Mathematics to be simply an extension of logic

[Nile3]

Hello,

I read that many people believe mathematics to be simply an extension of logic and therefore some or all of math to be reducible to logic. I thought this was an obvious fact for the longest time. I was wondering if there was any flaw with such an argument or what else there is which can make math not be partially or completely a subset of logic.

Thanks.

 

[MathematicalPhysicist]

existence of numbers is not a logical implication of logic, and thus the induction theorem or axiom is not a logical axiom or theorem.

 

[mathman]

Mathematics is an application of logic, not an extension.

 

[Stephen Tashi]

You could say that all drawing is the application of some substance to a substrate. That may be true, but the ability to apply a substance to a substrate doesn't give you the ability to draw well. Being good at applying logic doesn't guarantee you will be good at doing mathematics. To do mathematics you need to be inventive, or at least be able to appreciate inventiveness. Most of mathematics (perhaps all of it) is merely working out the consequences of definitions. (For example, from the viewpoint of mathematical logic, the real numbers aren't a collection of physical objects that have certain "properties". Instead the "properties" of the real numbers are simply part of the definition of the real numbers.) Coming up with useful definitions is an art.

 

[Nile3]

@MathematicalPhysicist: That's interesting. But I still don't know if there's more to math than logic from that perspective. I'm only informed that there's more to logic than what can be contained in logic at anyone point in time. We already knew that from Godel's theorem or the CH. But the math we can know of, is it all contained inside of logic? Or is there things in math which logic cannot do?

@mathman: Sounds good, thanks. Now the problem is, some people do believe there is more to math than what is contained in logic. The realists believe numbers are real things that can be sensed just like the eye see the light.

@StephenTashi: 1. You can draw a number in the air. 2. I don't see how stating that you need to be inventive to do math help understand if math is an extension of logic. Plus, settling all math questions as simply being an innate characteristic seem discriminatory in some way. Moreover, not all inventive persons do math.

Thank you for the answers.

 

[Stephen Tashi]

. I don't see how stating that you need to be inventive to do math help understand if math is an extension of logic..

Do you understand what logic is? In common language, people use the adjective "logical" to mean something that is "reasonable", "plausibe" or "true". However, this is not the technically correct definition of Logic.

Logic is only the study of reliable methods of reasoning. These methods of reasoning don't specify any particular topic to be reasoned about. The methods of logic do not involve determining the objective truth or falsity of anything in the real or the mathematical world. Logic only involves procedures for deducing the consequences of assumptions. It doesn't tell you what assumptions should be made in physics or mathematics.

When you study a particular set of assumptions (such as set theory or the axioms for the real numbers) this is doing mathematics, not doing Logic. The creation of particular assumptions isn't a matter of Logic.

If current methods of logic were powerful enough to tell how to prove or disprove any possible consequence of a set of assumptions then there wouldn't be any famous "unsolved" mathematical problems. Proving mathematical theorems involves applying Logic to a set of assumptions. However the current science of Logic doesn't explain how to apply methods of reasoning to get to a particular goal. It only explains how to evaluate a proof that a person has created. Creating the proof involves creativity and invention.

 

[yossell]

Nile3

The answer to your question depends on what you take a reduction of mathematics to logic to consist in.

There are many mathematical theories whose axioms are not reducible, recoverable or interpretable as theorems or axioms of logic. As MathematicalPhysicist said, logic doesn't imply the existence of any entities while, for example, the axioms of number theory and set-theory do imply the existence of (infinitely many) entities. (Though I'm not sure that the induction scheme mentioned by MathPhys is a good example, as it is really a conditional statement).

It's not entirely clear cut, because there's some debate about what precisely counts as a logical truth -- Frege and Russell thought that second order logic - quantification into predicate position -- did entail the existence of set-like entities.

However, you might think that mathematics doesn't really assert the existence of numbers and sets; rather, mathematicians are concerned with what would be true if a certain collection of axioms were true - the axioms defining number theory, or the axioms defining set-theory. From that point of view, there is a sense in which most modern mathematics, and certainly most standard modern mathematics, can be regarded as a kind of logic -- the art of drawing out ever more subtle and difficult logical consequences of a set of axioms.

