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I am reading about the 1900s revolution of math pioneered by figures such as Hilbert. I have seen many articles speak very fondly of these figures due to the fact they tried to study Mathematics through the perspective of Logic.

I do agree that it was great... but wasn't this perspective already put forward by Euclid when he tried to do axiomatic Geometry? What did the 1900s proof theory achieve that Euclid couldn't?

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  • $\begingroup$ Useful The Frege-Hilbert Controversy as well as Nineteenth Century Geometry. $\endgroup$ Commented Nov 14, 2022 at 12:53
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    $\begingroup$ The first relevant thing is the modern (post-Renaissance) use of symbols in mathematics that greatly improved mathematical practice. See also Leibniz’s Influence on 19th Century Logic. $\endgroup$ Commented Nov 14, 2022 at 12:54
  • $\begingroup$ In a sense, the power of Axiomatic method was rediscovered during the 19th Century and only at that stage it achieved a role in mathematics similar to what Euclid already achieved with it. $\endgroup$ Commented Nov 14, 2022 at 12:56
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    $\begingroup$ Your title refers to 1920 (and I wondered what you mean), your post to "the 1900s revolution of math". I think you should choose a better description for both, and be consistent. Maybe called it the "modern (Hilbert-style) formalization of math". $\endgroup$ Commented Nov 15, 2022 at 17:07

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For example, the very first proposition: Construct an equilateral triangle $ABC$, where one side $AB$ is given.
Euclid says

diagram

Draw a circle with center $A$ and radius $AB$. [By Postulate 1]
Draw the circle with center $B$ and radius $BA$. [By Postulate 1]
Let $C$ be a point where the two circles intersect.

But there is nothing in Euclid that proves the two circles intersect at all.

"Look at the picture, it's obvious" is not considered rigorous.

"A is inside the circle $BCD$, and $E$ is outside the circle $BCD$, so there must be a point of circle $ACE$ exactly on the circle $BCD$." There is no basis in Euclid to make that assertion.

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    $\begingroup$ And to make matters worse, it's not just that the proof has a gap. There's geometries that satisfy Euclid's postulates but where those two circles don't intersect, namely the collection of points in the plane with rational coordinates. (Though perhaps Euclid could quibble that this geometry is inconsistent with Definition 15, but once you start talking about the definitions which are full of statements like "A line is breadthless length" it becomes pretty clear why it's not rigorous.) $\endgroup$ Commented Nov 15, 2022 at 1:57
  • $\begingroup$ don't believe we have the actual books written by Euclid. Many of the definitions are in fact apocriphal. $\endgroup$
    – Rad80
    Commented Nov 15, 2022 at 10:26
  • $\begingroup$ That point was in fact discussed in the 5th century in Proclus' commentary on Euclid's elements. $\endgroup$
    – Slereah
    Commented Nov 15, 2022 at 13:39
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    $\begingroup$ Even if we don't have Euclid's original writing it seems unlikely that he was more rigorous and people just dropped that as they transcribed. $\endgroup$
    – msouth
    Commented Nov 15, 2022 at 23:48
  • $\begingroup$ @msouth: It's pretty far beyond unlikely - I would say that you can't prove those circles intersect without some definition of continuity, or an equivalent concept, and that just didn't exist before Newton and/or Leibniz invented Calculus. Even then, your proof would probably end up looking like a special case of the Jordan curve theorem, and the fully general case was only proved in the 19th century. It would be truly astonishing to me if Euclid had proved that with the tools available to him at the time. $\endgroup$
    – Kevin
    Commented Nov 16, 2022 at 19:22
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The main difference is that Mathematical logic and set theory did not exist at the time of Euclid. (The Logic of Aristoteles is still very far from mathematical logic created in 19th century). As a result, Euclid's Elements are not completely rigorous from the modern (and early 20th century) point of view: for example, Euclid did not state all axioms which he really used (the first complete from the modern view set of axioms for Euclid was formulated by Hilbert). There are actual gaps in Euclid's reasoning which in principle could be detected during the 24 centuries since Euclid but were not detected. For example, Cauchy's theorem on convex polytopes. It is a deep theorem which fills one of these gaps, but the gap was not even recognized before Cauchy.

The reason of this was that without rigorous mathematical logic, complete analysis of Euclid's axiomatic system was not possible. Understanding of relation between geometry and numbers at the time of Euclid was also very incomplete.

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The hope of Hilbert was that once correctly formalized, it would be possible to show that mathematics was sound and that any statement has a proof. Godel's incompleteness results put an end to that hope, at least for recursively enumerable theories strong enough to recover arithmetic.

A large part of Euclid's Elements, namely the geometric part, does not assume arithmetic but can actually be built on first order logic. Several axiomatics were devised circa 1900 to dispel a few gaps in the treatise of Euclid, mostly due to a lack of treatment of order on lines, corresponding to a point being between two other points. Amongst the many axiomatics developped at that time, the one introduced in the "Grundlagen der geometrie" by Hilbert is the most famous. The big result was yet to come.

Alfred Tarski gave a slightly different axiomatic for elementary geometry, of a more logical nature, and relying only on first order logic. He was then able to show that his theory of elementary geometry is

-- consistent: a well formed statement is either true or false, and cannot be both.

-- complete: a proposition (or its negation) stated in the language of first order logic always has a proof.

-- algorithmically decidable: there is an explicit algorithm that decides if a statement is true or false.

So, Alfred Tarski realized Hilbert's program in the context of elementary geometry and showed what mathematicians have anticipated for more than two millenia: the logical perfection of Euclid's geometry.

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    $\begingroup$ Can you expand just a bit to explain whether the "complete" of Tarski's setup is related to the "incompleteness" of Godel? I.e. does Tarski's system not have the complexity that invokes Godel or is complete used in a different sense? (I'm vaguely familiar with Godel's result that you will always be able to create an undecidable proposition given a certain amount of complexity in your system). $\endgroup$
    – msouth
    Commented Nov 15, 2022 at 23:55
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    $\begingroup$ @msouth And Tarski's result shows that first-order Euclidean geometry is not sufficiently complex for Gödel's agument. There is no undecidable proposition there. $\endgroup$ Commented Nov 16, 2022 at 1:27
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    $\begingroup$ @msouth: It is the same notion of completeness as Gödel's incompleteness theorems (but not Gödel's lesser-known completeness theorem!). The incompleteness theorems do not apply because you cannot use Euclidean geometry to do fully general arithmetic, so Gödel's construction is not possible. To my understanding, the principle difficulty is the lack of induction or anything resembling induction. $\endgroup$
    – Kevin
    Commented Nov 16, 2022 at 19:41

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