not identical to the meaning that I attach to a symbol then the meanings we each arrive at after manipulating that symbol are also likely to differ. Yet we have no way of detecting that difference - unless the different meanings lead to different choices of manipulation.

You are right: So what if my "meaning" for infinity is different to your "meaning" for infinity. If we are both constrained to the same set of manipulations then it doesn't matter if our meanings differ... we should both still output the same sets of symbols.

Hmm... your argument is good. It does help me to see why mathematicians are content to largely ignore the "meaning" with respect to a set of axioms - and why "it works" is regarded as a sufficient argument.

But I don't think it contradicts my argument. It still leaves mathematics as an arbitrary reflection of our biases with no means to reveal anything that isn't a direct result of those biases. And since mathematics does understand that axiomatic mathematics doesn't have a solid foundation...

Which still leaves me wanting more than "an arbitrary set of manipulations" with no inherent significance beyond that which we place there.

Cleverbeans wrote:This is not true. Non-Euclidean geometry being an easy first example.

I think the example you give shows that we do not always understand the rules/meanings we assume from the real world - and that careful examination can reveal our misapprehensions. I would say that the discovery of non-euclidean geometry is very much rooted in our real world experience.

The rest of what your post I agree with and is a good summation of both what I (naively) wanted to say and some of my position.
Last edited by Treatid on Fri Feb 07, 2014 9:47 pm UTC, edited 1 time in total.

Cleverbeans
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### Re: Axiomatic mathematics has no foundation

Treatid wrote:But by the same token, a computer does not check proofs, we may interpret the output symbols as a proof.

Actually every computer program is a proof.
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### Re: Axiomatic mathematics has no foundation

Treatid wrote:I agree that physics is no better founded than mathematics. My point was simply that physics doesn't claim to be well founded, whereas I was under the illusion that mathematics had loftier ideals. I even thought that mathematics might strive to reveal "universal truths". As it turns out, mathematics is just about finding better ways to count money.

There's nothing wrong with that, per se. It is my illusions that are at fault for regarding mathematics as something it isn't.

And your last point is extremely relevant. It isn't obvious that there is an alternative. This is a hard limitation on how knowledge can be expressed and communicated. Lacking any alternative, we work with what we have no matter how flawed. Doing nothing is not an option.

However, I think it is possible to construct something that doesn't depend on a fixed starting point that axiomatic mathematics requires.

I don't think the phrase "well-founded" means what you think it means. And math finds truths that are as universal as any truths we could hope to find.

The only limitations on math are those imposed by language itself. What were you planning on using to construct your "new math", if not language?
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### Re: Axiomatic mathematics has no foundation

Treatid wrote:So - we cannot communicate the meaning we have attached to a symbol. If the meaning you attach to a symbol is not identical to the meaning that I attach to a symbol then the meanings we each arrive at after manipulating that symbol are also likely to differ. Yet we have no way of detecting that difference - unless the different meanings lead to different choices of manipulation.

We have language to communicate those meanings. Sure, it's imprecise and un-rigorous. But it seems to work.

Treatid wrote:You are right: So what if my "meaning" for infinity is different to your "meaning" for infinity. If we are both constrained to the same set of manipulations then it doesn't matter if our meanings differ... we should both still output the same sets of symbols.

Hmm... your argument is good. It does help me to see why mathematicians are content to largely ignore the "meaning" with respect to a set of axioms - and why "it works" is regarded as a sufficient argument.

But I don't think it contradicts my argument. It still leaves mathematics as an arbitrary reflection of our biases with no means to reveal anything that isn't a direct result of those biases. And since mathematics does understand that axiomatic mathematics doesn't have a solid foundation...

I'll agree with most of this, but I think mathematics does reveal many things that surprise us. Everything that goes into the constructions of the rational numbers agrees with my bias, but the result that there are as many rational numbers as integers contradicts my biases. And of course,
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

And yet they're all equivalent.

Treatid wrote:Which still leaves me wanting more than "an arbitrary set of manipulations" with no inherent significance beyond that which we place there.
I think everyone wants this. Some people look to religion, some to mysticism, and some to mathematics. If you find it, more power to you. But, in my ever-so-humble opinion, it's not there to be found.

'Course, I am a proud member of the shut-up-and-calculate school of quantum mechanics, so I have my biases.
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### Re: Axiomatic mathematics has no foundation

Treatid wrote:However, I had the idea that there was a mathematics that existed independent of humans. That there was a realm of concepts not tied to our physical existence that could be explored. It turns out that modern mathematics doesn't even attempt to address anything that isn't intimately tied to our biases. I find this disappointing.

