Circularity in Formal Languages?

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Twistar
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Circularity in Formal Languages?

Postby Twistar » Tue Mar 11, 2014 6:14 am UTC

So this sort of comes out of the axiomatic mathematics has no foundation thread but it is along a different vein. I started reading about foundations of mathematics and formal languages and one thing has bothered me. Many of the definitions (or constructions? I don't really know the right word to use here) of formal languages involve mathematical notions such as "natural numbers", "finite", "uncountable", etc. These notions are used to describe things such as the alphabets of the formal language or something like that*. This seems circular to me since none of these terms are defined until you write down (in the formal language) the ZF axioms and prove the relevant statements about the natural numbers or uncountable sets. This question sort of gets at some of the ideas I'm looking at but doesn't have a totally sufficient answer. Note: I am very comfortable with the notions of "set" and "is an element of" relations being undefined 'primitives'. Of course we can't have a completely rigorous system from the get-go and there have to be some terms taken on faith which make sense in the metalanguage. However, a term like 'countable' has a very specific mathematical definition and it makes me uneasy to take that term to be understood in the metalanguage. I guess my goal is to define formal languages and ultimately mathematics in as few 'non-rigorous' terms as possible and I'm a little unsatisfied with what I've found so far. Am I just be overly pedantic? The line of reasoning I'm taking makes really good sense to me. The same way we axiomatized mathematics to as few axioms as possible I want to axiomatize (or define) formal logic in as few metalanguage terms as possible.

Does my question make sense? have other people here thought about this? Let me know if you have good resources I can look at that address this question. Thank you.


*Note I have only taken one logic course and it was very very basic. I haven't studied this stuff in detail so I might get lots of terms wrong.

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Re: Circularity in Formal Languages?

Postby lalop » Tue Mar 11, 2014 6:55 am UTC

With the disclaimer that I have only weak background in this:

I would treat the metalanguage as an informal, rather than formal, language that, since it's being used by mathematicians in practice, still retains most of the good properties of the formal language.

My impression has been that there's a mathematical realist tradition behind this, in which "natural numbers" and "countable" actually means something independent of our formalization, so we ought to be allowed to use them in our reasoning.

The only alternatives I can think of are infinite regress (meta-meta-meta-...-meta languages) or some form of meta-finitism (not very useful).

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Re: Circularity in Formal Languages?

Postby korona » Tue Mar 11, 2014 6:18 pm UTC

Yes, we use informal language to define what a formal language is. It may seem weird, but there is basically no other way to do that. I would argue that it is also not necessary to further formalize the definition of a formal language.

A lot of people who learn about formal logic are bothered by this; you're not an exception. The more you learn about logic the more sense doing this will make.

It may comfort you that once you have a formal language all math can be done by simple syntactic manipulations. You don't need any semantics at all. For example you don't have to interpret function symbols as functions, predicate symbols as predicates or the for-all quantifier symbol as an actual quantifier. You don't need a model theoretic universe. You can threat them as meaningless symbols and still derive all theorems by string manipulations. (At least this applies for first-order logic)

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Re: Circularity in Formal Languages?

Postby Twistar » Tue Mar 11, 2014 10:52 pm UTC

That all makes sense and I understand why it is necessary to use informal language to define a formal language. I guess I just want there to be as few informal terms as possible and I especially don't want any what I will call egregious informal terms. I guess the best example is countable and uncountable. I don't think these terms can make sense informally i.e. without reference to mathematical set theory.

So here's my question: Is it possible to build up to the concepts of countable and uncountable (formally) WITHOUT having to use those terms (informally) in the construction of the formal language in which you describe them?

I also ask the same question of the natural numbers. I'm more willing to allow natural numbers in the construction of formal languages because I guess I'm comfortable with those being informally defined but it is kind of a let down. I remember being really excited when I realized you can "derive" the natural numbers from the ZF axioms, but if it turns out you already have to make reference to them informally in the language in which you "derive" them then that is kind of a let-down.

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Re: Circularity in Formal Languages?

Postby Qaanol » Wed Mar 12, 2014 12:24 am UTC

Right off the bat we can define Dedekind infinite: a set is Dedekind infinite if it can be put in bijection with one of its proper subsets. That doesn’t reference the natural numbers at all, and it fits our intuitive idea of infinite: that adding or removing one element from an infinite set doesn’t change its size.

However the usual definition of infinite does reference the naturals: a set is infinite if it cannot be put in bijection with any natural number. There are subtle differences between infinite and Dedekind infinite that I will not get into here, and in a lot of ways Dedekind-infinite is a superior concept. Suffice it to say that every Dedekind-infinite set is infinite, but without something like the axiom of countable choice, there are models with infinite non-Dedekind-infinite sets.

We can construct the naturals with a bit of work. Let ∅ represent the empty set. We define “empty set” informally as the set containing no elements.

Let s(x) be the successor function: the successor of a set x is the set {x, {x}}. This is also an informal definition, or at least uses concepts that are only defined informally.

Now, it should be obvious that this construction generates the natural numbers as the successors of ∅. The difficult part is to encapsulate the set of natural numbers the way we intuitively think of them. We can’t just say “∅ and all its successors” because there exist nonstandard models.

Intuitively it is obvious that “The things you get when you iterate the successor function on ∅ for a while and eventually stop” are what we understand the natural numbers to be, but it is surprisingly difficult to formalize.

We can take the axiom of infinity, informally defined as “There exists a set containing ∅ such that the successor of each element of the set is also in the set,” to ensure there is some set containing what we want the naturals to be.

Then we can take the subset axiom, informally defined as “For any set, each subcollection of its elements is also a set,” to guarantee that the intersection of all sets containing ∅ and closed under the successor function exists. We still haven’t exactly constructed that minimal set, but I think this is about the best we can do.

Once we have the natural numbers, we can define finite as “A set is finite if it can be put in bijection with a natural number.” And infinite of course means not finite.
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Re: Circularity in Formal Languages?

Postby z4lis » Wed Mar 12, 2014 2:32 am UTC

So I always felt kind of uneasy about this for a long time. It never cleared up until I learned about different models of set theory, via topos theory, but you don't really need to do all that to understand what's going on. What set theory is really trying to do is create a miniature model M of objects that we can combine in ways that look sort of like sets, via certain rules. ZFC is just a list of rules that any system that mathematicians think a system of set-like things should do. For instance, we would want to be able to make sense of products, elements, functions, etc. It's a little like how a physicist might write down "laws of physics" that she thinks the world should follow. You and I, being humans who understand logic and the integers intuitively, can then use these external objects to reason about what these models should do, internally. The models might not, fundamentally, look or behave like what we envisioned, but we do know they share any similarity that we can deduce from the axioms.

So, this means that I can use induction (via the meta-level natural numbers) to reason about things within the model M. It may well be that the model has an object (set) N that behaves just like the natural numbers do, as far as the model is concerned. So these would be natural numbers inside the model M, and might not have anything to do with what we think are natural numbers. For instance, the "definition" of the natural numbers might be that the thing N should have something nontrivial in it (zero) and a successor function (plus 1). The object N might have a set-theoretic description that makes it wildly different form the natural numbers, but that's OK because I have wildly different notions of what "product" and "function" and "element" mean within the model. There is only a formal similarity, but that's the beauty of mathematics - that formal similarity is all I need to transfer any theorem proved in ZFC to this new, crazy model M.

