20

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:

Should a mathematical proof be 'convincing'?

I thought it would be more appropriate on here.

  • @Timkinsella I agree with this. In fact, I think there should be more than one definition of what is meant by a mathematical proof. If one claims that such a proof needs a mathematical community to verify it, then indeed we are hugely limited by the abilities of this mathematical community. On the other hand, in the real world, how could we decide we have a valid proof of some assertion without this mathematical community? So perhaps, the solution would be to have two separate definitions - a theoretical one and a practical one - since they seem so incompatible. – Stephen Aug 12 '18 at 17:30
  • Yes I agree. Also I think at least some of the disagreement about the proper definition of "proof" is due to confusing use and mention. en.wikipedia.org/wiki/Use%E2%80%93mention_distinction – Tim kinsella Aug 12 '18 at 17:55
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    "Convincing" is no perhaps the best word but Wittgenstein famously doubted that it makes sense to talk about following formal rules (for proofs or something else) outside of a practice that ensures mastering them, "no course of action could be determined by a rule, because any course of action can be made out to accord with the rule", see the rule-following paradox. Perhaps, "convincing" was meant to address such concerns, with nobody to convince the lone mathematician can do no proofs. – Conifold Aug 13 '18 at 0:31
  • @Conifold: On the contrary, computer formal rule verification is an easier problem. The rule-following paradox is exactly the distinction between blackbox and whitebox (should have been glassbox but I digress) testing in CS. – Joshua Aug 13 '18 at 15:09
  • @Joshua Computer formal rule verification is an easier problem if you have already come to an agreement as to the rules computers should follow when verifying rules. – Cort Ammon - Reinstate Monica Aug 13 '18 at 21:51
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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. Proof: A formal mathematical construction detailing a series of logical steps reaching a given conclusion.
  2. Proof: A social tool by which people convey a sense of logical progression to an audience.

(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. Audience: A body of real people, other than the author themself, who comprise a body of people able to understand, or interested in understanding, the logical-proof.
  2. Audience: A body of people either hypothetical or real, to whom the author writes with intent to convey understanding of the logical-proof.
  3. Audience: Any individual capable of understanding the logical-proof.

(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.

  • Would you have any reference that makes a similar argument to the one you are making? That would strengthen the answer and give a reader some place to go for more information. – Frank Hubeny Aug 12 '18 at 20:52
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    @FrankHubeny You're right, that would help. Sadly, I won't have a chance to update this answer for a while. I'll come back to it when I can. – Aza Aug 12 '18 at 21:05
  • How exactly would someone be given the right tools for mathematical construction and logical reasoning on their own, without social exposure? It was Wittgenstein's objection to Platonism that such idealization is nonsensical and confuses objects ("tools"), which are portable, with skills, which are not, and can only be meaningfully exercised within a community, "where correction is impossible, talk of correctness is out of place" for proofs or any rule following. – Conifold Aug 13 '18 at 4:11
  • @Conifold The question does not require that the mathematician was always alone, and I don't see that this answer makes such a requirement either. – pipe Aug 13 '18 at 13:33
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    @pipe That would be one way to make sense of the question but it makes it trivial. One can certainly write a proof according to the rules they internalized during prior social interaction, that would presumably make it "convincing" too. – Conifold Aug 13 '18 at 20:13
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Yes.

