Axiomatic mathematics has no foundation

For the discussion of math. Duh.

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Treatid
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Re: Axiomatic mathematics has no foundation

Postby Treatid » Thu Feb 13, 2014 5:38 am UTC

Twistar wrote:Cool it looks like we mostly agree then.

Yes - I think so. By analogy - there is no question over the general process of evolution; but still some room for maneuver over the details...

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..)

I agree that mathematicians agree that absolute truth is not possible. And that, to a degree, this is reflected in their behaviour.

Neither mathematics nor mathematicians can do what is impossible. Whatever it is that mathematicians do must be what is possible. In this regard, mathematics and reality have no choice but to be aligned to a large degree. However, I feel that this alignment is despite mathematicians rather than because of mathematicians.

This is specifically reflected in the structure of axiomatic mathematics. I propose that there are two elements to axiomatic mathematics:
i) The Turing Machine equivalent in which a set of symbols can be manipulated in a deterministic fashion.
ii) The perceived meaning of that manipulation for a given set of symbols.

Part i) is unambiguous, definite and completely meaningless by itself.
Part ii) is where axiomatic systems gain their significance and usefulness (meaning)... but lies entirely outside of mathematics, being utterly undefined by any formal mathematical system.

Any Universal Turing Machine can emulate/simulate any axiomatic system. ZFC is just a particular Universal Turing Machine (equivalent) and has no more or less significance than any other UTM. A UTM is just a sufficiently flexible language. It has no inherent meaning, no inherent significance.

When we attribute meaning to symbols we do so from outside mathematics. "set", "empty set", "contains", "cardinality", "choice" are not mathematically defined terms. Attaching these external meanings to mathematical symbols is arbitrary.

This is where I see mathematicians being inconsistent. The definite, but meaningless, manipulation of symbols is confused with the arbitrary assignment of "meaning" to give the impression that there is a definite manipulation of meaning.

The urge to hang on to an impossible ideal has lead mathematicians to seize upon "definite and unambiguous" manipulation of symbols as being a worthy goal in itself. This illusion is supported/justified by pretending that those symbols have inherent meaning - or that importing meaning from elsewhere can be done in a meaningful way.

3) I don't think math is doing that thing that you claim it shouldn't be doing i.e. point 1).

Obviously mathematics is not doing something that is impossible to do.

But... axiomatic mathematics by itself has no meaning. Literally.

So, meaning is imported into axiomatic mathematics. That meaning is then manipulated (via attached symbols) which supposedly results in something of some significance even though we don't, and can't, know what the original, imported, meaning actually was (beyond the happy accident of some shared experience).

Axiomatic mathematics requires that the symbols have an associated meaning in order to have any significance. This is the fundamental error of axiomatic mathematics. It is hanging on to the ideal that it is possible to specify the meaning for something.

It isn't that we can't know exactly what a set is... We can't know anything about a set at all (beyond its relationships to other objects that we (also) know nothing about).

Hmm... I'm sat here typing on a keyboard, looking at a monitor, with a glass of water by my side and telling you that it is impossible to know what any of those things are. You, no doubt, are reading this on a screen with a beverage in easy reach and wondering why I'm attempting to communicate if I think it is such a futile effort... It isn't that we can't communicate - it isn't that mathematics can't communicate... it is that mathematics is not understanding itself. An essential component of communication is to... communicate. Mathematics off-loads the definition of meaning and thereby abdicates responsibility for understanding how and why we perceive meaning in language.

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?

One issue is how floaty the floating point is... If the initial floating point is completely undefined - then any extrapolation from that point is also completely undefined. No progress has been made.

We could argue that the axioms of ZFC do have subjective meaning for humans and that this is sufficient to provide some degree of meaning to any extrapolation of the axioms. This argument has some merit... except that we can't state what that subjective meaning is. We can't measure or otherwise quantify how subjective that meaning is with existing tools. As much as we do perceive meaning with respect to words... we can't communicate that meaning (except by reference to common experience).

As such - no - this is not what I'm talking about with regard to relativistic mathematics. This is the break from 'conventional' thinking that is hard to grasp because we have lived for so long with the assumption that we can define a thing to some degree.

Mathematics divides the world into "things" and "relationships" ("states" and "functions"). It then attempts to describe various states and the mappings between them. This is (mostly) futile.

Language gives us no mechanism to describe something directly. Which means it gives us no mechanism to describe something. The only thing we are able to describe is the pattern of relationships surrounding a "thing".

This is how language works - and how thought works. We communicate patterns of relationships. We attach meaning to particular shapes of relationships. The nodes between relationships are completely beyond any possibility of us knowing directly.

Okay - so "a pattern of relationships" is a "thing"... we can communicate a pattern of relationships - we can communicate a "thing". And if a pattern of relationships gives us any degree of a foundation... then that is more than nothing and axiomatic mathematics does have something it can build on, albeit something that can never be completely fixed or absolutely understood. Hence axiomatic mathematics does work a bit.

Bear in mind that a relationship also cannot be defined except via a pattern of relationships. Thus, our basis for relativistic mathematics is "patterns of relationships". We simply do not have access to anything else with which to build. Language can only communicate relationships - and the constraints on language apply to any system of interactions of any type.

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.

(You are playing the argument - not the man (and being constructive in the process) - no offense seen)

I see mathematics as being a formalisation of language - a particularly rigorous and self-knowing approach to language. As such I agree that mathematics has the same scope and limitations as other languages (mathematics is probably a class of languages rather than a single language...). My problem with mathematics leaning on external systems is that there is a significant likelihood of hiding knotty problems in those external systems. Which is exactly what I think is happening. Inheriting meaning from systems external to mathematics means that meaning within mathematics isn't well understood. As good as mathematicians are at playing with symbols (very, very good, indeed), all that manipulation is only as significant as the meaning that is attached. And the meaning is left to other, much less rigorous and precise processes to determine.

More to the point - I don't think that there is any need to leave this essential component to other disciplines. I think that a full understanding of mathematics (language) is entirely within the remit of mathematics.

Descartes and solipsism put a damper on the idea of absolute knowledge being achievable... but obviously there is a third option between "absolute knowledge" and "nothing". By relying on other fields, I feel mathematics is abdicating its responsibility in this regard. If mathematics is a self-reflective language, then it should explore and understand precisely how communication of knowledge works and specify the capabilities and limitations of communication in as much detail as is possible. In this regard, at least, mathematics should be self-contained.

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Re: Axiomatic mathematics has no foundation

Postby Forest Goose » Thu Feb 13, 2014 11:20 am UTC

You seem to be jumbling up syntax, semantics, philosophy, and personal meaning.

When I write down a system of axioms:

1.) They are just formal strings that get manipulated according to formal rules; they aren't referring to anything.
2.) Given abstract objects that satisfy those axioms, those formal strings from 1.) will be "interpreted" by the objects in a way that truth is preserved.
3.) The state and nature of the abstract objects in 2.) is a matter of the philosophy of mathematics, not mathematics itself.
4.) When I write down a system of axioms, I may have an idea in of objects/relations in my head that I want to capture with those axioms; this idea may, or may not, end up getting captured well, but that too is not a mathematical matter.

For example:

I may think of how counting and adding work with real objects, thus I write down some axioms to codify it (let's say the axioms for a group). Those axioms are just strings, nothing "inheres" in them. There are various abstract systems, like the integers mod 2 and the reals, that will satisfy these axioms; theorems of group theory will hold in these systems. Things like the reals, rationals, etc. may, or may not, exist in some real sense, this is a debate of philosophy. Finally, I may decide that my axioms don't really match up with what I was thinking in my head, perhaps I was focusing on adding livestock and realize that there are no negative cows (whatever); this, also, has nothing to do with mathematics.

You seem to be upset that 4.) and 3.) aren't captured in 1.) and, as a result, seem to be disputing that 2.) makes sense. Personally, I think that you are using natural language terms in a loose fashion, but treating the conclusions as if they were formal well reasoned results. I'm not saying you don't have a point, but that you really need to be more precise, and clear, in what you are saying when you use terms like "meaning" with regards to this subject.
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Re: Axiomatic mathematics has no foundation

Postby quarkcosh1 » Thu Feb 13, 2014 9:28 pm UTC

The fact that axioms have no justification is a legitimate problem. The best solution to this problem I have seen is that the reason certain axioms are used and not others is because those axioms happen to lead to interesting results. Another way to look at the problem would be to look at how math developed in a historical context. The way math is taught and they way with developed historically have very different orders associated with them. For example some of the techniques used in calculus were created long before calculus was invented but while learning math in school you won't learn about any of them until you take a calculus class. There are techniques such as combining a bunch of rectangles to approximate area which I was able to figure out in middle school but were not actually taught to me until later despite their simplicity.

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Re: Axiomatic mathematics has no foundation

Postby Treatid » Fri Feb 14, 2014 1:52 am UTC

@Forest Goose: Let us suppose that I want to build a vehicle that creates a new land speed record... I build a fuselage that is milled to millionth of an inch accuracy. The engine is the most finely crafted piece of engineering ever created. The driver is trained from birth for just this one task. And the wheels are standard Robin Reliant road tyres.

If any part of the vehicle actually reaches 700 mph it will only be as shrapnel.

Mathematics is good at symbol manipulation. But symbol manipulation by itself is meaningless.

At some point, we need to associate meaning with those symbols for them to have any relevance.

You can argue that the assignment of meaning is external to mathematics, that mathematics is only concerned with the symbol manipulation aspect. Okay. Fine. Then mathematics is the engine in the above example. An engine that isn't used to do work is useless. An engine needs to be connected to something in order to be productive.

The whole is only as strong as the weakest component. Mathematics as a useful tool includes the perception of meaning. Whether mathematics regards itself as "only the engine" or "the whole car", it is only the whole car that is practically useful. And in this instance, meaning is not only not defined within mathematics... it isn't consciously defined within any human system. To the extent that the wheels exist at all, they are something that has appeared out of nowhere and whose properties we haven't the first clue about beyond looking vaguely roundish.

Mathematics and/or mathematicians have every right to draw boundaries wherever they like. It is perfectly reasonable to specify that inherently vague entities such as "meaning" are external to mathematics... but the consequence is that mathematics by itself is meaningless. If you ever try to visualise a symbol (e.g perceive an integer as an object) then you are not doing mathematics (as specified by this set of boundaries).

It is not possible to avoid or bypass the assignment of meaning. Mathematics can draw the boundary such that the assignment is external to itself, but anyone using mathematics has to assign meaning to symbols at some point in order for those symbols to have any relevance.

In either case, the understanding of mathematics is dependent on the meaning assigned to symbols. So long as meaning is unspecified, or just a tautology (see a dictionary), then the understanding of mathematics is also unspecified.

None of this is new. With all the bluster that I came into this thread with... I'm years late to the party (So far Hilary Putnam seems to me to address this most directly).

Most mathematicians ignore the problem. Without an alternative, a vehicle that moves at all appears better than no vehicle. This doesn't mean that the problem doesn't exist...

However, I think that there is an alternative. I think that the emergence of "meaning" can be specified in a formal fashion. The existing constraints are absolute. It isn't possible to create a fixed point. It isn't possible to define an object without ambiguity. But if one accepts the limitations and works within them, it is possible to present a formal structure that allows meaning to be understood as much as it is possible to do so.

I think that such a formal approach belongs within mathematics.

As for "using natural language in a loose fashion"... What do you think the alternative is? Do you think that you can precisely define the meaning of a word? How "formal and well reasoned" can your results be when you cannot specify the meaning of any word or symbol that you use?

You want me to be "more precise, and clear, in [...] using terms like "meaning"... this misses the whole point... There is no existing method to define "meaning" precisely. There is no fixed point from which unambiguous meaning can be constructed. I can only define my meaning for a word by reference to other words... whose meaning, in turn, can only be defined by reference to other words.

For natural languages and basic communication this is not a problem. There is a structure underlying the meaning of words, and our ignorance of the details doesn't prevent communication.

For formal language (with mathematics as the prime candidate), the details of that underlying structure are important. Meaning exists within the pattern of relationships (not in relationships, not in things... specifically, the pattern of relationships).

@quarkcosh1: (agreeing...) Mathematics has become like engineering. An engineer doesn't care why steel is stronger than wood. He tests it - finds it to be true and then uses the materials appropriately in his constructions. It matters that the bridge remains standing - not whether the materials and equations are fully understood in every detail. So - mathematics has resorted to doing stuff that works without worrying about why it works. Interpreting maths in a certain way tends to work... so the fact that the rationale for that interpretation isn't known is ignored.

As noted, I think that mathematics is missing a trick. There are reasons for why we, as humans, perceive meaning. Those reasons can be formally expressed. Descartes, Solipsism and Mathematics gave up too soon. The loss of the idea of "absolute truth" seems to have paralyzed thought. Our Galaxy, our solar system, our planet, us - we have evolved despite the lack of "absolute truths". Clearly there are functional mechanisms that don't rely on "absolute truth". We should be attempting to understand those mechanism rather than clinging to Unicorns, Santa Clause and the tooth fairy.

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Re: Axiomatic mathematics has no foundation

Postby Cleverbeans » Fri Feb 14, 2014 3:49 am UTC

Treatid wrote:At some point, we need to associate meaning with those symbols for them to have any relevance.