Sometimes, though, mathematicians can get into a fight as to whether or not an axiom is 'correct'. There used to be disagreement about whether the axiom of choice was an acceptable axiom. Insofar as mathematicians get involved in this discussion, we cannot understand what they are doing as simply doing logic, as they are not just drawing out the consequences of a set of axioms. Some set-theorists take sides on whether the continuum hypothesis is 'true.' But I would say it is doubtful whether such arguments play much part in modern mainstream mathematics. For a start, there are difficulties in using the notion of truth in the mathematical context without finding oneself committed to Platonism.

It's not clear the degree to which Godel's theorem is really relevant. A common formulation is that Godel's incompleteness theorem shows that any sufficiently strong consistent formal system is incapable of proving all the mathematical truths. But that statement involves the notion of mathematical truth, which is contentious. More formally, Godel's theorem just shows that, for any sufficiently strong consistent formal system, there is a mathematical statement P, such that neither P nor ¬P is provable. Whether this touches upon the reducibility of maths to logic depends upon your attitude towards P. For instance, many mathematicians are unmoved by the unprovability of CH from standard axioms of set-theory.

 

[Nile3]

@Yossell: Wow, that was amazing. How do we end up knowing anything at all? Is the only way to know if something is true or not simply to bridge the gap with abstractions? You know, I'm starting to think there may be more to this "mathematical realism" than meet the eye at the first sight. Do you have any good resources to keep digging deeper?@ Tashi: So I assume you weren't logic in the creation of the assumption that "The creation of particular assumptions isn't a matter of Logic". In that case, you did math because you were inventive. But not logic. But you used logic to deduce your inventive math applied not to logic. Therefore no creation of particular assumptions is made without logic. Therefore I was logical in my creation of the assumption you weren't logic.

That was just the obvious stuff that came out of your post...

 

[Mark44]

@ Tashi: So I assume you weren't logic in the creation of the assumption that "The creation of particular assumptions isn't a matter of Logic". In that case, you did math because you were inventive. But not logic. But you used logic to deduce your inventive math applied not to logic. Therefore no creation of particular assumptions is made without logic. Therefore I was logical in my creation of the assumption you weren't logic.

That was just the obvious stuff that came out of your post...

Someone isn't "logic" - they can be logical or illogical, depending on how they are reasoning.

Most of what you said above makes no sense to me, and it seems that you do not understand what Stephen Tashi was saying. A mathematician uses creativity to come up with one or more assumptions, and then uses logic to arrive at some conclusion about those assumptions.

Logic is not required to produce the assumptions. A good example would be the imaginary numbers, where the initial assumption is that there is a solution to the equation x² = -1.

 

[pwsnafu]

Late to the thread but

@ Tashi: So I assume you weren't logic in the creation of the assumption that "The creation of particular assumptions isn't a matter of Logic". In that case, you did math because you were inventive. But not logic. But you used logic to deduce your inventive math applied not to logic. Therefore no creation of particular assumptions is made without logic. Therefore I was logical in my creation of the assumption you weren't logic.

That was just the obvious stuff that came out of your post...

I get the impression that you simply don't know what logic is and how it is used in mathematics. What courses in Logic have you taken?

 

[lavinia]

I know nothing of formal systems but i can say that the thought process in mathematics is not only deductive which it what it would be if it were purely an extension of logic. Mathematical truth like much other truth is realized through insight. The logic comes later - like an after thought.

Also mathematical objects have subtle and complex structures much like Nature itself. These structure come from many sources again much like things in Nature. I do no see how these structures are extensions of logic.

 

[MathematicalPhysicist]

You could say that all drawing is the application of some substance to a substrate. That may be true, but the ability to apply a substance to a substrate doesn't give you the ability to draw well. Being good at applying logic doesn't guarantee you will be good at doing mathematics. To do mathematics you need to be inventive, or at least be able to appreciate inventiveness. Most of mathematics (perhaps all of it) is merely working out the consequences of definitions. (For example, from the viewpoint of mathematical logic, the real numbers aren't a collection of physical objects that have certain "properties". Instead the "properties" of the real numbers are simply part of the definition of the real numbers.) Coming up with useful definitions is an art.