As another example, consider a definition of "Euclidean geometry" given by all the postulates except the parallel postulate. This sets up a formal system Geo in which I can reason about any mathematical object that satisfies the axioms. While we might have a "natural" model of these axioms, Euclidean geometry, there are many other strange things that obey those axioms. Everything shown about the system Geo applies not only for the model of Euclidean plane geometry, but also for hyperbolic geometry, projective geometries, etc. ZFC is similar. It's just a formal system that lets us reason about sets, but whatever else satisfies those rules. So I would not imagine the natural numbers constructed via ZFC as "the natural numbers", but rather a formal object that behaves like natural numbers.

Because of this separation between the theory ZFC and the actual objects of mathematics, if there turned out to be an inconsistency in ZFC, while that would be interesting, it would probably not threaten what mathematicians do. We would just create a different formal system that hopefully did not have the problems of ZFC while also letting us express the desired mathematics. Think of it like an operating system. There are lots of things you want to do with your OS, and you hope it's bug-free. Maybe there turns out to be a bug that crashes the OS. You won't stop using your computer, but rather find a way around it, patch the OS, or just learn to live without that particular program or feature. Of course, you'll want to understand why the bug happened and to write an OS in the future that won't crash, but there are probably lots of awesome things you can do even with the annoying bug.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

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Re: Circularity in Formal Languages?

Postby Twistar » Wed Mar 12, 2014 4:21 am UTC

Ok yeah so these replies are making sense. I think what I really want is a sort of concise (or at least complete) list of all of the informal elements (or ingredients) that you need to put in to the formal language so you can get mathematics. The reason I want this is that I can then sit back and sort of judge whether I think all of those informal elements are legitimate and I think there's also kind of some interesting philosophical implications you could figure out from that perspective.

So I think the conclusion so far is I need to extend my list of necessary informal elements. My question now is do you think you can sum up in a forum post the ingredients that need to go into making a formal language that can describe mathematics and if not where would be a good place for me to start looking so that i can answer it on my own? (other than wikipedia..)

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Re: Circularity in Formal Languages?

Postby arbiteroftruth » Wed Mar 12, 2014 2:47 pm UTC

I'm no expert, but I believe that first order logic can be built with the only informal "ingredients" being modus ponens (direct implication) and universal quantification over strings in the language (ie, being allowed to use variables that can be replaced with whatever you like). Once you have those as undefined primitives, you can axiomatize the rest of first order logic in terms of strings involving symbols like ~ and &, where the axioms don't treat those particular symbols as variables. Set theory is then just a matter of axiomatizing strings that involve the "is an element of" symbol. And once you have set theory, you have all of math.

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Re: Circularity in Formal Languages?

Postby Forest Goose » Wed Mar 12, 2014 3:31 pm UTC

I think you can get all of zfc using just three variable symbols, actually, instead of needing countably many - I forget where that's from, though; I'll try and look up a source later if you're interested.

*I'm still looking; I remember reading it as a passing remark in a book on finite model theory; although, I'm fairly certain I'm remembering it correctly, thinking about it more, I feel like it must be either using a logic that is not FO, or that there was something more to it. I'll keep digging - I think someone borrowed my book, or some such.
Last edited by Forest Goose on Sat Mar 15, 2014 3:54 am UTC, edited 1 time in total.
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Twistar
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Re: Circularity in Formal Languages?

Postby Twistar » Wed Mar 12, 2014 7:43 pm UTC

Forest Goose wrote:I think you can get all of zfc using just three variable symbols, actually, instead of needing countably many - I forget where that's from, though; I'll try and look up a source later if you're interested.


Yes! I would LOVE that! That's exactly what I'm looking for.

edit: Uh but I guess that is just variable symbols? I'm not totally clear about how the logical symbols are defined and if they may need more powerful machinery. In particular the predicate logical symbols, like for all and there exists. Are concerns here also assuaged?

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Re: Circularity in Formal Languages?

Postby korona » Wed Mar 12, 2014 9:47 pm UTC

You can treat them as plain symbols without any underlying meaning. Proofs can be done on a fully syntactical level. Examples of such proof systems are:

http://en.wikipedia.org/wiki/Natural_deduction
Here the inference rules behave like you expect from informal proofs.

http://en.wikipedia.org/wiki/Hilbert_system
The Frege/Hilbert calculus has only one inference rule and a small number of axiom schemes. While it is very unintuitive it is still complete and sound for first-order logic.

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Re: Circularity in Formal Languages?

Postby radams » Mon Mar 17, 2014 11:28 pm UTC

Forest Goose wrote:I think you can get all of zfc using just three variable symbols, actually, instead of needing countably many - I forget where that's from, though; I'll try and look up a source later if you're interested.


This was first proved in Tarski and Givant "A Formalisation of Set Theory without Variables" (1987). Here's a good paper by Formisano et al on the subject that should show how it's possible; they write out the three-variable version of the Pairing Axiom.

http://www.dmi.unipg.it/~formis/papers/ ... sDraft.pdf

So we can give the axioms of ZFC using a finite alphabet. However, there are still infinitely many axioms. The axioms are a little longer now, too. Using four variables, the Pairing Axiom is 21 symbols long:

[imath]\forall x \forall y \exists z \forall w (w \in z \leftrightarrow w = x \vee w = y)[/imath]

The equivalent sentence with three variables that Formisano et al give is 261 symbols long.

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Re: Circularity in Formal Languages?

Postby Twistar » Thu Mar 20, 2014 5:08 am UTC

radams wrote:
Forest Goose wrote:I think you can get all of zfc using just three variable symbols, actually, instead of needing countably many - I forget where that's from, though; I'll try and look up a source later if you're interested.


This was first proved in Tarski and Givant "A Formalisation of Set Theory without Variables" (1987). Here's a good paper by Formisano et al on the subject that should show how it's possible; they write out the three-variable version of the Pairing Axiom.

http://www.dmi.unipg.it/~formis/papers/ ... sDraft.pdf

So we can give the axioms of ZFC using a finite alphabet. However, there are still infinitely many axioms. The axioms are a little longer now, too. Using four variables, the Pairing Axiom is 21 symbols long:

[imath]\forall x \forall y \exists z \forall w (w \in z \leftrightarrow w = x \vee w = y)[/imath]

The equivalent sentence with three variables that Formisano et al give is 261 symbols long.


Thank you and thank you for looking Forest Goose

I think this is what I was looking for. Is the only reason there are infinitely many axioms because you need an axiom of schema specification for each predicate?
I think I'm becoming more comfortable with these things.

Also,
I just read "Godel's Proof" by Nagel and Newman and my mind is sufficiently blown. It makes it a little more clear why mathematicians are so happy to move to non-finitistic things so quickly since they seem necessary to get things done.

A question on the theorem for some curiosity and to test my understanding of the proof. The proof involves proving some arithmetic statement which is undecidable (from the axioms of arithmetic) using meta-mathematical tools. Have mathematicians successfully used this type of reasoning to prove mathematical statements which are known to be undecidable within arithmetic? i.e. prove some tricky meta-math thing and then that corresponds to some arithmetic statement so we go ahead and take that statement as being true but furthermore the corresponding arithmetic statement is somehow 'interesting'?

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Re: Circularity in Formal Languages?