But this is outside the scope of mathematics. Many sciences choose to use mathematical descriptions of real world phenomenon, but there are entire branches of mathematics with no known application. If you study the history of math you'll find that it often takes hundreds of years before someone finds an application for a theorem. Mathematicians build intellectual tools, but it's up to an applied scientist to determine which tools if any make a good model for their particular application. This isn't to say that all math is done in a vacuum as many problems are motivated by the real world, it's just not required.

You seem to be unconvinced that math is a worth subject in it's own right without application, and frankly that's a popular opinion among laymen. Most people don't see any point to math past elementary school. Personally I consider math to be like music - it requires no other justification than the pleasure it brings to those that practice it. Most of the mathematicians I know will talk more about the beauty of the subject than the power of it, and that's how I see it too. I would highly recommend "A Mathematicians Apology" by G.H. Hardy to give you some insight into why pure mathematics is studied.

Also I would point that there are countless scientists who are using rigorous math for applications every day, they just aren't always mathematicians. You seem to be laboring under the assumption that somehow mathematicians should be doing all the science by themselves, but they can't and we defer to those who specialize in applications to find ways of assigning meaning to the symbols. In the meantime, someone has to sit back and shunt all the symbols around in the first place and I'm content to do just that.
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Re: Axiomatic mathematics has no foundation

Postby Forest Goose » Fri Feb 14, 2014 5:02 am UTC

I'll keep this short:

@treatid:

My problem isn't that you haven't defined "meaning" with perfect precision, it's that you're not using the word in the same sense across the board; sometimes you seem like you're talking about Platonism -vs- Formalism, sometimes you're talking about personal meaning, sometimes about an empirical correspondence, sometimes about logical meaning, and etc. This, pretty much, invalidates most of what you have to say right from the start; that's not to say that you might not have some sort of point, but if you do, you aren't actually making it.

But, to be perfectly honest, this reminds me a lot of discussing science (particularly quantum stuff) with people who read about science, but don't actually do science. Such discussions tend to treat the actual field as if they were a minor detail, and the philosophical interpretation of them as if they were the real subject - all of this is done by shuffling around associated words without any concept of what they refer to. You are doing the same thing with mathematics.

Answer the following questions:

1.) Do any of your arguments overturn any actual theorems, proof methods, etc.?
2.) Do they suggest any novel results, methods, etc.?
3.) Would any working mathematician, in any field of mathematics, need to do anything differently if you were correct?

If the answer is "no", then you are, at best, doing philosophy; and, unfortunately more likely, pontificating upon the nature of a subject matter that you, yourself , do not actually understand - I'm sorry if that sounds harsh, but you do not appear to have a solid grasp of either advanced mathematics or the foundations of mathematics (as in actual mathematical theory).

quarkcosh1 wrote:The fact that axioms have no justification is a legitimate problem.


Axioms do not need "justification", I'm not even sure that makes sense (mathematically speaking). We do have justifications for why we study what we study, but that's not mathematics, that's human interest - the theories we don't find interesting are none the worse off, theoretically, for it. You're kind of, personally, putting things in reverse: there's mathematics, we find some of it interesting; it is definitely not that there is interest and from it springs mathematics (which is, actually, what this whole debate seems to be about, to me).
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Re: Axiomatic mathematics has no foundation

Postby elasto » Fri Feb 14, 2014 5:14 am UTC

Cleverbeans wrote:You seem to be unconvinced that math is a worth subject in it's own right without application, and frankly that's a popular opinion among laymen. Most people don't see any point to math past elementary school. Personally I consider math to be like music - it requires no other justification than the pleasure it brings to those that practice it. Most of the mathematicians I know will talk more about the beauty of the subject than the power of it, and that's how I see it too.


On that note

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Re: Axiomatic mathematics has no foundation

Postby Treatid » Fri Feb 14, 2014 7:32 am UTC

Some 30 posts ago in this thread korona pointed out that my point was already known, and sizik linked to the Foundational crises in mathematics that covers aspects of this topic.

As much as some late comers to the thread might feel otherwise... this isn't "me versus the universe". There are significant foundational issues for mathematics. Since there are currently no direct solutions to these issues, most mathematicians just get on with mathing on the grounds that "it works".

I do appreciate being forced to clarify my ideas... to be encouraged to express my ideas clearly. I generally regard push-back as a positive thing. But at the same time I think it would be useful to move beyond re-iterating the same point that has already been covered within the thread.

Forest Goose wrote:My problem isn't that you haven't defined "meaning" with perfect precision, it's that you're not using the word in the same sense across the board; sometimes you seem like you're talking about Platonism -vs- Formalism, sometimes you're talking about personal meaning, sometimes about an empirical correspondence, sometimes about logical meaning, and etc. This, pretty much, invalidates most of what you have to say right from the start; that's not to say that you might not have some sort of point, but if you do, you aren't actually making it.

What if I told you that I am using the word consistently? That the perceived inconsistency is a product of your perception, not my expression?

I could regale you with the parable of the four blind men and the elephant... all describing aspects of the same object but with quite different perceptions/descriptions...

However, the meaning of "meaning" is a red-herring. What matters is that axiomatic mathematics cannot create meaning. Existing meaning can be extrapolated, but there is no mechanism to create new meaning within axiomatic mathematics. Furthermore, there is no way to create meaning within any language (where language is a set of symbol manipulations).

There is no requirement to understand what "meaning" actually is... so long as the tools available do not allow new meaning to be created then it is clear that we cannot have a complete understanding of any system that contains "meaning".

Where any meaning is perceived in mathematics - that meaning ultimately derives from outside mathematics. Since a component of mathematics is external to mathematics... mathematics cannot fully understand itself. A bit like what Gödel's second incompleteness theorem says...

But, to be perfectly honest, this reminds me a lot of discussing science (particularly quantum stuff) with people who read about science, but don't actually do science. Such discussions tend to treat the actual field as if they were a minor detail, and the philosophical interpretation of them as if they were the real subject - all of this is done by shuffling around associated words without any concept of what they refer to. You are doing the same thing with mathematics.

Ah yes - you don't understand what you are reading.... so the only possible conclusion is that I don't understand what I'm writing.

Answer the following questions:

1.) Do any of your arguments overturn any actual theorems, proof methods, etc.?

Yes.

To the extent that mathematics has no solid foundation, none of the extrapolations of mathematics have any foundation.

The pure symbol manipulation of mathematics holds fast. But by itself, that symbol manipulation has no meaning in the most literal sense.

2.) Do they suggest any novel results, methods, etc.?

Yes.

Specifically, a formal system that works within the known constraints of knowledge and provides a mechanism for specifying "meaning".

3.) Would any working mathematician, in any field of mathematics, need to do anything differently if you were correct?

Yes.

Axiomatic mathematics is predicated upon the idea that it is possible to specify an object to some degree. This illusion should be abandoned. The mechanical elements of axiomatic mathematics will still exist, of course. But the understanding of those systems will differ from current axiomatic mathematics' perception.

(I know you are going to choke on these three answers... but seriously... look up the Foundational crises in mathematics. Understand the implications of Godel's incompleteness theories. Mathematics cannot be well founded in the conventional sense.

In practice, no one is going to be abandoning axiomatic mathematics any time soon. Too much has been invested in it, and it is too useful for a minor detail like a lack of foundations to change the behaviour of most mathematicians.)

@Cleverbeans: I am afraid you have grasped the wrong end of the stick. Or at least perceived a boundary in entirely the wrong place.

I am well aware of the beauty and wonder of pure mathematics. It is, for example, to the credit of mathematics that it provides the tools to specify it's own limitations.

The set of all Turing Machines with all possible inputs is equivalent to the set of all possible axiomatic systems (with some redundancy). If mathematics dealt only with this mechanistic element of mathematics than it could claim to be a pure, self contained subject. But in practice we do choose particular axiom sets to work with. As much as it is tempting to minimize the impact of that choice it is like being "a little bit pregnant".

As soon as we make any kind of choice we are bringing elements into mathematics from some external system. Elements that cannot currently be specified or quantified by mathematics.

In practice, systems like Euclidean Geometry and ZFC are far, far removed from being simple random choices. The choice of axioms cannot (should not) be regarded as an irrelevant detail.
Last edited by Treatid on Fri Feb 14, 2014 7:56 am UTC, edited 3 times in total.

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Re: Axiomatic mathematics has no foundation

Postby Schrollini » Fri Feb 14, 2014 7:53 am UTC

Treatid wrote:(I know you are going to choke on these three answers... but seriously... look up the Foundational crises in mathematics.

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

?
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Re: Axiomatic mathematics has no foundation

Postby Forest Goose » Fri Feb 14, 2014 8:02 am UTC

What if I told you that I am using the word consistently? That the perceived inconsistency is a product of your perception, not my expression?


I would say that that's a bunch of bullshit and that you're trying to pass the burden of making sense off to your audience; perhaps you should take up debating literary theory instead?

I could regale you with the parable of the four blind men and the elephant... all describing aspects of the same object but with quite different perceptions/descriptions...


And I could regale you with the fact that I actually study the foundations of mathematics and mathematical philosophy...but I'm just some dipshit, obviously you know better.

However, the meaning of "meaning" is a red-herring.

There is no requirement to understand what "meaning" actually is


Surely you won't, then, continue to use that word and draw a bunch of conclusions:

What matters is that axiomatic mathematics cannot create meaning. Existing meaning can be extrapolated, but there is no mechanism to create new meaning within axiomatic mathematics. Furthermore, there is no way to create meaning within any language (where language is a set of symbol manipulations).

so long as the tools available do not allow new meaning to be created then it is clear that we cannot have a complete understanding of any system that contains "meaning".

Where any meaning is perceived in mathematics - that meaning ultimately derives from outside mathematics. Since a component of mathematics is external to mathematics... mathematics cannot fully understand itself.


...and fucking on.

This is just a re-statement of Gödel's second incompleteness theorem.


Which means nothing? Or is this just another example - as seen in math forums everywhere - of "I don't what the hell I mean, but Godel!"?

Ah yes - you don't understand what you are reading.... so the only possible conclusion is that I don't understand what I'm writing.


Yes. Moreover, you also don't understand the subject you are writing about...or, what? Was I supposed to conclude that I don't understand the subject I've actively researched for years?

Answer the following questions:

1.) Do any of your arguments overturn any actual theorems, proof methods, etc.?


Yes.

To the extent that mathematics has no solid foundation, none of the extrapolations of mathematics have any foundation.

The pure symbol manipulation of mathematics holds fast. But by itself, that symbol manipulation has no meaning in the most literal sense.


Rather than play Word Game Nonsense: The Forum Edition, list a theorem, provide a counter example/disproof on the basis of your ideas. Or did you mean, perhaps, that you have some philosophical objection to what everyone else has done? Because you see the truth; certainly not because you find the subject challenging and rather than actually buckling down and learning it, you want to dismiss everyone else's accomplishments.

2.) Do they suggest any novel results, methods, etc.?

Yes.

Specifically, a formal system that works within the known constraints of knowledge and provides a mechanism for specifying "meaning".


Then why all this bullshitery? Put up or shut up: provide your meaningless - undefined term that that is - formal system that is different than the formal systems that you just disproved and has meaning, which it can't, and...your reply is so ridiculous that I can't even be coherently sarcastic; I'll just move on.

3.) Would any working mathematician, in any field of mathematics, need to do anything differently if you were correct?

Yes.

Axiomatic mathematics is predicated upon the idea that it is possible to specify an object to some degree. This illusion should be abandoned. The mechanical elements of axiomatic mathematics will still exist, of course. But the understanding of those systems will differ from current axiomatic mathematics' perception.


I see...so philosophy, then? Not even good philosophy.

Also, copious hubris.

(I know you are going to choke on these three answers... but seriously... look up the Foundational crises in mathematics. Understand the implications of Godel's incompleteness theories. Mathematics cannot be well founded in the conventional sense.


Oh my God! I've totally never read up on that at all - especially not this Godel thing you mention! Seriously, continuously referring to an 80 year old result that is discussed in intro level logic texts like it's magic does not make you look educated; the opposite, actually. "Godel" is just as big a red flag to me as "Quantum" is, especially when there is no actual mathematics being presented.

In practice, no one is going to be abandoning axiomatic mathematics any time soon. Too much has been invested in it, and it is too useful for a minor detail like a lack of foundations to change the behaviour of most mathematicians.)


I was going to write something scathing here, but this quote is so arrogant that it speaks well enough on its own.
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Re: Axiomatic mathematics has no foundation

Postby Treatid » Fri Feb 14, 2014 8:59 am UTC

Schrollini wrote:
Treatid wrote:(I know you are going to choke on these three answers... but seriously... look up the Foundational crises in mathematics.

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

?

Yes - I changed my mind halfway through that (earlier) post (and since). (I'm surprised and gratified that you picked this up... I didn't think anyone was paying that much attention)

My first look at the foundational crises looked like it was focusing on different issues than were/are concerning me. As I read further, I discovered that pretty much exactly the points I was looking at were covered, and that other points which I had taken as being less relevant were just different perspectives on the same theme.

I did throw in an edit to that post to indicate that my perception had moved - but I didn't update the entire post.

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Re: Axiomatic mathematics has no foundation

Postby quantropy » Fri Feb 14, 2014 10:32 am UTC

I feel that I have to mention What the Tortoise Said to Achilles

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Re: Axiomatic mathematics has no foundation

Postby Treatid » Sun Feb 23, 2014 8:59 pm UTC

The current situation:

Axiomatic mathematics has no foundation within mathematics. But... Mathematics does lean on external systems to provide the needed foundation. This isn't necessarily ideal, but given the apparent alternative of curling up into a solipsistic ball it is the only practical option.

