### Inconsistency

Vladimir Voevodsky, one of the world’s best and most influential mathematicians, has stirred up a bit of a hornet’s nest with a video lecture suggesting the possibility that the Peano Axioms — the standard axioms for arithmetic — might be inconsistent.

Since the Peano Axioms are known to be consistent, it’s tempting to dismiss the whole lecture as either a prank or a shocking display of ignorance. The latter temptation is buttressed somewhat by Voevodsky’s bold misstatement of Godel’s Incompleteness Theorem, which plays a central role in the lecture. On the other hand, Voevodsky is smarter than almost anyone else on earth, which earns him the benefit of the doubt — maybe what he’s saying is subtler than it seems. On the other hand, some of those in the “shocking display of ignorance” camp are among the few people in the world who might be as smart as Voevodsky.

To believe that the Peano Axioms are inconsistent, Voevodsky must reject all of the known proofs that they are consistent. In particular, he must reject the simplest and most convincing of all those proofs, which goes like this:

1. The Peano Axioms, and therefore all of their logical consequences, are true statements about the natural numbers,
2. A collection of true statements cannot contradict itself.
3. QED.

If you were going to reject that argument, you’d pretty much have to reject part a). There are multiple ways to do that. You could deny that there is any such thing as “the natural numbers”, or you could deny that the Peano axioms are true statements about the natural numbers, or you could deny that all of the logical consequences of the Peano axioms are true statements about the natural numbers.

Denying the existence of the natural numbers is pretty much a non-starter. There is a class of people (mostly college sophomores) who refuse to admit the existence of the natural numbers but are nevertheless willing to debate the consistency of Peano Arithmetic (that is, the Peano Axioms and their consequences). But for a mathematician to deny the existence of the natural numbers would be as rare and fruitless as for a psychologist to deny the existence of conscious beings or for a physicist to deny the existence of physical objects. We can’t get by without them.

Besides, it is quite impossible to study Peano Arithmetic unless you know about the natural numbers in advance. For example, a “proof” in Peano arithmetic is a list of statements, each of which is either an axiom or follows from preceding statements. To understand that concept, you must know what a list is. To understand what a list is, you must have the concepts of “first”, “second”, “third”, etc. In other words, you need to know about the natural numbers! Anyone who claims to understand Peano arithmetic has already implicitly admitted that the natural numbers exist — and that you’ve got to be familiar with them before you can axiomatize them.

It would be a bit of an exaggeration, but not much of one, to call this assessment 100% noncontroversial among mathematicians. Almost (but not quite) everyone who studies math, including, I am almost sure, Voevodsky, accepts this account of what we do.

But if Voevodsky, like the rest of us, believes in the natural numbers, then it’s hard to see how he can doubt the consistency of Peano arithmetic. Surely the axioms are obviously true, but Voevodsky seems to be doubting that every logical consequence of the axioms is true. How can this be? If I understand him right (and I’m not sure I do), it’s because he thinks that some logical consequences of the axioms might be so complicated as to be meaningless, and are therefore neither true nor false.

This makes no sense to me at all, because complication does not by itself imply meaninglessness. More to the point, if some consequences of the axioms are meaningless, there must be a least complicated meaningless example. That least complicated example is only slightly more complicated than some other consequence which is meaningful. How can a slight increase in complexity introduce meaninglessness?

Again, I might have missed his point completely. If so, I’m in good company. There have been several threads in the May and June editions of the FOM mailing list (“FOM” stands for “Foundations of Mathematics”) where a mighty distinguished crew have expressed much the same confusion, and, in several cases, dismay.

For those who like this sort of thing, this is the sort of thing they’ll like. You can click on May or June above and select pretty much any thread with the word “Voevodsky” or “consistency” in it. I’m sure there’s content there I’ve failed to digest in full.

Edit: I should have said that in my (possibly flawed) understanding of Voevodsky, it’s not just the consequences of the axioms that can be too complicated to be meaningful, but some of the axioms themselves. For any property that can be expressed in Peano Arithmetic, there is an axiom saying (essentially) that if there are any numbers with that property, then there is a smallest one. V. seems to be saying (and again I want to stress that I’m not sure I understand him) that some of these properties are too complex to be meaningful, and therefore so are the corresponding axioms.

#### 75 Responses to “Inconsistency”

1. 1 1 Mike H

Your ‘a-b-c’ proof looks more philosophical than mathematical.

As for ‘a’, it’s clear that the Peano axioms are statements. It’s less clear that they are statements about anything in particular, let alone about what we intuit the natural numbers to be. Finally, even if they are “true statements about the natural numbers”, it does not follow that they are not self-contradictory – after all, there are many other things we know intuitively to be true that are full of contradictions.

Your ‘b’ sounds like a nice axiom, but how do you know it’s true?

… and as you point out in your later comments, any “proof” about the natural numbers uses the natural numbers. I would argue that this circularity is not a good reason to have faith in the proof’s consistency.

2. 2 2 Harold

I am interested in how much this is philosophy and how much mathematics. If the Peano axioms were not proved to be consistent, wouldn’t we just have to carry on as though they were? It may be of philosophical interest, but would it actually change anything in the practice of Math? Could this lead to something new, or would we be just where we were, but with awarenesss that we are on shakier ground than we thought?