Can you give an example of these concepts? I'm imagine that would clear up some confusion.

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### Re: Axiomatic mathematics has no foundation

Cleverbeans wrote:
Treatid wrote:But by the same token, a computer does not check proofs, we may interpret the output symbols as a proof.

Actually every computer program is a proof.

(mostly agreed with your previous post).

I see your point with regard to the Curry-Howard correspondence.

The question would be if it is a proof without a human regarding it as a proof? The computer doesn't "know" that it is a proof. A computer doesn't know what "proof" means.

Nevertheless, I do like the idea. Thank you for the reference.

The only limitations on math are those imposed by language itself. What were you planning on using to construct your "new math", if not language?

This is very much to the point.

What language cannot do is to define or specify a definite meaning.

People (mathematicians) tend to prefer absolute coordinate systems. The fixed plane of Euclidean Geometry on which lines and shapes are drawn, through the number line and number plane onwards are rigid, absolute coordinate systems within which all else fits. Having a fixed starting point from which to construct is handy, and it was a reasonable approximation of everyday experience (for a human on the surface of a planet).

But as hard as mathematics has tried, it isn't possible to define an absolute fixed point.

As much as a human's perspective might get the impression that fixed points are possible ("that mountain has been in that spot for all eternity - even my grandpa says it has always been there") - rigorous attempts to find or construct such a thing have failed.

What languages can do is to specify relationships between things. The language cannot tell us precisely what the things being related are (as an absolute) - but it can communicate that two or more symbols are related. This is what language does. It draws relationships. (obviously, the relationship itself can't be defined in an absolute sense... but its existence is sufficient).

Any attempt to start with a fixed point is doomed to failure. but a floating point (ahem) related to other floating points is exactly what language can describe. (A point here is not necessarily a geometric point).

Arguably Set Theory does exactly this. The empty set and the hierarchy of sets above that are just generic things, and the contents of a set are the related objects.

But set theory still carries some of the baggage from the absolutist view. From the start, ZFC is an axiomatically defined system - not a lethal flaw, but it does set the precedent of approaching things from the view that it is possible to have a fixed point. The "empty set" is another fixed point, although it could be regarded as an abstract 'every-man'. The Axiom of Choice is already a debated aspect... That the set of everything either doesn't exist or doesn't contain itself is too narrow a definition of "relationship".

What is needed is a system that is able to specify objects and relationships between objects without any concern for the actual structure of either the object or the relationship (because we cannot define either of those in unambiguous detail). In such a system, patterns of relationships are the primary information. This is exactly the sort of system that language is well structured to convey. Language has only ever communicated relationships between things.

Existing (axiomatic) systems have a pattern. They have states and relationships (functions) between states. The objects (systems) created by axiomatic mathematics can still be viewed. There's no point in a system if it has no relation to our existing systems.

Using structures of relationships to create meaning (with no fixed starting point) is how our brains are structured. The information in a neural network is contained within the pattern of connections. There is no single neural cell that encodes for "tree".

Humans (and mathematicians) have tended towards investing "meaning" into singular objects; an object is defined by its properties. But we know that we can't unambiguously define those properties.

An alternative is for meaning to derive from the pattern of relationships between things.

This is the alternative that I refer to when I mention "Relativistic mathematics".

Schrollini wrote:I'll agree with most of this, but I think mathematics does reveal many things that surprise us. Everything that goes into the constructions of the rational numbers agrees with my bias, but the result that there are as many rational numbers as integers contradicts my biases.

I think there is enough about ourselves, and sufficient contradictions in our self perception that surprises are inevitable even if we start from the most biased position. It is to math's credit that it reveals the ultimate implication of those biases even if it does nothing else.

And of course,
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

And yet they're all equivalent.

I laughed. Thank you for this.

Treatid wrote:Which still leaves me wanting more than "an arbitrary set of manipulations" with no inherent significance beyond that which we place there.
I think everyone wants this. Some people look to religion, some to mysticism, and some to mathematics. If you find it, more power to you. But, in my ever-so-humble opinion, it's not there to be found.

I would regard definitely knowing it isn't there as satisfying.

In which regard... mathematics has laid the framework such that it is possible to know without any question that a "fixed, definite meaning/point" cannot be constructed or communicated. So - I know that it is pointless trying to look for such a thing.