Postby Forest Goose » Thu Mar 20, 2014 6:59 am UTC

Godel's undecidable is, actually, true (if the system is consistent :-) ).

The Paris Harrington Theorem is true, but independent of PA; but can be proved from 2nd order arithmetic.
Goodstein's Theorem has the same feature. Kruskal's Tree Theorem is another example.

You may want to take a look Ordinal Analysis and Gentzen's Consistency Theorem (as a follow up on Godel Stuff).

Lob's Theorem may also be of interest to you; here's a cute introduction to it:
http://yudkowsky.net/rational/lobs-theorem. It says that if PA proves "If X is provable, X is true", then PA proves X.

Tarski's Undefinability Theorem may be worth mention; it states that you cannot define truth inside of arithmetic; that is to say that you cannot import truth from the metalanguage into arithmetic.

Lindstrom's Theorem may, also, hold some interest; it, basically, states that FO is the strongest logic that behaves nicely (in a way).

If you're interested in the topic, I'd suggest reading up on Model Theory and Logic. I'd specifically recommend A First Course in Logic by Hedmen and Set Theory by Jech.

-----
I'm not sure if that helps at all, I'm not 100% sure I'm answering what you're asking. But, if it is along these lines, I'd love to go into more detail.

*I added to my original post, not all of it is directly related to what you asked; but some of it may be of interest to you.

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@raddams: Thank you for that:-)

-----
A clarification:

When I say "is true" I mean in the standard model of PA. By the completeness theorem, if there is no proof of p in PA, then there is a model of PA in which ~p holds. So, there is a model of PA in which godel's sentence is false, but this model is not the standard one.
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Re: Circularity in Formal Languages?

Postby Treatid » Sun Jan 25, 2015 11:20 pm UTC

It is agreed that axioms can't quite boot strap themselves. This is annoying, but the gap can be filled with very little informal language.

We only need a few concepts in order to build the rest of mathematics.

What is an informal language? Is it fair to say that an informal language is partially defined, or partially understood? If the language was fully defined it would, presumably, be a formal language?

If we describe something using an informal language then there is a degree of uncertainty?

Suppose we rigorously define part of a system. If we allow the undefined parts of that system to take on any properties or magnitude (e.g. it could be infinite) then the unknown portion of the system can affect the known portion in unlimited ways.

That is, if the unknown element is unconstrained, then it could impact the rigorously defined portion such that the total system could be any conceivable system. Even an appropriate 'NOT' operation could invert an existing description.

It seems to me we must either describe the whole system without any ambiguity, or we must have a mechanism to constrain the uncertainty.

Given an arbitrary informal language, how might we constrain any degree of uncertainty or ambiguity within that language?

Minimising the number and scope of the informal concepts seems to be one approach to reducing ambiguity. Does this reduce the ambiguity? or concentrate it in one place? What tools can we use to measure the degree of ambiguity in an informal language?

Consistency of a set of axioms only applies once the system has been unambiguously stated. If an informal language permits sufficient ambiguity of meaning, then it isn't possible to construct a single definite system based on that language.

There is the very strong impression that well defined systems are specified. Yet that specification starts with an informal language that is inherently not fully defined. Even partial ambiguity (lack of definition) could change the system being described beyond all recognition. Is there a mechanism of working with partial definitions that I'm missing? Is there a mechanism for eliminating all uncertainty from informal languages?

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Re: Circularity in Formal Languages?

Postby Forest Goose » Mon Jan 26, 2015 9:18 am UTC

Treatid wrote:There is the very strong impression that well defined systems are specified. Yet that specification starts with an informal language that is inherently not fully defined. Even partial ambiguity (lack of definition) could change the system being described beyond all recognition. Is there a mechanism of working with partial definitions that I'm missing? Is there a mechanism for eliminating all uncertainty from informal languages?


Put up or shut up: show me a system that suffers such a defect, I have yet to see one that is adequately given. You are the equivalent of someone arguing that since we can't absolutely prove the physics of circuitry will follow the rules we've written that, therefore, we cannot, at all, trust computers - or that we should be highly skeptical of them...or, in your case, more like saying one shouldn't walk, lest the slide off the planet (all of physics being untrustworthy).

So, anyways, let's drop the speculation, show me a formal system, one from actual mathematics, that changes from all recognition, do this in a way that would be acceptable of a mathematics publication (not a ranting crank).

Have fun:p
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Re: Circularity in Formal Languages?

Postby Treatid » Mon Jan 26, 2015 4:01 pm UTC

You are absolutely right.

I can't prove that any given use of informal language is mistaken. Mathematics doesn't apply to informal languages.

But by the same token - there is no mechanism to show that a given interpretation is right, valid or exclusive of other interpretations.

Mathematicians choose to interpret informal language in a certain way. If someone comes along and has doubts about the precise meaning or interpretation, there is nothing mathematicians can do to clarify that doubt.

The whole purpose of introducing axioms to mathematics was that without knowing the full context of a statement - that statement can have any meaning.

We don't know the full context of informal languages. That is what makes them informal.

Mathematics has tried really hard to make axioms a self contained system. It has proven impossible to do. So mathematics invokes informal languages. The implication is that informal languages are able to do something that has been shown to be impossible within formal languages.

How can informal languages do something that is impossible for formal languages?

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Re: Circularity in Formal Languages?

Postby Forest Goose » Mon Jan 26, 2015 6:58 pm UTC

That's neat...but, how about instead of positing on and on and on, you show me an actual example of this that doesn't involve user error.

I mean, yeah, sure, philosophizing is fun and all (especially when you're ruminating over a subject you don't really know so, obviously, are precluded from actually doing), but unless you can point to some example of this, I'm not sure why I should care.

Like I said, it could be that the laws of physics are real real crazy or, maybe, there aren't even laws - or maybe we're in the matrix :roll: - but until someone can point to some actual consequent of that, I fail to see why anyone should regard it as more than a curiosity to ponder on a long car ride...surely you aren't here just to make my drive to work more amusing? For all that you've "proven math has no foundation" and is a "loop that <blah blah blah>" and has "no meaning" :roll: , you sure don't seem to be able to point out any failures or issues.

So, again, why should anyone care?

PS
You've taken the crankdom nosedive, by the way, which makes it even harder to trust you that this is all meaningful and important.

That nose dive being "you sound like x", x progressing as: "interested amateur" -> "confused amateur" -> "college freshman taking philosophy" -> "bitter your "paper" was rejected from ten journals" -> "postmodernist borrowing math terms" -> "crank" -> "exposed crank who is now going to "unmask" the absurdity of mathematics by showing "the emperor has no clothes"". You should, perhaps, dial it back down to level 1 - 3, anything past that is real bad, and you seem to be up to 7 - and, yet, I bet you're the one people look at as if they were wearing no clothes; at least I would if you were telling me such in person.

Have fun - still waiting for that publication grade expose that makes ZFC appear all "out of recognition":p
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Re: Circularity in Formal Languages?

Postby Treatid » Mon Jan 26, 2015 8:16 pm UTC

Consistency is the measure of the value of a system in mathematics.

If there is a contradiction within a system then that entire system is void.

Just one contradiction within a system means that every statement within that system can be contradicted. We don't have to individually show that every statement can be contradicted.

There is a contradiction with axioms.