Descartes famously doubted everything except that he had some form of existence. However, he went on to “derive” lots of other things from this. I think some of his reasoning is dodgy, but it seems to be necessary to accept something without proof in order to get anywhere. The natural numbers seem to be taking the place of “I Exist” in Descartes philosophy. Descartes himself almost equated the two:

“Yet when I turn to the things themselves which I think I perceive, I am so convinced by them that I spontaneously cry out, let whoever can deceive me, he will never bring it about that I am nothing, so long as I continue to think I am something; or make it true at some future time that I never existed, since it is now true that I exist; or bring it about that two or three added together are more or less than five, or anything of this kind in which I see a manifest contradiction.”

His doubt about the natural numbers was the samne as the doubt that in the future someone (God) would change the past so he never existed.

3. 3 3 Steve Reilly

Actually, I can go one better and prove that the natural numbers are inconsistent.

1)Look down at your hands. Count them. You will find that there are 2.

2)Yet your second paragraph above implies that you have 2 other hands.

3)This is inconsistent.

QED

4. 4 4 Steve Landsburg

Harold and Mike H: This is mathematics, not philosophy. The natural numbers are the starting point for mathematics, just as physical observations are the starting point for physics. It is standard practice through all of mathematics to start with the existence of at least the natural numbers and often much more (sets, for example). If you don’t do that, there’s pretty much nothing to study. There *are* mathematicians who choose to cripple themselves by assuming less, and they do quite respectable work examining the consequences of that self-crippling, but they are certainly out of the mainstream.

Mike: Of *course* any proof about the natural numbers uses the natural numbers. How could you possibly prove something about anything at all without mentioning that thing?

Harold: “If the Peano axioms were not proved to be consistent, wouldn’t we just have to carry on as though they were?” —- if the Peano Axioms weren’t proved to be consistent, that would mean we don’t know the natural numbers exist (otherwise their existence would provide a proof). If we didn’t know they exist, I expect we’d be less interested in studying them. That would presumably lessen our interest in the Peano Axioms. Though all of this is so very counterfactual that it’s hard to think about.

5. 5 5 Steve Landsburg

Steve Reilly: I appreciate the humor, but this seems as good a time as any to say this, because it’s going to come up in other comments: To say that the natural numbers are inconsistent is like saying that Brokeback Mountain is an electron’s favorite movie. It’s a category mistake. The natural numbers are not an axiomatic system, so the concept of consistency doesn’t apply. You can argue about whether they exist at all (though if you take the negative, you’ll be aligned against 95% or more of mathematicians), but it literally makes no sense to call them “inconsistent”.

6. 6 6 Ken B

Even smart people make blunders (cf Krugman, Paul). And there is so much math now people are often very compartmentalized. When I was a grad student one of the profs in the lounge shockingly misstated some basic fact about set theory, and my advisor warned me we’d be the only two in the room who knew. The Peano axioms are an extreme case of course.

7. 7 7 Jonathan Kariv

Let me start of by sayign that I haven’t actually watched the video (sound issue on the computer I’m using), and I’m sure I’d have trouble following it if I did. But my 2 cents anyway.

I’ve heard (vaguely) of people trying to prove this before. But figured that “proving the inconsistentcy of the peano axioms” was probably a simplificaion of what they where doing.

As Ken says, smart people do make mistakes. As Steve says, this stuff is hard enough to understand that we might be missign the point. Does VV actually point out an inconsistency that follows from the peano axioms? Or is it a non-constructive proof?

8. 8 8 Steve Reilly

Steve Landsburg, interesting. Actually, I had thought that a statement like “The natural numbers are consistent” was a sensible statement and “The natural numbers are inconsistent” was a meaningful, if wrong, statement. I hadn’t actually thought it was only axiomatic systems that could be consistent or inconsistent. See, even bad jokes can lead to knowledge.

9. 9 9 Harold

Is he saying the axioms are inconsistent, or merely that it is not proven they are consistent?

10. 10 10 math_geek

Why does something have to be meaningful to be true? That seems like such a surreal distinction.

11. 11 11 Roger Schlafly

Voevodsky was talking about a subject outside his core expertise, and he fails to accurately represent the mainstream views on the subject. Therefore, his lecture gives no reason to believe that those mainstream views are wrong. A lot of people have funny opinions of Goedel’s work. Just add Voevodsky to the list.

12. 12 12 Ken B

@math_geek: <>

13. 13 13 Ken B

@math_geek:”Ceci n’est pas un propostion.”

14. 14 14 math_geek

@Ken B. That’s fine, but i didn’t say that a statement had to be true or false. Only that meaningfulness wasn’t necessary to be true.

15. 15 15 Steve Landsburg

Harold:

Is he saying the axioms are inconsistent, or merely that it is not proven they are consistent?

He seems to be saying that he thinks there’s a pretty good chance they’re inconsistent, good enough that we should be worried about to do about it. From this I infer that he rejects the various proofs of consistency, though one possible reading of the video is that he is either unaware of those proofs or fails to understand them.

16. 16 16 Ken B

Tech detail alert. The proofs of the consistency of the PA all depend on further assumptions (axioms) beyond the list of the PA themselves. The first proof I know of used transfinite induction. There are other proofs. But in all cases they must rely on further axioms; you cannot prove the consistency of the PA using just the PA (as Godel proved). One could conceivably reject those further axioms.

