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Several questions such as the following have an answer with pictures in it.

It's the bias of this author that these pictures are not merely an illustration, but may be considered as an integral part of the proof. Yet I have the uneasy feeling that many "real" mathematicians do not wholeheartedly agree with this. Which is the reason why this question is raised.

A possible counter argument could be that the picture is observed by our eyes and that our eyes are not quite reliable in some sense. But mathematical formulas and text in a mathematical reasoning are observed by the same eyes, therefore the same counter argument would apply to a "common" formal proof. It's the same visual system that absorbs graphics, text and formulas. And ever since the ancient times, "algebra" (formulas) and "geometry" (pictures) have been going hand in hand.
When comparing geometry and algebra in this sense, courtroom-style ( > 60,000 lines ! ) algebraic proofs like those of the geometrically obvious Jordan Curve Theorem come into mind.

So, if graphical evidence doen't count as a proof, what is the real reason behind this?
Weaker statement: can graphical evidence eventually contribute to a formal proof?

Han de Bruijn
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    On this website, i like it when the person asking has gone to the trouble of graphing things to substantiate a claim. In one or two variables, it is generally possible to prove features of a diagram. Three is more difficult. – Will Jagy Jan 18 '14 at 18:55
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    If we can use arithmetic methods to prove geometric problem, why not the opposite? – LeeNeverGup Jan 18 '14 at 19:18
  • @WillJagy: Not so much trouble most of the time; I've been in computer graphics quite a while ago. – Han de Bruijn Jan 18 '14 at 20:08
  • @LeeNeverGup: A picture says more than a thousand words. Not? – Han de Bruijn Jan 18 '14 at 20:11
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    @HandeBruijn: http://www.amazon.com/Q-E-D-Beauty-Mathematical-Proof-Wooden/dp/0802714315 and http://en.wikipedia.org/wiki/Proof_without_words and http://www.amazon.com/Proofs-without-Words-Exercises-Classroom/dp/0883857006. How did you do some of the beautiful graphics? – Amzoti Jan 18 '14 at 22:16
  • @Amzoti: Would you please consider to formulate your (useful) comment as an answer? Then I can upvote and accept it. – Han de Bruijn Jan 20 '14 at 14:13
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    graphical evidence is not a proof simply by use of the word evidence. Evidence is in general not accepted as proof because it simply is not. So there is not much point to this discussion. – MWL Jan 20 '14 at 14:58
  • @Amzoti: How doing the graphics? I'm using a mixture of standard graphics as supplied by Delphi Pascal (version 6) : http://nl.wikipedia.org/wiki/Delphi_(software) . Augmented with my own developments, mainly consisting of a Contouring package. – Han de Bruijn Jan 20 '14 at 14:59
  • @M.Luethi: Sounds like : "we are satisfied with the status quo; so leave us alone" . Any progress in some other direction is blocked with such an answer. Yes, I know what the status quo is, yet want to investigate whether it's feasible to open other possibilities. – Han de Bruijn Jan 20 '14 at 15:07
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    @Han de Bruijn: The distinction between proof and evidence does not have anything to do with a status quo. I can assure you of that. – MWL Jan 20 '14 at 15:09
  • @M.Luethi: I truly wish you were right. But mathematics happens to be something done by human beings. – Han de Bruijn Jan 20 '14 at 15:16

2 Answers2

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Well, it certainly can work in an informal proof, since the purpose of an informal proof is rhetorical. Its purpose is to persuade a reader that there are good reasons to believe a formal proof exists, and to convey the intuition behind the proof strategy. But it's not part of a formal proof because a graph is not a formula in a formal language.

I don't care for pictures since A) the areas of my interest are non-numerical and don't lend themselves to graphing or visual depictions, and B) a few expressive formulae tell me volumes more than a picture.