The (known) problem:

Godel's second incompleteness theorem tells us that no system can prove itself consistent. Okay... but we can reference external systems that are able to prove a given system consistent. Except... those external systems cannot be proven consistent except by reference to yet further systems.

In other words, there is no conceivable way to define an absolute - a fixed point upon which to build definitions and descriptions.

If language only allows us to describe and define objects by reference to other words; then sooner or later we end up with a tautology, or just running out of words.

Hammering home the point:

Without a fixed starting point there is no way to describe an object. Notice that the problem isn't that we can't describe something precisely: It is that there is nothing with which to start a description of any kind. It is completely impossible to describe, in any degree, an object.

Erm... but we do describe things. The description may not be "absolute" but we do have an existing set of concepts upon which we can build.

Naively, it appears that Godel's second incompleteness theorem is directly contradicted by reality.

Moving forward:

The limitation on language is a well explored hard limit. It isn't possible to create a fixed reference point. Without a fixed reference point it isn't possible to construct a description of an object in any way.

(The limitations on language are general: they apply to any system of communication, of any kind... as such, universes that happen to contain sentient observers don't get a pass. "God did it" doesn't pass muster as an explanation).

At the same time, we perceive that we can describe objects with enough fidelity to build axiomatic mathematics.

Mathematics wins over perception every time.

We cannot describe an object at all. To the degree that perception differs from this rule, we would like to understand how the perception arises and what difference perceiving things differently (correctly) might make.

How does language work?:

Typically we define a new word by reference to other words. The (pre-existing) words convey the meaning of the new word. Except that there is no beginning word or words from which other words derive their meaning. If each word represents an object, and it is true that an object cannot be defined, then language cannot work.

If we remove specific meaning from symbols then all language is left with is the relationships between those symbols.

The only thing that language is able to convey is relationships.

Set Theory reflects this truth despite having its roots in the absolutist assumptions of axiomatic mathematics. The specific meaning of a set (and elements) is secondary to the relationships that the set implies.

However, a "relationship" is just as much an "object" as any other "object". There is no possible way to know any detail of a relationship any more than we can know anything about any other "object".

Additionally, while set theory allows many systems to be described, it doesn't provide a mechanism for understanding those systems as anything more than a purely mechanical process (i.e. it is a Universal Turing Machine Equivalent: able to emulate (most) other systems but providing no inherent meaning for those systems). To the extent that interpretation of ZFC Set Theory depends on the understanding of the (ultimately) natural language axioms, such interpretations are as vague as our understanding of the root of language.

Relativistic Mathematics:


Thus we have "objects" and "relationships" as the core elements of communication because there simply isn't anything else. Not that these elements are distinct... we really can't know anything about any instance of "objects" or "relationships", including whether they are distinct.

The only thing we can construct with these elements are networks of relationships. Clearly, then, it is the patterns of relationships from which all knowledge is constructed; again, because there is nothing else.

The set of all unique digraphs represent all the unique statements that can be made... although this isn't, by itself, any more informative than the set of all bit strings. It does tell us that all the "things" we think we are describing are actually sets of relationships... anything and everything you can conceive of is simply some product of a set of relationships.

We already know that any given symbol (statement) is meaningless without context (axioms). We also know that we cannot construct a definite starting point. This seems to represent a bit of an impasse.

Cogito Ergo Sum:

The observer is an essential component to understanding meaning (obvious, isn't it?). We cannot propose an observer who exists in some absolute reference frame... we still can't define anything in an absolute sense.

The only (sort-of) fixed point observer we can reference is "I". Furthermore, the observer must exist within the system being observed.

Provided that the observer is within a Universal Turing Machine Equivalent of sufficient size, the observer may still observe "other systems" by emulating them locally.

Now that we have a specific observer within the system we can compare and contrast against other possible observers within the system. Two observers that are identical within the system are indistinguishable and there is no need to worry about communication between them.

The observations of non-identical observers within a system will be different. Any communication between such observers will be constrained by the different meaning that their positions within the system confers upon a given pattern of relationships.

Given that mathematics is the formal end of linguistics (language and communication), it is now possible to construct efficient mappings between different observers and methods for each observer to interpret a given set of observations.

The possibility of knowing a single fact as absolute truth is gone. Even within this system it is impossible for an observer to 'know' every possible detail of the system they are in... but this is the limit of knowledge (communication).

Conventional mathematics cannot do what is impossible. As such, whatever mathematics currently does is consistent with this relativistic view - it really has no choice in the matter. In this respect, all of this should be familiar even if the pieces haven't previously been put together in this way.
Last edited by Treatid on Wed Feb 26, 2014 3:47 am UTC, edited 2 times in total.

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Re: Axiomatic mathematics has no foundation

Postby MartianInvader » Mon Feb 24, 2014 4:54 am UTC

I just wanted to correct a misconception you seem to have - around 99% of mathematicians don't really care about the foundational crisis, other than as an interesting story. As one of my professors put it, "If ZFC is proved inconsistent tomorrow, the logicians will sort it out and replace it with something else, but it won't change my work at all. All the same theorems will be true."
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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Re: Axiomatic mathematics has no foundation

Postby Schrollini » Mon Feb 24, 2014 5:11 am UTC

MartianInvader's professor wrote:"If ZFC is proved inconsistent tomorrow, the logicians will sort it out and replace it with something else, but it won't change my work at all. All the same theorems will be true."

And false, too. Ex falso quodlibet, and all that jazz. :lol:

Yes, I get you point, and it's a good one. I just couldn't resist.
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Re: Axiomatic mathematics has no foundation

Postby Forest Goose » Mon Feb 24, 2014 8:07 am UTC

Axiomatic mathematics has no foundation within mathematics. But... Mathematics does lean on external systems to provide the needed foundation. This isn't necessarily ideal, but given the apparent alternative of curling up into a solipsistic ball it is the only practical option.


Your first sentence doesn't actually mean anything unless you're talking in some grand philosophical sense - you're a step away from pomo rambling about theory laden perception and blah blah blah.

Godel's second incompleteness theorem tells us that no system can prove itself consistent. Okay... but we can reference external systems that are able to prove a given system consistent. Except... those external systems cannot be proven consistent except by reference to yet further systems.

In other words, there is no conceivable way to define an absolute - a fixed point upon which to build definitions and descriptions.

If language only allows us to describe and define objects by reference to other words; then sooner or later we end up with a tautology, or just running out of words.


You've confused provable and true: I can't prove ZFC is consistent using ZFC, ZFC is still consistent (,or not). But, you're whole motivation is goofy, we can have multiple consistent frameworks that, nevertheless, are not mutually consistent if we so deign. The issue here is that you are trying to attach some metaphysical significance to foundational mathematics that just isn't there - you're doing a more pomo-ish form of what happened when people considered Euclidean Geometry as The Geometry, as in the "true" and "real" theory. In short, you're working on solving a problem that not only isn't cared about, but doesn't even exist - this is what happens when you try to philosophize about topics you, yourself, don't know (which was the whole point of my example with pop quantum physics books earlier).

Without a fixed starting point there is no way to describe an object. Notice that the problem isn't that we can't describe something precisely: It is that there is nothing with which to start a description of any kind. It is completely impossible to describe, in any degree, an object.

Erm... but we do describe things. The description may not be "absolute" but we do have an existing set of concepts upon which we can build.

Naively, it appears that Godel's second incompleteness theorem is directly contradicted by reality.


The first set of sentences are just words, they don't actually mean (ha!) anything - I'm not even sure if you're talking about mathematical objects at this point (if so, it's not well executed; your notion of "describe" wouldn't really make sense of a system of objects satisfying axioms...). If you knew more foundational mathematics I might think you were having a philosophical issue with set theoretic models making sense since model theory uses sets; but that does not seem to be the issue you are having, nor is it an issue.

I don't see how Godel's Theorem, naively or otherwise, appears contradicted by reality. Moreover, what would that even mean? It's a result of logic so what is it that is supposedly appearing to contradict it, the reality of mathematical logic? Do you mean that you find it counter intuitive?

The limitation on language is a well explored hard limit. It isn't possible to create a fixed reference point. Without a fixed reference point it isn't possible to construct a description of an object in any way.

(The limitations on language are general: they apply to any system of communication, of any kind... as such, universes that happen to contain sentient observers don't get a pass. "God did it" doesn't pass muster as an explanation).

At the same time, we perceive that we can describe objects with enough fidelity to build axiomatic mathematics.

Mathematics wins over perception every time.

We cannot describe an object at all. To the degree that perception differs from this rule, we would like to understand how the perception arises and what difference perceiving things differently (correctly) might make.


Line by line:

1.) Nonsense: this only holds if you reject talking about things relative to other things, which is the whole point of a language, words to define words, etc. Language describes things in the universe by corresponding to them - you are neglecting correspondence, then complaining that you can't, essentially, use language without using language.

2.) Trying to make your point sound important and deep (especially the "God did it doesn't pass muster" part).

3.) Are you talking about math being used in science? It doesn't sound like it from the rest, but if not, then that's just empty nonsense. Mathematics is not trying to capture perceptual objects; it can be used to capture intuition, but even then, it doesn't need to do this. In short, this means nothing or is wrong.

4.) Mathematics isn't competing with perception, I think you are deeply confused.

5.) Insane or inane, probably both.

Typically we define a new word by reference to other words. The (pre-existing) words convey the meaning of the new word. Except that there is no beginning word or words from which other words derive their meaning. If each word represents an object, and it is true that an object cannot be defined, then language cannot work.

If we remove specific meaning from symbols then all language is left with is the relationships between those symbols.

The only thing that language is able to convey is relationships.


You are acting as if this is a problem, it isn't. The whole point of language is that it refers, this is not a defect, it is what makes it powerful and useful in the first place.

Set Theory reflects this truth despite having its roots in the absolutist assumptions of axiomatic mathematics. The specific meaning of a set (and elements) is secondary to the relationships that the set implies.


Yes, absolutist axiomatic mathematics, sounds legit, I mean obviously there's just one model of set theory and just one set of axioms...oh wait, that there isn't is a major bulk of the methods and results of set theory. Huh, fancy that.

By the way, there are not one single universe of objects that are "sets" anymore than there is a single group to which all elements belong. There are models of set theory, there are various groups in group theory, etc. You seem to be talking about axioms and models in a way that no one actually uses them...except philosophers who aren't good at math.

Additionally, while set theory allows many systems to be described, it doesn't provide a mechanism for understanding those systems as anything more than a purely mechanical process (i.e. it is a Universal Turing Machine Equivalent: able to emulate (most) other systems but providing no inherent meaning for those systems). To the extent that interpretation of ZFC Set Theory depends on the understanding of the (ultimately) natural language axioms, such interpretations are as vague as our understanding of the root of language.


This is the closest to sounding sane that you have gotten (also it is at odds with how you are using terms in the rest of your speech). However, that this is a problem continges not upon the normal ontological issues, but on some weird objection you have to language and reference. In essence, you sound like you are saying something sane, but the premise it is built on is nonsense.

Thus we have "objects" and "relationships" as the core elements of communication because there simply isn't anything else. Not that these elements are distinct... we really can't know anything about any instance of "objects" or "relationships", including whether they are distinct.

The only thing we can construct with these elements are networks of relationships. Clearly, then, it is the patterns of relationships from which all knowledge is constructed; again, because there is nothing else.

The set of all unique digraphs represent all the unique statements that can be made... although this isn't, by itself, any more informative than the set of all bit strings. It does tell us that all the "things" we think we are describing are actually sets of relationships... anything and everything you can conceive of is simply some product of a set of relationships.

We already know that any given symbol (statement) is meaningless without context (axioms). We also know that we cannot construct a definite starting point. This seems to represent a bit of an impasse.


You're either saying something very trivial, then pretending it is profound; or you are just rambling. At best, weak philosophy; at worst, junk.

The observer is an essential component to understanding meaning (obvious, isn't it?). We cannot propose an observer who exists in some absolute reference frame... we still can't define anything in an absolute sense.

The only (sort-of) fixed point observer we can reference is "I". Furthermore, the observer must exist within the system being observed.

Provided that the observer is within a Universal Turing Machine Equivalent of sufficient size, the observer may still observe "other systems" by emulating them locally.

Now that we have a specific observer within the system we can compare and contrast against other possible observers within the system. Two observers that are identical within the system are indistinguishable and there is no need to worry about communication between them.

The observations of non-identical observers within a system will be different. Any communication between such observers will be constrained by the different meaning that their positions within the system confers upon a given pattern of relationships.

Given that mathematics is the formal end of linguistics (language and communication), it is now possible to construct efficient mappings between different observers and methods for each observer to interpret a given set of observations.

The possibility of knowing a single fact as absolute truth is gone. Even within this system it is impossible for an observer to 'know' every possible detail of the system they are in... but this is the limit of knowledge (communication).

Conventional mathematics cannot do what is impossible. As such, whatever mathematics currently does is consistent with this relativistic view - it really has no choice in the matter. In this respect, all of this should be familiar even if the pieces haven't previously been put together in this way.


Oh look, more pomo style crap - I really like the talk about observers, etc. Perhaps you want to cite something about relativity in your discussion of set theory? You should consider discussing all of this with people who like to talk about "Critical Theory" and "something pretentious studies" while sipping coffee and stringing together philosophical nothings.