17. 17 17 Steve Landsburg

Ken B: Agreed, of course, except that I would not use the word “axiom” to describe a fact like the existence of the natural numbers. An axiom is a formal statement in a formal language.

18. 18 18 Ken B

@Steve: Yes but a formal *proof* will use an axiom. You mentioned proofs of the consistency of the axioms. The only proof I can even remember having seen used transfinite induction. So you need an axiom (or axiom schema, I cannot recall) for that. Brouwerites would reject tfi of course.

But my money’s on a simple blunder here. Roger Penrose wrote a whole book based on a simple misunderstanding of what is and is not an algorithm. It happens.

19. 19 19 Super-Fly

So, while conceding the fact that he’s about 3 bajillion times smarter than me, I have a few problems with the presentation.

He breaks it down by giving this trilemma (paraphrase):
1. Godel’s 2nd incompleteness theorem is wrong
2. We can have “transcendental” knowledge of things that are true, but provably unprovable
3. Arithmetic is inconsistent.

He dismisses statement 2 as philosophical gibberish. I could be conflating two issues, but isn’t that exactly what Godel’s *first* incompleteness theorem says? Paraphrasing: A consistent theory that can describe arithmetic necessarily contains statements that are true, but unprovable with that theory.

Again, I’m not an expert, and I could be wrong. I don’t see, though, how suggesting that there is knowledge that is true but unprovable is sophistry, but it’s reasonable to tease out the consequences of an alleged inconsistency in a foundational theory.

20. 20 20 Bob

“For those who like this sort of thing, this is the sort of thing they’ll like.”

Voevodsky does not accept your statement.

21. 21 21 nobody.really

You could deny that there is any such thing as “the natural numbers”….

Are you suggesting they dye their hair?

22. 22 22 Mike H

@Steve “Of *course* any proof about the natural numbers uses the natural numbers. How could you possibly prove something about anything at all without mentioning that thing? “

Here, you are deliberately missing your own point, let alone mine.

There is a difference between a proof talking about the natural numbers and using them. You yourself said something like “the concept of proof involves the natural numbers, since a proof is an ordered sequence of statements”.

Now let me assert more strongly that your a-b-c “proof” is philosophical, not mathematical. You merely assert that the Peano axioms are “true statements about the natural numbers.” Now what do you mean by “true”??

Either you mean “mathematically true” or something else.

If you mean “mathematically true”, then (a) is an empty statement, and (b) is false.

If you mean something else, maybe “epistemologically true” or “ontologically true” then you are talking philosophy, not mathematics.

Why not acknowledge that, like most mathematicians, you take the natural numbers as an item of pure and simple faith? Then, if the Peano axioms fall down, well, so be it. We survived Russel’s paradox, we’ll cook up something else.

23. 23 23 Roger Schlafly

Why not acknowledge that, like most mathematicians, you take the natural numbers as an item of pure and simple faith?

Because most mathematicians do not do that. The natural numbers have properties that are provable from self-evident truths. If you think that there is something wrong with one of those proofs, then say so.

@KenB: Penrose did not misunderstand an algorithm. He may be wrong for other reasons, but not that reason.

24. 24 24 Ken B

@Roger: It’s been a long time, but as I recall Penrose decided that the brain can operate non-algorithmically to discern truths. That seems to be wrong, and is based on the (wild) surmise that “oh no THAT cannot be the result of an algorithm”. I say surmise because we don’t know yet what it is based on.

25. 25 25 Roger Schlafly

@KenB: Penrose’s books on this subject are The Emperor’s New Mind and Shadows of the Mind. He may be wrong about how the mind works, but he is correct about what an algorithm is.

26. 26 26 Ken B

@Roger: I never said he did not know what an algorithm is. I said he did not know what is an algorithm. One can know what a fruit is without knowing a tomato is a fruit.

27. 27 27 Ken B

@Roger: Is 123211785444633341 a prime number?

I’d say you could get that wrong and still know what a prime number is.

28. 28 28 Mike H

@Roger “The natural numbers have properties that are provable from self-evident truths.”

I have no problem with the proofs. I have a problem when someone says the axioms are “self-evident” when they really mean “I take them on faith, since they can’t be proved”

29. 29 29 Steve Landsburg

Mike H: “I take them on faith, since they can’t be proved”

But they *can* be proved.

Proof 1: By Gentzen’s Theorem, the Peano Axioms are consistent. By Godel’s Completeness Theorem, they therefore have a model. The smallest of those models is (by definition if you like) the natural numbers, which, by virtue of being a model, must satisfy all the axioms. (Or, if you prefer, throw in a second order induction axiom and invoke Dedekind’s Theorem to get a *unique* model (at least within a given set-theoretic universe), so you don’t have to worry about showing that there’s a “smallest”).

Proof 2: Take the explicit von Neumann model of the natural numbers. You can directly verify that this model satisfies each of the Peano axioms.

30. 30 30 Ken B

@Steve and Mike H: To prove Gentzen’s theorem you need the axioms of set theory. (I cannot recall if you need choice, but anyone who thinks the axiom of choice is self-evident simply has too loose a defintion of self-evident.) And you need transfinite induction for Gentzen’s proof. So let’s amend Mike’s remark to refer not to the Peano axioms but to the axioms behind Gentzen’s proof whatever they might be. Steve’s answer does not apply now.