  • A statement only has meaning with respect to a set of axioms.
  • An axiom is a statement.
  • Informal language is not a fully defined system.
  • statements in informal language have known meaning.

"But it is okay for axioms to be inconsistent because philosophy and crack-pot and reasons."

It doesn't matter how useful you found that system. It doesn't matter how much you have invested in that system. Just one contradiction means the system isn't valid.

As axioms are currently constructed, every statement depending on axioms can be contradicted.

Edit: The very premise of axioms has to be broken in order to make them work. It has to be assumed that some statements have a known meaning independent of a defined context. How much of a red flag do you need?

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Re: Circularity in Formal Languages?

Postby arbiteroftruth » Tue Jan 27, 2015 12:05 am UTC

Treatid wrote:
  • A statement only has meaning with respect to a set of axioms.
  • An axiom is a statement.


Depending on how you want to phrase things, one of those two assertions is false. At the bottom of math, you have statements that have meaning with respect to basic human intuitions, not a set of axioms. Or you might phrase it by saying that those basic human intuitions are axioms, but not statements, because they're internal to the human mind.

We then accept those basic intuitions not on the basis of rigorous mathematical justification, but because if such basic concepts aren't valid then the entire human endeavor to cope with existence is screwed regardless of what we do next. So we might as well be optimistic and take those basic intuitions on faith and see where that gets us. *Then* we go about rigorously deriving everything else from those most fundamental intuitions.

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Re: Circularity in Formal Languages?

Postby Forest Goose » Tue Jan 27, 2015 4:46 am UTC

Treatid wrote:As axioms are currently constructed, every statement depending on axioms can be contradicted.


Provide me a contradiction in ZFC, please.

The real problem, though, is that you are conflating various senses of the phrases/terms you are using - especially "meaning", I don't think you know what it means. But, the first clue that your...analysis is flawed is that, if correct, it should also entail that your argument is meaningless. I want to be charitable and say there's something Wittgensteiny about that...but, then, I'm also aware that this chatter about meaninglessness followed about ten threads where you ranted about proving extremely difficult problems - while acting like the champion of mathematics - and, then, were soundly rejected by, well, everyone.

In other words, less clever philosophy, more "moan moan I'm bitter no one believes me about C=P=NP=T (so atoms know 3SAT...or whatever)...but I'm good at math, my highschool teacher told me so, thus, math is wrong".

But, yeah, please pony up on that contradiction.

Or, hey, stop playing at being a philosopher, buy a few math books, and work through them - it won't make any of this right, but it might get you back on track. (inb4 "But I've studied all the maths and am real real good at them! Seriously guys, I know C=P=T!")
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Re: Circularity in Formal Languages?

Postby elasto » Tue Jan 27, 2015 12:35 pm UTC

Thought experiment for you Treatid: Let's imagine there's only one person in the whole universe, and he happens to be a mathematician.

While it's true that axioms are defined in language which is notionally informal, this mathematician 'knows exactly what he means' when he says to himself a statement like: 'Axiom: The empty set exists'. To him it's as formal as anything can be. To assume that he's mistaken in his understanding of his understanding would be to doubt anything and everything of the world, and that way madness lies.

This mathematician can now construct all of mathematics without contradiction - proving that your proof of math's fragility is misguided. Your problem (as I think we've stated in repeated threads) is that you can't prove formally that if person A says something to person B that person B will end up with exactly the same internal representation of it as person A does.

Fortunately, as FG says, in practice that kind of disconnect simply doesn't occur - at least not permanently - and definitely not with statements as unambiguous as 'let's assume that the empty set exists'.

Sure, when mathematical proofs come out (eg. Andrew Wiles' proof of FLT) it can sometimes take a lot of effort to understand them. But that's down to their sheer length and novelty, and not a result of mathematical language being 'too informal for the job'. It works just fine thankyouverymuch.

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Re: Circularity in Formal Languages?

Postby Forest Goose » Tue Jan 27, 2015 2:33 pm UTC

^
I agree with everything you say, but did want to point out that you don't, actually, need to know what "the empty set exists" means, not to do produce proofs, at least. We can write all the axioms as strings and proceed formally; hell, we can even remove creativity from the mix and just try every possible sequence of valid strings and rules till a proof of the theorem at hand.

The point being that, while no one works this way, nor should, to get results requires next to nothing philosophically. And this is where Treatids confusion comes in - meaning has no bearing on terms like contradiction and consistency. The only time this doesn't apply is once we've stepped out of first order logic, but, even then, there's nothing that requires the viewer guess at what is meant.

Of course, one may not always understand what mathematical language means, or one may get it wrong - but this is the fault of the viewer. As for what math means to me, internally, that's a human thing, and while beautiful, is not required for math to proceed (hell, we don't even need to have the same models in mind for math to work)

Of course, I think taking this as one's philosophical perspective for what math truly is, formalism purely, is like being Christian because Pascals Wager - but, like quantum physics, your interpretation and philosophy don't matter, that's all just inspiration and faith :-)
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Re: Circularity in Formal Languages?

Postby Treatid » Tue Jan 27, 2015 4:13 pm UTC

Let me get this straight.

We have two distinct systems with distinct rules.

We have the real world from which we derive intuitive thought. And we have mathematics which is founded on intuitive thought.

The rules of mathematics and the real world are different. In the real world, meaning is intuitive - it doesn't have to have a defined context - it is simply understood.

In mathematics, meaning requires a defined context.

This is what I understand when you tell me that the rules of axiomatic mathematics don't apply to axioms themselves.

And you are okay with this? Despite the fact that the whole purpose of axiomatic mathematics is that intuitive meaning isn't satisfactory?

arbiteroftruth: You present a choice - either axioms are valid or we have nothing. This isn't the case. We are (relatively) sentient and we can learn. Just because one approach to knowledge turns out to be wrong doesn't mean we should give up. When a house falls down we learn from our mistakes and build a better house. Perhaps put the roof on top next time.

Intuition is sometimes helpful in pointing us towards a line of investigation. Sometimes that line of investigation was misguided. Intuition is frequently wrong. We have formal systems because we know we can't rely on intuition. Using intuition as the sole basis for your system of thought is silly. You might as well start a religion.

We don't need to fill the gaps with God or Intuition. All we need to do is realise that we have taken a wrong turn, back track, and see what other avenues are open.

Forest Goose: Okay. ZFC Set Theory: The empty set. The empty set is defined as a set that contains itself. This contradicts the axiom of regularity. All statement following from ZFC Set Theory can thus be contradicted.

"No, no, no." You say. "That isn't what the empty set means."

Really?

Prove it.

"Axioms lie outside formal mathematics. They are intuitive."

You mean that you can't demonstrate or otherwise prove a given meaning? Yes. Quite right. Which means you can't disprove my chosen meaning. My meaning is exactly as valid as your meaning.

"But you are just being perverse."

If you cannot prove that your meaning is the only possible meaning - then you can't claim that it is the only possible meaning. This isn't a matter of democracy. It isn't a question of how many people have voted for a given meaning. You either have a defined meaning or you don't. Using informal languages and/or intuition does not give you a single definite, unambiguous definition.

elasto: It looks like you are assuming that a mathematician's thoughts are automatically true provided they aren't tested against an outside perspective. You seem to be assuming that thought can somehow behave differently to language. This is the same fallacy as axiomatic mathematics. Sweeping the inconvenient bits into an unknown and claiming to have explained something is not science. We want to be able to describe the system we find ourselves in (our reality). If we find something that is immune to our description - then there is a fault in our description - not a fault in reality.