You should also define what is "useful" to you.

But it does seem as an art and it doesn't matter if you do pure maths or applied maths (and for me the distinction is blurred anyways).

 

[MathematicalPhysicist]

Nile3

The answer to your question depends on what you take a reduction of mathematics to logic to consist in.

There are many mathematical theories whose axioms are not reducible, recoverable or interpretable as theorems or axioms of logic. As MathematicalPhysicist said, logic doesn't imply the existence of any entities while, for example, the axioms of number theory and set-theory do imply the existence of (infinitely many) entities. (Though I'm not sure that the induction scheme mentioned by MathPhys is a good example, as it is really a conditional statement).

It's not entirely clear cut, because there's some debate about what precisely counts as a logical truth -- Frege and Russell thought that second order logic - quantification into predicate position -- did entail the existence of set-like entities.

However, you might think that mathematics doesn't really assert the existence of numbers and sets; rather, mathematicians are concerned with what would be true if a certain collection of axioms were true - the axioms defining number theory, or the axioms defining set-theory. From that point of view, there is a sense in which most modern mathematics, and certainly most standard modern mathematics, can be regarded as a kind of logic -- the art of drawing out ever more subtle and difficult logical consequences of a set of axioms.

Sometimes, though, mathematicians can get into a fight as to whether or not an axiom is 'correct'. There used to be disagreement about whether the axiom of choice was an acceptable axiom. Insofar as mathematicians get involved in this discussion, we cannot understand what they are doing as simply doing logic, as they are not just drawing out the consequences of a set of axioms. Some set-theorists take sides on whether the continuum hypothesis is 'true.' But I would say it is doubtful whether such arguments play much part in modern mainstream mathematics. For a start, there are difficulties in using the notion of truth in the mathematical context without finding oneself committed to Platonism.

It's not clear the degree to which Godel's theorem is really relevant. A common formulation is that Godel's incompleteness theorem shows that any sufficiently strong consistent formal system is incapable of proving all the mathematical truths. But that statement involves the notion of mathematical truth, which is contentious. More formally, Godel's theorem just shows that, for any sufficiently strong consistent formal system, there is a mathematical statement P, such that neither P nor ¬P is provable. Whether this touches upon the reducibility of maths to logic depends upon your attitude towards P. For instance, many mathematicians are unmoved by the unprovability of CH from standard axioms of set-theory.

I am not sure I understand your objection to my example, even if the theorem/axiom is structured by a conditional it's still not a logical axiom, a logical axiom is any tautology in first order classical logic. I mean this axiom doesn't apply for the real numbers for example.

 

[yossell]

@Mathematical Physicist
You originally wrote

existence of numbers is not a logical implication of logic, and thus the induction theorem or axiom is not a logical axiom or theorem.

My point was this: since the induction theorem does not imply the existence of numbers, the conclusion of the `thus' does not follow from the fact that the existence of numbers.

even if the theorem/axiom is structured by a conditional it's still not a logical axiom, a logical axiom is any tautology in first order classical logic.

(Aside: Not all logicians agree that the limit of logic is first order logic -- although first order logic can take us very far. Modal logic, temporal logic, infinitary logics are just some extensions of first order logics that logicians regard as part of their domain. In particular, since useful mathematical concepts such as 'there are finitely many', `is a well ordering of' are not expressible in first order logic, the full induction axiom of Peano arirthmetic (as opposed to the weaker first order schema, there are many who argue that logics allowing second order quantification should be part of their domain.)

Strictly speaking, the induction axiom applies only to a certain kind of mathematical object: it is the *natural numbers* that obey the induction axiom. Accordingly, a regimentation of the induction axiom would contain a predicate `Nx' which would act to restrict the domain of the quantifiers. Now, how to define `Nx'? Well, why not say that to be a number just is to be part of a structure which satisfies the peano axioms. To be a number just is to be one of a collection X, on which there is a relation R which satisfy the following principles: X has an R-least member; each member of X has successor; etc. etc. As we have quantification into predicate position, X and R, this turns out to be a sentence of second order logic. From this point of view, the induction formula is simply part of the definition of what it is to be a number.