Disclaimer: I personally think the PA are self-evident. That is why Peano suggested them as axioms.

31. 31 31 Ken B

A little perspective for those readers who did not study math.
It was once believed that all of math could be shown to be provable just from pure logic. An initial form of set theory — full of self-evident truths — was used both formally and informally as a convenient basis for mathematics. Then Russell showed this naive set theory was inconsistent. Mathematics has never fully recovered. Various replacements for naive set theory have been proposed. The most common is called ZFC. This is a set of “self-evident” axioms ZF and one that SEEMS obvious C. C leads to some very odd results, so not everyone accepts it. More to the point C is independent of ZF. So if ZF is consistent so is ZFC and so is ZF not-C.

Which world do we live in, ZFC or ZF not-C? Or some other world? A hard question for the “but the truths are self-evident” types to answer.

In either case the Peano axioms are not the foundations of mathematics. The PA can (as Steve notes) be proven from whatever set of axioms you do choose as the foundation. That does not establish their truth.

Steve says the PA must be true because X created a model. Indeed. But X’s model is created using a particular set of axioms such as ZFC. Are those axioms true, and how do you prove it?

I think that is the cart before the horse. We CHOOSE the foundations in order to GET the PA because we really believe the PA,
and if I invented a new theory KBC full of “self-evident” axioms that did not give PA we wouldn’t use it as the foundation of mathematics.

32. 32 32 Æternitatis

Maybe it really just was a blunder? It would be no greater than the Grothendieck’s famous prime. http://en.wikipedia.org/wiki/57_(number)

33. 33 33 Ken B

I poked about at a couple of the comments in the May thread and found this from Joe Shipman:
“The real howler was Voevodsky’s presentation of 3 alternatives as
exhaustive, completely overlooking the obvious 4th alternative of a
hierarchy of systems of increasing strength, such that the strongest
system acceptable to him is stronger than Peano Arithmetic and proves
its consistency.
Voevodsky acted like his failure to see the well-orderedness of
epsilon_0 as an acceptably obvious axiom was all there was to say,
ignoring the existence of second order arithmetic, even weak forms of
which prove the consitency of PA. ”

This seems to fit the part where V complained about weird sets with one free variable, and argue because the sets are so weird we cannot reason about them. (For example a free variable formaula describing a set with the charming property we cannot prove any particlar n does or does not belong to. But prove means prove in first order.) But doesn’t second order remove some of that weirdness? (Question for Steve or any logic profs out there).

34. 34 34 Mike H

Fascinating reading on Gentzen’s theorem. Me likey.

Nonetheless, as KenB pointed out, while the Peano Axioms seem self-evident enough, pointing out “this axiom just says so-and-so. Obvious, isn’t it?” is not a proof – it requires a step of faith (or at least inductive logic, which is the same thing) to move from the sense of obviousness to belief.

Even besides these, Steve’s a-b-c proof has serious holes as a “mathematical” proof, as I pointed out. “Truth” in Mathematics means “follows from the axioms”. “Truth” in philosophy can’t even be expressed in mathematical terms.

35. 35 35 Steve Landsburg

Mike H:

“Truth” in Mathematics means “follows from the axioms”.

So you’re telling me that before anyone wrote down axioms for the natural numbers (say, a couple of hundred years ago), it was not true that 2+2 = 4?

36. 36 36 Mike H

@Steve – this is why I asked you what you meant by “true” in your a-b-c “proof”. We can assert that 2+2=4 is intrinsically true of the universe, but this ontological assertion is then a statement of faith. We can assert that 2+2 has always equalled 4 so far, then we are making an epistemological assertion, and if we assert that it’s always going to be true, again we make a leap of faith. Or, we can write down some axioms.

If “true” doesn’t mean “follows from axioms”, then your “proof” is not mathematical. Mathematics can’t talk about any other kind of “truth”. Here’s why.

Godel’s proof (at least the way it was presented to me in uni) works by encoding mathematical statements as numbers. Then, we construct a predicate PROVABLE(x), and encode it as a number. Finally, we write down Epimenides’ paradox : x = “NOT PROVABLE(x).” If x is provable, the system is inconsistent. If not, the system is incomplete. However, we don’t get a contradiction, because it’s not a priori true that “PROVABLE(x) OR NOT PROVABLE(x)”.

However, if TRUE(x) could be expressed mathematically, a mathematician of Godel’s calibre could construct a number y such that y = “NOT TRUE(y)”. However, here we do have “TRUE(y) OR NOT TRUE(y)”, hence “y=NOT TRUE(y)” is a full-blown contradiction. We either become constructivist or not, but either way, we should reject the idea that truth can be expressed mathematically.

Hence, your ‘a-b-c’ proof is either non-mathematical, or else ‘a’ is empty and ‘b’ is wrong.

Next time a mathematician asserts that some mathematical statement is “true”, ask how they know. They’ll almost always point to a derivation from axioms. Hence I say “mathematical truth” equates to “provable from axioms”.