As for the "but you need to use your eyes to read formulae too" argument, I think it's just plain silly. There's a world of difference in ease of mistaking, say, an 89 degree angle for a 90 degree angle, and in the ease of mistaking $A\to B$ or $\mathcal{P}(x)$ for other typographical strings. There are loads of examples of deceptive diagrams, but I can't think of many examples of deceptive typography...

Malice Vidrine
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  • I do not agree with the last sentence in your answer. Take for example the Proof of lemma in: http://math.stackexchange.com/questions/510202/how-prove-this-inequality-frac2ab4-ab-frac2bc4-bc-frac2/634839#634839 . Don't know how other people react on this, but I am becoming dizzy when my visual system has to absorb such a thing. – Han de Bruijn Jan 18 '14 at 20:04
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    Being intimidated by a big block of text isn't really the same as say, not being able to tell the difference between 5's and 1's by a slight change in typesetting. Careful examination and good bookkeeping could let you go through a large formula and verify it says what you think. The fact remains that text is largely designed to make use of features we can easily distinguish, where as no amount of working memory can necessarily tell the difference between a good graph and a deceptive one. Also, curious what the downvote is for... – Malice Vidrine Jan 18 '14 at 20:20
  • Click on downvote arrow and read: This answer is not useful. – Han de Bruijn Jan 20 '14 at 10:19
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    Not useful how? In that you don't agree with it? Or do I fail to address the question somehow? – Malice Vidrine Jan 20 '14 at 10:25
  • Quote: "Loads of examples of deceptive diagrams". Come on! For example the difference between $89^o$ and $90^o$ is an issue every two years old geometer is aware of. Try to come up with something better in the first place. – Han de Bruijn Jan 20 '14 at 14:10
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    Here's an example of a misleading geometric "proof." The trick is exactly that something that looks like a $180^\circ$ angle is actually not quite a straight line. – Viktor Vaughn Jan 20 '14 at 15:37
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    Here are some more. – Viktor Vaughn Jan 20 '14 at 15:39
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    @Han: So because I didn't come up with an obscure example of a misleading diagram it's not useful? The point is being missed then, and you seem to be basing your entire evaluation of my answer on the paragraph that disagrees with a remark of yours, rather than the part that forms the main answer to your question. I think from the comments that your definition of "useful" is "pats you on the back". – Malice Vidrine Jan 20 '14 at 17:53
  • @MaliceVidrine: Maybe there is something to say like "the pot is calling the kettle black", but since MSE is not a place for debating, I'd rather finish the discussion here. In short: you've told me nothing that I do not already know. – Han de Bruijn Jan 20 '14 at 21:15
  • It's not really a saying that works here. I don't care that you don't like my answer, but a downvote is for an egregiously bad or misleading answer. I am all for improving my answer if it fails to address your question in some clear way, since I would rather give a good answer than a bad one, but so far you haven't actually said how I failed to do so. – Malice Vidrine Jan 20 '14 at 21:51
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It depends on the goal of your proof:

  • if you just want to convince someone, then a graphical proof is a good option (well, to convince a non-scientific :-).
  • But if you want to construct a mathematical proof that can be verified and build upon, then you need to use a formal language. Formal means that it doesn't let place to interpretation: everything is defined. And I don't know of any formal graphical language.

To be fully honest, their are always some assumption in a formal proof. Example: let "$n \in \mathbb{N}$" implicitly assume lots of definition on what is the set $\mathbb{N}$ and probably on operations (+ - $\times$) defined on it.

So this is not black or white. But graphical proof is either too far from being formal or, if it actually doesn't let place to interpretation, then most probably it is easy to prove using formal language. Example:

  • the graphical proof for the sum of odd numbers is very useful to get an intuitive understanding. But if a proper proof is needed, it is quite easy to deduce the equivalent (formal) recursive proof.
  • On the other hand, the missing square puzzle (tx to @SpamIAm for the link) shows how the lack of formal context can lead to a wrong conclusion. Here it is implied that the upper bound is the same at the start and the end (and so the area under it), which is actually wrong.
Juh_
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