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As everybody knows, there are thousands and thousands of theorems in mathematics having a proof that is accepted without any doubt because it follows impeccable and rigorous mathematical reasoning.

I would like to know which the limits are (if any) when thinking about preparing a formal deduction for some mathematical proof:

  • Is every existing theorem proof (made by mathematicians up to the moment) formalizable inside one of the most important axiomatic systems (PA, ZF, ZFC, ...)?
  • Is there any theorem with an accepted mathematical proof that escapes all efforts so far to formalize it?
  • Is there any evidence that humans use some kind of mathematical reasoning that is impossible to formalize?

NOTES:

  1. I know that some proofs are not accepted by some mathematicians (e.g. intuitionists). Currently I am interested in normal logic, not the intuitionistic and other special logics.
  2. I am not referring to open problems of mathematics, but to theorems for which a proof already exist and is accepted generally by mathematicians. There are two examples that interest me greatly:

    • Fermat Last Theorem, a very simple arithmetic sentence that has an extremely difficult and long proof. As far as I know, it is an open question whether it is provable inside PA or not, but this doesn't worry me at the moment. I just would like to know if the existing proof can be formalized in one of the most important axiomatic systems (ZF, ZFC, ...).

    • Gödel's G sentence. In the introduction to "On Formally Undecidable Propositions Of Principia Mathematica And Related Systems", Gödel states that this sentence is true in the context of the Principia Mathematica (which I take as equivalent to being true in the standard model of natural numbers). He says that the sentence has been effectively proven using meta-mathematical arguments. These arguments are, of course, outside PA. The question is: is it possible to formalize those sophisticated meta-mathematical arguments in one of the most important axiomatic systems (ZF, ZFC, ...)?

  3. I know also that you can reach the same conclusion (theorem sentence) following different "paths" from the axioms in a formal proof (i.e. there are many different formal proofs for the same theorem). Where I am curious the most is about "replicating" the mathematical proof formally as best as possible, so that the mathematical proof serves as a sketch for the formal deduction. I would like to know if every step in a mathematical proof can have the corresponding piece in the formal deduction.

ADDITIONAL NOTES (April 9th):

At this moment I am not curious about the philosophical aspect regarding the limit of human reasoning, just wondering about existing theorems produced so far by mathematicians. The history has provided a vast amount of mathematical knowledge already. If there is no evidence that the reasoning techniques used so far by mathematicians for proving theorems are beyond the usual formal systems (say ZFC), then that could be a good reason to think that no human will ever produce such a proof.

So I would like to know if there is practical evidence based on the study of existing theorem proofs. It has been said that the “proof of Fermat's Last Theorem […] is a long way off from being formalized in a system like ZFC”. I would like to know if the reason is one of the following:

  1. Impassable points (there are theorems on the way up that logicians are unable to formalize so far). In the process of formalizing all theorems on which Wiles proof is based (or even Wiles theorems themselves), there is (perhaps) some theorem that, as much as logicians try, they don’t find the way of translating it into a formal deduction because it uses a kind of mathematical reasoning that can’t be formalized (because ZFC falls short or because logicians don’t find the formal “path” in ZFC for going from certain point of the proof to the following one). Has this ever happened?
  2. It is just a matter of patience and mechanical work (until all the theorems on which Wiles proof is based are formalized, which will happen eventually because formalizing theorems is just tedious mechanical work and people is working on this). Is this the case? Is the process of producing a formal deduction from a mathematical proof a straightforward process (although tedious). Can this “translation” process be guided directly by the deductions used in the mathematical proof or (on the contrary) does it put logicians into constant challenge for producing the formal proof?
  3. Lack of interest?

In short, I would like to know if the formalization process is just mechanical work (guided by the mathematical proof) or if some parts require lots of investigation, and even put logicians at dead ends.

ADDITIONAL NOTES (April 12th):

For expressing my doubts clearer, let's take some existing theorem TH and suppose that all the theorems and theories on which this theorem is built (let’s call these "external theorems") are already formalized in the default formal system (ZFC + FOL). If my very basic (amateur) knowledge about axiomatic systems is good enough, the formal deduction for TH will be built directly on the (already existing) formal deductions for the external theorems. Then what is left for formalizing TH is formalizing the arguments in the theorem itself. Let's also suppose that the natural-language mathematical proof for TH has no gaps and it is correct (by extensive peer review).

With those suppositions, I would like to know what kind of difficulties one can face when formalizing the proof:

  • Has it ever happened (with some existing theorem) that logicians have reviewed the mathematical proof, have found it intuitively correct (as in every peer review) and they are unable to translate it into a formal proof as much as they try? (without having to rewrite completely the argument, of course).