Knowing that "an integer", "a set", "a system" and every other mathematical object cannot ever be fully described in an unambiguous manner is itself satisfying. To be able to draw a hard limit on what can (or cannot) be known is, itself, an achievement.

'Course, I am a proud member of the shut-up-and-calculate school of quantum mechanics, so I have my biases.

Magnanimous wrote:
Treatid wrote:However, I had the idea that there was a mathematics that existed independent of humans. That there was a realm of concepts not tied to our physical existence that could be explored. It turns out that modern mathematics doesn't even attempt to address anything that isn't intimately tied to our biases. I find this disappointing.

Can you give an example of these concepts? I'm imagine that would clear up some confusion.

If it were that easy...

I imagine a conceptual space. This conceptual space contains every conceivable possibility - including those conceivable only by much more significant inhabitants of this conceptual space than humans.

I imagine a tool that can classify that space, that can determine whether two apparently distinct possibilities are, in fact, the same possibility from a different perspective. This tool could theoretically distinguish and order that entire conceptual space into individual elements (and all the permutations and combinations of elements) permitting each possibility to be inspected and savoured with no repetition and no significant information omitted. Naturally the tool can identify and exclude impossibilities.

Obviously such a tool cannot encompass the entirety of that conceptual space from the constraints of a finite universe. But it should maintain as much functionality as possible within that constraint.

To be honest, I'm working backwards. Relativistic maths makes enumeration of all unique possibilities fairly easy - which is most of the above wish-list. Something similar can be done by enumerating all possible Turing Machines - except that it is rather more difficult to construct a unique enumeration with no functional duplication. But the set of all conceivable possibilities is equivalent to the set of all conceivable Turing Machines (with all conceivable inputs) (plus or minus some duplication).

Farabor
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### Re: Axiomatic mathematics has no foundation

Nothing has an absolute foundation when it comes to anything which requires communication. How do you define a word? By reference to other words. Circular logic. It just so happens that as social animals our brains have the ability to construct shared meaning. Even grunting/pointing/other forms of communication require a shared agreement as to what they are.

If you're searching for some kind of platonic ideal that transcends language itself (Because that's the only thing that could be 'absolute')....good luck. The rest of us will get on with the business of exploring the wonders and beauty of our ever growing mathematical systems that we grow in our own shared experience.

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### Re: Axiomatic mathematics has no foundation

It does sound a bit like platonic ideals, though expanded. What would relationships between individual elements look like? I'm wondering exactly what you mean by distinguishing and ordering.

More to the point though, I'm not seeing ambiguity in formal mathematics. Our understanding of math comes from the isomorphism between abstract logic and concrete physics. If everyone is in the same universe and observing the same patterns of causation, defining those patterns won't necessarily be ambiguous. The concept of 1 is fundamental. In my view axioms aren't a magical foundation, they're a consequence of the universe existing.

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### Re: Axiomatic mathematics has no foundation

Magnanimous wrote:It does sound a bit like platonic ideals, though expanded. What would relationships between individual elements look like?

Objects and relationships cannot be defined in any detail.

But we can, of course, attach a symbol to "object" and "relationship". So - a relationship will look like the symbol we attach to "relationship". This is just business as usual. We can't directly see "an integer", "infinity", "addition" - but we do attach symbols to these things. We then manipulate these symbols and pretend that this manipulation corresponds to manipulating the associated concepts.

I'm wondering exactly what you mean by distinguishing and ordering.

"exactly what you mean"...

This is the absolutist way of thinking. It is never, ever going to be possible to define "exactly what you mean" for anything.

The most definite illustration I know of is Newtonian Mechanics versus General Relativity. Both these viewpoints describe (roughly) the same thing. But the two viewpoints are totally incompatible. Even though General Relativity is often presented as sets of euclidean spaces (reference frames) with mappings between them, General Relativity as a whole is in no way an extension of Newtonian mechanics. It is an entirely distinct system and it isn't possible to understand GR in terms of Newtonian Mechanics. Attempting to do so leads to brain frazzle, as a good many physics students will attest.

That doesn't mean it is impossible to communicate. It just means that we can't communicate absolutes. General Relativity has no fixed foundation (ish), yet still manages to construct a coherent (and successful) description. It does mean that any assumption connected with there being a definite meaning must be left behind.

More to the point though, I'm not seeing ambiguity in formal mathematics. Our understanding of math comes from the isomorphism between abstract logic and concrete physics.

This would be more compelling if we had a concrete physics. While I'll allow that the universe itself is fairly definite... we don't have a complete understanding of that universe. An isomorphism with an undefined object hardly gives mathematics a firm foundation.