By your argument - everyone's thoughts are automatically correct so long as they are never communicated.

Forest Goose 2: Pure symbol manipulation.... We can run a computer program and say that it is proof of the computer language. This is an empirical proof rather than a logical proof - but maybe there is a connection between the two types of proof.

Now all we need to do in order to prove our abstract theory/model is to map the computer language to the theory. Once we have done that then the Theory must also be true.

Unconstrained mapping allows us to map any program to any theory (or any theory to any program). So the programs that crashed - the programs that disprove themselves - can all be mapped to any given set of symbols (theory). Alternatively, every Theory is equivalent to every possible computer program.

Shifting the problem around doesn't eliminate it. Whether the definition is asserted before the symbol manipulation or after the symbol manipulation - the definition is, in fact, asserted. The same problem exists at both ends. How do we know that one mapping (or one definition) is correct? How do we exclude alternate mappings?

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Re: Circularity in Formal Languages?

Postby arbiteroftruth » Tue Jan 27, 2015 4:26 pm UTC

Treatid wrote:Let me get this straight.

We have two distinct systems with distinct rules.

We have the real world from which we derive intuitive thought. And we have mathematics which is founded on intuitive thought.

The rules of mathematics and the real world are different. In the real world, meaning is intuitive - it doesn't have to have a defined context - it is simply understood.

In mathematics, meaning requires a defined context.

This is what I understand when you tell me that the rules of axiomatic mathematics don't apply to axioms themselves.

And you are okay with this? Despite the fact that the whole purpose of axiomatic mathematics is that intuitive meaning isn't satisfactory?


Yes. Because the appropriate response to "intuitive meaning isn't satisfactory" is to try to minimize our reliance on intuition, not to eliminate our reliance on intuition. Trying to completely eliminate reliance on intuition is impossible for exactly the reasons you describe, so we don't do it. We take only the most basic of intuitions (like "logical deduction works"), where reliance on intuition is as close to satisfactory as it can be, and then start trying to be formal with those intuitions as the starting point.

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Re: Circularity in Formal Languages?

Postby Gwydion » Tue Jan 27, 2015 4:41 pm UTC

Treatid, I'm not sure I understand your entire response there, but I'll take one part that stands out, namely your response to FG saying that ZFC is inconsistent. If you choose to define the empty set in the way that you did, namely as being a set that contains itself, then that axiomatic system which also includes the axiom of regularity (as written and commonly understood) would be inconsistent as you said. This doesn't invalidate ZFC or any other axiomatic systems, it merely reflects the fact that your chosen axioms (as defined) are inconsistent. No one is suggesting that ZFC is the *only* axiomatic system, and in fact there are infinitely many others, but it happens to be one in which, when you use the axioms as written and commonly understood, seems to produce some remarkable results from seemingly simple origins. Currently, we believe that ZFC is consistent because an inconsistency has never been found; this fact makes it no more or less applicable to the real world, no more or less "meaningful" in that sense, but does make it mathematically meaningful in that statements derived from it logically are also consistent with this model.

Demanding that someone "prove" that your definition of the empty set is wrong has no value. Your definition is not the same one being used in the ZFC model, and by that standard it is wrong, in the same way that I am wrong in saying that the axiom of infinity means that a set can have at most 10 elements because that's how many fingers I have. This may reflect a real-world concept, but this "axiom" leads to a contradiction with the other traditional ZFC axioms, so that axiomatic system is not useful mathematically. To take an example of changing an axiom and still producing a consistent system, non-Euclidean geometry is often used as an example - by modifying the parallel postulate and combining it with the other axioms, other consistent systems like hyperbolic geometry can be created. Neither of these are "right" or "wrong" - they are both consistent axiomatic systems from which different results follow.

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Re: Circularity in Formal Languages?

Postby Sizik » Tue Jan 27, 2015 4:57 pm UTC

Can we rename Treatid to Tortoise?
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Re: Circularity in Formal Languages?

Postby Forest Goose » Tue Jan 27, 2015 5:50 pm UTC

Treatid

The problem, as I keep saying, is that you don't actually know, or even sort of understand, mathematics, thus, you are just stringing together words...like people talking about particle physics over drinks after reading a popsci book, or watching Star Trek, or reading a bunch of Wikipedia.

For example, the axiom of the empty set is not "the empty set exists", then everyone just knows what "empty set" means. The axiom of the empty set is "There exists a set E such that for all sets X not X is an element of E", with each of those logical words functioning in a specific way. In other words, your whole "but I declare the empty set contains itself" doesn't even get off the ground - or you arent talking about ZFC. Either way, your objection misses the point and misunderstands the question - as do you.

For example, your dichotomy of empirical proofs and regular proofs is your own - you seem to be making that distinction because if you don't, your argument just crumbles - the problem is that it isn't a real distinction, there's no content, you've just said, essentially, "proofs that show my argument are junk are a different type...so yeah, since they are, you're wrong"...real classy, that. Maybe you can join up with some creationists and discuss the distinctions between micro/macro evolution later... :roll:

I could go on...and on, but I'll stop, seeing as you have yet to provide an example or to counter my argument, or to even display a basic understanding of terms that you're using.
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Re: Circularity in Formal Languages?

Postby z4lis » Tue Jan 27, 2015 10:40 pm UTC

You can't disprove the definition of a word.
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Re: Circularity in Formal Languages?

Postby alessandro95 » Wed Jan 28, 2015 1:47 pm UTC

Treatid wrote:see what other avenues are open.


Can you define unambiguously any of those other avenues? If you can't they're just as bad as the axioms.
Can you do working mathematics in any of those other avenues? If you can't they actually worse than the axiomatic approach.
Isn't it about time you show us a working alternative, if you are so sure that there is one?

Treatid wrote:This is an empirical proof rather than a logical proof


I don't know what the meaning of those terms is (or what your meaning for those terms is), can you please explain it in a way so that they're meaning is context free and doesn't rely on informal intuition?
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Re: Circularity in Formal Languages?

Postby Treatid » Wed Jan 28, 2015 9:15 pm UTC

We are in agreement that the existing use of axioms isn't as ideal as we would like. The Foundational Crises in mathematics came along well before me. In pointing out a lack of perfection I'm not bringing anything new to the table.

The significant arguments for the status quo that I am seeing are "Yeah... but what's the alternative? Something is better than nothing. It seems to be working fine despite the issues".

I'll briefly reiterate my opinion that using informal languages to state axioms is a direct contradiction of the first premise of axioms: That a statement only has specific meaning with respect to a set of (defined) axioms. One contradiction in a system means that every statement within that system can be contradicted.

As such, axioms working must be an illusion. And axioms are nothing; they are no better than any other system that contains a contradiction. In any case, we disagree on the magnitude of the problem - but not on the existence of the problem.

[sorry the following is so long - but how brief can an introduction to an alternative to axiomatic mathematics be?]

But what is the alternative

Plainly we do perceive meaning within language. That meaning might be a bit hard to pin down, but there is something there. We can obviously learn and improve our understanding. Nihilism isn't an option.

Personally I think that we can know and communicate everything that can be relevant to us. There is nothing unknowable that can apply to us.