With this in place, we can now regard much work on the Natural numbers as simply working which conditionals of the form -- if there is a set X and relation R satisfying Peano axioms then so and so' are true.

What this idea won't get you is that mathematical objects exist -- mathematics becomes the working out of logical conditionals of the above form; but logic doesn't entail any of the pure existence theorems of mathematics -- it doesn't entail that there *are* an infinity of primes; only that, if there is an X and an R satisfying Peano axioms, then there are an infinity of primes in X. Since your original objection mentioned the existence of numbers, you appeared to be endorsing an objection of this form.

However, many are not happy supposing the existence of numbers.

 

[Nile3]

Well, this is an interesting discussion. But it seems that no matter the path of thinking I take from the ideas you give me, I end up thinking "What makes the math in the physical world, which we describe with math?" This is redundant. Is there a way out?

 

[Mark44]

Well, this is an interesting discussion. But it seems that no matter the path of thinking I take from the ideas you give me, I end up thinking "What makes the math in the physical world, which we describe with math?" This is redundant. Is there a way out?

How about this? We (humans) create the mathematics that models or approximates what we observe in the physical world.

 

[Nile3]

How about this? We (humans) create the mathematics that models or approximates what we observe in the physical world.

That's simply physics. Which we describe with math.

Mathematical truth like much other truth is realized through insight. The logic comes later - like an after thought.

 

[Bacle2]

I think Stephen Tashi clinched it by mentioning that Logic( Let's assume for now that we're working with sentence logic) does not have any content of its own; it is a tool,and it is as good as the premises one works with
(This is why I think the magazine called "Reason" is so absurd; reason does not vouch neither for nor against capitalism nor socialism nor any other belief system. Logic is a collection of technicques/methods that are designed to be "truth-preserving" ,i.e., so that if you start with premises that have been determined to be true and apply correctly the rules of sentence logic ,you will end up with a true sentence . The best display of this is in the truth tables for the conditional in which T-->F is F .) The essential concept is that of a valid argument,an argument in which if the premises are true,the conclusion cannot be false.

And there are ways of knowing that Mathematics is not a subset of sentence logic, since sentence logic is decidable --just use a truth table, while Mathematics is not. Moreover, the truth-value in sentence logic is arbitrary in the sense that it does not depend on the structure of the sentence; sentences here are atoms, whose internal structure has no affect on the sentence's truth value. I don't think this is reflective of , nor in agreement with Mathematics.

The issue becomes much more complicated when you start working with Predicate Logic.

 

[Mark44]

That's simply physics. Which we describe with math.

Mathematical truth like much other truth is realized through insight. The logic comes later - like an after thought.

I don't see how this makes sense - that the logic comes later. Do you have any particular example in mind?

 

[jpz]

Logicism has long failed, since the formulation of Godel Incompleteness theorem, although we still have a relentless attempt to place logic at the foundation of mathematics. And whether it is or not, depends on what axioms you consider to be part of logic. Logic as traditionally been considered, does not assume equality. However, with a (somewhat ad hoc) model of logic, one that utilizes equality, one can generate the Peano (true math) axioms. Even then, there are still axioms in math, that cannot be defined with traditional logic.

In any event, Godel marvelous theorem shows, essentially, that in mathematics (and perhaps in any large formal system), the proof for at least one given truth, is never based on the rules that exist to define that problem. This directly implies that there exist axioms, which have gone undiscovered and thus, that humans have some creative /intuition-like ability which allows us to 'realize' these axioms. It will be long time before the common man appreciates the importance of Godel.

Mod note: Deleted off-topic paragraph and link to web site.[/color]

 

[Stephen Tashi]

... that humans have some creative /intuition-like ability which allows us to 'realize' these axioms.

Are there any mathematical examples of this?

 

[robertjford80]

Mathematics is an application of logic, not an extension.

Applied math is an application of logic, pure math is an extension of logic. Numbers are just words whose definition is non-vague. You can make some very certain deductions with non-vague objects. It's much more difficult to make deductions with vague objects.