If everyone is in the same universe and observing the same patterns of causation, defining those patterns won't necessarily be ambiguous.

More bad news... I mentioned General Relativity earlier... observation is relative. When you say "the same pattern of causation" you are assuming that i) there is a fixed, definite pattern of causation, ii) that is observed the same way by all parties.

But beyond that: we are still limited by language. We cannot communicate a definite object in an unambiguous way.

Have a look at the "Twin Earth thought experiment" from Hilary Putnam: http://en.wikipedia.org/wiki/Twin_Earth_thought_experiment

The only way to communicate unambiguously is for both parties to have identical foundations. The only way this can happen is if both parties are identical... in which case communication is redundant.

The concept of 1 is fundamental.

I'm not sure what you mean by 'fundamental' here.

The concept of '1' cannot be defined. You can define the relationships that 1 has with other numbers... but you cannot define '1' itself. With a sufficient number of relationships you may feel that you have a pretty good handle on what '1' means...

In my view axioms aren't a magical foundation, they're a consequence of the universe existing.

Current axiomatic systems used in mathematics are definitely a direct consequence of our (human) existence within this universe. I agree with this.

However, as unambiguously as those systems are defined mechanically (we can create deterministic systems (computers) with those axioms); the "meaning" of those systems is poorly defined at best. Even if we accept that our experiences are similar enough for most purposes, there is no possible way to ensure identical meaning. There is no conceivable way to even assess whether your concept of '1' matches my concept of '1'.

It seems to me to be futile to attempt to establish unambiguous meaning when we know that it can't be done.

Pretending to give the symbol "infinity" a definite meaning (c.f. ZFC) that everyone agrees with is an interesting party trick... but it isn't profound or enlightening. It is just futile.

Farabor wrote:Nothing has an absolute foundation when it comes to anything which requires communication. How do you define a word? By reference to other words. Circular logic. It just so happens that as social animals our brains have the ability to construct shared meaning. Even grunting/pointing/other forms of communication require a shared agreement as to what they are.

Entirely agreed.

Yet axiomatic mathematics still attempts to define things in an unambiguous fashion. Mathematics tries to define "integers", "infinity", "line", "set" and all the other mathematical objects... when it is known that such definition is impossible.

If you're searching for some kind of platonic ideal that transcends language itself (Because that's the only thing that could be 'absolute')....good luck. The rest of us will get on with the business of exploring the wonders and beauty of our ever growing mathematical systems that we grow in our own shared experience.

Except that mathematics is still trying to define objects that cannot be defined. My beef is that mathematics is not trying to explore the beauty and wonder of ever growing mathematical systems. It is doubling down on circular logic - on tautology.

Existing mathematics is structured to build upon the previous foundations. But there is no possible fixed foundation. Which means that existing mathematics has no fixed meaning. Mathematics pretends that "an integer" is a definite, defined object. At the same time, mathematics knows that "an integer" can never be unambiguously defined. This hypocrisy is not "getting on with the business of exploring the wonders and beauty of mathematics". It is insanity. Granted - the insanity works to a degree... But as long as mathematics tries to define objects, it is attempting the impossible - and as such, a waste of effort.

I think it is more productive to recognise the limits of what is knowable and work within those limits. I'm not after unattainable knowledge, as axiomatic mathematics is. I'm after exactly what is attainable... without prejudice.

Schrollini
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### Re: Axiomatic mathematics has no foundation

I realize that this is largely irrelevant to the post, but this is something that irritates me to no end:

Treatid wrote:The most definite illustration I know of is Newtonian Mechanics versus General Relativity. Both these viewpoints describe (roughly) the same thing. But the two viewpoints are totally incompatible. Even though General Relativity is often presented as sets of euclidean spaces (reference frames) with mappings between them, General Relativity as a whole is in no way an extension of Newtonian mechanics. It is an entirely distinct system and it isn't possible to understand GR in terms of Newtonian Mechanics. Attempting to do so leads to brain frazzle, as a good many physics students will attest.

As a (former) physics student, no. Just no. Newtonian mechanics clearly emerges as the low-curvature limit of GR. And not by accident. This was a crucial step in Einstein's derivation of GR! He had worked out what the relationship between energy and curvature should be, but needed to set the conversion factor between the two. He did this by showing that GR reduced to Newtonian gravity in the low-field limit, thereby deriving his unknown constant from Newton's constant.