As I'm sure you can tell, I think little of most of what you have to say. This isn't because I'm just being a jerk: you don't know mathematics, you aren't proposing mathematics, you aren't even doing something mathematics like - you are spewing empty rants, passed off as faux insightful and quasi-philosophical, and pretending to see some deep truth that others may not yet have. Personally, I find this kind of nonsense quite offensive; it's like an out of shape smoker saying that baseball players aren't athletes because of their personal take on Greek and etymology, and, then, proposing to define the word in a way that makes them feel validated.
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Re: Axiomatic mathematics has no foundation

Postby madaco » Tue Feb 25, 2014 2:38 am UTC

I thought Godel's theorem only showed that non-trivial consistent systems couldn't prove themselves consistent?

And allowed for trivial ones to do so.
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Re: Axiomatic mathematics has no foundation

Postby LaserGuy » Wed Feb 26, 2014 2:42 am UTC

Yes, sufficiently simple systems can be proven to be self-consistent.

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Re: Axiomatic mathematics has no foundation

Postby Treatid » Wed Feb 26, 2014 3:25 am UTC

MartianInvader wrote:I just wanted to correct a misconception you seem to have - around 99% of mathematicians don't really care about the foundational crisis, other than as an interesting story. As one of my professors put it, "If ZFC is proved inconsistent tomorrow, the logicians will sort it out and replace it with something else, but it won't change my work at all. All the same theorems will be true."

I find this attitude odd. I understand that:

i) Players only need to know that everybody is playing by the same set of rules. For the most part, the justification for those rules is secondary to playing the game (by a consistent set of rules).
ii) Most mathematicians are only dealing with a small subsection of mathematics at any one time. It simply isn't practical to be worrying about every other branch of mathematics, even foundational mathematics.
iii) It isn't (hasn't been) obvious that there is any choice: with no clear solution the only option is to do what works - no matter how flawed.

However, your post (and other comments I've seen) give me the impression that point iii) has been internalised to the extent that it is assumed to be the only possibility. "The system has always been flawed, therefore the system will always be flawed and there's no point in worrying your pretty little head about it."

This isn't personal - it is an impression that has built up over a number of posts from several people...

However, this isn't an issue of whether ZFC is consistent or not. The whole of axiomatic mathematics has no foundation*... and there is no simple substitution that will make the problem go away.

*Technically, mathematics is founded on our perception of the physics of this universe. So - there is a foundation - we just don't know exactly what that foundation is.

madaco wrote:I thought Godel's theorem only showed that non-trivial consistent systems couldn't prove themselves consistent?

And allowed for trivial ones to do so.

You are right - my mistake. A system may not be both consistent and complete. I was being consistently wrong in my references to consistency.

I was taking this: "For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent." overly broadly.

...

IF we accept that:

a) language can only communicate relationships.
b) an observer must be a part of the system they are observing.

Digression:

Given that an object has no inherent properties of its own, our understanding of an object is entirely a product of relationships. Given that relationships also have no inherent properties, the only information we actually have are patterns of relationships.

There is some low hanging fruit we can take from this... The greater the number of diverse relationships an object has, the more certain we are likely to be in our perception of the nature of that object. For common human experiences where we have a great deal of relationship information we have a large context of relationships which gives a lot of reference points from which to triangulate any specific perception (don't worry that each reference point is using the others for definition - yes there is a degree of tautology - but it is good tautology).

People can (sometimes) convey nuanced meaning regarding common subjects. This can lead to a sense that we really do have a precise (even absolute) understanding of an object.

However, move outside common human experience and the rich context is liable to be diminished.

Small scale physics (Quantum Mechanics) is a long way from normal experience. If 'length', 'dimension', 'charge', 'time' cannot be defined in absolute terms, then describing anything with reference to those perceived qualities may not be as productive as hoped.

To be fair - if ever there was an object that was clearly only the product of its relationships it is 'spin'.

On the other hand... a fundamental theory of physics that uses concepts that can only be emergent to describe fundamental behaviour may be a bit confused. 'Length', 'dimension', 'charge', and every other 'object' are (must be) emergent perceptions. Assuming dimensions, position, velocity and all the other baggage of macroscopic human experience when those terms cannot be inherently defined in any way (except as placeholders within a network of relationships) isn't going to be as enlightening as intended.

A truly fundamental theory of physics can only be a network of relationships that changes. The terms that we are familiar with must arise from our observation within this system... but they cannot be fundamental elements in their own right.

Observers:

An implicit observer has always existed in mathematics. If in no other regard, humans are the ones doing mathematics.

Godel makes it explicit that the language used to describe a set of axioms is part of the system being described (a system cannot be described without a language, so no system can be considered independently of language). It is clear (once stated) that an observer must also be within the same system as the language.

A naive view of a language is as a set of symbols (that describe relationships between objects). Axiomatic mathematics is good at manipulating these symbols in deterministic ways. Where axiomatic mathematics fails is in understanding those manipulations. As fine as the symbol manipulation is, it is genuinely meaningless by itself. The perception of meaning is entirely external to axiomatic mathematics.

Yet we, as humans, do perceive meaning in language.

We know that the individual symbols have no inherent meaning. There is no reason to suggest that a relationships is inherently meaningful. Even a pattern of relationships is not inherently meaningful (we can enumerate the unique digraphs - but each graph is individually as meaningless as an arbitrary bit-string).

Okay, so meaning must be emergent. (well duh - anything we perceive that isn't a simple relationship is an emergent entity). Meaning is also subjective - as much as humans are similar enough that they appear to have much 'meaning' in common, each observer has their own perception of meaning that must necessarily differ to some degree from all other observers (if observers are sufficiently identical to perceive identical meaning then there is nothing to distinguish the observers - they may as well be considered the same entity).

However, to specify meaning requires specifying each of the qualities that meaning is related to. 'Meaning' is only 'meaning' when related to 'sentience', 'knowledge', 'ignorance', 'life' and the myriad other aspects that we perceive. There is no way to define this set of inter-related concepts in an axiomatic way. But emergent behaviour is comprehensible and tractable even if we lack good tools at the moment (if anyone knows of work being done on emergent behaviour beyond Stephen Wolfram - please let me know).

As such, it isn't necessary for mathematics to rely on some black box (external to mathematics) to provide meaning. The definite foundation that axiomatic mathematics would like to exist is not possible. But the knowledge and language that are available are very much amenable to direct scrutiny and understanding.

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Re: Axiomatic mathematics has no foundation

Postby Forest Goose » Thu Feb 27, 2014 8:49 am UTC

I find this attitude odd. I understand that:

i) Players only need to know that everybody is playing by the same set of rules. For the most part, the justification for those rules is secondary to playing the game (by a consistent set of rules).
ii) Most mathematicians are only dealing with a small subsection of mathematics at any one time. It simply isn't practical to be worrying about every other branch of mathematics, even foundational mathematics.
iii) It isn't (hasn't been) obvious that there is any choice: with no clear solution the only option is to do what works - no matter how flawed.


, Or most people are of the opinion that issues can be cleared up and resolved; and that most issues are likely to involve massively large cardinals, and other such, that have no impact (either way) on a whole bunch of useful mathematics. There isn't some single monolithic foundation in the sense of a single axiomatic theory.

However, your post (and other comments I've seen) give me the impression that point iii) has been internalised to the extent that it is assumed to be the only possibility. "The system has always been flawed, therefore the system will always be flawed and there's no point in worrying your pretty little head about it."


...and you're here to save us from the flaws?

However, this isn't an issue of whether ZFC is consistent or not. The whole of axiomatic mathematics has no foundation*... and there is no simple substitution that will make the problem go away.


You have not, in any sense, shown this.

*Technically, mathematics is founded on our perception of the physics of this universe. So - there is a foundation - we just don't know exactly what that foundation is.


No, it isn't. Some mathematics is inspired by the desire to solve physics problems, that's not a foundation. I don't think you know what exactly foundations are/is - nor have you used this word in the same sense consistently. In short, you're just making noises that sound like points.

You are right - my mistake. A system may not be both consistent and complete. I was being consistently wrong in my references to consistency.


Presburger Arithmetic is consistent and complete.

...the rest of your post is mainly rambling. It looks like you're putting language at the bottom and surprised you can't derive the world from it; or something like that. You certainly aren't talking about mathematics, not even the philosophy of it. Are you into postmodernism?
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Re: Axiomatic mathematics has no foundation

Postby Magnanimous » Thu Feb 27, 2014 9:06 am UTC

Treatid wrote:Digression:

Given that an object has no inherent properties of its own, our understanding of an object is entirely a product of relationships. Given that relationships also have no inherent properties, the only information we actually have are patterns of relationships.
Why do patterns of relationships give us information when patterns of objects don't?

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Re: Axiomatic mathematics has no foundation

Postby DavCrav » Thu Feb 27, 2014 11:03 am UTC

Forest Goose wrote:Presburger Arithmetic is consistent and complete.


So is the theory of algebraically closed fields. Just thought I'd throw in some actual mathematics to try to stem the tide of pseudo-philosophical manure in this thread.

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Re: Axiomatic mathematics has no foundation

Postby philoctetes » Thu Feb 27, 2014 1:33 pm UTC

I'm actually enjoying this discussion (for which I must greatly credit Forest Goose), which is why I feel kind of guilty at the following attempt to kill it (also, I'm going to be a bit of a dick about it):

Treatid, let's give you a small case study. I'll present a small and simple proof taken from Wikipedia in the woefully-deficient "old math", and you can apply your - what is it exactly your thing is? system? - to it, and show us the massive gain in explanatory and overall improvement in mathematics that results.

I quote directly:

The pigeonhole principle states that if n items are put into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item.

Hand-shaking theorem:
If there are n people who can shake hands with one another (where n > 1), [...] there is always a pair of people who will shake hands with the same number of people.

Proof:
As the 'holes', or m, correspond to number of hands shaken, and each person can shake hands with anybody from 0 to n − 1 other people, this creates n − 1 possible holes. This is because either the '0' or the 'n − 1' hole must be empty (if one person shakes hands with everybody, it's not possible to have another person who shakes hands with nobody; likewise, if one person shakes hands with no one there cannot be a person who shakes hands with everybody). This leaves n people to be placed in at most n − 1 non-empty holes, guaranteeing duplication.

Please, go ahead and work your magic.

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Re: Axiomatic mathematics has no foundation

Postby Treatid » Thu Feb 27, 2014 4:47 pm UTC

Magnanimous wrote:Why do patterns of relationships give us information when patterns of objects don't?

What is the difference between a pattern of objects and a pattern of relationships?

A single object is just an object - no pattern. Many objects that are in no way related are just many objects - no pattern.

"Many objects that are related" are the same thing as a "set of relationships between objects".

If we have a number of objects sprinkled on a euclidean plane then it might appear that there is a pattern to the objects without relationships between those objects. However, those objects are related to each other via their relationship with the euclidean plane.

Perhaps you are suggesting that something like the Euclidean Plane is an object with specific shape (properties)?

One way to view a euclidean plane is as a set of points (objects) with a specific relationships between them such that the overall distribution of points has the characteristics associated with a typical euclidean plane. From this perspective, a euclidean plane is a set of infinitely many objects related in a specific pattern. Anything drawn on the euclidean plane is related to a subset of those points and related to other objects within the euclidean plane through those points.

This is one of the principles behind ZFC, of course. Using just sets (and the empty set, and the axiom of choice), almost all other axiomatic systems can be constructed. Which is to say that ZFC is (nearly) a Universal Turing Machine Equivalent. Any Universal Turing Machine Equivalent can emulate any other Turing Machine Equivalent (i.e. everything we know we can describe).

The feature that allows Universal Turing Machines to describe everything we can know is that everything we can know can be constructed from sets of relationships (between objects). If (e.g.) dimensionality were a true primitive, then a system which didn't include that primitive would not be able to describe systems that were based on that primitive. The only primitive needed to describe everything is "relationship". I can't think of a way to construct the concept of 'relationship'. As far as I can tell, 'relationship' just has to be accepted as a given... there's no way to construct the concept without reference to structures that already contain the concept.

Clearly, talking purely in terms of relationships isn't ideal at all scales. Describing biology only in terms of Quantum Mechanical equations isn't an attractive proposition. 'Dimensions', 'lines', 'numbers', 'functions', ... are useful shorthand for particular patterns of relationships. It is, however, useful to know that these perceived properties are emergent features.

For instance, when perceiving a number line (integer, real or imaginary plane) as a set of relationships between objects it becomes obvious why singularities such as 'division by zero' arise: We know that it isn't possible to define a fixed point upon which to build axiomatic mathematics. Yet number systems pretend that 'zero' is a fixed point. The very definition of number systems start with an impossibility. It is then little surprise that such systems throw out the occasional anomaly.

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Re: Axiomatic mathematics has no foundation

Postby Schrollini » Thu Feb 27, 2014 5:38 pm UTC

Treatid wrote:The definite foundation that axiomatic mathematics would like to exist is not possible.

Methinks you are projecting slightly here.
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Re: Axiomatic mathematics has no foundation

Postby Twistar » Thu Feb 27, 2014 6:27 pm UTC

Treatid wrote:For instance, when perceiving a number line (integer, real or imaginary plane) as a set of relationships between objects it becomes obvious why singularities such as 'division by zero' arise: We know that it isn't possible to define a fixed point upon which to build axiomatic mathematics. Yet number systems pretend that 'zero' is a fixed point. The very definition of number systems start with an impossibility. It is then little surprise that such systems throw out the occasional anomaly.

