If there were only one single mathematician in the world, would s/he be able to produce a mathematical proof?
This question was motivated by the Math stackexchange question:
I thought it would be more appropriate on here.
This is a semantic problem. There are multiple operating definitions of "proof," and the textbook you're reading from fails to distinguish between them. That's okay in a practical sense, but it's not okay if you're asking a question like this one.
So let's split it up. There are lots of other potential operating definitions that are compatible with the one in the textbook, but here are two relevant ones:
(1) is, obviously, pure math. There's no reason why someone could not construct a "proof" in this sense, on their own, given the right tools. This part of the question is easy to answer. (2), however, is social, and requires an audience to be present. Now we have to ask what an audience is, in this context. There are three main readings I can see:
(1) obviously excludes the author; in this case, the author would not be able to write a (social) proof for themself, even if they could convince themself. In (2), the author could construct a proof, by writing it out as if it were going to be read by another person. In (3), the author could be their own audience.
There are other "audience" definitions possible, but these are the important ones I can think of. There are also other "proof" definitions possible. I definitely won't claim this is complete.
This is what it comes down to, though: how do you choose to define "proof"? And, if your definition requires an audience, how do you define "audience"? If both those permit a single person to write a proof for themself, then it's possible.
Some will say that a proof is defined simply in purely technical and syntactical terms: a set of statements that conforms to a certain set of syntactical transformation rules. As such, you could even have proofs with no mathematicians at all.
However, even if you add the 'convincing' part in there, I would say that when people produce proofs, they are doing this to convince themselves of the truth of something as much as any else.
So, either way, a single person can still produce proofs.
I, for one, don't agree with the definition given by the author of your logic book. I think your question here highlights one of the many problems with it.
Formal proofs are the Platonic objects to which refer the sorts of proofs you find in textbooks and research papers. In this sense the relationship is no different from that between any mathematical object and the names and typography we use to refer to that object. You might as well say a manifold is a psychological or sociological notion, simply because the author of a textbook on manifold theory might fail to communicate to his readers the definition of a manifold. This is clearly absurd.
If you’re very verbose, you can write down your axiom schema, perhaps starting with David Hilbert’s six rules of logic or the axioms of Natural Deduction, the axioms of equality, set theory or Peano arithmetic, and so on. Many mathematicians use automated proof checking to verify the correctness of a proof, or even proof-assistant software. If the one mathematician on Earth is also a competent programmer, he or she will be able to write a simple program to parse a formal proof and verify its validity through pure symbol manipulation. When you specify which theorem or axiom justifies each transformation of the well-formed formula on each line of the proof, there’s no ambiguity about what it means for the proof to be correct or incorrect given its premises and axioms.
I posted a simple little proof to Math.SX a few days ago. It tries to be simple and elementary, spelling out the steps that the other answers and comments left the reader to work out for themselves. The first comment it got was, “The best answer for us folks who aren’t so math savvy.”
There are a lot of gaps in it, and it assumes that the audience is comfortable with high-school algebra. I suspect that a lot of people who aren’t regular readers of a math website would have the same reaction to seeing me go from ln u + ln v - ln u to ln (u·v/u) that I do to skimming a short proof from a branch of mathematics I haven’t used in twenty years. I don’t bother to explain why that’s valid. Nor do I explain why you add constants of integration when taking the indefinite integral of both sides of an equation, or why you can move the two constants to a single constant on one side.
I could. If I’d spelled out every step of the transformation from ln u + C₁ = ln v + C₂ to ln u = ln v + C, it might have gone something like this:
We’re implicitly, although the question didn’t specify one, working in a mathematical structure called a field. Since we defined u and v as functions of some variable x and need the natural logarithm to be a valid operation for all values of u and v, Given the context, it’s likely that x is a positive real number, and u and v are positive. Neither the original question nor my proof ever spelled this out, and other interpretations are possible, but that’s what a mathematician writing in English in the early twenty-first century would assume the question meant by default. So I stay somewhat agnostic: I implicitly assume that the range of ln u and ln v is some field, but not which, and I don’t spell that assumption out.
