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I just read a description of what is a mathematical proof in my mathematical logic textbook, and I'm a bit puzzled by it. It goes like this:

A mathematical proof is a finite sequence of mathematical assertions which forms a valid and convincing argument for the desired conclusion from stated assumptions.

Why "convincing"? What does this mean? Sadly, the text doesn't comment much on this. It only says that "convincing" is a psychological notion, and so from the point of view of formal proofs is unsatisfactory.

This of course makes sense, but what bothers me is the notion that mathematical proofs should be convincing. This seems to suggest that a logically valid mathematical proof can somehow be rejected because it was "unconvincing"?

Edit

Added the full page for context here: https://image.ibb.co/gjso1U/proofsss.png


A related question:

If there were only one single mathematician in the world, would s/he be able to produce a mathematical proof?

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    $\begingroup$ A logically valid mathematical proof is perfectly convincing (except to people to don't understand the logic, of course) so it would not be rejected by people who understand it. $\endgroup$ – littleO Aug 11 '18 at 22:21
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    $\begingroup$ People make mistakes. People are not infinitely wise. Some aspects of a problem can be very subtle. Ascertaining that a proof is logically valid is in itself a sort of proof. $\endgroup$ – copper.hat Aug 11 '18 at 22:36
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    $\begingroup$ You have to convince people that it is logically valid, for starters. $\endgroup$ – Morgan Rodgers Aug 11 '18 at 23:22
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    $\begingroup$ @DanielWainfleet The problem that arises then is that the validity of their "proofs" becomes irrelevant. All it takes is for them to be convinced, no matter how fallacious their argument could be. $\endgroup$ – Stephen Aug 12 '18 at 12:58
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    $\begingroup$ See the famous paper "Proofs as Social Processes" by DeMillo, Lipton, and Perlis (originally in Communications of the Association for Computing Machinery)--and the many subsequent letters to the Editor in response. $\endgroup$ – E. Douglas Jensen Aug 13 '18 at 19:30
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The primary reason to write down a proof is in order to communicate with other mathematicians. If mathematicians acquainted with the relevant literature cannot understand your argument (i.e: do not find it convincing) then I think the author has failed to write down a proof.

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    $\begingroup$ So, basically, if you come up with a proof too complicated for your cohorts to understand, you haven't got a proof? $\endgroup$ – Stephen Aug 11 '18 at 22:27
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    $\begingroup$ @Stephen I don't want to get too philosophical about it, but my opinion is that if the author finds a proof convincing and it is logically valid, it is likely that others will find it convincing as well. If additionally others find your argument to be convincing and still deem it to be logically valid, then it will probably be widely accepted as a proof. $\endgroup$ – Andres Mejia Aug 11 '18 at 22:42
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    $\begingroup$ The answer reflects the most important aspect of proofs in daily math, but isn't the question more about what a 'proof' is as a mathematical object in the context of mathematical logic? $\endgroup$ – Steffen Plunder Aug 11 '18 at 22:52
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    $\begingroup$ A related question: $ \ $ If there were only one single mathematician in the world, would s/he be able to produce a mathematical proof? $\endgroup$ – Stephen Aug 12 '18 at 12:13
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    $\begingroup$ I think the perfect example of this is Shinichi Mochizuki's supposed proof of the ABC conjecture. He self-published 500+ pages in five papers which the community as a whole still hasn't been able to decipher and has yet to be declared correct or incorrect (as far as I'm aware) $\endgroup$ – Kai Aug 12 '18 at 21:14
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Everything about a mathematical proof is psychological. Including the valid and convincing part. Who gets to decide if it is valid and if it is convincing? The primary audience for a proof is the writer of the proof. Presumably the writer thinks it is valid and convincing. If the proof is convincing enough, then others may agree.

Understand that what is valid and convincing depends on time and place. The geometric proofs of ancient mathematicians were thought then to be valid and convincing enough at the time, but Later mathematicians have found that those proofs were not rigourous enough. For a nice discussion about this read the MSE question 80930 "What is the modern axiomatization of (Euclidean) plane geometry?".