Yes, GR looks radically different from Newtonian mechanics, but the two viewpoints are totally compatible.
Treatid wrote:More bad news... I mentioned General Relativity earlier... observation is relative. When you say "the same pattern of causation" you are assuming that i) there is a fixed, definite pattern of causation, ii) that is observed the same way by all parties.

Wut? GR certainly preserves causality locally, and it preserves it globally as long as there are no closed time-like curves. (No self-respecting cosmology should allow such a thing .) All observers will agree on the pattern of causation; that's one of the few things preserved by GR!

Now back to our regularly-scheduled confusion....
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### Re: Axiomatic mathematics has no foundation

Treatid wrote:Current axiomatic systems used in mathematics are definitely a direct consequence of our (human) existence within this universe. I agree with this.

However, as unambiguously as those systems are defined mechanically (we can create deterministic systems (computers) with those axioms); the "meaning" of those systems is poorly defined at best. Even if we accept that our experiences are similar enough for most purposes, there is no possible way to ensure identical meaning. There is no conceivable way to even assess whether your concept of '1' matches my concept of '1'.

It seems to me to be futile to attempt to establish unambiguous meaning when we know that it can't be done.

Pretending to give the symbol "infinity" a definite meaning (c.f. ZFC) that everyone agrees with is an interesting party trick... but it isn't profound or enlightening. It is just futile.
Is this specifically about communication then? Because to me this sounds like brain-in-a-vat and philosophical zombies and problem of consciousness and so on. In the context of only one mind, would you say meaning can be absolutely defined?

If you're essentially saying that the uncertainty that comes from physical senses extends to mathematics because axioms aren't foundational in the cogito ergo sum sort of way, that sounds reasonable. I'm just wondering what your idea of "meaning" is, if it's not a consequence of relationships between concepts.

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### Re: Axiomatic mathematics has no foundation

Magnanimous wrote:Is this specifically about communication then?

Moreso about the limits of knowledge. What can be communicated is an integral part of that...

Because to me this sounds like brain-in-a-vat and philosophical zombies and problem of consciousness and so on. In the context of only one mind, would you say meaning can be absolutely defined?

No (meaning can never be absolutely defined) - I think the same rules apply internally to the mind as apply externally. At the extreme, a 'thing' is itself - and that is arguably definite... but nothing external to that 'thing' can know the 'thing' in an absolute, unambiguous sense.

Even a mind thinking about itself cannot be definite about its own nature in an absolute sense. Thinking is still constrained by the rules of language.

If you're essentially saying that the uncertainty that comes from physical senses extends to mathematics because axioms aren't foundational in the cogito ergo sum sort of way, that sounds reasonable.

This is definitely "I think, therefore I am" territory. But I'm working my way towards a more positive conclusion.

There is a hard limit on knowledge. The limit is formally expressed in the limits of language... but sensory input can be considered just another language in this context. In standard solipsism, we get to the point where everyone else could be a figment of my imagination (and/or senses) and then say "fuck it... either starve to death because you don't believe in an external reality, or pretend that we can know 'truths' because "it works, bitches".

I think there is a third way. We know that definite, complete, unambiguous knowledge about anything is impossible to convey (and consequently pointless to have or pursue). So it seems... inefficient... to use a system predicated upon this possibility (axiomatic mathematics).

I'm just wondering what your idea of "meaning" is, if it's not a consequence of relationships between concepts.

As noted, we cannot unambiguously define any single concept.

What we can do is create networks of relationships between "stuff". We have no idea what "stuff" is. The only thing that language lets us communicate is relationships.

In practice, any "concept" is, itself, specified by a set of relationships to other concepts, which are specified by relationships to other concepts. I previously suggested (in my dictionary illustration) that this leads to tautology - which is true... but since this is inevitable and unavoidable this cannot be the criticism that "tautology" is frequently perceived to be.

So - what we have is a bunch of 'placeholders' each of which has relationships to other 'placeholders'. The conventional view is to perceive that meaning is inherent in the 'placeholders'. The truth is that the meaning resides within the shape of the relationships. We can distinguish 'placeholders' from each other because the local pattern of relationships is distinct from the pattern of relationships around other 'placeholders'.

This is the only way to construct meaning (whatever the precise meaning of "meaning" is...).

As such, in practice, all language (including mathematics and thought) works this way. As much as we think we might have defined the integer '1'... what we have actually done is specify a set of relationships around a place holder. Our concept of '1' derives entirely from that pattern of relationships and its similarities and differences to other placeholders.