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 partially agree with this. See my comment above. – Stephen Aug 12 '18 at 17:35
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    We are looking for comprehensive and referenced answers, not private opinions. Could you add something beyond "I would say", like references to philosophers who studied the issue. Aside form that, it is unclear how the "lone mathematician" would even survive, let alone develop logical, linguistic or mathematical skills to write down proofs, whatever those are. The OP question as stated makes no sense, part of the answer has to make sense of it before it can be answered at all. – Conifold Aug 12 '18 at 23:57
  • I'll second your last paragraph. In getting my math degree, I would frequently detour to write out a proof for myself so that I could be certain, even if it wasn't pertinent for the work I was publishing. – Mister B Aug 13 '18 at 20:32
  • @Conifold You're right ... I should not have started with 'I would say...', since it's fairly obvious that many times we do construct proofs for our own benefit and increased knowledge. As such, I don't think this needs a reference to a professional philosopher either. As far as the implausibility of the scenario goes: I agree, this is completely unrealistic, and personally I am worried how language can survive with a single person. But that's the nature of philosophical thought experiments, isn't it? I mean, should I reject Searle's Chinese Room argument based on its utter implausibility? – Bram28 Aug 15 '18 at 15:45
  • @Bram28 Rejecting the premises of the Chinese Room is a valid stance in its own right. And there are several philosophers who would do the same to this thought experiment. So I think you miss the force of Conifold's point. Also, you don't have any argument (or even a citation) for what [s]ome will say about the nature of proof. – Canyon Aug 16 '18 at 20:53
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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.

  • So not only are mathematical objects themselves Platonic, but also proofs of their properties devised according to formal systems with all of their contingent choices and conventions? I doubt Plato would approve, even if points and sets are Platonic Euclid's or Zermelo's lists of axioms hardly are. Proofs would have to be paths from the contingent to the eternal, and hence a mix. – Conifold Aug 13 '18 at 3:13
  • @Conifold fair enough. but then i don't think anyone in plato's time could have been thinking about "proof" in the right way. for evidence of this you can consider how deficient Euclid's axioms and proofs appear to a modern reader. the notion of proof became less and less muddy as mathematicians groped toward formal logic. this is how things always proceed in math -- the right definition of an object only becomes clear after you've been working with it for some time. – Tim kinsella Aug 13 '18 at 4:46
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    I am afraid this is the mythical "rational reconstruction" of history adopted by textbooks for expediency. Euclid's axioms were "deficient" only because he was not interested in axiomatic systems of today, and mathematics did not "grope" towards today any more than evolution "groped" towards humans. How things would appear to a modern reader only matters to the modern reader, that does not make it "the right way". But even if it was, are you suggesting that there is "the right" list of axioms of set theory, along with "the right" list of deduction rules that produce "the right" proofs? – Conifold Aug 13 '18 at 5:11
  • @Conifold it's hard to be interested in something whose advent comes two millennia after your death. but i suspect that if he had known what modern geometry and logic have had to say about the parallel postulate, he would have been very interested. and i suspect he would have regarded those developments as progress. – Tim kinsella Aug 13 '18 at 5:30
  • We'll just never know, will we? Nor would we know what today would have had in store for him if the history happened to take a different path. Progress does not imply a present day teleology, it is implausible because there is too much randomness in history and two many alternatives not taken. But even if we accept it, our enlightened today does not endorse the idea of a privileged set theory, let alone of a privileged set of axioms for it. So not only are platonic proofs in doubt, but even platonic objects behind them are. – Conifold Aug 13 '18 at 5:49
2

Yes: At the Very Least, Trivial Proofs

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.

But Only the Most Trivial Proofs Are Like That

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:

  • ln u + C₁ = ln v + C₂ (By Previous Lemma)
  • ln u + C₁ + (-C₁) = ln v + C₂ + (-C₁) (Additive property of equality)
  • ln u + [C₁ + (-C₁)] = ln v + C₂ + (-C₁) (Associativity of addition)
  • ln u + 0 = ln v + C₂ + (-C₁) (Definition of additive inverse)
  • ln u = ln v + C₂ + (-C₁) (Definition of additive identity)
  • ln u = ln v + [C₂ + (-C₁)] (Associativity of addition)
  • Let C = C₂ + (-C₁) (Definition)
  • ln u = ln v + C (Substitution property of equality)

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.

What about Srinivasa Ramanujan?

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.”