This doesn't make sense.. without getting into a big debate about division by 0: the reason we don't define division by 0 has absolutely nothing to do with defining a "fixed point upon which to build axiomatic mathematics." Number systems don't pretend 'zero' is a fixed point. The natural number are defined in terms of the axioms of ZFC and thus 0 is defined. The only thing the axioms take for granted is the existence of the empty set and the notion that one set can be an element (or inside of) another set. This second notion may be what you mean by "relationships between objects". All you have to take for granted is one object: the empty set, and one type of relationship between objects: "is an element of" and you get all of math. Which of these two things do you draw issue with?
Also, maybe look up the difference between syntax and semantics as they regard to formal languages

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Re: Axiomatic mathematics has no foundation

Postby Treatid » Thu Feb 27, 2014 9:15 pm UTC

Schrollini wrote:
Treatid wrote:The definite foundation that axiomatic mathematics would like to exist is not possible.

Methinks you are projecting slightly here.

:) A little straw-manning makes the argument that much easier.

I agree that in theory, mathematics has accepted the lack of a solid foundation. Nevertheless, axiomatic mathematics is structured around the idea of it being possible to define axioms. The specific manipulation of symbols can be well defined without worrying about what the axioms mean in any larger sense. However, mathematics without meaning is... meaningless. The attribution of meaning to symbols is an essential component of mathematics (as used by people). The attribution of meaning is outside the cope of mathematics - and yet is vital to understanding mathematics.

Moreover, the repeated attempts to construct and justify absolute systems suggests that mathematicians haven't given up the attempt to find a definite foundation despite the evidence. Number systems may be considered a legacy of older approaches to mathematics, but the empty set is a fixed point in ZFC. And I'm reasonably sure that the "axiom of choice" is yet another appeal to absolutism. I think that the ability to choose an element from a set in all cases pre-supposes an absolute position/knowledge against which to make that choice.

Given that the axiom of choice has been controversial, it suggests to me that there isn't an obvious way to construct the axiom of choice from other primitives. Were it possible/easy to construct the axiom of choice from component parts, I assume it would be more compelling than simply asserting it.

As such, either:
i) the axiom of choice can be constructed from first principles and any Universal Turing Machine can fully implement ZFC.
ii) all Universal Turing Machines contain the axiom of choice as an implicit (or explicit) component.
iii) Universal Turing Machines cannot fully emulate ZFC because ZFC has an impossible axiom (or because there are classes of system that UTMs can't emulate).

Given that the axiom of choice in ZFC is only essential for infinite sets, actually testing whether a UTM can fully emulate ZFC is problematical (as I understand it).

As such, I don't think I'm being too unfair in my projection. Yes, elements of mathematics accept the fundamental limitations, but it is questionable whether mathematics has grasped that limitation and started working within it rather than despite it.

philoctetes wrote:Treatid, let's give you a small case study. I'll present a small and simple proof taken from Wikipedia in the woefully-deficient "old math", and you can apply your - what is it exactly your thing is? system? - to it, and show us the massive gain in explanatory and overall improvement in mathematics that results.

It is possible to take a set of symbols, specify a set of manipulations for those symbols, get a set of resulting symbols and call those resulting symbols true with respect to the axioms.

But the resulting symbols are just that: symbols. They don't have any inherent meaning. Calling them 'true with respect to the axioms' doesn't tell you anything.

Handshakes and pigeon-holes are not concepts that mathematics defines or understands. Even if you abstract them further; while any shred of meaning is attached to the symbols - that meaning is outwith mathematics.

The process of attaching meaning to symbols is not formally defined. In many cases people (and mathematicians) have sufficient agreement over the meaning of symbols and the appropriate manipulations that the output also appears meaningful.

For concepts that are familiar, attaching meaning to symbols can appear pretty straightforward. What happens out towards the edges? What happens when the system behaves significantly differently to the systems we are familiar with? How do we attach meaning to those symbols? or extract meaning from those symbols?

Euclidean Geometry worked fine for the best part of two thousand years without the fifth postulate. The group assumption was sufficiently consistent that it wasn't necessary to state the fifth postulate (or realised that it needed stating).

Existing mathematics works. Most of the time our interpretation of meaning is fairly consistent with the symbol manipulation. Mathematics is a useful tool.

But... while the assignment of meaning is not formally defined, neither can the interpretation of meaning be formally defined. And the further mathematics gets from common experience, we need more than our informal use of language to understand the significance of any results.

The proof you provided probably feels like it is a well defined, unambiguous proof. However, if you presented the proof to a trans-dimensional alien you would not convey anything. You could show the symbols and how they interact... but without some information about what each of the symbols is supposed to mean, nothing is conveyed.

The proof you provided means nothing from a pure symbol manipulating perspective.

That we do perceive that the proof has significance is due to factors that mathematics tells us nothing about. I think that we should try and understand how and why we perceive meaning and significance (particularly with respect to mathematics). I think that we have, or can create, the tools to do this in a formal fashion that is appropriate to mathematics. Since, thus far, mathematics has abdicated all responsibility for creating a formal basis for meaning; anything is better than nothing.

Twistar wrote:
Treatid wrote:For instance, when perceiving a number line (integer, real or imaginary plane) as a set of relationships between objects it becomes obvious why singularities such as 'division by zero' arise: We know that it isn't possible to define a fixed point upon which to build axiomatic mathematics. Yet number systems pretend that 'zero' is a fixed point. The very definition of number systems start with an impossibility. It is then little surprise that such systems throw out the occasional anomaly.

This doesn't make sense.. without getting into a big debate about division by 0: the reason we don't define division by 0 has absolutely nothing to do with defining a "fixed point upon which to build axiomatic mathematics." Number systems don't pretend 'zero' is a fixed point. The natural number are defined in terms of the axioms of ZFC and thus 0 is defined. The only thing the axioms take for granted is the existence of the empty set and the notion that one set can be an element (or inside of) another set. This second notion may be what you mean by "relationships between objects". All you have to take for granted is one object: the empty set, and one type of relationship between objects: "is an element of" and you get all of math. Which of these two things do you draw issue with?
Also, maybe look up the difference between syntax and semantics as they regard to formal languages

You are making my argument for me: "The only thing the axioms take for granted is the existence of the empty set and the notion that one set can be an element (or inside of) another set."

I'll throw in the C bit of ZFC... as another (possibly) fixed point.

The empty set has a property: that of being empty. Emptiness is an absolute - a fixed point.

Not having a fixed starting point is a bitch. Trying to construct something without having a foundation is hard. Substitute 'the empty set' for 'a set' and it becomes obvious very quickly how much of a fixed point the empty set actually is.

"is an element of" might have been okay... except that the definition of 'contains' is such that the 'everything' set doesn't contain itself (if I'm remembering the right set theory). That a set cannot contain itself (directly or indirectly) implies certainty about a property of relationships. In particular, it implies a definitive hierarchy of sets... something that can only be constructed if there is a fixed starting point (the empty set).

I've already suggested that the axiom of choice may also imply a fixed reference frame in order to function for infinite sets...

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Re: Axiomatic mathematics has no foundation

Postby LaserGuy » Thu Feb 27, 2014 10:37 pm UTC

Treatid wrote:But the resulting symbols are just that: symbols. They don't have any inherent meaning. Calling them 'true with respect to the axioms' doesn't tell you anything.


Sure it tells you something. It tells you whether the theorem is either true or false within the context of that model. Mathematics is really nothing but the manipulation of the symbols. Assigning meaning to those symbols is not mathematics. Quite the opposite, in fact; the benefit of a high level abstraction is that the symbols could mean different things in different contexts, but the underlying theorem still remains true. You can say that the symbols represent handshakes or pigeon-holes or tennis matches or murder-suicide pacts. The meaning is irrelevant. You can attach whatever meaning you want, and use the same symbols to represent it as long as the underlying mathematical properties remain the same.

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Re: Axiomatic mathematics has no foundation

Postby Schrollini » Fri Feb 28, 2014 6:23 am UTC

Treatid wrote:For instance, when perceiving a number line (integer, real or imaginary plane) as a set of relationships between objects it becomes obvious why singularities such as 'division by zero' arise: We know that it isn't possible to define a fixed point upon which to build axiomatic mathematics. Yet number systems pretend that 'zero' is a fixed point. The very definition of number systems start with an impossibility. It is then little surprise that such systems throw out the occasional anomaly.


I let this go by, since it seemed a minor point. But the more I think of it, the more it seem symptomatic of the problem here.

Division by zero is perfectly well defined...

...in the real projective line.

It's not defined in the field of the real numbers, but that's because the field axioms say that the additive identity of a field has no multiplicative inverse.

You're so busy railing against "absolutist mathematics" that treats zero as a "fixed point", that you haven't noticed that mathematicians have developed a number of different systems, and zero's properties can only be specified relative to a given system. Isn't that exactly what you wanted?

I can hear you getting ready to say
But all of those are based on ZFC. That's the absolutist basis to which I object!

Not all mathematics assume ZFC! Much only assumes ZF. Some assumes ZF and the negation of C. And there are other set theories out there, other axioms that may be added, and even non-set-theoretic approaches. No mathematical result is stated absolutely; all exist only relative to the system in which they were derived. And there are many many systems to choose from.

Go ahead a blather about "meaning", whatever that means, all you want. But your description of axiomatic mathematics as "absolutist" displays such a lack of understanding of mathematics that it's hard to take anything else you say seriously.
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Re: Axiomatic mathematics has no foundation

Postby Forest Goose » Fri Feb 28, 2014 7:15 am UTC

Which is to say that ZFC is (nearly) a Universal Turing Machine Equivalent. Any Universal Turing Machine Equivalent can emulate any other Turing Machine Equivalent (i.e. everything we know we can describe).


There are two possibilities:

1.) You have no clue what you are talking about
2.) You are not, in fact, speaking English, but some other language that is, shockingly, like English, but with all the words defined differently.

I mean, come on, that sentence makes no sense.

The feature that allows Universal Turing Machines to describe everything we can know is that everything we can know can be constructed from sets of relationships (between objects).


What does it mean to "describe" here? This is also confusing given your whining about meaning and bit strings; does it follow that everything we know has no meaning? What about noncomputable definable numbers, how does knowledge of their various properties fit into this scheme? Personally, I think you're just doing mathematical philosophy the way every postmodernist trying to look smart does it: like a parrot speaks English. But, please, prove me wrong.

If (e.g.) dimensionality were a true primitive, then a system which didn't include that primitive would not be able to describe systems that were based on that primitive.


If "dimensionality" were a primitive, wouldn't it be a primitive of some system? Or do you mean to say that "systems that don't have "dimensionality" as a primitive, don't have "dimensionality" as a primitive"? That's rather trivial, so what do you mean? Do you mean to say that there are things that are just primitive? Not primitives in some theory/system, but literally just baldly primitive? WTF?

The only primitive needed to describe everything is "relationship". I can't think of a way to construct the concept of 'relationship'. As far as I can tell, 'relationship' just has to be accepted as a given... there's no way to construct the concept without reference to structures that already contain the concept.


Oh, I see! You're, essentially, talking about an axiomatic theory of relationships. Of course, you speak moon-man with a sprinkle of math terms, so I can't tell if you are saying that this is the problem or if this is your solution...either way, since axioms can't be defined and aren't well founded (so you say), I guess this is meaningless (by your own ideas).

Clearly, talking purely in terms of relationships isn't ideal at all scales. Describing biology only in terms of Quantum Mechanical equations isn't an attractive proposition. 'Dimensions', 'lines', 'numbers', 'functions', ... are useful shorthand for particular patterns of relationships. It is, however, useful to know that these perceived properties are emergent features.


...Yes, this is clearly mathematics related. I'm an extreme variant of Platonist exactly because QM isn't useful for Biology...no, wait, that doesn't make any sense. Is there an analogy here? Did you have a couple browser tabs open and part of something else ended up here? Do you think bio is mathematics? ???

For instance, when perceiving a number line (integer, real or imaginary plane) as a set of relationships between objects it becomes obvious why singularities such as 'division by zero' arise: We know that it isn't possible to define a fixed point upon which to build axiomatic mathematics. Yet number systems pretend that 'zero' is a fixed point. The very definition of number systems start with an impossibility. It is then little surprise that such systems throw out the occasional anomaly.


Division by zero in a math discussion is like Nazis in Goodwin's Law - in other words, you automatically lose the debate.

I agree that in theory, mathematics has accepted the lack of a solid foundation.


No, it hasn't. At least, not in any of the ways you mean - supposing this means anything.

However, mathematics without meaning is... meaningless.


No, it's mathematics. I'm not saying that formalism is right (or that it isn't); but that mathematics does not need to refer to extramathematical content.

Moreover, the repeated attempts to construct and justify absolute systems suggests that mathematicians haven't given up the attempt to find a definite foundation despite the evidence.


Yes, I guess those pesky mathematicians have, yet, to stumble on your wonderful posts - but once they do, watch out! I, for one, will be keeping an eye out for articles with titles like "Entire Edifice of Mathematics Torn Down by XKCD Forum Posts" and "Treatid Undermines All Those Academics That Wouldn't Publish His Ideas"...sorry, I think I got telepathically crossed with your thoughts for a moment. Back to reality, you do realize that you have yet to demonstrate anything, I mean literally, like not a single nontrivial point.

but the empty set is a fixed point in ZFC.