37. 37 37 Alexander Kruel

Further information can be found here:

“Voevodsky’s seemingly polemic statement concerning the potential inconsistency of PA in fact seems to amount to the following: all the currently available proofs of the consistency of PA in fact rely on the very claim they prove, namely the consistency of PA, on the meta-level. (S. Awodey adds that it is a consequence of Gödel’s results that all proofs of consistency of PA must use methods that are stronger than PA in the meta-language.) So if PA was inconsistent, these proofs would still go through; in other words, there is a sense in which such proofs are circular in that they presuppose the very fact that they seek to prove. I raised a similar point before in comments here: if PA was unsound (which it would be if it were inconsistent), it might be able prove its consistency even if it is actually inconsistent (which also means that Hilbert’s original goal may have had a suspicious circular component all along). Now, we know by Gödel that PA cannot prove its own consistency, but the proofs of the consistency of PA available all seem to presuppose the consistency of PA (on the meta-level), so it boils down to roughly the same situation of epistemic uncertainty.”

38. 38 38 Roger Schlafly

“Truth” in Mathematics means “follows from the axioms”.
So you’re telling me that before anyone wrote down axioms for the natural numbers (say, a couple of hundred years ago), it was not true that 2+2 = 4?

You seem to be trying to distinguish truth from our knowledge about truth. You could even say that the Poincare Conjecture was true a couple of hundred years ago, even tho no human knew the proof. People just did not know back then how it followed from the axioms.

39. 39 39 Mike H

@Roger “You seem to be trying to distinguish truth from our knowledge about truth”

What I am trying to do is point out that Steve’s ‘a-b-c’ proof is not mathematical, as he asserts, but philosophical. Since sometimes, when people say “true”, they mean “follows logically from axioms”, I address this possible meaning of “truth” in mathematics. If you don’t accept that “truth” ever means that, fine. My case is even easier to make. I’d just have to say “There is no way to express truth mathematically, hence his argument (which depends on assertions about the nature of truth) is not mathematical.”

40. 40 40 Roger Schlafly

@MikeH: Yes, I agree that the a-b-c proof is not mathematical. It is philosophy. V’s attack on consistency is also more philosophy than math.

41. 41 41 Steve Landsburg

Mike H:

Next time a mathematician asserts that some mathematical statement is “true”, ask how they know. They’ll almost always point to a derivation from axioms. Hence I say “mathematical truth” equates to “provable from axioms”.

I think I have enough experience with mathematical discourse to say this is flat-out wrong. They will almost never point to a derivation from axioms. Instead they will point
to an *informal* but (at least seemingly) incontrovertible argument.

Ask a number theorist whether Fermat’s Last Theorem is true. In almost every case, they will say yes. Ask him how he knows, and he’ll point to the arguments of Frey/Serre/Ribet/Wiles. Ask him whether it follows from the Peano Axioms, and he’ll say he has no idea. (As far as I’m aware, nobody has the foggiest idea whether FLT follows from the Peano Axioms.) Ask him what axioms underlie the Frey/Serre/Ribet/Wiles argument, and he’ll probably say he has no idea. Figuring out what axioms would be necessary in order to formalize the Wiles argument is an active ongoing area of research. The fact that the argument has not been formalized, that it might never be formalized, and that we don’t know what axioms it rests on, does not stop people from accepting the argument and believing that it’s taught them something true.

If you think that Wiles argument is an argument from axioms, then you need to explain why people are working on trying to *figure out* what axioms would be necessary to formalize the argument.

42. 42 42 Ken B

@Steve: I think you have misread Mike H. He is making a cynical observation about mathematicians, not an ontological point. He is accusing the breed of ducking the question of how you know something is true by just saying “it follows from the axioms”. He is suggesting mathematicians are a little too cavalier in their claims to certainty. In particular he asserts they — in this case YOU — try to claim for their philosophical arguments (“a-b-c”) the same level of certainty as their formal proofs.

At least that is what his wording suggests to me.

43. 43 43 Mike H

Thank you, KenB.. :-)

It’s worse than that – not only do we cavalierly treat our informal derivations as formal proofs, we also treat our formal proofs as evidence of “truth”.

I can’t see any logical justification for these leaps of faith.

(but that doesn’t mean I think they are wrong… :-)

44. 44 44 Steve Landsburg

Ken B and Mike H: I have a question for you, to clarify what you’re saying.

Suppose I work in purely formal Peano arithmetic, where I have the symbols 0 (which, in the standard model, will be interpreted as the number 0) and S (which, in the standard model, will be interepreted as the successor function.

I write down the expression S0 and the expression SS0, which, in the standard model, will be interpreted as the numbers 1 and 2.

I note that these two expressions are *different*.

My specific question to you: Is the observation that these expressions are different an “act of faith”?

If, by your definition, the answer is yes, then of course everything we do with formal systems involves an enormous number of acts of faith. So the “act of faith” label can’t be avoided by saying “I’m deriving something from axioms”.

If, by your definition, the answer is no, then you’ve admitted that you can *count the S’s* without having committed an act of faith. That in turn means that you can *use the natural numbers* without committing an act of faith.

So — according to your definition, does it require an act of faith to note that the expressions S0 and SS0 are different?

