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When teaching geometry it is common to use pictures/figures to "show" the problem and its solution. It's also common to say things like "more than one figure can be shown which demonstrates the same problem".

Is there a formal connection between geometric figures and geometric proofs?

It has been suggested that geometric figures are just another notation for the geometric relations we write in a geometric proof e.g.

enter image description here

is just a different notation for $A \perp B$.

To what extent has this notational equivalence been exploited formally and in an educational setting?

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  • $\begingroup$ Pictures provide evidence and guidance, not proof. There's no such thing really as proof by picture. I would say it's pretty common in geometry since you can frequently draw the objects you are describing. Educationally, many students at that level can understand the picture better than the words so that is why it is typically used. $\endgroup$
    – Chris C
    Mar 10, 2016 at 19:28
  • $\begingroup$ I disagree. It's my understanding that to the ancient Greeks, proofs and pictures (i.e., constructions) were the same thing. See also "proofs without words" which is a beautiful series. $\endgroup$
    – Pat Devlin
    Dec 13, 2016 at 12:09
  • $\begingroup$ This is a really interesting question, because in mathematics there absolutely are, for example knot diagrams and Reidemeister moves or 1d cobordisms provide exactly such a diagrammatic language of proof. But in education, I feel like the pictures we draw in single and multi-variable calculus are explicitly "not proofs." That said, the square of orthogonality, the three points connected by line segments, and the hashes and arc hashes of equivalence seem to provide a formalized diagrammatic language. $\endgroup$
    – Nate Bade
    Apr 18, 2018 at 20:05
  • $\begingroup$ Not sure about diagrammatic proof, but I don't think it is impossible. Would also add that it is normal in EE, MechE, and business to refer to diagrams or tables that have to be interpreted to solve the problem. This is more efficient than writing the words to describe nodes of a circuit. Also, it is a skill that students need to develop. So referring to a diagram with angles and letters and asking student to prove something (without a description) is fine. [I would expect the answer normally to be written, not drawn though.] $\endgroup$
    – guest
    Apr 18, 2018 at 20:46

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I think you will find this paper very interesting. It addresses your first question about any formal connection between geometric figures and geometric proofs and argues the two are inseparable.

The paper explores whether Euclidean geometric proofs are axiomatic-deductive as usually believed or diagrammatic as in the sense "one reasons in the diagram."

The author concludes that "[a] Euclidean demonstration is not, then, diagram-based, its inferential steps licensed by various features of the diagram. It is properly diagrammatic. One reasons in the diagram, in Euclid, that is, through lines, dia grammon, just as the ancient Greeks claimed."

Diagrammatic Reasoning in Euclid's Elements by Danielle Macbeth

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For mathematical monologue, dialogue, discourse, in short for communication including proofs ("A proof is any completely convincing argument" [Errett Bishop: "Schizophrenia in contemporary mathematics", Amer. Math. Soc. Colloquium Lecture, Seventy-eighth summer meeting, University of Montana, Missoula, Montana (1973)]), every language is suitable. This includes abstract logic-based languages as well as ordinary spoken language like German, English, Latin, geometrical drawings and even gestures. The snooty Bourbaki-attitude admitting only a very narrow spectrum of languages may be welcomed by those who have problems with other languages. But with respect to the failure of this approach to recognize the fatal mistakes of modern set theory (see for instance https://philosophy.stackexchange.com/questions/51038/will-mcduck-go-bankrupt) this language cannot be recommended, and "machine checked proof" is not a mark of quality.

"Pure mathematics and science are finally being reunited and, mercifully, the Bourbaki plague is dying out." [Murray Gell-Mann: "Nature conformable to herself", Bulletin of the Santa Fe Institute 7 (1992) p. 7]

Of course a rigorous geometrical proof is as correct as any other. For people with corresponding sensual perception it is even better.

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