So: There exists an assumption that it is possible to have an absolute. Whether the absolute is a fixed starting point, a fixed reference frame, or a fixed definition. This assumption is embodied in the current use of axioms.

Without the use of informal language, the construction of axioms is a closed loop. To define something we first need a definition.

This tells us that the concept of an absolute is mistaken to the point where we can't define it. (Perhaps we could start with a fixed definition (or other fixed reference) - but this also leads to contradictions that show that whatever "absolute" might be - it isn't relevant to anything we can communicate).

Which is to say - absolutes don't exist.

Which leaves 'Relatives'.

Our world is a network of relationships. The meaning we perceive for a given part of that network depends on its relationship with the rest of the network.

This isn't very surprising. Category Theory recognises that the bits between relationships are irrelevant. Only the relationships matter. (Unfortunately, Category theory then spoils it by trying to specify axioms - but it is heading in the right direction).

In an axiomatic/absolute view of the world, a thing has significance by itself. It may be considered as 'true' without regard to the rest of the system.

In a Relative view of the world, a concept changes when the rest of the system changes. As such, it is impossible to consider anything in isolation. Concepts/things are only meaningful in the context of the whole system. That doesn't make simplification or abstraction totally impossible - but it does tightly constrain such things.

At a macroscopic level, we have planet Earth as an anchor. While it remains true that everything is 'defined' by everything else; we have some elements of our experience that seem to be pretty fixed. It is this experience of relatively stable elements of our world that support the belief that there could be a fixed reference frame. So long as we work within the scope of those fixed entities, then pretending that they are fixed is viable.

General Relativity showed that the assumptions of a fixed reference frame do not hold true at large scales.

How do we build a system from first principles

We can't build a system a piece at a time. A piece only gains some significance in relation to other pieces.

We must describe the whole system - only then can we see if what we have described is meaningful. It is the way the pieces fit together and interact that defines a system - the actual pieces are largely irrelevant.

So, when I start with a 'network of relationships', we are starting with a label. There is no assumption that you know what a 'network of relationships' is.

We also have 'change' which applies to that 'network of relationships'. You should infer that a network-of-relationships is capable of change. Our macroscopic conception of change is informed by the vast network of relationships within which we live. This more fundamental definition of change is simply what a network-of-relationships (whatever that might be) is capable of.

You can start off with different labels - but changing the labels doesn't change the system.

In theory, we could connect as many concepts as we like together in this way. However, these systems are entirely abstract. A is related to B and C. B is related to A and C. C is related to A and B. For as many letters of the alphabet as we wish.

Unless we know what B and C are, we have no clue what A is. We haven't specified anything yet.

Given our two concepts: A network-of-relationships and 'change', we can only begin to understand one by understanding the other.

Okay. Let us pretend that a network-of-relationships is what languages can describe. This isn't an entirely arbitrary choice, but the justification for this choice won't become apparent until we have the whole system.

We can now use our normal conception of a network of relationships to think about a 'network-of-relationships' and, in turn, 'change'. We know how to change one network of relationships into another network of relationships, so we know what 'change' must now be.

Since 'change' is defined by our conception of network-of-relationships - it doesn't matter very much what our conception of a network of relationships is. A different conception of a network-of-relationships would result in a different conception of 'change' - but the system described would still be the same.

However...

Let us assume that there are things that language can do. And things that language can't do.

There is no question that language can describe networks of relationships. Mathematics describes networks of relationships all the time. Without worrying about whether a particular network conforms to ZFC axioms, we can draw a network of relationships on a piece of paper without any ambiguity. Graph Theory and Category Theory do this quite explicitly, but any mathematical equation represents some aspect of a network of relationships.

At the same time, we know that we cannot define anything in an absolute sense.

If there are no absolute definitions - then what else can language describe beyond a network of relationships?

We ascribe meaning to different concepts. I will suggest that those concepts are, themselves, networks of relationships. Language only ever describes a network of relationships. Without absolutes - there is nothing else that can be described.

If we want to describe the universe, we have to assume that the universe can be described. If the only thing that language can describe are relationships, the the universe has an aspect that corresponds to our notion of relationships. By the same token, we can describe relationships because the universe has an aspect that is relationships.

Note that if an aspect of our system is a network of relationships, and all we can describe are networks of relationships - we have nothing else we can describe to map to C, D, E and F in our earlier construction. We used up language on A. There is nothing left for the rest of the alphabet.

We can only describe what language allows us to describe. Without axioms - without absolute definitions - we are extremely tightly constrained on what we can describe. To whit: we can only describe networks of relationships and changes to them. Which means that rather than there being an infinite number of ways to describe a given system, there is, in fact, only way. This makes life easy. There aren't any choices to make.

Even if you still think that axioms are valid - this is an alternative approach to knowledge. The premise that all things are relative has only one path. Don't take my word for any of the above. Simply start with the idea that context is king (shouldn't be hard - that is exactly where axioms start - "a statement only has meaning with respect to a specific context"). That context is everything else within the system. Go from there.

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Re: Circularity in Formal Languages?

Postby arbiteroftruth » Wed Jan 28, 2015 10:54 pm UTC

So I have a network of relationships, and I'm allowed to describe things in terms of their relationships to everything else in the network, correct?

Am I allowed to conceive of a thing in the network such that the relationship between that thing and any other thing in the network is always the same?

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Re: Circularity in Formal Languages?

Postby alessandro95 » Wed Jan 28, 2015 11:06 pm UTC

Treatid wrote:That a statement only has specific meaning with respect to a set of (defined) axioms.


Kleene (1950) wrote:As the first step the prepositions of the theory should be arranged deductively, some of them, from which the others are logically deducibile, being specified as the axioms.
...
Then it should be possible to perform the deduction treating the technical terms as words in themselves without meaning.


Kleene (1950) wrote:From the standpoint of the metatheory, the object theory is not properly a theory as we formerly understood the term, but a system of meaningless objects subject like the positions in a game of chess, subject to mechanical manipulations like the moves in chess.






Treatid wrote:In an axiomatic/absolute view of the world, a thing has significance by itself. It may be considered as 'true' without regard to the rest of the system.


Hilbert (1928) wrote:It is by no means reasonable to set up in general the requirement that each separate formula should be interpretable taken by itself


quotes from "introduction to metamathematics", Kleene (1950), one of the standard beginners book on the subject.




Treatid wrote:Without the use of informal language, the construction of axioms is a closed loop. To define something we first need a definition.


Aren't you using an informal language to describe your networks of relations? It actually seems to me to be much more hand-wavy and informal than the usual axiomatic approach



Treatid wrote:Okay. Let us pretend that a network-of-relationships is what languages can describe.


I'm pretty sure I can describe the axiomatic approach or a specific formal system using a language, does that make it a network of relationship?





Treatid wrote:We can now use our normal conception of a network of relationships to think about a 'network-of-relationships' and, in turn, 'change'. We know how to change one network of relationships into another network of relationships, so we know what 'change' must now be.


So basically, a network of relationship is anything that can be described by a language, and we can change anything describable into anything else describable?
How do we know how to change one network of relationship into the other?




Treatid wrote:We can only describe what language allows us to describe. Without axioms - without absolute definitions - we are extremely tightly constrained on what we can describe.