 

[robertjford80]

Logic is only the study of reliable methods of reasoning.

This definition just moves the problem around. First, we wanted to know what logic was, now we want to know what 'reasoning' is.

When you study a particular set of assumptions (such as set theory or the axioms for the real numbers) this is doing mathematics, not doing Logic.

In order to justify this claim you would need a definition of math and a definition of logic and then you would need to show why working with your definition logic is excluded.

 

[robertjford80]

Hello,
I was wondering if there was any flaw with such an argument or what else there is which can make math not be partially or completely a subset of logic.

Thanks.

To find out if there is a flaw in the argument, check to see if the theorist has a definition of math and logic which does not simply move the problem around. It is almost certainly the case that they do since almost all theorists do that. What you're likely to find is that they define math or logic in terms of something equally mysterious then they leave those terms undefined.

 

[robertjford80]

Strictly speaking, the induction axiom applies only to a certain kind of mathematical object:

I'd be interested if anyone can tell me what a mathematical object is. It seems that 2, triangles, variables, lines, circles are mathematical objects. It's not easy to see what these have in common. My guess is that they are things such that there definitions are highly precise.

What this idea won't get you is that mathematical objects exist -- mathematics becomes the working out of logical conditionals of the above form; but logic doesn't entail any of the pure existence theorems of mathematics -- it doesn't entail that there *are* an infinity of primes; only that, if there is an X and an R satisfying Peano axioms, then there are an infinity of primes in X.

Very good point.

many are not happy supposing the existence of numbers.

What do you mean by existence? I believe everything exists but everything has a different type of existence. So there are things that exist in space (physical), things that did exist in space (historical), things that could exist in space (abstract), things that will exist in space (inevitable), things that cannot exist in space (delusions), sensations, and things that affect space (mental and divine).

 

[robertjford80]

Logicism has long failed, since the formulation of Godel Incompleteness theorem, although we still have a relentless attempt to place logic at the foundation of mathematics.

No it hasn't. All Godel proved was what we already knew: you can't prove axioms. Many axioms are so obvious that no sane person would doubt them. The fact that you can't prove axioms doesn't matter. We can still carry on with the business of deriving highly instructive consequences from a set of axioms.

Even then, there are still axioms in math, that cannot be defined with traditional logic.

Can you give some examples? My guess is that we cannot define them now but we will eventually. You need to prove that they are in principle undefinable with logic which is very hard, especially when you don't even have a definition of logic.

In any event, Godel marvelous theorem shows, essentially, that in mathematics (and perhaps in any large formal system), the proof for at least one given truth, is never based on the rules that exist to define that problem.

So? This is no big deal.

 

[robertjford80]

Here's a very helpful quote from Russell's My Philosophical Development published in 1959

The primary aim of Principia Mathematica was to show that all pure mathematics follows from purely logical premisses and uses only concepts definable in logical terms.} This was, of course, an antithesis to the doctrines of Kant, and initially I thought of the work as a parenthesis in the refutation of ‘yonder sophistical Philistine’, as Georg Cantor described him, adding for the sake of further definiteness, ‘who knew so little mathematics’. But as time went on, the work developed in two different directions. On the mathematical side, whole new subjects came to light, involving new algorithms making possible the symbolic treatment of matters previously left to the diffuseness and inaccuracy of ordinary language. On the philosophical side, there were two opposite developments, one pleasant and the other unpleasant. The pleasant one was that the logical apparatus required turned out to be smaller than I had supposed. More especially, classes turned out to be unnecessary.

 

[jgens]

All Godel proved was what we already knew: you can't prove axioms. Many axioms are so obvious that no sane person would doubt them. The fact that you can't prove axioms doesn't matter. We can still carry on with the business of deriving highly instructive consequences from a set of axioms.

Gödel showed something very different than this with his incompleteness theorems. He established two things: One that any sufficiently strong* axiom schema is either incomplete or inconsistent; and two that any sufficiently strong axiom schema capable of proving its own consistency is necessarily inconsistent. If you cannot tell the difference between the incompleteness theorems and "proving axioms" then you probably should not be proselytizing about the role of logic in mathematics.

