Does Graphical evidence count as / contribute to a Proof in Mathematics?

Question

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

• How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$
• How prove this inequality $(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$
• How prove this inequality $\frac{2}{(a+b)(4-ab)}+\frac{2}{(b+c)(4-bc)}+\frac{2}{(a+c)(4-ac)}\ge 1$
• How to prove this inequality(7)?

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?

Answer 1

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...