Are you complaining about existential axioms? Why isn't the axiom of infinity a problem? Did you know that there was such an axiom?

And I'm reasonably sure that the "axiom of choice" is yet another appeal to absolutism. I think that the ability to choose an element from a set in all cases pre-supposes an absolute position/knowledge against which to make that choice.


Usually, as I've seen, choice is rejected because people are uncomfortable with the fact that we can't actually describe how we made the choice. In other words, AC seems to introduce sets without any clear explanation of where they come from (and no method to explain that existing). You're free to your opinion, I just find it odd.

Given that the axiom of choice has been controversial, it suggests to me that there isn't an obvious way to construct the axiom of choice from other primitives. Were it possible/easy to construct the axiom of choice from component parts, I assume it would be more compelling than simply asserting it.


AC was proved independent, half of that was done by Godel; I find it strange that you don't know this, yet know that all of mathematics is wrong. By the way, you should look up Shoenfield's Theorem: there's plenty of stuff, big stuff, so that if there is a proof of it in ZFC, then it can be proved in ZF.

As such, either:
i) the axiom of choice can be constructed from first principles and any Universal Turing Machine can fully implement ZFC.
ii) all Universal Turing Machines contain the axiom of choice as an implicit (or explicit) component.
iii) Universal Turing Machines cannot fully emulate ZFC because ZFC has an impossible axiom (or because there are classes of system that UTMs can't emulate).


Allow me to try and speak back to you in the same style as you do, perhaps that will help:

Ref orange yellow, Turing Turing Godel: therefore, you are wrong, ZFC is consistent because consistency consists (insert name dropping that I don't understand) of emergent emergencies. Since nothing means nothing means something, hence, because, recursive: you're wrong. Also, digraph.

Given that the axiom of choice in ZFC is only essential for infinite sets,


I was not aware that ZF~C only had finite models, live and learn.

It is possible to take a set of symbols, specify a set of manipulations for those symbols, get a set of resulting symbols and call those resulting symbols true with respect to the axioms.

But the resulting symbols are just that: symbols. They don't have any inherent meaning. Calling them 'true with respect to the axioms' doesn't tell you anything.


What you actually mean is: I'm calling them meaningless because I find it challenging, thus, rather than put in the man hours everyone else has, I'll just discredit the whole field. You talk about math like a creationist does about evolution; you don't like it, you don't understand it, but you want to talk in your own way about the same domain.

For concepts that are familiar, attaching meaning to symbols can appear pretty straightforward. What happens out towards the edges? What happens when the system behaves significantly differently to the systems we are familiar with? How do we attach meaning to those symbols? or extract meaning from those symbols?


If you actually knew mathematics, you'd be well aware that systems don't generally do exactly what we expect - tons of mathematics is extremely counterinutitive and intellectually slippery.

Euclidean Geometry worked fine for the best part of two thousand years without the fifth postulate. The group assumption was sufficiently consistent that it wasn't necessary to state the fifth postulate (or realised that it needed stating).


For thousands of years human sacrifice was acceptable, therefore I'm correct in asserting that we're, as in current people, all monsters. Right? That's the logic of this sentence, isn't it?

Existing mathematics works. Most of the time our interpretation of meaning is fairly consistent with the symbol manipulation. Mathematics is a useful tool.


If meaning can't be defined because it lacks a foundation, how exactly do you know this?

But... while the assignment of meaning is not formally defined, neither can the interpretation of meaning be formally defined. And the further mathematics gets from common experience, we need more than our informal use of language to understand the significance of any results.


So, there is not just meaning, but an interpretation of meaning now too? Perhaps we should just start talking about "abstractions of semantics of models of relations of senses of semiotics of symbolisms of metaphors of interpretations of meanings"? I mean why stop now, throw some more vague synonyms together so it's all really really confused instead of just really confused.

However, if you presented the proof to a trans-dimensional alien you would not convey anything. You could show the symbols and how they interact


Who the Hell cares? Rocks, clouds, turtles, and some fish wouldn't understand it either. You are confusing the content of language with language - just because my dog doesn't know English well enough to understand "I love you" does not mean that I don't love my dog, nor does it make that sentiment continge upon English. You are, as I mentioned, much more suited to talking with people who read pretentious French Philosophy from a few decades ago.

The proof you provided means nothing from a pure symbol manipulating perspective.


No, it means a proof of the theorem.

That we do perceive that the proof has significance is due to factors that mathematics tells us nothing about. I think that we should try and understand how and why we perceive meaning and significance (particularly with respect to mathematics).


You are talking about what here, the psychology of the philosophy of the foundations of mathematics?

I think that we have, or can create, the tools to do this in a formal fashion that is appropriate to mathematics.


So you'll use a meaningless formal theory to formally give meaning to formal theories, like the formal theor...yet again, I can't even be sarcastic due to the ridiculous circularity in your ideas.

Since, thus far, mathematics has abdicated all responsibility for creating a formal basis for meaning; anything is better than nothing.


Mathematics is quite fine, thank you. But, if you up your syllable count, get a little more political, and get a smidge more intellectually arrogant, then donning a black turtle neck will make you plenty of friends among confused undergrad philosophy students with an ax to grind against rigor. Seriously, go for it, this kind of pseudophilosophy exists just for that reason.

The empty set has a property: that of being empty. Emptiness is an absolute - a fixed point.


...huh?

Not having a fixed starting point is a bitch. Trying to construct something without having a foundation is hard. Substitute 'the empty set' for 'a set' and it becomes obvious very quickly how much of a fixed point the empty set actually is.


I tried your suggestion, it is no more obvious; maybe a little less.

"is an element of" might have been okay... except that the definition of 'contains' is such that the 'everything' set doesn't contain itself (if I'm remembering the right set theory). That a set cannot contain itself (directly or indirectly) implies certainty about a property of relationships. In particular, it implies a definitive hierarchy of sets... something that can only be constructed if there is a fixed starting point (the empty set).


Are you complaining about V?

By the way, you can have set theory without the axiom of foundation.

I've already suggested that the axiom of choice may also imply a fixed reference frame in order to function for infinite sets...


But that would violate the Lorentz Invariance of the Annihilation Operator of Woodin Cardinals! (Why not? I want to use completely unrelated terms from completely unrelated fields too!)
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Re: Axiomatic mathematics has no foundation

Postby snowyowl » Fri Feb 28, 2014 9:57 am UTC

Given that the axiom of choice has been controversial, it suggests to me that there isn't an obvious way to construct the axiom of choice from other primitives.


Um, yes, this is well-known and was proved correct by Kurt Gödel and Paul Cohen. It's the reason that the Axiom of Choice (AC) is even remotely interesting; if AC could be proved from other axioms, it wouldn't need to be an axiom itself. What did you think the importance of AC was, if you didn't know that?

You seem to have a fairly weak grasp of axiomatic set theory, for someone trying to overthrow ZFC. Yes, we're always happy to educate you about the fields we specialise in (life pro tip: people love talking about themselves). Yes, your ideas can and should be evaluated independently from your merits as a person and a mathematician. And yes, sometimes fatal flaws crop up in systems that have been thought bulletproof for centuries. But we've been here before, and we'll come here again:

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Re: Axiomatic mathematics has no foundation

Postby jestingrabbit » Fri Feb 28, 2014 10:04 am UTC

Schrollini wrote:It's not defined in the field of the real numbers, but that's because the field axioms say that the additive identity of a field has no multiplicative inverse.


Just to be clear here, a naive reading of the axiom of the existence of multiplicative inverses does not guarantee the existence of a multiplicative inverse of 0. Using the other axioms, you can prove that a0 = 0 for all a. This implies that there cannot be a multiplicative inverse, so long as the field has more than one element, which is typically assumed by stating that 0!=1.
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Re: Axiomatic mathematics has no foundation

Postby Twistar » Fri Feb 28, 2014 4:33 pm UTC

LaserGuy wrote:
Treatid wrote:But the resulting symbols are just that: symbols. They don't have any inherent meaning. Calling them 'true with respect to the axioms' doesn't tell you anything.


Sure it tells you something. It tells you whether the theorem is either true or false within the context of that model. Mathematics is really nothing but the manipulation of the symbols. Assigning meaning to those symbols is not mathematics. Quite the opposite, in fact; the benefit of a high level abstraction is that the symbols could mean different things in different contexts, but the underlying theorem still remains true. You can say that the symbols represent handshakes or pigeon-holes or tennis matches or murder-suicide pacts. The meaning is irrelevant. You can attach whatever meaning you want, and use the same symbols to represent it as long as the underlying mathematical properties remain the same.


Ok, Treatid. I had a hunch this is what you were talking about but wasn't sure.
I'm becoming more sure that you're worried about 2 things. I'll state them then address them.

1) The thing you're more worried about (I think) is that you realize it is possible to prove theorems in mathematics but you see that as pure string manipulation. What bothers you is that mathematicians give names to these theorems and then use them to describe real life situations, or physics or biology models and they did this without mathematics telling them how to they should apply their theorem to the real world situation. In other words, you think mathematics should give a prescription for how to apply mathematics.

2) You're weirded out that axiomatic mathematics has to assume things like the existence of the empty set and the 'is an element of' relation. You say things about 'fixed points' when I talk about empty sets. I think you understand the idea that once you DO assume these things you can rigorously and unambiguously derive the rest of mathematics and definitely at least understand this from a symbol manipulation point of view. This is all mathematics asks of you. It asks that you recognize that IF you assume this small amount of things you get this big amount of stuff and that is cool.

Now to address the points.
1) This is major: It is not the job of mathematics to prescribe meaning to it's own statement. It must only provide the framework in which the symbol manipulation can be done. It is the job of people who apply mathematics (this is the job of physicists, chemists, biologists, people in the checkout line, historians, archaeologists, stock brokers, carpenters, and even some mathematicians) to figure out which statements from mathematics model some situation that is of importance to them. The way I think about this is some real world situation. The "applier" of mathematics takes the relevant real world things and lets them be represented by mathematical objects of his or her choosing and then does mathematical manipulations and ends up with some new mathematical objects. The "applier" then has to interpret what these NEW mathematical objects mean in the real world and now they have made some real world prediction.
I teach an intro physics class and early on I put up this diagram:
Physics Problem
\/ convert problem state into math (i.e. set up the problem)
Math Problem
\/ do some math calculation (i.e. solve the problem by taking some integral)
Math Solution
\/ convert the math solution back into some physics interpretation (i.e. we found out the car's speed was increasing rather than decreasing)
Physics Solution

The first and last arrow represent doing physics. The middle arrow represents doing math. I think you are trying to say mathematics should prescribe how to do the first and last arrow and that is wrong. That is someone else's job.

I think you are going to get extra confused because I am using physics as an example and you have said earlier things like math is based on the physics that we see or something yet that is also wrong. Physics is built almost entirely on mathematics. One thing you may be interested to know is that people have been trying to "axiomatize" physics. This is one of Hilbert's problems. However, this would be different from axiomatizing mathematics. Mathematics has already been axiomatized. An axiomatization of physics would be a prescription for how to turn any physical situation into mathematical symbols which you can then manipulate using math. In other words, it would be a prescription of the first arrow above. However, we must be clear that these would be axioms of PHYSICS not MATH. Math should not purport how to apply or use math, it should just make sure it keeps proving theorems.

2) I think understanding the first point will make you slightly more comfortable with the second point. The first point is not a question of mathematics and the second one is so right now you are conflating two things, one which is math, and the other which isn't math, and it makes your points very confusing. Once you understand the first we can leave the non-mathematics out of the discussion and just talk about the math.

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Re: Axiomatic mathematics has no foundation

Postby Treatid » Sat Mar 01, 2014 12:23 am UTC

A couple of general points:

I'm not trying to show that ZFC, in particular, is flawed. I'm not even trying to show that axiomatic mathematics is flawed... The foundational crises already happened. The... problems... exist. Aspects of mathematics continue despite those problems, other aspects try to adapt to the problems (with varying degrees of success).

Twistar and LaserGuy have suggested that the boundary of mathematics can be drawn around pure symbol manipulation. There is no problem with symbol manipulation, so excluding the assignment of meaning from mathematics means there isn't a problem within mathematics.

This is a perfectly reasonable argument. It would seem to leave Euclidean Geometry, Number Theory, Group Theory, Category Theory and a few other categories of mathematics looking for a new home... But I'm perfectly happy with the idea that assigning and interpreting meaning isn't any part of mathematics.

It is specifically the assigning and interpreting of meaning that my comments relate to. I feel that, whether inside or outside of mathematics, this aspect is an unavoidable component of using mathematics. Given that a chain is only as strong as its weakest link - the use of mathematics is only as effective as the least understood component of that use.

What I'd like to do is to develop an explicit understanding of how meaning can arise given the known constraints on communication and knowledge. Hence I wanted to talk about Relativistic Mathematics with the observer as an explicit part of the system observed and each observer necessarily being a different reference frame.

Twistar wrote:1) The thing you're more worried about (I think) is that you realize it is possible to prove theorems in mathematics but you see that as pure string manipulation. What bothers you is that mathematicians give names to these theorems and then use them to describe real life situations, or physics or biology models and they did this without mathematics telling them how to they should apply their theorem to the real world situation. In other words, you think mathematics should give a prescription for how to apply mathematics.