One could argue that since language (and mathematics) has been doing this all along, there isn't any reason to change old habits.

On the other hand, I feel that if our intentions were more in line with our tools (language) then we would be able to use that tool (language) more efficiently.

Instead of trying to do the impossible and define 'placeholder', we could understand that the only thing we can specify unambiguously is a pattern of relationships. Instead of trying to define "integer", "set", "infinity" which cannot be done, we can define, read and compare patterns of relationships.

In reading about Fermat's Last Theorem, one of the cool aspects of the solution was that in the process, two branches of mathematics that were perceived to be quite distinct were shown to have similar patterns.

Schrollini wrote:I realize that this is largely irrelevant to the post, but this is something that irritates me to no end:

oops.

As a (former) physics student, no. Just no. Newtonian mechanics clearly emerges as the low-curvature limit of GR. And not by accident. This was a crucial step in Einstein's derivation of GR! He had worked out what the relationship between energy and curvature should be, but needed to set the conversion factor between the two. He did this by showing that GR reduced to Newtonian gravity in the low-field limit, thereby deriving his unknown constant from Newton's constant.

1/x reduces to zero in the limit. This does not mean that 1/x = 0.

In my experience, concepts like the twin paradox are frequently mis-understood because assumptions from Euclidean/Newtonian systems are carried over that are not appropriate to Relativity.

Yes, GR looks radically different from Newtonian mechanics, but the two viewpoints are totally compatible.

They must be compatible to the extent that they are both fairly good approximations to the same observations...

Wut? GR certainly preserves causality locally, and it preserves it globally as long as there are no closed time-like curves. (No self-respecting cosmology should allow such a thing .) All observers will agree on the pattern of causation; that's one of the few things preserved by GR!

So far, we haven't actually proven that our personal universe has any self-respect. My initial comment was a suggestion that we should not pre-judge the qualities of the universe in case we blind ourselves to the actual perversities of reality.

However, while I did want to allow other possibilities, I do agree that the pattern of causation probably has a sufficient degree of objectivity.

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### Re: Axiomatic mathematics has no foundation

So Treatid, I'm not exactly sure what you're driving at here. I agree with a lot of what you're saying. For example, the way I few foundational (ZFC) mathematics is like this. First you need the linguistics department to figure out what a language is. You have words in your languages, and grammar and syntax which tell you ways you can connect those words etc. The linguistics people are the ones who define and debate what we mean when we say things like "meaning" and semantics. I don't claim to be an expert on that stuff but I know it is their field. Now, mathematics is a certain type of formal language which means it has words or objects and ways to connect them. The fundamental object in mathematics is a set. It is well recognized that in ZFC the SET is undefined. BUT once we take the existence of sets and the empty set and the rules of the language we can state the axioms of mathematics in that language and then you get all of math. Now I think you will agree with everything I have said so far. But here is where you differ: you are not satisfied with this. You think mathematics should somehow be unambiguously and self-consistently defined. There shouldn't be undefined things like "sets". But I think you also recognize it may be impossible to have such an unambiguous language due to the limitations of communication itself?
Either way, I think the key issue here is that you are misunderstanding the goal of math. You talk about this realm of concepts and it is very reminiscent of Plato's "forms". Essentially, you think there are things which are absolutely true and absolutely false. I think this is a very misguided way to think about things. There are many philosophical arguments which imply that humans are not capable of understanding absolute truths about the universe. The best we can do is come up with logical constructions which "work the best". EVEN IN MATH. I think it is very important for math to recognize that it does not deliver absolute truth (contrary to what many people may believe about it).

If you drop the notion that math should communicate absolute truth and adopt the notion that math should generate useful* constructions I think you will find the axiomatic approach to mathematics satisfying and beautiful. One idea which amuses me but I don't want to push too heavily is that the axioms we choose for mathematics reflect something about our psychology and how we organize/perceive our experience in this universe.

*useful defined socially or by some other possibly more rigorous but not absolute definition.

Treatid
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### Re: Axiomatic mathematics has no foundation

Twistar wrote:So Treatid, I'm not exactly sure what you're driving at here. I agree with a lot of what you're saying. For example, the way I few foundational (ZFC) mathematics is like this. First you need the linguistics department to figure out what a language is. You have words in your languages, and grammar and syntax which tell you ways you can connect those words etc. The linguistics people are the ones who define and debate what we mean when we say things like "meaning" and semantics. I don't claim to be an expert on that stuff but I know it is their field.

