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I'm a little confused when I read that mathematical logic is actually a very recent field (1800's - 1900's), regarding the foundations of mathematics and so on.

By this I refer to writing proofs in classical propositional logic systems (Hilbert-style, natural deduction, sequent calculus, etc), extending up to first-order logic, using that to define theories like ZFC which (to my knowledge) is largely seen as a modern-day foundation for mathematics. Likewise for Peano axioms, Peano arithmetic, and so on.

I get confused on this because haven't we been doing mathematics for thousands of years? How were we doing proofs before? How were we doing mathematics with no "foundation"?

Were we just blindly using arithmetic and real numbers and so on without really defining what they were or how they worked? Did Euler and Gauss do all their advanced number theory stuff on these informal foundations? What about Newton and Leibniz inventing calculus? All of this years before we start talking about logic and model theory? How did people convince each other that this stuff actually worked especially once we start getting into concepts like infinity?

I don't really understand the timeline of it all or why logic was such a late subject. Did we choose to formalize it so late for some specific reason? What were we trying to solve or achieve? How was it being done before? Why did it take so long before we started asking questions about mathematical foundations?

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    $\begingroup$ "How were we doing proofs before? How were we doing mathematics with no "foundation"?" See Euclid's Elements: it is full of "proofs" and it is "founded": there are axioms. $\endgroup$ – Mauro ALLEGRANZA Oct 29 '18 at 8:18
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Mathematical logic is a quite modern discipline : it emerged in the mid-19th century with Boole, Peirce and Frege.

Logic instead, is quite ancient : we can date it at least from Aristotle (384–322 BCE) and the Stoics.

In ancient Greek mathematics we have poofs and axioms : see Euclid's Elements and Archimedes' treatises.

Thus, the axiomatic method is at the core of mathematics since the beginning.

For a recent study on the Greek origins of proofs, see Reviel Netz, The Shaping of Deduction in Greek Mathematics (2003).

For the ancient world (Greece compared to e.g. China) the works of G.E.R. Lloyd are releveant; see e.g. The Ambivalences of Rationality (2017) and The Ideals of Inquiry (2014).


"Foundational" aspects are present since the origins of science, mathematics and philosophy; see Aristotle's Posterior Analytics as well as Descartes' Mathematics and Physics.

But the concepts of method, rationality, certainty and so on evolved over time, as well as "tools" and "languages" : thus, it makes little sense to ask "how it is possible to do math prior to $\mathsf {ZFC}$".

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  • $\begingroup$ I am still confused though, if the "axiomatic method" was the core, I still don't see how we did certain kinds of math like calculus without defining things like natural numbers, real numbers, irrational numbers, calculus and so on. What sparked the need for more rigorous logic and definitions? I don't understand why it's such a recent thing compared to everything else. $\endgroup$ – user525966 Oct 29 '18 at 13:36
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    $\begingroup$ @user525966: "I still don't see how we did certain kinds of math like calculus without defining things like ..." At the risk of stating the obvious, there are thousands of 17th and 18th century math books digitized and freely available on the internet that you can look at to see exactly what was done. Also thousands (maybe tens of thousands) of journal papers from many dozens of math journals whose 18th century volumes can be found digitized and freely available on the internet. $\endgroup$ – Dave L Renfro Oct 29 '18 at 14:48
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    $\begingroup$ @user525966: For a rare book in English (virtually all at this time were in German or French) from the mid 1800s that discusses concerns with rigor from the perspective of that time, see this translation (published in 1843) of Ohm's The Spirit of Mathematical Analysis, and its Relation to a Logical System (original German version published in 1842), especially the translator's Introduction on pp. 1-17. $\endgroup$ – Dave L Renfro Oct 29 '18 at 14:56
  • $\begingroup$ I still don't see what actually kick started the need to formalize the mathematical logic? $\endgroup$ – user525966 Oct 30 '18 at 1:29
  • $\begingroup$ @user525966 - Logic was already there: during 19th Century, the development of math, mainly algebra, give birth to the idea that math "tools" can be usefully applied to logic. See George Boole and The Algebra of Logic Tradition. A forerunner was Leibniz, in the 17th, with his Characteristica universalis. $\endgroup$ – Mauro ALLEGRANZA Oct 30 '18 at 7:09
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Here is a simple summary of Mauro Allegranza's excellent answer. Mathematics was done since Euclid by "usual logic". What we call "mathematical logic" is a later invention, a part of the 19th and 20th century program to formalize mathematics (and logic). And of course, this 19-20 century program is only a continuation of the attempts to formalize logic, starting at least from Aristotle. But the technical term "mathematical logic" is reserved for this modern formalization.

Mathematics was done at the time of Euclid (and most of it nowadays) using "ordinary" (non-formalized) logic.

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  • $\begingroup$ You left out the issue asked by the OP about why the 19th century saw the attempts at a rigorous foundation for calculus that had not been so strongly perceived as necessary before, e.g., weird features of Fourier series. $\endgroup$ – KCd Oct 31 '18 at 13:53
  • $\begingroup$ @LCd: this is addressed in my answer to another similar question: hsm.stackexchange.com/questions/7891/… $\endgroup$ – Alexandre Eremenko Oct 31 '18 at 16:52
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I'm not sure it is a choice, more likely a historical accident.

It took from Thale to Euclid to establish what we now call Euclidean geometry as the foundation of logic (and indeed Elements was a textbook of logic in the middle ages), and it pretty much survived intact up till late 18th early 19th century (e.g. Newton actually used Euclidean geometry not calculus to come up with his inverse square law, despite his later lies/half-truths in his quarrel with Leibniz) because it has work well enough. It wasn't until non-Euclidean geometry come into being that this was challenged -- it took 2000 years to finally have someone (Bolyai and Lobachevsky) brave enough to stand up publicly to challenge the parallel postulate, and sparked the crisis that ultimately change the foundation of modern mathematics away from Euclidean geometry to logic (and later set theory etc.).

This is not to say logic wasn't there in Euclidean geometry. For example, Euclid's Elements has a number of reductio ad absurdum proofs. But ultimately its foundation is geometry, e.g., a number is represented by line segments in Elements.

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  • $\begingroup$ "Euclidean geometry as the foundation of logic" really ? $\endgroup$ – Mauro ALLEGRANZA Oct 29 '18 at 8:27

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