*Sufficiently strong in this context roughly means containing PA. The exact conditions under which the theorems hold are a little different than this, but this should give you a rough idea about how strong your axiom schema need to be.

 

[Nick O]

You can model many arithmetic operations with logic, so long as you define a few propositions as being true. Peano's Principia Arithmetices uses logical operations and such defined propositions to do this.

I am not a mathematician, so I can't say that all mathematics can be thus modeled. The consensus in this thread (if I am correctly interpreting the snippets I have read) seems to be "no", and I will defer to that.

 

[robertjford80]

Gödel showed something very different than this with his incompleteness theorems. He established two things: One that any sufficiently strong* axiom schema is either incomplete or inconsistent; and two that any sufficiently strong axiom schema capable of proving its own consistency is necessarily inconsistent. If you cannot tell the difference between the incompleteness theorems and "proving axioms" then you probably should not be proselytizing about the role of logic in mathematics.

*Sufficiently strong in this context roughly means containing PA. The exact conditions under which the theorems hold are a little different than this, but this should give you a rough idea about how strong your axiom schema need to be.

This is not a problem. Show that Gödel's theorems should give someone who uses a calculator something to worry about.

I'm also not quite sure that you're using inconsistent in the same way that I do. To me inconsistent means: 'leads to contradictions.' What contradictions do Peano's Axioms lead to? You might even have a different meaning for 'incompleteness'. To me incompleteness means that you cannot prove the axioms. That's no big deal and no cause for worry. So again, why are these theorems such a big deal? They are massively overhyped.

 

[pwsnafu]

So again, why are these theorems such a big deal? They are massively overhyped.

Mostly historical reasons, it was Hilbert's second problem that started all this. When Russel's paradox came to light, there was panic in the mathematical community that mathematics was going to fall apart. But that was 100 years ago.

You might even have a different meaning for 'incompleteness'. To me incompleteness means that you cannot prove the axioms.

That is not what incompleteness means. A system is complete if every true theorem (i.e. true statements which is not axiom) is provable from the axioms. A good example is Goostein's theorem. Goodstein sequences are clearly sequences of naturals, and yet PA can't prove that they all terminate.

 

[jgens]

Show that Gödel's theorems should give someone who uses a calculator something to worry about.

You really ought to be more specific about which theorems of Gödel you mean. But assuming you mean the incompleteness theorems, then they have little applicability to calculator computations. Then again so does nearly the entirety of mathematics so that really says very little.

I'm also not quite sure that you're using inconsistent in the same way that I do. To me inconsistent means: 'leads to contradictions.'

This is, more or less, the correct notion of inconsistency.

What contradictions do Peano's Axioms lead to?

Probably none because we generally assume that PA is consistent. However the incompleteness theorems tell us the following two things:

1. PA is either incomplete or inconsistent.
2. If PA were consistent, then it could not prove its own consistency.

You seem to be confused about the meaning of incompleteness, so on this point refer to my response below.

You might even have a different meaning for 'incompleteness'. To me incompleteness means that you cannot prove the axioms.

My meaning of incompleteness is the standard one and yours is completely off the mark. What incompleteness roughly means is that there are statements expressible in our axiom schema that are neither provable nor unprovable. Note that within any particular such scheme the axioms are vacuously provable, and as a result, this is very different from not being able to prove axioms. I repeat: If you cannot understand this difference, then do not proselytize about the role of logic in mathematics. If you want the difference explained to you, then start a new thread about the matter.

So again, why are these theorems such a big deal? They are massively overhyped.

I agree that these theorems get over-hyped and as pwsnafu has already said the primary interest in these theorems is historical. What the theorems guarantee, however, is that any set of axioms strong enough to express most of modern mathematics will necessarily have undecidable statements*. So effectively the theorems tell us to stop looking for an all encompassing set of axioms for mathematics and instead just look for something sufficient.

*These undecidable statements are not the axioms of our set theory. Rather they are things like the Whitehead Problem or the Continuum Hypothesis or the existence of large cardinals.

 

[jpz]

Are there any mathematical examples of this?