45. 45 45 Ken B

@Steve: You are conflating. I do not agree with Mike H, as I believe my comments have made clear. (I believe the Peano Axioms ARE obviously true, and that we will and should pick the foundations we want to get stuff like that which we know is true. Whether we also want AC is another question …)

My answer to Mike is perhaps more modest than yours but still leaves me on firmer ground than he is. My answer is this. “Some truths I can derive from pure logic, but even in mathematics this is not generally possible. So I exlicitly identify a small group of axioms (and axiom schemeas) whose truth I presume. From them I draw entirely certain deductions. I quite agree that if my axioms are false som might be my deductions but my axioms are 1) quite clearly true to most people 2) explicit 3)operationally accepted by my critics who rely on many of them implicitly. I think that makes my position as a mathematician considerably firmer and clearer than anyone else’s in any other discipline. And I won’t strike off your head if you don’t believe.”

46. 46 46 Ken B

@Mike H: I will leave Steve’s questions for you except I want to poke a hole in Steve’s argument. Is it necessary to *count* the S characters in SSSSS0 and S0 to know they differ? I suggest you can build a finite state machine which is not actually capable of counting and yet which can distinguish these strings.

47. 47 47 Roger Schlafly

Steve: If you think that Wiles argument is an argument from axioms, then you need to explain why people are working on trying to *figure out* what axioms would be necessary to formalize the argument.

They are trying to demonstrate that FLT is a legitimate mathematical theorem, and to sharpen the statement of it.

48. 48 48 Mike H

@Steve clearly you need faith in your own powers of observation. This is not as trivial as it may sound.

@KenB[1] I was always taught that when someone says something is “obvious”, you should for a simple proof. Sometimes, this reveals that they don’t have any logical reason for the assertion. To continue to believe it requires some kind of faith, no? Now, you’ve given good reasons to believe the Peano axioms. However, I’m sure you’ll agree that these are not really logical reasons. They’re epistemological. You are putting your faith in the critical abilities of experts, and people’s ability to recognise abstract statements about numbers. I won’t say this is unreasonable, but I will say it’s a step of faith.

PS – I’d be interested to hear some of these “truths [you] can derive from pure logic”. I can follow Descartes as far as “I think therefore something is”, but I can’t quite see how his “I am” comes after that.

@KenB[2] it’s not necessary to count the S’s to observe that S0 and SS0 are different. You could, for example, measure their albedo instead. Or rely on some vague intuitive sense.

49. 49 49 Mike H

Mike H:

mathematics doesn’t have statements about “truth”

You keep saying this, but I can’t figure out why. The mathematics with which I am familiar contains many statements about truth. I’ve published quite a few papers in math journals that made assertions about truth, and the editors didn’t seem to think they were off-topic. My professors in grad school talked a lot about truth, and I’ve never heard anyone say that the Chicago Math Department doesn’t teach real math. When I go to math conferences, I hear people talking about truth all the time; nobody ever interrupts and says “Okay, time to stop talking about truth and get back to math”.

What experience, or what other than experience, leads you to say that “mathematics doesn’t have statements about truth”?

50. 50 50 Ken B

@Mike H: Model theory has statements ABOUT truth. Tautologies are derivable from pure logic.

51. 51 51 Mike H

Suppose we were able to express, as a predicate, “X is a true statement” in some axiomatic system. We would need this predicate to have certain qualities we expect of truth, eg “For all X, TRUE(X) OR ~TRUE(X)”, “X iff TRUE(X)” and so forth.

If this system were sufficiently powerful, we could construct the statement X = “NOT TRUE(X)”, in the same way Godel’s proof constructs the statement Y = “NOT PROVEABLE(Y)”.

However, if X=NOT TRUE(X), an Epimenides paradox means there is a formal contradiction within the system. Hence, no system that has predicates of the form TRUE(X) and is sufficiently powerful for Godel’s methods can be consistent – or no sufficiently powerful consistent formal system can reason about the truth of its own statements.

That’s what I mean.

52. 52 52 Steve Landsburg

Mike H: I agree with you that we cannot express truth as a predicate in a formal system (this is essentially Tarski’s theorem).

It does not follow that mathematics is not concerned with truth or that “mathematics doesn’t have statements about truth”, as you keep saying. The whole point is that mathematics is not a formal system. Formal systems cannot talk about truth (in the sense of “talk about” that we’re considering here). Mathematics can.

53. 53 53 Mike H

Mathematics is not a formal system

Would you deny that mathematical proofs are supposed to be “rigorous”? That is, that proofs are supposed to be, in the limit as rigor approaches infinity, formal proofs in an axiomatic system?

Perhaps, instead of saying “your a-b-c proof is philosophy, not mathematics” I should say “your a-b-c proof is completely non-rigorous and should be cleaned up or rejected”? You could start by defining what you mean by “true” and “natural numbers” in part (a). Otherwise, I can’t see where you get the assertion from.

54. 54 54 Steve Landsburg

Mike H:

Would you deny that mathematical proofs are supposed to be “rigorous”? That is, that proofs are supposed to be, in the limit as rigor approaches infinity, formal proofs in an axiomatic system?

I reject your “That is”. Of course mathematical proofs are supposed to be rigorous. That doesn’t mean they have to look anything like formal proofs in an axiomatic system. By way of example, I give you almost any paper in almost any mathematical journal, but let’s take
Wiles’s famous paper as a specific case. You have failed to confront the facts that a) every mathematician who has read and understood the paper accepts it as a rigorous argument and b) no mathematician in the world, including Wiles or anyone else, has any idea what axioms you would need in order to formalize this proof. This seems to be a clear counterexample to your assertion. Why isn’t it?

