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