I don't think mathematics can afford to rely on linguistics to provide a firm groundwork for mathematics. Or perhaps I think that Mathematics is the formal end of linguistics. As such, I don't think mathematics should be leaning on other disciplines to provide a framework... because, as a rule, those other disciplines are even more poorly defined than mathematics.

Now, mathematics is a certain type of formal language which means it has words or objects and ways to connect them. The fundamental object in mathematics is a set. It is well recognized that in ZFC the SET is undefined. BUT once we take the existence of sets and the empty set and the rules of the language we can state the axioms of mathematics in that language and then you get all of math.

"it is impossible to do X; BUT if you could do X, then...."

It is absolutely impossible to specify the meaning of a set of axioms. Which means that anything based upon a set of axioms also does not have a specific meaning.

I have presented this as a criticism of mathematics - which by itself is unfair. Mathematics cannot do what is impossible. However, I do feel that much of the time mathematics ignores that impossibility and pretends that axiomatic systems do have (or can have) specific meaning. e.g. the hierarchy of infinity is supposedly defined with respect to ZFC... despite such definition being impossible.

Now I think you will agree with everything I have said so far. But here is where you differ: you are not satisfied with this. You think mathematics should somehow be unambiguously and self-consistently defined.

I do think that mathematics should be consistent. I don't think it should be unambiguous.

Mathematics cannot be unambiguous. But mathematics pretends that it can (be unambiguous). It is this hypocrisy that I think is mistaken.

Again - mathematics does recognise that there is an issue here and goes with what works rather than a platonic ideal that it knows it cannot achieve. Yet at the same time it hasn't abandoned that ideal. Axiomatic mathematics is still structured around the idea that it is possible to define something unambiguously.

So - mathematics is practical, recognises that there is a hard limit and does what works rather than try to force an ideal that cannot happen. This practicality over idealogy seems laudable... except that I feel that if the ideology is shown to be wrong (impossible) it should be abandoned altogether - not just worked around.

There shouldn't be undefined things like "sets". But I think you also recognize it may be impossible to have such an unambiguous language due to the limitations of communication itself?

I think that there should be no attempt to do the impossible. The limitations of language tell us we cannot define anything in the manner that axiomatic mathematics attempts. It isn't that there shouldn't be undefined objects... it is that we shouldn't be trying to define objects when we know that it is impossible to do so.

... obviously we do want to create understanding. we do want a system that enlightens and empowers us. I want us to recognise the limitations that we have and work within those limitations.

Either way, I think the key issue here is that you are misunderstanding the goal of math. You talk about this realm of concepts and it is very reminiscent of Plato's "forms". Essentially, you think there are things which are absolutely true and absolutely false.

Not at all - As nearly the opposite of that as it is possible to be. I think that it is impossible to define anything in an absolute sense.

When I talk about a "realm of concepts" I am specifically limiting that realm to what is possible. Absolutes of knowledge are not possible.

I think this is a very misguided way to think about things.

I agree.

There are many philosophical arguments which imply that humans are not capable of understanding absolute truths about the universe. The best we can do is come up with logical constructions which "work the best". EVEN IN MATH. I think it is very important for math to recognize that it does not deliver absolute truth (contrary to what many people may believe about it).

More strong agreement.

But more than this... our thinking should be in line with our tools and what is possible. It isn't sufficient for mathematics to recognise a limitation. Mathematics should embrace that limitiation... understand that limitation and work within it.

Mathematics does "what works" because it hasn't abandoned the platonic ideal despite knowing that the ideal is fundamentally impossible.

"what works", must already be a reflection of what is possible. But mathematics falls between two stools. It aims for an impossible target and then haphazardly does what it can in light of the impossibility of reaching that target. This leaves mathematics as a misguided mess whose slight saving grace is that "bits of it work".

Imagine if, instead, mathematics was fully consistent with the limitations of language. If mathematics used language in the most efficient way - aiming only to do what is possible. There would be no need for compromise. No ad-hoc "because it works".

If you drop the notion that math should communicate absolute truth and adopt the notion that math should generate useful* constructions I think you will find the axiomatic approach to mathematics satisfying and beautiful.

My (idealistic) notion of mathematics is that it should be a pure subject. Engineering can be satisfied with what works. Mathematics should (I feel) be consistent.

One idea which amuses me but I don't want to push too heavily is that the axioms we choose for mathematics reflect something about our psychology and how we organize/perceive our experience in this universe.

This idea has a great deal of value. It is the entirety of what justifies existing mathematics. I do think that existing mathematics is almost entirely a reflection of our biases.