(I note, for the benefit of others, that the word “proof” has two completely different meanings in this discussion. One is “proof” as used in the everyday practice of mathematics, the other is “proof” as used in formal logic. I am sure you understand this difference, though other readers might be confused by it. We are talking about proof in the first sense.)

55. 55 55 Mike H

All right, what do you think rigor is?

56. 56 56 Ken B

I think Mike H is missing an important point. Formal systems and axiomatics are TOOLS that mathematics uses. Carpentry is not a saw, sculpture is not a chisel.

57. 57 57 Ken B

Here’s a simpler example than Wiles’s proof. We have seen on this board some of the bizarre results the axiom of choice gives. This raises the question — which mathematicians discuss and disagree about — “is the axiom of choice true”? Formal theories don’t answer the question since it is know the AC is independent of the other axioms.

58. 58 58 Steve Landsburg

Mike H:

All right, what do you think rigor is?

I think I know it when I see it. That’s of course an unsatisfying answer, but it becomes more satisfying when I observe that pretty much all mathematicians agree on when they’ve seen it and when they haven’t, even in the absence of a formal definition.

59. 59 59 Steve Landsburg

KenB:

Formal systems and axiomatics are TOOLS that mathematics uses. Carpentry is not a saw, sculpture is not a chisel.

Perfect! I will steal this analogy!

60. 60 60 Mike H

“I’ll know it when I see it” Would you say your ‘a-b-c proof’ is rigorous? The latest person to edit the relevant wikipedia article says “Rigor is a gold standard for mathematical proof… [it] can be defined as amenability to algorithmic proof checking”

Hence, at least one other person agrees with me that in the limit as rigor approaches infinity, a proof becomes a derivation within a formal system.

Since your a-b-c proof talks about the natural numbers and truth, it can’t be turned into a derivation within a formal system. Therefore it can’t be made rigorous by the definition given (by me and) wikipedia.

@KenB : yes, but the place we use these tools is when we want to prove something very very very carefully. My critique is against Steve’s ‘a-b-c “proof”‘ which is not a proof at all. His proof is not mathematical, because it doesn’t use these tools, and it’s theoretically impossible to tighten up his proof to use them.

61. 61 61 Steve Landsburg

Mike H:

Since your a-b-c proof talks about the natural numbers and truth, it can’t be turned into a derivation within a formal system. Therefore it can’t be made rigorous by the definition given (by me and) wikipedia.

I’ll say this again: Any formal system presupposes the natural numbers, because it presupposes things like lists of statements. So my a-b-c proof assumes less than any proof using a formal system assumes. I think it’s an odd definition of rigor that says that when you assume less, your proof becomes less rigorous.

62. 62 62 Mike H

Three points of confusion :
* The rigor or otherwise of a proof has nothing to do with the assumptions made in the proof. Rather, rigor has to do with how careful the logic is, how easy it might be to formalise the proof.
* The definition of ‘formal system’ may presuppose natural numbers, but that doesn’t mean any particular formal system does.
* Even when a formal system does presuppose natural numbers, that doesn’t mean every proof in the system does.

On the other hand, if a formal proof talks about some concept, that means the concept must be expressible within the formal system that contains the proof.

Your ‘a-b-c’ talks about the ‘truth’ of ‘statements about natural numbers’. If your ‘a-b-c’ could be made rigorous, then it could be encoded into a formal system. That formal system would be able to express statements about the natural numbers, and would be subject to Godel’s methods. It therefore could not simultaneously express statements about ‘truth’.

Therefore, any argument either
* does not talk about ‘truth’
* does not talk about ‘natural numbers’, or

In your ‘a-b-c’ you are “Tackling the Problems of mathematics using ideas from philosophy”, rather than the other way round :-)

63. 63 63 Mike H

PS – here’s a formal system that does not presuppose the natural numbers.

Axioms : P
Rules of inference : (none)

Admittedly, it doesn’t presuppose anything very much. Note that we may ourselves need to presuppose some things inexpressible in this formal system to reason about it, but the formal system itself can’t be used to talk about natural numbers, and is immune from Godel’s methods.

64. 64 64 Roger Schlafly

Steve, your a-b-c proof may assume less, but it is not really a proof unless it can be formalized.

65. 65 65 Roger Schlafly

Steve, your a-b-c proof may assume less, but it is not really a proof unless it can be formalized.

66. 66 66 Steve Landsburg

Mike H:

PS – here’s a formal system that does not presuppose the natural numbers.

Axioms : P
Rules of inference : (none)

Here are two examples of proofs in your formal system:

Proof A:

1. P.

Proof B:

1. P.
2. P.

How do you tell whether these proofs are different?

67. 67 67 Steve Landsburg

Mike H: Let me be more precise here.

You say your formal system consists of the axiom “P” together with no rules of inference.

But those are not all the components of a formal system. A formal system also consists of proofs, where a proof is defined to be a list of statements, each of which is either an axiom or follows from previous statements according to the rules of inferenece.