  • Is the translation quite straightforward or, on the contrary, logicians have to spend days and days of investigation trying to find the way of translating an argument into specific ZFC axioms and FOL inference rules?

I guess that formalization nowadays is a sufficiently mature field and has already provided good experience on the kind of difficulties one can face. Sorry for my insistence. I am not a mathematician, just an amateur of foundations of mathematics, and perhaps I am asking too obvious a question. I have had this curiosity for years and I would like to know.

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A very quick note on your last question:

  • Is there any evidence that humans use some kind of mathematical reasoning that is impossible to formalize?

You mention Godel's Incompleteness Theorem, and some people (most notably the well-known mathematician Roger Penrose) have based an argument off this result to argue that mathematicians reason in a way that transcends any formal method. Roughly, the argument goes like this:

"Suppose our reasoning is captured by some formal system $F$. Given that this is a formal system, we can use Godel's method to construct a Godel sentence $G$ for this system: a sentence that is true but unprovable by this system. Hence, there is something that I can prove but this system cannot. So, I cannot be that formal system $F$. Since this argument can be made for any formal system, my reasoning cannot be captured by any formal system."

The big problem with this argument is that you can do the Godel construction only for systems that are consistent (for, if they are inconsistent, they can prove everything). So, you'd first need to prove for any formal system $F$ that it is consistent. And that can be really hard to do! In fact, Godel's Second Incompleteness system proves that no (complex enough; think PA or above) consistent formal system can prove its own consistency. And, so far we have no good reason to believe that we human mathematicians can prove the consistency of any formal system. We have no proof of the consistency of ZFC for example, and for some of the most simple Turing-machines we (at least so far) can't figure out their halting behavior. Because of that, it certainly looks like that if our reasoning is captured by some formal system, we cannot prove that it is consistent, and hence the whole line of reasoning above will fall apart.

In fact, if there is any reason to believe that our reasoning is not captured by a consistent system like ZFC it is that our reasoning may well contain inconsistencies. Of course, it is exactly through processes of rigorous checking and evaluation (think peer review) that those inconsistencies are typically revealed and we correct our errors. As such, you could make the argument that the reasoning by the mathematical-community-at-large probably is consistent. Indeed, the attempts to formalize mathematical proofs into formal proofs in systems like PA or ZFC are a way to keep that reasoning consistent, and so in that sense you could make the following argument:

"the reasoning by the mathematical community can be captured by formal methods given that we use exactly those formal systems to rigorously accept those claims. Indeed, until we do, it is not accepted as a proof by the community"

But, I don't think that's quite right either: you mention the proof of Fermat's Last Theorem, which indeed has clearly been accepted as proof and yet is a long way off from being formalized in a system like ZFC. Also, I know of no occasion where any formalization effort to that level of formal detail has ever revealed a mistake in our reasoning that wasn't revealed beforehand simply by peer review, so it is not as if doing explicit formal proofs in those systems is at the core of mathematical reasoning. Indeed, most mathematicians do math without ever creating such detailed formal proofs. So, I don;t think this second argument really works either.

In sum, I would say this is an open question. But no, I would say we don't have any evidence that mathematical reasoning cannot be captured by a formal system, or even what I would consider a good argument. In fact, there are good arguments that our reasoning can be captured by some formal system. I find Turing's argument that any 'systematic way of figuring stuff out' can be captured by a formal method quite compelling. You could also make the argument that anything the brain is doing is computable, and can thus be transformed into a formal system. Of course, we do lots of reasoning using tools from the environment (most notably symbolic systems to express our thoughts, and of course formal systems themselves), and so I don't think we can reduce all of mathematical intellect to naked brains alone, but those other factors may well all be computable and thus ultimately formalizable as well. But, this is all pretty speculative at this time.

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  • $\begingroup$ Dear Bram, I appreciate very much your answer. I find it very interesting. You seem to believe that all mathematical reasoning could be formalized. However, I would like to know if there is practical evidence based on the study of existing theorem proofs. You say that the “proof of Fermat's Last Theorem […] is a long way off from being formalized in a system like ZFC”, which leaves me with my initial curiosity. I have added an additional note to my question above. $\endgroup$ – Manuel Quijada Apr 9 '20 at 17:35
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Is every existing theorem proof (made by mathematicians up to the moment) formalizable inside one of the most important axiomatic systems (PA, ZF, ZFC, ...)?

Yes, essentially - or rather, that's been incorporated into what we claim is the case (we don't actually write out the formal proof itself, generall).