Unless we have very different ideas of language i can describe axioms in a language




Ok, let's say that sounds like a possible alternative, a very hand wavy and informal one to be honest, but still an alternative, can you show me how to do basic arithmetic in a network of relations?
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Re: Circularity in Formal Languages?

Postby Twistar » Wed Jan 28, 2015 11:12 pm UTC

The obligatory link for every time we have this discussion

Treatid, It's like this. When you build a house you build a foundation. Once you have the foundation you can build whatever you want on top of it. The house is mathematics and the foundation is the axioms of mathematics. Typically, houses are built on solid earth. So the entire structure from top to bottom is entirely rigid. This is satisfying, nothing is going to happen to the house.
However,
You could just as well have built the foundation on a boat and built the house up on that. Now it isn't rigid all the way down. In fact the house is built on moving flowing liquid. But... there is still a house there. You can live in it. This is more what mathematics is like. Yes, you are correct that the axioms of mathematics don't stand on solid, rigorous, rigid ground. Instead they stand on intuition which is certainly much more fluid and wishy-washy than cold hard logic. No one is denying that

I think your problem is that you want the system to be entirely rigid and solid the whole way through. You are probably hoping that math can tell us 'truths' about the universe. But you have set too lofty of a goal for mathematics (or any rational endeavor as I will soon explain). It is no surprise that axiomatic mathematics fails to live up to your overly lofty goals. I went down your same line of reasoning many years ago but drew different conclusions. I realized that axiomatic mathematics (or more generally axiomatic thought) works very well once you have the axioms, but I also realized that there was no rational or logical way to choose whether an axiom is true or not. The question then, is how do we, as humans, decide which axioms we will accept and which axioms we will reject. The answer is that we just pick one. What do we make that choice on? The short answer is we make the choice based on faith. The long answer is that we use some arational* psychological part of our minds to guide us. Once we do that obviously all of mathematics follows.

My stance can be summed up as follows: "All human knowledge is contingent upon some piece of knowledge taken on faith." This statement will probably upset you and a lot of other people, but as for myself I haven't been able to convince myself that it is not true, so rather than be upset by it I have become comfortable with it and come to accept it as part of the human condition. So yes, mathematics is built on a foundation which is floating on water (faith,) but that is not a problem as you claim it is. You will find that your "relative mathematics" also relies on context and ideas which you, at some level, take on faith.

In any case I'm curious what you and other have to say in response to this.

*arational - not a typo. To be contrasted with "irrational," by arational thinking I mean thinking which is not necessarily rational but neither is it irrational. For example, it is rational to to buy a car which you like more and is cheaper than the alternative. It would be irrational to buy a car which you like less and is more expensive. If both cars cost the same and have the same features but a different body, then you must make an arational choice about which one you like more to determine which to buy.

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Re: Circularity in Formal Languages?

Postby z4lis » Thu Jan 29, 2015 12:54 am UTC

We can only describe networks of relationships and changes to them. Which means that rather than there being an infinite number of ways to describe a given system, there is, in fact, only way. This makes life easy. There aren't any choices to make.


What.

A is below B

B is above A

Only one is right, but which one is it? Stay tuned.
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Re: Circularity in Formal Languages?

Postby Forest Goose » Thu Jan 29, 2015 4:38 am UTC

Treatid wrote:Blah Blah Blah...foundational crises I read the Wikipedia lede about...blah blah blah...my math didn't work in the prior threads because contradictions...blah blah blah...philosophical words...blah blah blah...networks of relationships...blah blah blah...lack of any examples of how these neat sounding pseudophilosophical terms work or that stringing them together this way is expressing anything...blah blah blah...still no examples, or real arguments, showing that there is even a problem...blah blah blah...I don't appear to have actually studied anything pertaining to the foundations of mathematics, but you should listen to my large rambling lectures telling you the true foundations of a subject I don't really know - it's alright, students (I'm the self-proclaimed lone comprehender of truth and language, so "students"), while I'm sure I could master them in a night, I don't need to because they're all wrong anyhow...blah blah blah...surely none of this is egomaniacal, surely none of this is condescending to those who devoted their lives to it and know way more than me about the subject...


That's what I read when I look over your paragraph - not seeing anything worth taking serious; hell, you're describing your system in more informal terms than all of the informalness you're bitching about - speaking of that, demonstrate your depth: explain to me just what philosophical commitments axioms actually require, or even what we are forced to deal with informally (and why that's bad, or even how we do it), use ZFC as a reference point.

What bugs me, here, though, isn't that you are absolutely clueless, but that you have such a supercilious smug approach - almost like you are unaware that you have, seemingly, little real math skill; and even less pertaining to foundations. Nope, instead you show up like the kid who watched the Matrix fifteen times to tell all the black belts how they've been doing Kung Fu wrong all these years - despite not being able to jog without getting out of breath.

This is not a debate between equals, this is not the outsider wearing down the stodgy old guard, this is not Socrates being berated while he calmly shows the underpinnings aren't as sturdy as we thought...this is Charlie and the lawyer and you are the man going on about bird law.

Oh, and don't take this as me trying to say "I'm smarter than you, I don't have to listen, so you're wrong.", nope, you're the smug one here. I, actually, think I'm quite a bit stupider than most people - however, I have spent, literally, 30,000+ hours studying the foundations of mathematics and the philosophy of mathematics...so, yeah, when you've clearly nothing more than an afternoon on Wikipedia and some bad memories of your ideas being rejected backing you up, it's a little insulting for you to lecture, without any seeming doubt, about how my whole field is not just wrong, but ought, clearly, be replaced by your nebulous lunacy.
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Re: Circularity in Formal Languages?

Postby Treatid » Fri Jan 30, 2015 6:31 pm UTC

Axioms have been around practically forever. And they aren't just some piece of mathematics. A vast amount of modern thought incorporates the assumptions of axioms. The ideas of logic and proof are touch stones for all of science.

Axiomatic thought impacts so much, and so many people have worked with, in and around axiomatic thought, that it isn't credible that there is some serious flaw. Even less so that the flaw be pointed out by some smug sounding git on a random forum.

Granted the implementation of axioms isn't perfect. But expecting some platonic ideal is unrealistic. Whatever slight lack might exist in the ideal form can easily be filled. We know what we intend even if we can't quite articulate it in exact terms.

If ever there was grounds for skepticism, then an attack on the validity of axioms is it.

And yet the 'proof' of axiom's complete invalidity is readily accessible to anyone with a passing familiarity with mathematics and a little bit of self honesty.

The elements of the proof are well known and understood. There is nothing esoteric or controversial in the individual statements:

1. Any contradiction in an axiomatic system means that every statement of that system can be contradicted. Such a system is null.

Some people may feel that the words and symbols need to look more mathematicky. Feel free to substitute symbols if it makes you happy. But mapping one set of symbols to another set is an entirely arbitrary process. Either you have a sense for what those symbols mean, or you don't (pretending to be formal within informal language isn't productive).

Anybody familiar with axioms knows that a single contradiction means that the system described is null; invalid; not an actual description of anything.

There is no controversy about this. It is well established, standard mathematics. There are no edge cases to worry about. One contradiction: instant death.

2. A statement only has meaning with respect to a (defined) set of axioms.


The first premise (axiom) of axiomatic systems. Reject this and you are rejecting axiomatic system.

3. Axioms are, themselves, statements.

A couple of people have attempted to argue that axioms aren't like other statements. That is an... interesting... level of denial.