It becomes a matter of communication and cooperation between the writer and reader of a proof. A proof by itself has no power to convince the reader who does not want to be convinced, no matter how rigorous or valid the proof may be. The proof may be rejected for several different reasons. There are many examples of this in mathematics history. A good example is the controversy over Cantor's theory of sets.

As for the last hypothetical question:

If there were only one single mathematician in the world, would s/he be able to produce a mathematical proof?

I think it is obvious that mathematician M could do whatever M wanted and nobody else could have mathematical objections. If M is convinced about the validity of a mathematical proof, then that is all that is required of a proof. After all, who decides if any given proof is "fallacious", if not mathematician M? Of course, M might detect a mistake in a proof and correct it. Thus, validity is time dependent. The entire idea of a "proof" is an artifact of history. Without context, what is the raison d'etre and meaning of writing and reading proofs?

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  • $\begingroup$ Re your last paragraph: $ \ $ Okay, but then wouldn't the validity of their "proofs" become irrelevant? All it would then take is for them to be convinced, no matter how fallacious their argument could be. Surely, a mathematical proof can't be reduced to that? See the link in the OP for some interesting arguments. $\endgroup$ – Stephen Aug 13 '18 at 8:57
  • $\begingroup$ @Stephen would mathematicians be convinced by a fallacious proof? $\endgroup$ – Carsten S Aug 13 '18 at 12:12
  • $\begingroup$ They are humans - they might miss the fallacious assertions/steps. The more they are to verify that there aren't any logical errors, the more confident they can be that there indeed aren't any. A single mind can easily fool itself in believing all sorts of things. $\endgroup$ – Stephen Aug 13 '18 at 12:21
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Mathematical proofs often, if not always, have some gap, and consequently, we have the 'convincing' part of the definition. Note that the author indicates gaps in his/her proof as he/she talks about the rule that from $a=b$ and $b=c$ we can deduce $a=c$. I would take such basics of equational reasoning as assumed for some arithmetic, but I don't regard the partitioning of natural numbers into "even" and "odd" as something assumed for natural numbers. So, personally, I'm not so bothered by a proof getting rejected if it gets found as 'unconvincing'.

Consider the text that you cited where there existed a purported proof that if the square of a natural number is even, then the natural number is even. The author showed that the assumption of $n$ as odd leads to $n^2$ as odd. I have no contention with that, and that does show that such an assumption, along with the assumption of $n^2$ as even, leads to a contradiction. But, the author jumped to the conclusion that $n$ must be even. I don't find that convincing as a proof, because one might believe that there exists a third category of natural numbers other than odd or even, and some members of that category have even squares. Or someone might not have proved that such a partition exists for the arithmetic at hand. So, as a proof, I don't find it convincing, unless I missed some part of the text where the author first established that every natural number is exclusively either odd or even.

When I said that I don't find the proof convincing, I'm not just saying that the proof has not gotten formalized. When I said that I didn't find the proof convincing, I mean that were the proof to get formalized it would need non-logical notions not implied by the text in addition to what it already contains. Though I don't find the partitioning of natural numbers into odd and even numbers as by any means unachievable, it's not something that arises purely from logic or equational reasoning, but partly has to do with the nature of natural numbers. The reasoning of the proof may be regarded as valid in the sense that it draws a correct conclusion and no step moves from a true step to a false step, but the reasoning is inadequate in that it doesn't provide adequate details for deducing it's final claim. Specifically, it is correct to say that "if a natural number $n$ cannot be odd, then $n$ must be even", but that I don't think that consists of an axiom of the arithmetic of the natural numbers. So far as I can tell, the proof needs the concept of even and odd numbers as forming a partition of the natural numbers, and the author did not prove that, nor even spell it out as a needed lemma for the proof. Thus, the proof has a non-logical gap, and if someone did not see how that gap can get filled in, or believe that filling in that gap poses a serious challenge, rejecting the proof seems perfectly reasonable.

