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I'm interested in understanding the logical structure of mathematics - I want to know how it all fits together. To this end, its interesting to ask whether any authors systematically distinguish between 'theorems' (which have 'proofs') and what we might call mathematical 'beliefs' (which have what we might call 'evidence').

Let me try to explain the difference. By the way, this question is inspired by Peter Smith's answer here and Asaf Karagila's comment thereunder.

We'll use the following example. Begin with the group axioms, and for good measure, lets throw in the assumption that the domain of discourse is infinite (thereby obtaining the 'infinite group axioms'). We can do this by adjoining a new function symbol $S(*)$ together with the following axioms.

  1. $\forall x\forall y[S(x)=S(y) \rightarrow x=y]$
  2. $\exists x \forall y[S(y) \neq x]$

Okay, that will be our example. Now lets get to the point.

Observe that the following is a theorem of the infinite group axioms: 'for all $x$ and all $z$, there exists $y$ such that $xy=z$.' Furthermore, we know its a theorem because it has a formal proof from the axioms.

Now consider the following statement: 'for all $x$ and all $y$ we have $xy=yx$.' Almost nobody believes that this is a theorem of the infinite group axioms - surely there exist infinite groups that are not Abelian! Thus, the sentence 'commutativity cannot be deduced from the group axioms' is a belief. And, different people will offer different evidence in favor of this belief.

For instance, I might proceed as follows. First, I formalize the statement that 'commutativity cannot be deduced from the infinite group axioms' in the language of set theory. Then, I prove it using ZFC, perhaps by constructing an infinite group that is not Abelian. Or perhaps using other means. In any event, I have used the ZFC axioms to prove the formalized-in-the-language-of-set-theory version of the statement 'commutativity cannot be deduced from the infinite group axioms.' So, this is evidence for my belief. But, it is not proof of that belief. After all, ZFC might be inconsistent, or even if its consistent, nonetheless it may prove statements about arithmetic that are false. So I repeat, this is 'merely' evidence. And, someone may produce stronger evidence, perhaps by appealing to only a fragment of the ZFC axioms, or perhaps by formalizing the statement 'commutativity cannot be deduced from the infinite group axioms' in the language of arithmetic and proving it using only PA. If they're able to do this, then their evidence is stronger than mine.

So whereas all proofs are, in some sense, equally good, the same cannot be said of evidence. Not all evidence was made equal. If you appeal to a weaker theory, then your evidence is stronger. Of course, there's a socially determined cut-off point beyond which nobody will doubt your evidence. That's fine - we're still better off calling it 'evidence,' rather than 'proof', as I argue below.

Okay, so that's the idea. But, do any authors purposely and systematically make this sort of distinction? And, does it help clarify the logical structure of their writing? Also, if any logicians or philosophers of mathematics have written about this sort of thing, I'd be interested in reading their stuff.

Addendum. It is tempting to argue that, if we prove the consistency of a first-order theory using a sufficiently weak foundations, then this is more than just evidence of its consistency, it is proof. We might therefore try to reclassify certain beliefs and start calling them theorems.

There's at least 2 reasons to fight this temptation.

Firstly, I don't think there's a clear place to draw the line between safe and risky foundations. Thus, I don't think there's a clear place to draw the line between beliefs that must remain beliefs, and beliefs that we're so sure of that we can start calling them theorems. So, instead of trying to find the perfect location for this hypothetical line, and then dogmatically forcing our somewhat arbitrary decision upon others, I think we're better off just leaving beliefs and theorems as separate concepts.

This brings me to my second point, which is this. Proving a theorem is methodologically different to accumulating evidence in favor of a belief. In the former case, we can immediately begin trying to prove the theorem of interest directly from the axioms. Whereas in the latter case, we must first choose a foundational theory in which to proceed, we must formalize our belief as a sentence in the language of that theory, and only then do we begin trying to prove something.

Thus, I contend that the crucial difference between beliefs and theorems is not how certain we are of their truth; but rather that, methodologically, we deal with them differently. Hence the possibility that emphasizing the belief/theorem distinction could help to reveal the deeper logical structure of mathematics, which is what I'm interested in.

By the way, the word 'belief' isn't a great fit for what we're meaning here. Perhaps it would be better to call them 'blurgles' to avoid this issue. So under this view, theorems have proofs, and blurgles admit evidence.

