As I understand it, the Pythagorean Theorem, which defines the metric for Euclidean space, is said to be strictly mathematical in the sense that it is derived from a set of purely theoretical axioms (and hence differs for non-euclidean geometries).

I get confused however, when I look at proofs for the Pythagorean Theorem, since they all involve drawing triangles, lines, or squares in some way or another. Since drawing a picture is a strictly empirical act, i.e. one that can only be performed in the physical world, isn't any 'proof' of the Pythagorean Theorem actually just a mode of physical observation? To draw triangles or squares or angles you have to use a canvas which is already empirically Euclidean, and so to me it seems you're basically just getting out what you put in, observation-wise.

My question is then: in what way is the Pythagorean theorem is actually derived from purely theoretical axioms?

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    You have several misconceptions here. (1) There are proofs that do not use drawings but are purely algebraic. (2) A drawing in a proof does not mean that the proof is empirical. In modern mathematics it is mostly used as an abbreviation, since it is expected from specialist that they are able to make a proof formal, if necessary. (3) The Pythagorean theorem does not define a metric. – Jishin Noben Feb 13 at 23:09
  • That's like saying algebraic proofs are empirical since written symbols require consistency therefore we need to relatively measure the size and position of pen strokes on paper. – Cell Feb 13 at 23:18
  • It is not a "physical observation" simply because it is not proved by the "effective" measuring of the sides of the triangle. "in what way is the Pythagorean theorem is actually derived from purely theoretical axioms?" Through its proof. – Mauro ALLEGRANZA Feb 14 at 7:15
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    Logic and reasoning is not something that "lives in the mind" only; it needs way to be communicated and shared : language and symbols. Symbols are today mainly "algebraic style" but also diagrams are part of symbolic reasoning. – Mauro ALLEGRANZA Feb 14 at 9:06
  • There are non-geometric proofs anyway. – Richard Feb 14 at 21:20

No, they do not. First, even if picture proofs were empirical it does not mean that it can not be derived by other, non-picture, means. Just because we can surmise 1+1=2 from our experience with common objects does not mean that it is empirical either. "An a priori science is one whose knowable truths are all knowable in an a priori way, allowing that some may be knowable also in an a posteriori way", as Giaquinto puts it.

More importantly, writing/typing a sentence/formula is also a strictly empirical act, and no "purely theoretical axioms" can be expressed without that. So if a picture proof makes the theorem empirical then all of mathematics is empirical. A "picture" (diagram) is simply an alternative means for expressing geometric relations, it can be converted into symbolic form, if one so wishes, as Hilbert famously did.

The non-empiricity (in the usual sense) of diagrammatic reasoning itself was already pointed out by Kant, who coined a specific new category for this type of reasoning, synthetic a priori. Kant also suggested that we have a special faculty for this sort of thing, pure a priori intuition. That was and remains controversial, but it turned out that he did not need to. The detailed analysis of Euclidean reasoning by Manders showed that diagrammatic reasoning involves nothing epistemologically different from the familiar symbol manipulationin axiomatic systems, whatever "apriority" status one wants to assign to it. See his classical work Euclidean Diagram (published in Mancosu edited volume, freely available).

Subsequently, formalizations of Euclidean geometry were developed that use diagrams directly. The most attractive one, due to Mumma, is called Eu. Some of the axioms are directly stated in a diagrammatic form, and its proofs closely mimick Euclid's diagrammatic demonstrations, much more closely than Hilbert's symbolic axiomatics. One can see multiple descriptions and expositions of it on his homepage. For philosophical discussions see his Synthese papers Proofs, Pictures, and Euclid, Constructive Geometric Reasoning and Diagrams, and Azzouni's That Some Diagrammatic Proofs Are Perfectly Rigorous. Here is from Mumma's The role of geometric content in Euclid’s diagrammatic reasoning:

"Manders’ general philosophical concern in the paper iswith mathematical justification — i.e. what is required for it, and the nature of the techniques developed to meet these requirements. His specific topic is the role of diagrams in Euclid’sElements... The result is a compelling analysis which reveals that diagrams have a principled, theoretical role in the Elements. Only a restricted range of a diagram’s properties are permitted to justify inferences for Euclid, and these self-imposed restrictions can be explained as serving the purpose of mathematical control.

[...] My case... consists in an account of the informal proofs of Euclid’s elementary geometry, an account based on recent formalizations of Euclid’s diagrammatic proofs. The account confirms the Rav/Leitgeb view in that it identifies an irreducible role for geometric content in the reasoning. Yet the role played by geometric content is highly constrained, and the constraints are explicated in formal terms. Inferences grounded on an understanding of geometric content occur within a sharply defined formal structure [of Eu]. Accordingly, geometric proof is relativized to a formal framework, a feature that Leitgib identifies with formal proofs. What is and isn’t provable is relative to the formal syntax of the framework and the rules for how it can be used."

One can generalize this to reasoning about non-Euclidean geometries as well, by altering how the diagrammatic information is read. This was done already by Lobachevsky, see Did Lobachevsky Have A Model Of His "imaginary Geometry"? by Rodin, and even before him by Saccheri and Lambert.


I would say that mathematics is itself abstract and assumed to be consistent. Unless you subscribe to a form of solipsism I would argue that mathematical relations exist independent of your mind and body. We may discover the relations through observation of the physical world, and we may also use physical objects to demonstrate an abstract mathematical relation. But if you close your eyes and imagine the geometry you can still demonstrate and proof the Pythagorean theorem.

You can also prove the Pythagorean theorem using linear algebra and calculus, if you are comfortable using the underlying formulae and relations.

If your concern is more in the realm of epistemological solipsism, such as "the only way I personally know mathematics to be consistent is through continued observation and demonstrations", you may be right. The only way you know anything at all to be consistent is because you have observed things before and now assume that the pattern will hold. This argument is impossible to support or dispel with logic because it necessarily throws out the rule of inference.

  • If you have any references they may support the answer and give the reader a place to go for more information. For example, would you have a reference to proofs in linear algebra and calculus for the Pythagorean theorem? Welcome to Philosophy! – Frank Hubeny Feb 14 at 0:36
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    @FrankHubeny Frohman, Charles The Full Pythagorean Theorem, From the University of Iowa Department of Mathematics Web Resource, Iowa City, Iowa, 2010 – Max Feb 14 at 0:57

Maybe what is confusing you is the distinction between a particular triangle and the proof that is universal.

When we prove using mathematical algebraic language the Pythagorean theorem, we want to say that all right triangles have that propriety. It is a universal statement. The picture one uses to grasp some intuition and write the algebraic proof, is just one instance of that, is just one right triangle. But the proof is not about that right triangle, it is about all right triangles.

In some sense there is an analogy with the way someone could do universal statements about some empirical data, like in physics. Maybe is that.

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