That allows me to add, subtract, multiply, divide, and take additive and multiplicative inverses. In particular, it lets me construct the constant -C₁ such that C₁ + (-C₁) = 0.
A formal, step-by-step, proof from this point on where every step is justified by exactly one standard axiom might look like this:
Even after that handwaving ahead of time about how I’m assuming all the terms here are elements of some field, there’s still some hand-waving left. I said that C is a constant, but to actually prove that, I might show that it’s the difference of two constants, a special case of applying a pure function to constant inputs. And of course I use a lot of shorthand and take for granted a lot of mathematics that not every mathematician in history would have accepted, for example, that ln u and ln v are well-defined.
I didn’t write out the steps in that level of detail, because I was writing for humans (who would have found it tedious and patronizing), not a computer. A proof written to be as easy as possible for a computer to parse algorithmically would have looked very different. But, even if I were writing a proof checker for my own use, as the last mathematician on Earth, I’d find it a lot more interesting to figure out how to make it smart enough to fill in more of those gaps itself than I would to write every proof that way.
Nobody ever does that. It would take forever, waste a lot of space, and annoy everybody.
He’s the most famous example of an autodidact who wrote down amazing results in notebooks, which other mathematicians don’t consider “proofs.” This is probably because he worked out the derivations and notes on a piece of slate, wrote down what he considered the important results, and then erased the slate to save money. He wasn’t really trying to “prove” anything to some mathematician in Britain yet.
If the only mathematician in the world learned that way, his personal writings would probably be like that: not really intended to convince anyone else. (One philosopher’s comment on the earliest known proof in Greece, that the area of both halves of a circle split by a line through its center is the same, is that the real brilliance of it was not the argument, but realizing that a proof was needed.) Are his notebooks “proofs?” The fewer steps he skips—or the more that the steps he skips fit into a pattern of regular examples that a reader could figure out to reconstruct his thought process—the more proof-like they are.
I think the important question is, what happens if the world’s only mathematician takes on a student? Even someone who finds his lost notebook years after he died (which also happened to Ramanujan). If the student thinks his writings constitute a convincing argument that the teacher hasn’t made a mistake somewhere, even when they results are counterintuitive, and they don’t need to be taken on faith, it’s probably a sort of “proof.”
Other answers here are wrong because they assume the existence of mathematicians and proofs in the absence of social practices capable of making those descriptions true.
In Philosophical Investigations 200 Wittgenstein gives a suggestive example:
It is, of course, imaginable that two people belonging to a tribe unacquainted with games should sit at a chessboard and go through the moves of a game of chess; and even with all the mental accompaniments. And if we were to see it, we'd say that they were playing chess...
If we stumbled upon the tribe as they moved bits around the board, we would indeed think they were playing chess. It looks just like chess! (This is certainly what the other answerers would say.) But we would be wrong to do so. Chess is a game, and these people aren't playing a game. They have no idea what games are.
The obvious objection: chess is a formal system, so they are of course playing chess, even if they don't realize it. But is chess a formal system? It certainly has formal components---rules---but is it just a formal system? If it were, anything isomorphic to it would be chess as well. The example goes on:
...But now imagine a game of chess translated according to certain rules into a series of actions which we do not ordinarily associate with a game -- say into yells and stamping of feet. And now suppose those two people to yell and stamp instead of playing the form of chess that we are used to; and this in such a way that what goes on is translatable by suitable rules into a game of chess. Would we still be inclined to say that they were playing a game? And with what right could one say so?
These questions aren't rhetorical. We obviously aren't inclined to say that they were still playing a game, and if we did so, it would be purely because we had a theoretical ax to grind. That theoretical ax would be: there is some essence to chess, which is a purely abstract thing, so I can say, a priori, that everything isomorphic to chess is chess. Since the common use of the term is in conflict with that idea, you must have some better idea of what chess is in order to correct the masses. But what could that better idea possibly be? Do you really know "what chess is" better than I do? What is it that you are supposed to know, that I do not? The only answer I can come up with is the idée fixe, "Chess is entirely abstract and anything isomorphic to it is chess as well". But what is the foundation for such a belief?