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goblin GONE
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  • Your second "infinite axiom" is self-contradictory. There is some $x$ such that $S(x)$ is not defined, because it is not equal to any of the elements of the universe. – Asaf Karagila Jul 21 '13 at 00:27
  • @Asaf, i think it is fixed now. – goblin GONE Jul 21 '13 at 00:36
  • Yes. Now it's better. It says that $S$ is an injection which is not a surjection. Equally you can write a schema which says that there are at least $n$ different elements in the universe, for every $n$. But if you doubt all your foundations, than what are the integers and what are schemata? So you probably took the right approach in adding a new symbol. – Asaf Karagila Jul 21 '13 at 00:38
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    I don't think he's ever done this in writing, but when talking to you, Hugh Woodin sometimes mentions statements "that are true". He clarifies "note that I said that this is true, not that we have a proof of it." You may be interested in Theoretical mathematics, by Arthur Jaffe, and Frank Quinn, Bulletin of the AMS, 29 (1), (1993), 1-13. It is available online on the Bulletin's page. Also, read the responses, that were published in volume 30, pp. 159-211. Particularly recommended is the beautiful piece by William Thurston, On proof and progress in mathematics. – Andrés E. Caicedo Jul 21 '13 at 00:41
  • @AndresCaicedo, thanks for the references - I am reading 'On proof and progress in mathematics' as we speak. The author puts things in very nice terms, such as the 'definition-theorem-proof (DTP) model of mathematics.' – goblin GONE Jul 21 '13 at 01:31
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    You seem to be saying that, if I formally deduce a statement from the axioms for infinite groups, then that statement is a theorem, but if I formally deduce something from the axioms of ZFC, then that is only a belief. I suspect I'm misunderstanding you, so could you please tell me where my error lies? Or, if I'm correctly understanding you, then where is the borderline between those axiomatic systems that produce (via formal deductions) theorems and those that produce only beliefs? – Andreas Blass Jul 22 '13 at 01:37
  • @AndreasBlass, if you formally deduce something from the axioms of ZFC, then that is a theorem of ZFC. And unless you made a mistake, no one can ever take that theorem away from you. One thousand years from now, if a day comes when ZFC is found inconsistent, nonetheless your theorem will still be a theorem of ZFC. Now perhaps the intuitive reading of that theorem is that 'the infinite group axioms are consistent'. So, your theorem is evidence for this belief. – goblin GONE Jul 22 '13 at 02:08
  • But, you have not proved that the infinite group axioms are consistent. Rather, you have proved that a formalized version of this belief follows from the ZFC axioms. This is very good evidence! But, it is not proof. Indeed, a day may come when ZFC is discovered to be inconsistent, and perhaps a week later it is discovered that the infinite group axioms are, too! In this hypothetical future, your theorem is still a theorem of ZFC, but the belief it was being used as supporting evidence for turned out to be wrong! – goblin GONE Jul 22 '13 at 02:08
  • I hope you will forgive the ludicrousness of this example. I am not arguing that there's any real possibility that the infinite group axioms are inconsistent, nor that there's a high probability that ZFC proves false sentences about the consistency of formal systems. Rather, I am trying to illustrate the conceptual difference between a belief (or 'blurgle', if you prefer) and a theorem. – goblin GONE Jul 22 '13 at 02:08
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    I still don't see a qualitative difference. If I prove in ZFC that the infinite group axioms (IG) are consistent, then I have a theorem that might become worthless if ZFC turns out to be inconsistent. If I prove in IG that $\forall x,z\exists y,(xy=z)$ (as in your question), then I have a theorem that might become worthless if IG turns out to be inconsistent. I see a quantitative difference; it's easier to imagine inconsistency of ZFC than of IG. But I see no qualitative or essential difference. – Andreas Blass Jul 22 '13 at 15:05
  • @AndreasBlass, okay I will need a bit of time to write something. – goblin GONE Jul 22 '13 at 15:44
  • @AndreasBlass, here's a good way of looking at it. We can only ever prove theorems of the form $T \vdash \phi,$ never theorems of the form $T \not\vdash \phi$. For example, suppose you prove $\mathrm{IG} \not\vdash \bot$. Well, what you really proved was $\mathrm{ZFC} \vdash (\mathrm{IG} \not\vdash \bot),$ or $\mathrm{PA} \vdash (\mathrm{IG} \not\vdash \bot),$ or whatever. Does that clarify the distinction? – goblin GONE Sep 11 '13 at 01:04
  • So what I'm saying is statements like $\mathrm{IG} \not\vdash \bot$ and $\mathrm{PA} \not\vdash \bot$ are legitimate beliefs, however they can never be proved. We can prove statements that give us more confidence in our beliefs, like $T \vdash (\mathrm{PA} \not\vdash \bot),$ but ultimately this rests on our confidence in theory $T.$ – goblin GONE Sep 11 '13 at 01:16

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When you dig deep enough, you have to run into belief, or "evidence". The reason is that you have to stop at some point and say to yourself "Okay, my foundations seems somewhat solid. It matches how I think mathematics should be working, even if it's not an extremely well-suited foundations, it does work, as long as I believe in something."