Specifically, in modern mathematics, when we claim that we've proved a theorem we're claiming a bit more. We also implicitly claim that the "natural language" proof we present can actually be translated into ZFC (unless we specify some other system - ZFC is the "default").

There are of course a few subtleties:

  • This is a huge leap from merely asserting that we have a convincing argument for the statement's truth (whatever "mathematical truth" is exactly). It's really the major coup of the axiomatic approach to mathematics: that we've pinned down a formal system (namely, ZFC in classical first-order logic) which is agreed upon by the vast majority of mathematicians as a final testing ground for whether an argument needs further hypotheses (even if one believes that large cardinal axioms are true, say, one needs to explicitly say if one uses them). I think this is partly due to the growing awareness of the philosophical and practical difficulties with mathematical Platonism and its relatives. Regardless of why, eventually the attitude that good mathematics is formalizable became sufficiently dominant.

  • The formalization process is in general extremely difficult, even when there's nothing really wrong with the natural language proof. Natural language reasoning hides a lot of stuff under the hood, so to speak, and when building a formal proof from a natural language proof we have to tease all of that out. In particular, we're nowhere near a formal proof of FLT. (Godel's incompleteness theorem, however, has been formally proved - and FWIW is really not that mysterious at all. Also, note that while the proof of $G_T$ cannot be carried out in $T$ for appropriate theories $T$, the theorem "If $T$ is appropriate then $T$ does not prove $G_T$" is provable in very weak systems such as $I\Sigma_1$, a tiny fragment of $PA$.) Also, one major point in favor of explicit formalization (as opposed to "we can but we're lazy") is error detection: see e.g. here (and more generally the issue that a lot of published mathematical arguments have "nontrivial gaps").

  • There's no claim of optimality here. Indeed, for almost everything ZFC is massive overkill (and while a drawback from some perspectives, this is part of the reason it won the "foundations battle," at least for now). Moreover, determining better axiomatic bounds is extremely complicated. For example, there are theorems in logic which tell us that the axiom of choice hypothesis cannot be necessary for the proofs of certain "sufficiently simple" sentences (e.g. Shoenfield absoluteness). To take a more specific example, with Fermat's last theorem the situation is roughly the following. If one glances at the proof very briefly, one might be worried about the possible role of large cardinals (specifically inaccessibles, or Grothendieck universes). However, with relevant background it's (apparently) clear that these are utterly unnecessary - the point being that we really don't need the full category-theoretic apparatus which those large cardinals are used for (see e.g. here). The big proof-theoretic question for FLT is whether it's provable in PA. The general suspicion at this point is yes, and my understanding is that McLarty and McIntyre have (separately? together?) developed an outline of how this would go, but it's certainly nontrivial (to put it mildly). For reasonably-concrete theorems, the search for better axiomatic bounds (upper and lower) belongs to reverse mathematics.


Is there any theorem with an accepted mathematical proof that escapes all efforts so far to formalize it?

I'm not sure. FLT certainly escapes all efforts so far, but that's because there haven't been any - nobody thinks (so far as I know!) that we're anywhere close to being able to actually do that, even granting that the natural-language proof is fine. A better candidate would be a theorem towards whose formalization there has already been substantial (unsuccessful) effort for reasons other than regress ("to formalize this we have to formalize that, and that means we have to formalize those, but the way we formalized these wasn't actually optimal it turns out so ...").

I'm not aware of anything like this. I'm confident that a conceptual barrier to formalization - "we have no idea how to formalize this bit" rather than "oh wow this is going to suck" - would cast any proof back into doubt, as long as that barrier could itself be convincingly communicated (if I tell you I can't formalize something I need to convince you that I'm not just bad at my job before you'll be worried).


Is there any evidence that humans use some kind of mathematical reasoning that is impossible to formalize?

No, I don't think so. "Impossible to formalize" is a grotesquely high bar, to the point that I'm not even sure what would constitute evidence for such a claim.

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  • $\begingroup$ Hi Noah, thank you very much for your answer. It comes close to the kind of answer I was expecting but it leaves me with doubts because I still don’t understand the nature of those extreme difficulties you talk about. On the one hand you say that when a mathematician claims that they have proved a theorem, it is taken for granted that it can be formalized in the "default" formal system (ZFC + FOL). That would be more or less as if they were saying: "Hi logicians, this is one more theorem for you to entertain translating it into a formal deduction". $\endgroup$ – Manuel Quijada Apr 12 '20 at 13:05
  • $\begingroup$ However, on the other hand, you say that The formalization process is in general extremely difficult, while I would have expected to read just laborious work. I have added an additional note to the question text above trying to express my doubts more clearly. $\endgroup$ – Manuel Quijada Apr 12 '20 at 13:05

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