Chess isn't the pieces, or the motions, or the rules, or the thoughts, or the set of possible game-states. Chess is a game involving all of those things.
Math is a lot like chess. We have rules for transforming sentences, as opposed to rules for transforming the game board. Math is not a set of rules, although it is governed by rules. Math is not a set of sentences, although it is expressed in sentences. Math is not a bunch of objects (numbers, sets, functions, whatever), although it deals with those objects. And so on. There's no boiling it down to something where you could say "And anything isomorphic to that is math!", because there's no boiling it down at all, in that sense.
(By the way, these statements aren't one man's opinion; they're observations, open to anybody, about the way we use the word "math".)
In the end, though, what is and is not a proof is a bit fuzzy. If you like, someone can have "proved" 2+2=4, just by taking 2 rocks, and then taking 2 more, and ending up with 4 rocks. "Demonstration" seems an apt word. If that is all you mean by proof, then yes, anybody could do it.
On the other hand, if you mean a formal proof, then certainly not. Say they make a geometrical construction, the earliest sort of proof. If they have no experience of the application of shapes in a "math-y" way, why wouldn't we think the "mathematician" isn't just making art? (By hypothesis they don't have a tradition of that application. Otherwise they wouldn't be the only mathematician. After all, measuring and calculating are parts of math as much as proving and following rules of inference are.) Or say they write what looks like a formal proof. How could they possibly say "We get from this step to the next one by applying the distributive property of addition"? They wouldn't know what the words meant.
(I didn't go through the link.)
Philosophy is often friendly to Mathematics. Since your question is admitted here for higher level consultation after a long treatment in Mathematics SE, I think here also we will have to use Mathematics as well as Philosophy. Mathematics can make it more reasonable and acceptable than Philosophy in this case, I think.
Let's think about the different possibilities regarding the convincing mathematician and the person/s to be convinced.
Let's check the plausibility of your usage--'a single mathematician' in a real life situation. Do you think this title (mathematician) can be evaluated qualitatively or quantitatively with precision?
Even if the framing of questions for evaluating is impossible (as there is nobody above him to evaluate, according to your condition), let's suppose that the mathematician got 81% marks (which is just above the cut off mark for regarding him as a mathematician) and others got 80%, 79%, and so on. If this is the case, we are treating all the others as non-mathematicians, aren't we? In such a situation, would it be difficult for the mathematician to produce a mathematical proof?
Sometimes, (not necessary) knowledge about abstract ideas would be the only thing that differentiate a mathematician from a non-mathematician. Even in such situations there would be no problem if the proof is of a concrete idea.
Even if all the non-mathematicians are comparatively poor at maths there would be no problem in producing his (the lone mathematician's) proof. But they must wait until it is convinced. Some of the works of famous mathematicians (scribbled as their notes) come under this category.
If truth, even though it is told by an idiot, is truth. It may take a long time to concede. Here you can replace the word truth by proof. Proof is proof...until it is proved wrong. Otherwise the term 'Proof' should be defined using quantitative terms...but it will create problems. When doing so we will get two categories only. We would lose some very valuable findings if there were only two categories i.e., 1. Proofs that has been convinced/accepted 2. Proofs that has been dissuaded/rejected. The proofs that is not conceded/convinced will come under the second category and they will never be reconsidered. So it is not rational.
For more clarification let us think over this through the following perspective (also):
The mathematician you are talking about must be living in your (this) period. He cannot be a person living in millions of years back. If you say 'No', the criteria for mathematicians must be not the same as of now. If we treat it in the same way, we cannot deny the same possibility in future also. Then the criteria may vary. 'Your condition' can be applied if very brilliant mathematicians lived before and after the lone mathematician. In that case also we should take a layman's proof as proof if it is not approved.
All these possibilities points to one answer...
If there were only one single mathematician in the world, would s/he be able to produce a mathematical proof?
The answer is an unconditional YES, by the very definition of the word "mathematician". Without proofs there's no math so even if only one person is doing it, that person would be producing mathematical proofs.
Perhaps an analogy can make it more evident: Change the the question to ”If there were only one single musician in the world, would s/he be able to play music?”