Similarly, you have to wake up and believe that you are alive, and that you exist somehow. You don't have any proof of that. You only have highly circumstantial evidence. Your conscious mind tells you that its getting signals of an outer world, and it thinks that it is alive. But you can never know for sure. If you ever experiment with hallucinogens then you can easily assert the truthfulness of this statement. We conclude that life is out there and we are out there by some relatively long and somewhat consistent stream of input.

Mathematics is the same. You have to believe that if after some stream of input (in the case of first-order logic and $\sf ZFC$ we are talking about nearly a century of very smart people digging into the dirt) that it sounds like a good foundations for the mathematics you are doing. Then you just have to believe that it is consistent, but in that case group theory and the unprovability of commutativity from the basic axioms of group theory, are all granted because we know what is out there.

Of course you are free to feel that first-order logic and $\sf ZFC$ are not good foundations. You are free to choose other foundations. Type theory, category theory, second-order arithmetic, Kripke-Platek set theory, $\sf CZF$, $\sf ETCS$. You can even reject everything else and conclude that mathematics must be based on solid provability which we can actually write down to the last detail and become an ultrafinitist.

But all that is up to you, and the philosophical outlook that you have on mathematics. But you really just have to remember that at some point you really have to stop pussyfoot around this question and start doing mathematics, or else you're not a mathematician you're a philosopher (which is not a bad thing for itself, but it's a whole other thing).

Asaf Karagila
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  • Asaf thanks for your answer. Just to clarify, I don't think its very likely that ZFC proves a false sentence in the language of arithmetic. My interest in this topic comes more from the point of view of: 'how should we think about and/or write mathematics' as opposed to some sort of uber-skepticism regarding the usual foundations. Furthermore, I used the words 'belief' and 'evidence' in the question because they were the best words I could find at thesaurus.com, not because I was trying to make some sort of an uber-skeptical philosophical point. – goblin GONE Jul 21 '13 at 01:25
  • Also, here's a fun little observation. Suppose we're 99% certain that ZFC proves only true statements in the language of arithmetic, and we're 99.9% certain that PA proves only true statements in the language of arithmetic. In that case, a proof of the consistency of a formal system using PA is $10$ times as good as a proof of the consistency of that formal system using ZFC, holding all else equal. From this point of view, the project of reverse mathematics is very much worthwhile, even if we're very much not doubtful of ZFC! – goblin GONE Jul 21 '13 at 01:36
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    @user18921: In all honesty, I don’t think that the hypothesis here is really meaningful, so I don’t see this as any kind of justification for reverse mathematics. (Mind you, I’m not knocking reverse mathematics; I just think that it’s its own justification.) – Brian M. Scott Jul 21 '13 at 04:18
  • @BrianM.Scott, wait are you saying that the hypothesis that PA could be unsound is unmeaningful, or are you making the stronger claim that the hypothesis that ZFC is unsound wrt arithmetic is unmeaningful? If you're only making the first claim, then yes, I tend agree; although, if I were ever to become a teacher, I would not try to push that agreement onto my students as some sort of a dogma. – goblin GONE Jul 21 '13 at 13:27
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    @user18921: I don’t think that there’s any chance that either is unsound, but that wasn’t my point. I don’t think that it really makes any sense to quantify (un)certainty in that way, let alone to manipulate the resulting quantities arithmetically. – Brian M. Scott Jul 21 '13 at 20:39
  • @BrianM.Scott, I am not sure either way. I actually asked a question about it a long time ago, but received no responses. – goblin GONE Jul 21 '13 at 20:42
  • @user18921: That question was about consistency, not soundness; I do grant a (very remote) possibility of inconsistency. However, my objection to quantifying (un)certainty still applies. – Brian M. Scott Jul 21 '13 at 20:52
  • @BrianM.Scott, I believe that soundness implies consistency. So, are you saying that you grant a very remote possibility to ZFC being inconsistent, and but no possibility to ZFC being consistent but unsound? In any event, I see no reason why quantifying the probability of unsoundness versus quantifying the probability of inconsistency should be fundamentally different problems. – goblin GONE Jul 21 '13 at 21:02
  • @user18921: Ignore my earlier comment about soundness: I wasn’t awake yet! I don’t understand your last sentence: I thought that I made it clear that I don’t consider them different. – Brian M. Scott Jul 21 '13 at 21:07
  • @BrianM.Scott, yes we're in agreement then. In the sense that, I see no reason why they should be different. Anyway, I have yet to read an argument for or against the position that these kinds of uncertainties can be quantified. Is there any particular reason why they should be unquantifiable? – goblin GONE Jul 21 '13 at 21:14
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    @user18921: The simplest possible reason: for me they are unquantifiable. I literally cannot quantify them: I have no idea what such numbers would mean. Perhaps you can, and perhaps I can infer something about your beliefs from the numbers that you assign, but they tell me nothing about the theories. – Brian M. Scott Jul 21 '13 at 21:21
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This answer is related to the comments of Andreas Blass above, but I think it more directly addresses the point you're trying to make. Let's use YFFS as an abbreviation for "your favorite foundational system".