Also, sometimes mathematicians having different requirements for what constitutes a proof. Have you heard of intuitionistic logic or constructivist mathematics? Some mathematicians won't allow for a use of the law of the excluded middle in mathematical proofs. So, they can reject certain proofs.

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  • $\begingroup$ Very helpful, thanks. Regarding the partition of N into even and odd numbers, it was established in the number theory part of the course. I suppose this is why it was assumed here. $\endgroup$ – Stephen Aug 12 '18 at 1:03
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    $\begingroup$ You can't use notions like "odd" and "even" unless you define them first (and in this case the definitions are trivial and well-known, of course). I sort of see what you are trying to say, but the example that "there might be some numbers that are neither odd or even" seems about as UNCONVINCING as it is possible to get, IMO. $\endgroup$ – alephzero Aug 12 '18 at 12:05
  • $\begingroup$ @alephzero I don't believe I tried to prove such a "might" claim. Also, I don't think that the definition of even and odd numbers taken together imply such a partition existing. At the very least, we can define even and odd numbers in the context of the real numbers, but there exist plenty of real numbers which are neither odd, nor even. $\endgroup$ – Doug Spoonwood Aug 12 '18 at 22:05
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Suppose the definition excluded the word "convincing" and thus was:

A mathematical proof is a finite sequence of mathematical assertions which forms a valid argument for the desired conclusion from stated assumptions.

Now suppose you have a sequence of mathematical assertions that meets the definition above.

Now if another individual reads and agrees that your sequence of mathematical assertions is valid, and thus agrees it constitutes a proof by the above definition, then one could argue that you have created a "convincing" proof, and thus including or excluding "convincing" in the definition in this scenario would not matter.

But suppose instead however, you cannot find another individual who agrees that your sequence of mathematical assertions is valid. The key question here is then whether what constitutes a "valid sequence of assertions" is a matter of objective fact, or a matter of social agreement. If the latter, then it becomes impossible to have a "valid" yet "unconvincing" sequence of assertions (since then a sequence of assertions would only be "valid" if there is social agreement, which requires "convincing"). In this case, again, adding "convincing" to the definition would not affect whether a sequence of assertions is classified as a proof or not. However if the validity of an argument is an objective property that exists independently of people, then one could argue that an unconvincing, yet valid argument could indeed exist, which I believe would merit being labelled a "proof".

Therefore I have shown that depending on your philosophy of mathematics either (1) adding "convincing" to the definition doesn't change the content of the definition or (2) a proof could exist that is not "convincing". As such, whether (1) or (2) is adopted, the word "convincing can be safely removed from the definition of a proof.

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    $\begingroup$ Yes, this is mostly why this description baffles me. I have always considered a proof to be an objective truth. And now I'm hearing, well in fact, not really. I think I'll always consider a proof to be an objective truth, but sometimes, the community will fail to recognise the objectivity and validity of a given proof (no matter how clearly it is laid out). This doesn't make the said proof invalid, it just means the community has failed to recognise it (provided it was clearly laid out, indeed, without major logical gaps). $\endgroup$ – Stephen Aug 12 '18 at 9:39
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    $\begingroup$ I am not convinced by your argument, as I don't believe (1) is a valid deduction. $\endgroup$ – András Salamon Aug 12 '18 at 9:52
  • $\begingroup$ @AndrásSalamon, I'm a proponent of the philosophical position (2), and thus I can assure you my arguments are valid, even if not convincing ;) $\endgroup$ – Kenshin Aug 12 '18 at 10:00
  • $\begingroup$ @AndrásSalamon why not? any counterexamples? $\endgroup$ – Stephen Aug 12 '18 at 10:00
  • $\begingroup$ @Stephen Validity has a definition in logic purely in terms of the truth values of the statements. It would technically be a valid argument to say something like "The sum of 2 and 2 is 4, therefore, the square of any even number is even." But, I doubt anyone would call such a 'mathematical proof'. So, I think the word 'convincing' does something here. I think my example also serves as a sort of counterexample that you wanted in response to Andras Salamon's objection. $\endgroup$ – Doug Spoonwood Aug 12 '18 at 22:13
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No, you don't want to sell anything. No reason to convince anyone. Proofs that leave the reader bedazzled and scratching his head, after robotically following and confirming every single step to be correct, are the best, IMO. But that's a matter of taste, I guess.