Yes - at least, I think something more robust than 'intuition' should connect mathematics to the real world. I'm not insistent that the borders of mathematics be drawn in any particular place... I'm happy for this to be outside mathematics. However, I'm not after a vague philosophical discussion on what it means to be alive. I'm looking at a system that has the same rigour as mathematics to the extent that the limitation on communication permit that.

The last several posts give the impression that I'm trying to find fault with conventional mathematics. Such fault finding pre-dates me. I've been discussing aspects of the consequences of those faults because that is the bit I've been getting feedback on (and it is interesting in its own right).

2) You're weirded out that axiomatic mathematics has to assume things like the existence of the empty set and the 'is an element of' relation. You say things about 'fixed points' when I talk about empty sets. I think you understand the idea that once you DO assume these things you can rigorously and unambiguously derive the rest of mathematics and definitely at least understand this from a symbol manipulation point of view. This is all mathematics asks of you. It asks that you recognize that IF you assume this small amount of things you get this big amount of stuff and that is cool.

(I have to say - I really appreciate your post. I'm assuming that being a teacher is some aspect of it... Your reflection of my position gives me a good sense of how successfully I'm communicating and your attitude gives me the impression that you would like me learn (as opposed to wanting me to be wrong). Thank you.)

IF it were possible to assume something, then I would be happy to construct the rest of mathematics on that assumption.

I have no trouble with constructing the set of all bit-strings, then constructing all the possible ways of mapping those bit-strings to themselves. This set of constructions represents every possible statement and every possible system constructed on those statements. And has no significance by itself.

In this regard, it is trivial to construct all of mathematics (and everything else) from some primitives.

We require language to express even this mechanistic component of axioms. My problem is that you can't describe a set of axioms to me and be certain that I invest the same meaning into that description as you do.

Supposing I construct a system and my first axiom is 'an impossibility'. Without some formal definition of language, I can't know for any given statement whether that statement represents something that is possible. "An unstoppable force meets an immovable object" is probably not a good basis for an axiomatic system.

We know, beyond a shadow of a doubt, that it is not possible to construct a fixed point upon which to build. Asserting that you do, in fact, have a fixed starting point is simply starting with 'an impossibility'. No doubt it lets you construct all sorts of interesting systems...

1) This is major: It is not the job of mathematics to prescribe meaning to it's own statement.

As noted, I have no problem with this view. However, it is a view that doesn't fix anything. It allows mathematics to abdicate responsibility... but every use of mathematics requires an assignment of meaning.

You list some disciplines who's job you suggest is to assign meaning... and given that each discipline has its own requirements there is certainly an argument for each discipline using mathematics in a different way. But it seems redundant for every discipline to have to decide from scratch how to use the mechanistic components of mathematics in the real world.

The first and last arrow represent doing physics. The middle arrow represents doing math. I think you are trying to say mathematics should prescribe how to do the first and last arrow and that is wrong. That is someone else's job.

And if everyone assumes that it is someone else's job?

Quantum Mechanics assumes an absolute reference frame right from the start. But we know that an absolute reference frame is utterly, completely, and in all ways, impossible. Quantum Mechanics is a very successful theory, so having an impossible axiom isn't a complete disaster, but I doubt that it is completely benign.

I think you are going to get extra confused because I am using physics as an example and you have said earlier things like math is based on the physics that we see or something yet that is also wrong.

Difference between the actual physics of the universe and the subject of physics. We use and understand mathematics with respect to ourselves as having evolved within this universe. It is in this sense that mathematics must be grounded in the physics of the universe.

Physics is built almost entirely on mathematics. One thing you may be interested to know is that people have been trying to "axiomatize" physics. This is one of Hilbert's problems. However, this would be different from axiomatizing mathematics. Mathematics has already been axiomatized. An axiomatization of physics would be a prescription for how to turn any physical situation into mathematical symbols which you can then manipulate using math. In other words, it would be a prescription of the first arrow above. However, we must be clear that these would be axioms of PHYSICS not MATH. Math should not purport how to apply or use math, it should just make sure it keeps proving theorems.

My agenda here is that I need unambiguous terms in order to describe physics (in so far as that is possible). If we are to describe fundamental aspects of the universe then it serves us to be sure that the language we use is appropriate. If we should find ourselves in a situation where we are trying to describe fundamental behaviour in terms of emergent properties it could lead to confusion.

The only thing that can be communicated (and therefore all that can be known) is relationships. Whatever a photon, electron or quark might be - we can never describe them directly to any degree. We can only describe the patterns of relationships. 'Dimensions', 'Charge', 'spin', 'position' are observed, but cannot be fundamental qualities. In order to comprehend physics, we must understand the symbols used to describe physics. 'Dimensions' are not defined by mathematics. There are systems in mathematics that we interpret as being dimensional... but we have no formal mechanism for ensuring that the assignment of meaning is appropriate (beyond the results or such assignment being sufficiently clearly out of line with our expectations).

snowyowl wrote:
Given that the axiom of choice has been controversial, it suggests to me that there isn't an obvious way to construct the axiom of choice from other primitives.


Um, yes, this is well-known and was proved correct by Kurt Gödel and Paul Cohen. It's the reason that the Axiom of Choice (AC) is even remotely interesting; if AC could be proved from other axioms, it wouldn't need to be an axiom itself. What did you think the importance of AC was, if you didn't know that?

You seem to have a fairly weak grasp of axiomatic set theory, for someone trying to overthrow ZFC. Yes, we're always happy to educate you about the fields we specialise in (life pro tip: people love talking about themselves). Yes, your ideas can and should be evaluated independently from your merits as a person and a mathematician. And yes, sometimes fatal flaws crop up in systems that have been thought bulletproof for centuries. But we've been here before, and we'll come here again:

Thank you for the confirmation.

As much as I started this thread in a naive way... I was put straight fairly quickly. There are known limitations which are expressed in various components of the foundational crises. I'm not trying to re-invent that wheel. However, I do perceive a gap between the limitations on knowledge and our perception of knowledge (and meaning). While it is impossible to establish a fixed point upon which to build; our everyday experience shows that this limit on absolute knowledge does not prevent our existence and perception of meaning. There is clearly a mechanism whereby specific perception of meaning arises. Whatever that mechanism - it can be understood in a detailed, non-hand-wavey, manner. I call this particular wheel Relativistic Mathematics.

Schrollini wrote:I let this go by, since it seemed a minor point. But the more I think of it, the more it seem symptomatic of the problem here.

Division by zero is perfectly well defined...

There was an interesting thread hereabouts recently involving division by zero and infinities. There are several systems options that accommodate division by zero.

And your reference to the real projective line is spot on. That is very nearly an example of a non-absolutist system. However, the system is still predicated on Real numbers. In a true ring, there is no way to distinguish between any points on that ring. Further, the diameter of the ring cannot be distinguished from the perspective of a given point within the ring.

As it is, the numbers within the system are assumed to be distinguishable and infinity is not just another number on a ring.

You're so busy railing against "absolutist mathematics" that treats zero as a "fixed point", that you haven't noticed that mathematicians have developed a number of different systems, and zero's properties can only be specified relative to a given system. Isn't that exactly what you wanted?

It would be useful to have a clear idea of what a fixed point is:

i) A fixed point is one that cannot be constructed or derived from primitives. e.g. The Empty Set and The Axiom of Choice.

I'll also take the opportunity to describe a tautology set:

ii) A closed set of symbols such that any given symbol in the set may be described by reference to the other symbols, but without any fixed reference.

Everything that can possibly be communicated fits into one of these two groups. And it is impossible to define a fixed point.

As much as tautologies have a bad name, it is the only means of communication we have. We cannot define a fixed point. While we can assert a fixed point for the purposes of axioms - any such assertion represents an impossibility.

If a system is structured such that the definitions are hierarchical with whatever primitives of type i) at the base, then that system is absolutist. It seems to me that most, if not all axiomatic systems are constructed in this way (pretty much by definition).

If a system is structured such that meaning derives from the relative position within a network of relationships then that system is Relativistic.

(Genuinely, thank you for that collection of references. I've had a quick flick through them and will be taking a more in-depth look).

LaserGuy wrote:Sure it tells you something. It tells you whether the theorem is either true or false within the context of that model. Mathematics is really nothing but the manipulation of the symbols. Assigning meaning to those symbols is not mathematics. Quite the opposite, in fact; the benefit of a high level abstraction is that the symbols could mean different things in different contexts, but the underlying theorem still remains true. You can say that the symbols represent handshakes or pigeon-holes or tennis matches or murder-suicide pacts. The meaning is irrelevant. You can attach whatever meaning you want, and use the same symbols to represent it as long as the underlying mathematical properties remain the same.

I agree with pretty much everything you say here. Even to the point where multiple meanings can be applied to the same set of mathematical manipulations.

That still leaves the assignment of interpretations having nothing more definite than intuition as a guide. Intuition has not, historically, been a completely reliable guide.

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Re: Axiomatic mathematics has no foundation

Postby Schrollini » Sat Mar 01, 2014 3:33 am UTC

Treatid wrote:And your reference to the real projective line is spot on. That is very nearly an example of a non-absolutist system. However, the system is still predicated on Real numbers. In a true ring, there is no way to distinguish between any points on that ring. Further, the diameter of the ring cannot be distinguished from the perspective of a given point within the ring.

i) How did you get on rings?
2) A ring has both additive and multiplicative identities, so why would it be less "absolutist" than a field?
c) Diameter?

Treatid wrote:It would be useful to have a clear idea of what a fixed point is:

I feel like we've had a discussion about the importance of definitions before....

Treatid wrote:i) A fixed point is one that cannot be constructed or derived from primitives. e.g. The Empty Set and The Axiom of Choice.

What's a "primitive"? Are primitives "fixed points"? Are fixed points "primitives"? Are all axioms "fixed points", since they aren't derived from anything? The empty set is derived from the axiom schema of specification. Does that mean that the axiom schema of specification is not a "primitive"? Is it still a "fixed point"?

Treatid wrote:I'll also take the opportunity to describe a tautology set:

ii) A closed set of symbols such that any given symbol in the set may be described by reference to the other symbols, but without any fixed reference.

What's a "set"? Do you mean a thing defined by the ZF axioms? What's it mean for a set to be "closed"? (I'm guessing that you don't mean it includes all its limit points, but that's the only definition I'm familiar with.) What's a "description"? Do those descriptions have to exist, or is it enough that they "may" exist? If it's the latter, how do you show that those descriptions cannot exist in the case of fixed points? If it's the former, are those descriptions part of the set? What's "fixed reference"?

If I have a fixed point A, can I make it into a tautological set {A, B} with the description of A being B and the description of B being A? If not, why not? If so, why is this distinction meaningful?

Can you give us an example of a tautological set?
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Re: Axiomatic mathematics has no foundation

Postby PM 2Ring » Sat Mar 01, 2014 8:19 am UTC

Treatid wrote:Euclidean Geometry worked fine for the best part of two thousand years without the fifth postulate. The group assumption was sufficiently consistent that it wasn't necessary to state the fifth postulate (or realised that it needed stating).

WTF?

Euclid himself stated the fifth postulate in Elements. Note that Euclid didn't invent Euclidean Geometry - he systematised and added rigour to the geometrical / mathematical knowledge of the time. See Wikipedia.

The ancient geometers recognised that the 5th postulate is more complicated than the other 4 postulates, and Elements doesn't use the 5th postulate until the 29th proposition. Those first 28 propositions are very important parts of geometry, but you need the 5th postulate to do stuff like similar triangles & Pythagoras' theorem, which are essential for practical applications of geometry and are the basis of trigonometry.

The clunkiness of the 5th postulate has inspired many geometers since ancient times to try to prove it as a theorem from the other 4 postulates, or to try to find some simpler postulate that would allow a proof of the 5th in terms of this new postulate & the other 4. Wikipedia lists a variety of statements equivalent to the parallel postulate, but none of these are really any simpler (or more self-evident) than Euclid's formulation of the 5th.

Eventually, some mathematicians decided to attack the 5th with a reductio ad absurdum approach, hoping to arrive at a contradiction by assuming the 5th was false. However, they were unable to do so. Eventually it was realised that the truth of the 5th is independent of the other 4 postulates, and that geometries that assumed the 5th is false can be consistent, hence non-Euclidean geometry was born. Sort of. :) Actually, people had been doing spherical geometry for centuries by then, but it was seen as a mere derivative of Euclidean geometry, not a legitimate geometry in its own right.

...

I'm not quite sure what you're getting at in this thread. But it seems to me that you're saying "Mathematics has traditionally assumed that it had an absolute (God-given) foundation, but that was undermined by things like Russell's paradox & especially Gödel's theorems. Yet the rest of mathematics seems to ignore these issues and continues to operate as if it still has rock-solid foundations! Mathematics as a whole ought to renounce its absolutist pretensions and embrace a relativist approach!!"

The rest of us are saying "But those foundational issues do not actually undermine the validity of the rest of mathematics, so mathematicians (in general) are not particularly concerned with such issues. They may contemplate and discuss such things when they're in a philosophical mood, but they can generally ignore foundational issues while they are actually doing mathematics."

Here's a crude analogy. Mathematics is like a house, and the traditional wisdom is that a house should have solid foundations, or it's in danger of collapse. But people like Gödel showed us that it's impossible to have absolute foundations. The rest of the mathematical world said: "Ok. So we're living in a houseboat. kthxbye."