  • I have to disagree with your general argument, especially in the last part of your text. Any serious proof should have a high level of rigor, and in absence of such a rigor, the reader should make sure that any encountered gap is indeed fillable, and absolutely not just believe the author could have easily fill in the gaps - in math, that is a recipe for disaster. The only exception I see here is when one is reading a proof that has already been verified previously and accepted - such as those in textbooks/books. – Stephen Aug 13 '18 at 12:12
  • If the reader can’t see any obvious way the gaps could be filled, the reader should then either complete the proof (if they can), or indeed, acknowledge that what they have been presented with is not a proof, but rather a possible draft of one. I’m not saying one should write in any little detail when composing a proof, but rather that one should be rigorous and complete enough to avoid ending up with major logical gaps that aren’t obvious to fill in. – Stephen Aug 13 '18 at 12:12
  • @Stephen Proofs are almost all written for people who’ve taken similar courses and been trained on the same standard examples. That’s who’s expected to be able to fill in the gaps themselves. – Davislor Aug 13 '18 at 18:46
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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.

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(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?

"Yes".

  • Quote: "Even if all the non-mathematicians are comparatively poor at maths there would be no problem in producing his proof."___ But why do you think this? Are you saying every mathematician can reproduce any other proof provided it was originally produced by a mathematician? Or that they could understand any maths concept and write down a sound explanation of it? – Stephen Aug 13 '18 at 20:07
  • Qte: "Sometimes knowledge about abstract ideas would be the only thing that differentiate a mathematician from a non-mathematician."__ Again what makes you think this? My experience is very different. I've come across many smart people who couldn't solve rather straightforward mathematical problems simply because they lack the appropriate background. It's not just a matter of abstract thinking (in fact not all mathematics is abstract), but even if it were, the lack of this ability would simply put an end to any hope the layman could have at grasping an abstract concept, let alone prove it. – Stephen Aug 13 '18 at 20:18
  • It's possible I've missed the point you're making (and please do correct me if I have) but are you arguing that "even if a layman could eventually produce a proof-like argument, then certain the lone mathematician will also be able to"? – Stephen Aug 13 '18 at 20:28
  • @Stephen: To your first part: No, never. I didn't means so. The authority to produce the proof is only the mathematician. I edited that part to avoid confusion. – SonOfThought Aug 14 '18 at 13:10
  • @Stephen: To your second part: Quote " I've come across many smart people who couldn't solve rather straightforward mathematical problems simply because they lack the appropriate background. " This remark actually affirms that the ideas I mentioned is true. So there is no problem in producing the proof before such people. But as I said, you will have to wait till it is conceded. Actually I meant : "Even in such a situation......" That is why I used the word--'sometimes'. I have edited this part also. – SonOfThought Aug 14 '18 at 13:12
0

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?”

  • Welcome to S.E. I think the last question you asked doesn't suit here. Playing music is not like producing proof. – SonOfThought Aug 15 '18 at 8:01
  • You can do mathematics, and use it to solve problems, without producing a single proof. Proofs allow mathematical progress, but they are not required to do mathematics. – Stephen Aug 15 '18 at 11:56
  • @Stephen I think there's a big difference between using mathematics to solve problems and doing mathematics, which is what mathematicians do. Using software doesn't make you a programmer. – Sam Aug 16 '18 at 10:02
  • @SonOfThought I'm not saying playing music is like producing proofs, but playing music is what a musician does (let's agree on that, I'm aware of late Beethoven), so even if there's only one left, he or she must play, otherwise you wouldn't use the word "musician" to describe him/her. – Sam Aug 16 '18 at 10:06
  • @Sam We won't progress much with word games. Mathematicians use/do (or whatever word you want) mathematics to solve problems. This doesn't require proving anything. E.g. solving a differential equation to find a displacement vector doesn't require proving anything - it requires doing mathematics. Hence, a mathematician can do mathematics all his life without needing to prove a single assertion. – Stephen Aug 16 '18 at 10:41

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