4. Informal languages (and the elements thereof) are not defined in the axiomatic sense.


This is why they are called informal. This is probably the area most open to doubt or ambiguity. We do perceive meaning in a given language. It feels like we know what certain words or phrases mean. This sense of meaning can (and is) confused with definition. But the axiomatic definition of definition is clear and unambiguous. Informal languages are not defined in the axiomatic sense.

5. Initial axioms are expressed using informal language(s).

We have five statements. Each statement by itself is well understood. There is no controversy.

...

Now put them together. This isn't esoteric, higher order, abstract philosophy. It is just ordinary axiomatic mathematics. You put some statements together and see if they are consistent. No need to look in the back of the text book for the answer. No need to wait for some authority figure to tell you it is okay to think.

Perhaps this one specific formulation of axioms doesn't work - but a little twerking will produce a consistent system? Play around. Change the assumptions.

...

Axiomatic thought is pervasive. Even if the evidence might, possibly, maybe, point towards some degree of inadequacy within axioms, it isn't immediately obvious what a world without axioms looks like.

What are the practical results of abandoning axiomatic thinking?

In immediate practical terms - there is almost no consequence.

Axioms haven't suddenly vanished in a puff of logic. If axioms are impossible now - then they were always impossible. Nothing has been lost. Axioms never contributed to anything.

Practical uses of mathematics work. They don't need to be justified by axioms. That they work is their justification.

Absolute definitions, or course, don't exist. The Euclidean plane doesn't exist. Our local, subjective understanding of a flat piece of paper exists. Our conception of lines and points and triangles as actual lumps of ink on paper, or lit LEDs on a screen, is fine. The mathematical abstractions took away all the essential components and left... nothing. The mathematical concept of a dimensions told us nothing. We associated our physical experience of dimensions and manipulated that direct experience. No-one ever knew what a mathematical dimension was.

The sets of relationships that mathematics describes are valid to the extent that they correlate with real world experience.

The major change in realising that axioms don't exist; is in the way we understand the world around us. Again, we haven't removed a previous understanding - we have simply realised that part of our attempt at understanding was an illusion with no substance. No doubt this is one of the choke points for many people. The perception of having worked with axioms is compelling. Compelling to the point that mathematicians have ignored the evidence that has been available for at least a hundred years, in favour of believing in the illusion of axioms.

An objection to my last post was that I was using informal language. Well yes. That is the point. Absolutes as imagined by axiomatic thought don't exist. There are no formal languages. You can't construct formalism. We communicate using informal languages. There isn't an alternative. You can't construct a language that 'can be understood by any conceivable observer on sight'.

The assumption that there is some absolute - a fixed starting point, an absolute reference frame - is deeply ingrained. It will take a long time to adjust to an entirely relativistic view.

Another view that came up in response to my last post was the idea of 'rigidly defined areas of doubt and uncertainty'.

We can only know what we can describe. If we cannot fully describe it - we cannot know it.

If we cannot describe something, then we can't distinguish it from the other things we cannot describe. As far as we are concerned, all the things we can't describe are a single indistinguishable blob. There simply isn't any point in worrying about the things we can't describe.

It may not be immediately obvious, but only things we can describe can have an impact on us. If it is impossible for us to know (describe/communicate) something - then that thing is not part of our reality.

As fundamental properties, dimensions, spaces, particles, waves, and any other supposed elemental property is irrelevant to us. They are null concepts.

On the other hand, we have a physical experience of distance, electricity, light and sex. We can understand these things in considerable detail. Not as absolutes - but as interacting elements of a total system. This may not be the way you wanted to understand things - but no amount of wishing for the impossible will make it possible.

arbiteroftruth
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Re: Circularity in Formal Languages?

Postby arbiteroftruth » Fri Jan 30, 2015 6:44 pm UTC

Treatid wrote:1. Any contradiction in an axiomatic system means that every statement of that system can be contradicted. Such a system is null.


Agreed.

Treatid wrote:2. A statement only has meaning with respect to a (defined) set of axioms.


False. Informal language has meaning. Not rigorous, "axiomatic" meaning, but meaning nonetheless.

Treatid wrote:3. Axioms are, themselves, statements.

...

4. Informal languages (and the elements thereof) are not defined in the axiomatic sense.

...

5. Initial axioms are expressed using informal language(s).


Agreed.

But your argument fails because your second premise is false.

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Forest Goose
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Re: Circularity in Formal Languages?

Postby Forest Goose » Fri Jan 30, 2015 7:51 pm UTC

^
5 isn't true either. Every axiom of ZFC is in the language of set theory, which is a formal language.

You can, rightly, point out that, at some point, we need to use informal terms to get the process started - but that begins, and ends, with logic; and, at that level, it is describing how to manipulate the symbols, which is, once you get used to it, quite simple and basic. That we have to use informal terms to say things like what a string of symbols are, or that conjunction is commutative, or etc. doesn't really seem a problem.

@Treatid

2 is false too - this isn't the time of Euclid. Statements in formal languages don't have "meanings", that's not really how they work - the best sense of meaning I can see existing in axiomatic mathematics is in terms of models, and, in that sense, one major part of foundations is that ZFC has tons and tons of models. Heck, there's a model of ZFC + ~Con(ZFC), even; have fun with that:-)

But, at any rate, axioms don't give "meaning" to statements in formal languages - there are theories that don't even have axioms, actually.

4: neither are formal ones. However, theories in informal languages do use the axiomatic method, so there's that - heck, you're using the axiomatic method as we speak :shock:

By stating 3, you're giving yourself some problems, friend: one of the biggest bits of informal language is defining what a statement is - since you are, apparently, okay with statements (and, clearly, with using logic), I'm confused as to what you're disagreeing with...or, what, are you reductio'ing logic and language, supposedly? If that's how I'm supposed to read what you're doing, then my reply to you is "Flugsnatz,,,;goob-li=li-gock". Why? Because I declare my string defeats yours, and with logic and language gone, I can do so. In other words, you're either hanging yourself or admitting you reject everything, in which case anything goes - have fun on the horns of that dilemma :roll:

-----

As always, it's pretty clear you don't actually know anything about the foundations of mathematics. What is clear, however, is that someone is real grumpy that their proof of C = P = NP = T using graph theory didn't work - you keep grinding that axe, but seem to be getting duller and duller and duller...

Final thoughts: This:

Now put them together. This isn't esoteric, higher order, abstract philosophy. It is just ordinary axiomatic mathematics. You put some statements together and see if they are consistent. No need to look in the back of the text book for the answer. No need to wait for some authority figure to tell you it is okay to think.


Really cements my feelings that you are upset with academics rejecting your absurd "proofs"; you know, from earlier, back when you were King of this ridiculous foolish endeavour us authority-figure-looking-to mathematicians pursue.

And this:

Perhaps this one specific formulation of axioms doesn't work - but a little twerking will produce a consistent system? Play around. Change the assumptions.


Describes how I wish I could approach playing around with metamathematics...who hasn't wanted to twerk at large cardinals every now and again? (I don't have the ass, sadly)
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

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Dopefish
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Re: Circularity in Formal Languages?

Postby Dopefish » Fri Jan 30, 2015 8:38 pm UTC

(There exists forum magic that makes twerking the byproduct of the term for small adjustments to things, just in case you weren't aware.)


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