(It just has to be correct. For some people, that's convincing enough. But when it's not, that's where the fun starts.)

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    $\begingroup$ This is a direct answer to the question. $\endgroup$ – trollkotze Aug 12 '18 at 7:44
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    $\begingroup$ @Leucippus Not everyone would necessarily agree with this answer, in this golden age of Maths Education, but this is imho an answer to the question. $\endgroup$ – Arnaud Mortier Aug 12 '18 at 9:52
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Proofs should be convincing, because convincing mathematicians is how we verify the validity of most proofs.

Perhaps the wording of that quote is a little less... rigorous than it could be. The proof being "valid" is what matters; being "convincing" is secondary. But it must be convincing (the reader can follow the steps and see that they seem to logically lead to the conclusion) for other mathematicians to realize that the proof is valid.

Ideally, you'd want to just deterministicly verify that the proof shows inconclusively through that a certain conclusion follows from some set of accepted axioms. That is the theory of a mathematical proof, but most proofs are not rigorous enough to achieve that without some potential for human error.

With computers, automatic theorem proving provides a higher level of rigor. But it is also significantly more complicated, and relatively few proofs are verified this way. Fundamentally, the issue is that to automatically verify things this way, the proof can't skip any steps at all, even "obvious" ones.

So when a human writes a proof for human consumption, they must convince the reader that each step is logically valid, even if a computer wouldn't be convinced.

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  • $\begingroup$ "(2+2)=4, therefore the square of an odd number is odd" is valid. So, does validity enough to count as "what matters"? $\endgroup$ – Doug Spoonwood Aug 12 '18 at 22:16
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Only the most simple proofs are ever written down step-by-step, written out in a fully formal way, so a computer program would be able to mechanically verify each line. For example, no one writes out a proof that if $x = y + 3$, $x - y = 3$, from axioms. If someone did, outside some introductory text where that was the whole point, any editor would tell them to simplify it and save space. A proof is more like a summary showing how it could be done.

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It's important to remember the context.

The vast majority of mathematical proofs are not formal derivations from a set of axioms and a set of deduction rules. Rather, they are a sequence of informal statements along the lines of, "Let $G$ be a graph. Without loss of generality, we may assume that $G$ is connected. For any $x\in V(G)$..."

A (correct) axiomatic proof is already convincing, since any reasonable mathematician can check that each axiom was correctly instantiated and each deduction was correct. It is the other, more common, kind of proof where "convincing" is a non-trivial property. Since the proof is not axiomatic and cannot be literally verified, the only thing it can hope to do is to convince other mathematicians.

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  • $\begingroup$ I'm not sure I agree with the sentence beginning "A (correct) axiomatic proof is already convincing ..." Faced with an extremely long formal proof, I could go through and verify each line, but after finishing I wouldn't be totally confident that my eyes hadn't glazed over and missed something somewhere along the line. This is of course where computer verification comes in - the problem of verifying the whole proof being reduced to the problem of verifying something much simpler - but is nontrivial. $\endgroup$ – Noah Schweber Aug 13 '18 at 12:25
  • $\begingroup$ @NoahSchweber Perhaps is should replace “can check” with “can, in principle, check”. But the bigger point is that the overwhelming majority of proofs don’t fall into this category, anyway. $\endgroup$ – David Richerby Aug 13 '18 at 12:30
  • $\begingroup$ Sure, but I think the gulf between "in principle convincing" and "convincing to mathematicians as they are now" is meaningful. As you say, this is currently a rare phenomenon, but I could see it becoming more and more common in the not-too-distant future. (I mean, I don't disagree too much, I'm certainly a fan of formal proofs, but I think there is a meaningful issue here.) $\endgroup$ – Noah Schweber Aug 13 '18 at 12:36

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