The distinction you're making is a valid one. It's stronger to exhibit a formal proof from the group axioms than to prove one exists in YFFS. In the first case, you have the proof, while the second relies on YFFS "telling the truth" about the group axioms and their consequences. This would be an issue if doing group theory was about deriving consequences of the group axioms.

It's not! A group is a mathematical object (a set, together with a binary operation, satisfying the group axioms) living in our imagined mathematical universe (which can be encoded/interpreted in YFFS). A theorem about groups is a theorem of mathematics (which can be formalized in YFFS), about these mathematical objects.

Similarly, number theory is not about looking for formal consequences of PA - it's about using all the tools of mathematics to investigate the properties of $\mathbb{N}$, which is a particular mathematical object.

There are a few reasons to prefer this wholistic approach (proving theorems of YFFS about groups) to the restricted approach (proving theorems of $T_{\text{Groups}}$, the first-order theory of groups):

  1. Most interesting theorems about groups can't be expressed in the first-order language of groups. Okay, as you pointed out in your question that you linked to above, you can move to a different language to capture different scenarios. So to quantify over homomorphisms, you can add a new sort for functions and assert that the things occupying this sort are supposed to be functions etc. etc. But then you have to start from scratch proving theorems in this totally disconnected logical setup - there's no underlying framework to connect them.

  2. Not only are the different situations involving groups disconnected from one another, you also miss out on connections with other areas of mathematics completely.

  3. Proving much of anything directly from $T_{\text{Groups}}$ and its related theories is very difficult, not to mention tedious!

My point is that what you call beliefs are exactly the same things as what most people call theorems (of mathematics), and rightly so.

Now, there are some people who are interested in theorems that are "stronger" than theorems (of mathematics). The project of reverse mathematics is about understanding how much of mathematics (i.e. how much of YFFS, which for reverse mathematicians is usually second order arithmetic) is really necessary to prove a theorem. The less you use, the more robust the theorem, in the way you noted. But this is still using a foundational system, not proceeding directly from $T_{\text{Groups}}$. I don't know much about the automated theorem proving community, but I could imagine that a more restricted setting could be useful here.

Alex Kruckman
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The most canonical way to prove -- prove, not just believe -- that something does not follow from some axioms is to construct -- i.e., prove the existence of -- a model of the axioms in which the "something" comes up false. Therefore, the role that YFFS plays in the answer above is that of a semantics, meta-theory or "model-construction" language. We're definitely not better off calling an independence result "evidence" for a "belief" rather than proof of a theorem -- because all these claims that something does not follow are just proved in a different theory -- in a meta-theory, model theory, or model-construction language.

Some might still object that everything that is proved from the axioms is following purely syntactically, whereas independence proofs are being done "semantically" instead, through reasoning about models. Even that semantic meta- reasoning, however, also becomes purely syntactic inference from axioms if we do it from/using a model-construction language (typically ZF) that is itself formally axiomatized.

So the notion that we need to demote independence results to "beliefs" is just wrong. The only thing that needs to be so demoted is our "belief" in the consistency of ZF, and that is arguably just not worth arguing. This insistence that people just shut up about questioning the consistency of the semantic framework that everybody is already using anyway (doubts or none) is motivated by (in addition to the fact that everybody is already using it anyway) the following two considerations: 1) If non-provability (i.e. consistency) proofs in general require some sort of semantic ascent, then, having used ZF to prove that PA is consistent, we would then have to use something even more complex than ZF to prove that ZF is consistent, and this ascent would be perpetual -- we would never "get there". 2) If you achieve a consistency proof without ascending (if one of these theories should somehow construct a model of itself, or otherwise prove its own consistency) then by the incompleteness theorems, that theory is inconsistent.

In practice the burden of proof is always on the side that would allege that YFFS is inconsistent, because (as long as YFFS is, like ZF, a recursively axiomatizable first-order theory), it follows by the completeness theorem that if that inconsistency is happening, it must be provable, while (by contrast) if YFFS is consistent, that must (within the same theory) not be provable (so we shouldn't worry about being asked to do the impossible).

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  • Honestly, I think you're just dodging the issue. (I mean a respectful way). The culprit is this phrase: 'The only thing that needs to be so demoted is our "belief" in the consistency of ZF, and that is arguably just not worth arguing.' You don't know that ZF is consistent (or sound), and neither do I, because to know you would have to prove it, and you cannot prove it w/o using a stronger, more objectionable system, which itself cannot be proven consistent (or sound) w/o yet a stronger system. Thus at the end of the day, we're left with a lot of mathematical beliefs. – goblin GONE Aug 22 '19 at 09:16