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Why I asked this question : https://gallica.bnf.fr/ark:/12148/bpt6k90195m/f54.image

p 50-51, in course of Cauchy, a proof of the intermediate value theorem. Now, that's not a proof.

And I learned that Cauchy had made a lot of mistakes, so this is not a proof, of a change of standard between our era and that of Cauchy.

But the question stay : has the standard of mathematical proofs changed over time ?

Dattier
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Dattier
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Yes and no. It is better to say that there were always several different standards. Most proofs of Euclides, Archimedes and Apollonius are on the level of modern standards, though gaps in those proofs can be found, and some were found in the ancients time. At the time where calculus was invented, many mathematicians understood that the proofs using new methods are not to the same standard of rigor as the proofs of Archimedes. It took more than two centuries to clean all this and to establish calculus on the same degree of rigor as in the work of Archimedes. Archimedes himself also used sometimes non-rigorous arguments, and he perfectly understood this, and said this explicitly. And this is what happens nowadays too. So at any time, there are several different standards of rigor.

Alexandre Eremenko
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  • But David Roberts says : "Yes. Euclid assumed things he never even stated as axioms.

    He assumes in proposition 1 that circles intersect in points. But Euclid's stated axioms are satisfied by $\mathbb Q^2$ , IIRC. More generally, mathematicians used to assume things that followed from physical intuition."

     – Dattier Nov 03 '18 at 19:44
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    @Dattier: And what do you think modern mathematicians do? How many of them can state (without looking to books) the complete set of ZFC axioms? – Alexandre Eremenko Nov 03 '18 at 23:28
  • @David Roberts is a modern mathematician – Dattier Nov 04 '18 at 10:43
  • Let's say there were several different standards, the strict, the loose, the heuristic, etc. Did any of them change over time? The formal standard (rigorous proof must be formalizable) could not have existed before 20th century, for example. – Conifold Nov 04 '18 at 22:28
  • @Conifold: The notion of "formalizable" indeed evolved with time. This is normal, everything evolves. Euclid understood (or introduced) the idea, and did his best. Some gaps and omissions in Euclid indeed were corrected in 19-20 century. Which is not surprising. Surprising is how much of his book remains unchanged. There is no parallel to this longevity in other areas of knowledge. – Alexandre Eremenko Nov 05 '18 at 00:20
  • @Conifold: Does anything that Archimedes or Apollonius claimed they proved need a correction today? – Alexandre Eremenko Nov 05 '18 at 00:22
  • The problem is that Euclid did not introduce any such idea (Aristotle tried but it did not work, his formal logic was too weak), and his proofs do not have the supposed gaps and omissions. There is a large body of literature that shows gap interpretations you are alluding to to be artifacts of 19-20th century formalizing translations of what he is doing, see e.g. Manders Euclidean Diagram. Similarity in theorem statements does not entail similarity in how proofs are done. – Conifold Nov 05 '18 at 00:32
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    I agree that Euclid's proofs are rigorous, once one includes the facts that are taken as granted in the axioms for his geometry. I haven't read a lot of old mathematics, but one sees in the 19thC people struggling with the idea that they have to justify mathematics in terms of physical reality (cf Kant on geometry and intuition. Reading Dedekind is wonderful as he really 'gets' it, as abstract mathematical objects really start to turn up in peoples' thinking). Conversely, things that are physically 'obvious' are taken to be true in the course of proofs rather than stated with other assumptions – David Roberts Nov 05 '18 at 01:46
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    @DavidRoberts The idea of "justifying" non-logical assumptions, stating them explicitly, and reducing all inferences to formally logical ones is the invention of 19th century that is the current norm. Hence current misconceptions about gaps, rigor, and what Euclid was doing. This norm is currently challenged by computer proofs, "theoretical mathematics" and proponents of diagrammatic reasoning. On the Dedekind-inspired change see Buldt-Schlimm . – Conifold Nov 05 '18 at 02:06
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Not an answer, as Eremenko’s can hardly be improved upon. But, courtesy of M. Audin (2016):

— It's still irksome, Cartan said one day. You read a theorem, its proof. Then there is a remark that says “the theorem we have just proved is not always true”.

— You read that in Goursat?

— No, that one is in Bertrand.

— What is the theorem?

— Cauchy’s theorem. And, he shows his colors before proving it: “this theorem holds in general, but it often fails”.

— And he proves it?

— Yes. Then he gives counterexamples.

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Francois Ziegler
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    Exactly! Of course, it sounds ridiculous or hilarious if we insist on an ultra-strict, orthodox, contemporary reading... which is what lends it its charm. At the same time, by this point in my life, I can easily understand how such words (or French equivalents) could be understood in a different, distant time. – paul garrett Nov 04 '18 at 22:42
  • @paul garrett: This discussion has me wondering whether several decades from now people will similarly be forcing new standards on things written now, such as when someone says the best way to solve such-and-such problem is blah-blah (they certainly don't mean this literally; it's probably not even possible to determine "the best way", whatever that means), or "can anybody solve this problem", or "I don't know what to do" (after someone has actually done something, such as ask for help), etc. (I got these examples in a few seconds by looking at Math Stack Exchange questions just now.) – Dave L Renfro Nov 05 '18 at 11:32