It's easy; just a google a list of mathematical axioms

 

[jpz]

No it hasn't. All Godel proved was what we already knew: you can't prove axioms. Many axioms are so obvious that no sane person would doubt them. The fact that you can't prove axioms doesn't matter. We can still carry on with the business of deriving highly instructive consequences from a set of axioms.

No, Godel proved there are an infinite number of mathematical axioms, and they are not derivable through any axiomized form of mathematics. Instead, they rely on some insight/intuitive ability.

Can you give some examples? My guess is that we cannot define them now but we will eventually. You need to prove that they are in principle undefinable with logic which is very hard, especially when you don't even have a definition of logic.

No, you can never define all mathematical axioms in one list. Every time you add an axiom, there will be another one required to prove something about that set. I don't have to prove it; that's what Kurt Godel was for.

 

[robertjford80]

So again, why are these theorems such a big deal? They are massively overhyped.

I agree that these theorems get over-hyped and as pwsnafu has already said the primary interest in these theorems is historical. What the theorems guarantee, however, is that any set of axioms strong enough to express most of modern mathematics will necessarily have undecidable statements*.

What started this debate was someone asserted that Gödel's theorems prove that logicism is false. I objected. I can't remember if you took the counterposition but it seems that we are now in agreement that Gödel's theorems do not falsify logicism.

What incompleteness roughly means is that there are statements expressible in our axiom schema that are neither provable nor unprovable.

How is this different from my assertion that all's Gödel's theorems prove is something that we already knew: 'that you can't prove axioms'?

Note that within any particular such scheme the axioms are vacuously provable, and as a result, this is very different from not being able to prove axioms.

You're going to have to explain how 'vacuously provable' is different from unprovable. Axioms are by definition unprovable. You just assume them for the purposes of making deductions. The axiom, a=b, b=c, therefore a=c is not provable, it is assumed.

 

[robertjford80]

The consensus in this thread (if I am correctly interpreting the snippets I have read) seems to be "no", and I will defer to that.

Why are you deferring to a majority opinion when the proponents cannot even say what mathematics and logic are?

 

[Nick O]

Why are you deferring to a majority opinion when the proponents cannot even say what mathematics and logic are?

Odd, the quote shows a name that is not mine when viewed Tapatalk. I can't say whether it does the same on a web browser.

The key point there was that I had only skimmed the thread, and that I am not qualified to have an opinion on this. I mentioned that I may have misinterpreted the discussion as well.

 

[jgens]

What started this debate was someone asserted that Gödel's theorems prove that logicism is false. I objected. I can't remember if you took the counterposition but it seems that we are now in agreement that Gödel's theorems do not falsify logicism.

We were never in disagreement about the issue. I never took a position on that claim. This is REALLY easy to verify by reading the thread and checking the usernames associated to each post. In the future please do so.

How is this different from my assertion that all's Gödel's theorems prove is something that we already knew: 'that you can't prove axioms'?

If you absolutely insist on hijacking this thread instead of starting your own, then here goes: Once you fix a particular axiom schema for mathematics, the axioms trivially prove themselves, so the theorems of Gödel obviously say something very different than this. One of the things they roughly assert is that (assuming consistency) you can make statements that cannot be proved or disproved from the axioms. Examples of this can be found with the Whitehead Problem and with the existence of large cardinals. These results are not axioms in ZFC and they are undecidable in ZFC. These are the kinds of statements the Incompleteness Theorems refer to.

You're going to have to explain how 'vacuously provable' is different from unprovable.

Vacuously provable means there is a proof for them (within the axiom schema). Unprovable means there no proof or disproof for them. Huge difference.

Axioms are by definition unprovable.

They are unprovable in the abstract yes. Within a fixed system of axioms the axioms verify themselves. The Incompleteness Theorems are in the context of fixed systems of axioms.

Why are you deferring to a majority opinion when the proponents cannot even say what mathematics and logic are?

In the future please do not fill my username into someone else's quote. I could be wrong about this, not being a mod and all, but I am fairly certain that is a fantastic way to get banned.

 

[Evo]

Closed pending moderation.