Twistar
Posts: 445
Joined: Sat May 09, 2009 11:39 pm UTC

### Re: Axiomatic mathematics has no foundation

Treatid wrote:But more than this... our thinking should be in line with our tools and what is possible. It isn't sufficient for mathematics to recognise a limitation. Mathematics should embrace that limitiation... understand that limitation and work within it.

Mathematics does "what works" because it hasn't abandoned the platonic ideal despite knowing that the ideal is fundamentally impossible.

"what works", must already be a reflection of what is possible. But mathematics falls between two stools. It aims for an impossible target and then haphazardly does what it can in light of the impossibility of reaching that target. This leaves mathematics as a misguided mess whose slight saving grace is that "bits of it work".

Imagine if, instead, mathematics was fully consistent with the limitations of language. If mathematics used language in the most efficient way - aiming only to do what is possible. There would be no need for compromise. No ad-hoc "because it works".

If you drop the notion that math should communicate absolute truth and adopt the notion that math should generate useful* constructions I think you will find the axiomatic approach to mathematics satisfying and beautiful.

My (idealistic) notion of mathematics is that it should be a pure subject. Engineering can be satisfied with what works. Mathematics should (I feel) be consistent.

One idea which amuses me but I don't want to push too heavily is that the axioms we choose for mathematics reflect something about our psychology and how we organize/perceive our experience in this universe.

This idea has a great deal of value. It is the entirety of what justifies existing mathematics. I do think that existing mathematics is almost entirely a reflection of our biases.

Cool it looks like we mostly agree then. So here is I think the crux of the disagreement and I think it is mostly our subjective perceptions. You say mathematics aims for "an impossible target." I'm not exactly sure what you mean by this but I'll take a stab at it.

I think that you think:

1) mathematics purports to convey absolute meaning in some way.
2) Due to limitations of linguistics this is impossible
3) therefore math is doing something that it shouldn't be doing and
4) thus you propose mathematics should instead aim to be "fully consistent with the limitations of language. If mathematics used language in the most efficient way - aiming only to do what is possible."

Here is where I disagree. I don't think 1) is true. Mathematics doesn't purport to convey absolute meaning. Mathematics knows its limitations and works within those limitations. Maybe naive mathematicians think mathematics reflects some absolute truth about our reality but I have a feeling that idea has gone by the wayside among category/set theorists and philosophers of mathematics (I think generally the idea that humans can know truth has gone by the wayside..)
2) We agree that it is impossible to convey absolute meaning
3) I don't think math is doing that thing that you claim it shouldn't be doing i.e. point 1). However, I do agree that math shouldn't be doing 1)
4) I think that currently math does exactly what you're saying with one caveat. There are formal definitions for what it means for a theory (read: math) to be complete and for a theory to be consistent*. Godel showed that if the theory is sufficiently complex it is impossible for it to be both consistent and complete. This is a constraint placed on mathematics by the rules of linguistics. I think the usual conclusion here is that we assume our theory is consistent (until proven otherwise) and realize that it is incomplete and that there will be sentences which can't be proven to be true or false (with respect to the axioms). So your goal of having a consistent mathematics is maybe loftier than you think. Disclaimer: I am really pretty clueless about this Godel incompleteness theorem stuff and I have to defer to more of an expert.

I want to address some other things you're saying.
"the hierarchy of infinity is supposedly defined with respect to ZFC... despite such definition being impossible."
Why do you have an issue with this? You say defined with respect to ZFC. Is this not exactly what you're talking about with "relativistic mathematics?" I feel like everything in mathematics is defined relative to ZFC (or some other axiomatic system). Thus you can take those axioms and undefined things (sets) as the sort of floating point to which everything is compared. what's the issue here?

Also you say that you don't think mathematics should lean on other fields to provide a framework. I think this is a little bit of mathematical hubris speaking. Perhaps a hubris similar to that which leads mathematicians to think mathematics finds absolute truths (not meaning to offend you here..) I think you may have to realize that mathematics is on just as shaky ground as the English language is. As languages they have the same rules and limitations. The difference is that mathematics plays within those rules in such a way that it can very efficiently be used to discover certain types of things that people find interesting while english plays within those rules in such a way that it allows people to communicate efficiently in every day activity and certain kinds of conversations. It's the same way binary and computer code allows computers to communicate information in the shortest amount of time possible. different languages for different purposes.

I'm curious to hear your response

*just to be clear the definitions don't rely on some absolute meaning, they are instead defined within the construct of the linguistics.