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Re: Axiomatic mathematics has no foundation

Postby snowyowl » Sat Mar 01, 2014 1:14 pm UTC

Treatid wrote:As much as I started this thread in a naive way... I was put straight fairly quickly. There are known limitations which are expressed in various components of the foundational crises. I'm not trying to re-invent that wheel. However, I do perceive a gap between the limitations on knowledge and our perception of knowledge (and meaning). While it is impossible to establish a fixed point upon which to build; our everyday experience shows that this limit on absolute knowledge does not prevent our existence and perception of meaning. There is clearly a mechanism whereby specific perception of meaning arises. Whatever that mechanism - it can be understood in a detailed, non-hand-wavey, manner. I call this particular wheel Relativistic Mathematics.

I'd like to apologise for my earlier post. It came across as very rude, and suggested you weren't paying attention to what the other commenters were telling you, which was not only incorrect but hypocritical of me. In future, I'll stay out of controversial threads when I'm tired.
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Re: Axiomatic mathematics has no foundation

Postby Twistar » Sun Mar 02, 2014 10:41 pm UTC

So Treatid, Here is my stance on the things you are talking about.
1) Where should we draw the boundary of mathematics? I am perfectly comfortable putting a box around math as being entirely symbol manipulation. You start with some symbols (axioms) and rules for manipulating them and you go from there. In my opinion, this is a very natural boundary since it is so self-contained and there is no ambiguity of precisely the sort you are worried about. It is very clear cut. Now admittedly, I am saying all of this on the assumption that all math is based on the ZFC axioms and under that assumption my demarcation line makes perfect sense. I.e. mathematics is very cleanly self-contained and there is literally no ambiguity in math beyond the as yet unresolved uncertainty about the completness/consistency of that axiomatic system due to Godel's theorem. However, we must recognize in reality that there are different versions of the foundation of mathematics that lead sometimes to different mathematical results such as category theory instead of set theory.*
Next:
"I think something more robust than 'intuition' should connect mathematics to the real world."
I entirely disagree with this statement, and I think this might be the crux of this entire thread. Intuition is extraordinarily robust. It is perhaps the most robust thing about humanity. It is intuition that got mathematics to where it is today. What I would say instead is that intuition is not rigorous or predictable. Let me expand upon this and tell a little bit about myself to see if we can draw some common ground. I am a first year grad student in a physics phd program but I also got a math bachelor degree. I considered getting a math phd but I decided not to. Perhaps the biggest reason I chose not to is that I didn't want to be responsible for creating new mathematics. i.e. proving new theorems that hadn't been proven. The reason I didn't want to do that is that there is no guarantee I will be able to do it. Me finding the proof of some theorem is not predicated on some rigorous system which I can follow to get the proof, instead it is predicated on my creativity, intuition and insight to pick the right things and put them together in the right way. I guess it was a lack of faith in my own ability to do that that kept me from pursuing the math degree. However, not surprisingly, it turns out getting a physics degree ALSO relies on creativity, intuition and insight. In fact, in my opinion, if you want to be good at anything you do you need these things and there is no prescription for how to achieve them and this can seem daunting.**
Now how does this apply to this system? It sounds to me like you want something that will prescribe how people should imbue meaning to mathematics (or in language I prefer how people should apply mathematics to "real life" situations.) It might be possible to create such a system to some degree but my argument is that you could never create a system to replace the human intuition and creativity which are presently used to do this task since intuition and creativity are more powerful than any rigorous system.
Now, it is possible that this is a matter of opinion, and, even in light and comprehension of what I have said, you still think there should be some rigorous way to figure out how we should apply mathematics to a real life situation. However, if we want to go into more debate about this I think there are some strong pragmatic reasons for accepting my definitions and points of view (obviously that is why I hold them). So I have questions for you on what I've said:
1)I think we are on the same page with regard to the symbol manipulation. How do you feel about the hard demarcation line I have drawn for what should constitute mathematics? You have already said you are fine saying the assignment of meaning does not have to be part of mathematics. Can you use the definition I have proposed for the confines of mathematics for the rest of this discussion?
2)creativity vs. rigorous prescription. Do you get what I'm saying here? How do you feel about it? You've been saying things like there should be a way of doing this. Why do you think there should be such a way? I think that statement is a matter of opinion so why do you hold that opinion? Also, I think it is almost impossible that there could be any such system. Why do you think it is possible?

Ah I forgot one more important thing about all of this. In the way I define mathematics the want-to-be-fixed-points that you talk about such as the empty set and the "is an element of" relation are technically NOT part of mathematics precisely because of their ambiguity or lack of rigorous definition. The motivation for my definition of mathematics is that it is the subset of human rationality which is entirely rigorous therefore it doesn't make sense to allow un-rigorous concepts into mathematics.


Summary
Basically I am trying to kill a lot of birds with one stone by defining mathematics a certain way and saying it is ok to leave the other things you are worried about to non-mathematics and furthermore saying that you should give more credit to non-rigorous things like intuition and creativity.

*
Spoiler:
I think this is a good framework within which to have this discussion, however we must recognize that there are other more sophisticated axiomatic systems that foundational mathematicians work with. Here is how I think about it and someone should please correct me if I'm wrong: There are different sets of axioms that you can start with (i.e. picking different axioms or using categories instead of sets or something) but if I understand correctly all of these differently ways of starting lead to systems which are self-contained in the way I have described and in this sense it makes sense to think of these different starting points as leading to "different mathematics". It would seem that all of the different foundations lead to most of the same math. For example, if someone proposed some foundation that didn't allow us to do calculus anymore it would be rejected as not useful (note that I didn't say not mathematical. it is still mathematical, it is just not useful.) However, the different foundations do at times lead to different mathematical results and this is why they are interesting. For example, I know that some tensor spaces can't be defined in set theory because they require sets which are forbidden by the set theory axioms so we must move to category theory.


**Sorry for this sort of aside, but I think it is kind of important. That's why I wrote it down.


edit: Oh yeah and I want to point out one inconsistency I see in stuff you have been saying. I don't know what you're talking about with all of the euclidean geometry. euclidean and non-euclidean geometry and everything "math" we have been talking about is somehow defined and described within the symbol manipulation we have been talking about. If we redefine math this way nothing would need to "find a new home". It is all contained in the system I have described. You're getting a little attacked for these points. I haven't looked too much but it looks like the attacks are legitimate but I think it's pretty unrelated to your main point.

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Re: Axiomatic mathematics has no foundation

Postby Treatid » Wed Mar 05, 2014 12:27 am UTC

There appears to be a disconnect between what people know and what they understand.

There is no conceivable way to define anything.

Among the (possibly infinitely) many things that can't be defined are axioms.

It isn't a question of whether a particular set of axioms is consistent. It isn't a question of which set of axioms to prefer as our foundation.

Axioms cannot be defined. At all. In any way. They can't be approached in the limit or approximated. Axioms are a fiction.

Using natural language and our physical experience in the real world does not impact on this result in any way. Inside or outside of mathematics there is no way to define anything as axiomatic mathematics understands 'definition'.

Obviously this doesn't make all our concepts and ideas disappear in a flash of logic. The concepts still have significance to us. It is merely futile to attempt to define those concepts using an axiomatic structure.

Everything we think we know and perceive are emergent qualities. You can't dissect a human down to the component parts and ask of those bosons and leptons which one is the 'human' bit. No more can you understand an integer, an empty set, a line or any other concept by examining its component parts. And if it doesn't have component parts - then we simply cannot understand it.

Our perspective is biased by our existence on a sodding great lump of planet. As humans on the face of a planet it is tempting to perceive that planet as a fixed point (or the local mountain range, or that set of stars that always form the same pattern). For practical purposes it works to allow ourselves the illusion that there are knowable fixed points.

Where experience, custom and technology support it; assuming a fixed computer architecture, numbering system or meaning for a set of (pseudo-)axioms can work, to a degree. Albeit all existing (pseudo-)axiomatic systems are necessarily tautologies.

However, if we want to describe fundamental physics, starting with undefinable concepts is not going to get us very far.

Trial and error works. When you were a sperm, you found the egg just by wriggling the thing at the back. Shame about the other billion sperm.

Both mathematics and physics can continue to work by throwing equations at things and seeing which ones seem to make some sort of sense. And while mathematicians continue to believe that they can define axioms, that is the only way forward. It seems so ape-man.

So, we can't define a fixed point. We can't define axioms. Without axioms there is no logic, no proof. But we already knew that. So why has no-one claimed all six of the open Clay Institute Millennial prizes by pointing out that the questions are all undefined? How can anyone pretend that there is some significance to the question of whether P=NP when "P", "=", and "NP" cannot be defined? I can understand accountants, engineers and students going along with the illusion that unicorns axioms exist... but anyone who knows the first thing about axioms should be aware that they can't be stated.

@Schrollini: Yes - we have been here before. This thread is a direct follow on.

You (not personal - all mathematicians) make assumptions. These assumptions are expressed in the structure of set theories and number systems (and all other axiomatic systems). Not least of these assumptions is that it is possible to define/state a set of axioms. In theory, you know that such definition is impossible... yet you ignore that possibility in favour of the mantra "it works". If your worldview incorporates a fundamental impossibility then it is skewed.

It isn't sufficient to build something new upon an existing conception of the world when that existing conception is so distorted. This is particularly relevant to small scale physics where you might end up describing behaviour in terms of emergent properties that haven't yet emerged at those scales.

I absolutely understand the desire to have concrete definitions and meanings for words. But that simply isn't possible. Yet we can construct agreement and a large degree of consistency in the way we use words. Axiomatic mathematics isn't dead - it never existed. But mathematics does exist. It isn't doing what it thinks it is doing - but nor is it doing nothing. There is something to be understood. There is a way to understand it. The first step to understanding is realising that there is no spoon axiomatic mathematics.

@PM 2Ring: Agreed with respect to Euclidean Geometry.

With regard to houseboats: the equivalent is that you don't have any timber, bricks, nails, mortar, shingles, glass or any other building blocks. Without a fixed starting point there are no axioms, there is no proof: there is only tautology.

Mathematics works because we choose the bits which work. The only justification for mathematics is that we choose to find significance in certain things. This works. We try things out... some don't appear useful, others do appear useful. It is fine that mathematics does what works. It is not okay that anybody should think that what works has any relationship with axioms.

Twistar wrote:1)I think we are on the same page with regard to the symbol manipulation. How do you feel about the hard demarcation line I have drawn for what should constitute mathematics? You have already said you are fine saying the assignment of meaning does not have to be part of mathematics. Can you use the definition I have proposed for the confines of mathematics for the rest of this discussion?

Sorry to move the goal posts on you. I mistakenly accepted the idea that something (symbol manipulation) could be rigorously defined by axiomatic mathematics. My own experience with programming made it very hard for me to separate the idea of deterministic computer programs and an unambiguous definition of axioms.

Somebody linked to the idea that a computer program is a set of axioms and each subsequent state is a proof following from those axioms. But it still isn't possible to define a set of axioms. And without a set of axioms there are no proofs. Computer programs exist, of course. But a computer program is not, in any sense, similar to a set of axioms.

Similarly, whatever mathematics does is real... it just isn't axiomatic mathematics.

The trouble now, is that without axioms as the border guard of mathematics it becomes almost impossible to determine where to draw a boundary. The symbol manipulation becomes entirely arbitrary - there is no logic or structure that constrains it. As such, our choice of symbol manipulation, and any consistency are properties that we choose to bring to the symbol manipulation. We are intimately involved in mathematics. You cannot separate the observer from the process.

2)creativity vs. rigorous prescription. Do you get what I'm saying here? How do you feel about it? You've been saying things like there should be a way of doing this. Why do you think there should be such a way? I think that statement is a matter of opinion so why do you hold that opinion? Also, I think it is almost impossible that there could be any such system. Why do you think it is possible?

Your description of Intuition seems to me to be consistent with 'trial and error'. Internally or externally, we test ideas against other concepts we are already familiar with. A few ideas show something interesting during testing. Trial and error works. And I don't see us ever removing intuition altogether.

Again... I went off track. I'm not seeking a mathematics that can simply be enumerated by a computer. I'm seeking a mathematics that isn't crippled by assuming impossible things as its core. I'm looking for a mathematics that works. One that is consistent with the known limitations of communication.

'Meaning' and 'Significance' are specifically properties connected to an observer. No observer, no meaning. There is no way to separate the observer from what they observe. Doing so smacks of, yet again, trying to achieve an impossible fixed point. As such, there is no danger of removing humans from mathematics. So long as humans are observers, they will be an intimate part of mathematics.

I accept the premise that if we could define a couple of basic assumptions then great swathes of 'mathematics' falls out - it appears beautiful how much can be accomplished with so little. Unfortunately we can't define, state or otherwise specify any assumptions of any kind. One small assumption is as impossible as... well... an impossible thing. Wishing, hoping and turning a blind eye will not change the fact that we cannot define a set of axioms. And without a set of axioms, we can't define anything else.

The axiomatic concept of definition is a pipe-dream. Utterly unattainable.

Yet we do have a system that is more than simple chaos. We do have structure. We have created the modern world... all without having any axioms upon which to build. The loss of something we never had really isn't that big a deal. So why cling to the idea of axiomatic mathematics? Why not try to understand the working systems we have without pre-supposing a flat impossibility?

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Re: Axiomatic mathematics has no foundation

Postby doogly » Wed Mar 05, 2014 1:32 am UTC

Treatid wrote:So, we can't define a fixed point. We can't define axioms.

Well, some of us can.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?


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