Note that a proof is a *list* of statements. You *cannot even define the notion of proof* unless you admit that you know what a list is. You therefore cannot work with a formal system — not even one as simple as yours — unless you already know quite a bit about the natural numbers. Therefore it cannot be true that the formal system “comes first” in the sense that everything we know about the natural numbers can be reduced to it. This is a far simpler and more elementary observation than Godel’s Theorem, but I claim it’s all I need to refute your position.

68. 68 68 Mike H

I agree that you cannot work with a formal system without understanding the natural numbers – that you can’t even define a formal system without some concept of a set, and you certainly need some concept of an ordered set to define proofs in the formal system.

However, this does not at all mean that any particular formal system can be used to reason about natural numbers or sets. It’s important not to confuse these two points. Hence, your point does not invalidate my proof at http://www.thebigquestions.com/2011/06/08/inconsistency/#comment-27702 (or less rigorously at http://www.thebigquestions.com/2011/06/08/inconsistency/#comment-27685) that your ‘a-b-c’ can’t be made rigorous.

And now, to change the topic completely….

Your point that “You therefore cannot work with a formal system — not even one as simple as yours — unless you already know quite a bit about the natural numbers. Therefore it cannot be true that the formal system “comes first” in the sense that everything we know about the natural numbers can be reduced to it” is very interesting. It may not be completely true – after all, we can formalise the machinery we need to reason formally, and I’m not convinced we need the full power of the natural numbers for that… however, in practice, you’re right. We don’t (in practice) approach the natural numbers formally. Instead, we pick them up through a mixture of social conditioning, empirical observation and intuition, with a little smattering of logic thrown in. We end up with an understanding of natural numbers that works pretty well for day-to-day life. You and I alike have taken a step of faith to believe that this thing we understand goes beyond pragmatism – that the natural numbers are really real, intrinsic to the universe, not merely a creation of man. I haven’t heard you acknowledge this as a “step of faith”, but whether you just believe in the natural numbers, or trust some informal chain of reasoning that leads up to it, There’s still at least one step of unreasoning faith in some axiom, observation or proof. There’s nothing wrong with that, I don’t see how we could function otherwise.

69. 69 69 Steve Landsburg

Mike H:

You and I alike have taken a step of faith to believe that this thing we understand goes beyond pragmatism – that the natural numbers are really real, intrinsic to the universe, not merely a creation of man. I haven’t heard you acknowledge this as a “step of faith”, but whether you just believe in the natural numbers, or trust some informal chain of reasoning that leads up to it, There’s still at least one step of unreasoning faith in some axiom, observation or proof. There’s nothing wrong with that, I don’t see how we could function otherwise.

I think we basically agree on this. I recoil from the phrase “step of faith”, since I think of a step of faith as something that is optional, whereas a basic awareness of the natural numbers seems to me to be unavoidable for human beings. But I repeat — I think we mostly agree.

70. 70 70 Mike H

Yes, mostly people don’t know why they believe what they believe. Feel free to dismiss my use of the phrase “step of faith” as semantics :-) However, I find it helpful for me to be aware of what things I believe because they follow from other beliefs, and what I simply believe. I suspect most people are not aware of these things. Sadly, this includes many who claim to be logical an rational, not taking anything “on faith”.

This has been an interesting discussion. I learned some useful things.

PS – As you no doubt know, I still don’t accept your ‘a-b-c’ :-)

71. 71 71 Ken B

@Steve re different proofs: I can use a finite state machine that distinguishes different lengths. I need not be able to do all of Peano arithmetic. I need STRICTLY LESS computational power.

72. 72 72 bbzippo

Hi Steven,

Consistency of PA means that it has _some_ model.
You claim that _the_ natural numbers is a model of PA. How do you know? Have you checked every axiom (including the induction scheme) against all natural numbers? And what are _the_ natural numbers anyway?

It is undisputed that claims about the consistency of PA may only rely either upon intuition or upon ZFC (or other “non-finitistic” means).
Your claim seems to be intuitive, do you agree?

I did not listen to Voevodsky’s lecture, but there is nothing wrong about discussing the possibility that the foundations are inconsistent. On the other hand, I know many highly ranked mathematicians who have poor undertsanding of the foundations and frequently misinterpret and abuse them.

73. 73 73 Steve Landsburg

bbzippo:

And what are _the_ natural numbers anyway?

_The_ natural numbers are the unique model of second order Peano arithmetic.

74. 74 74 bbzippo

Yes, second-order arithmetic is categorical.
But how do you know that it is consistent?

Once again, I know only two correct ways to claim consistency of PA:
1. We intuitively accept N as a model of PA, thus we believe PA is consistent.
2. We prove that PA has a model using transfinite induction, thus we reduce the question about consistency of PA to the question about consistency of another system.

Well, if you are familiar with Feferman’s “general arithmetizations” there is a third way, but it’s tricky…

Really, all I wanted to say that your claim “N is a model of PA hence PA is consistent” was a bit informal.

And I don’t agree Voevodsky’s claim that “consistency of PA is a genuine open problem”.

Consistency of PA is no mistery but it is not too trivial either.

75. 75 75 Steve Landsburg

bbzippo:

Really, all I wanted to say that your claim “N is a model of PA hence PA is consistent” was a bit informal.

I agree with this. I also claim that “I think, therefore I am”, and that’s equally informal. But I find it completely convincing, informal as it may be. I find “N is a model of PA hence PA is consistent” to be almost equally convincing.