Is there some good book or YouTube channel that make a good comparison/distinctions between the mathematics before René Descartes, with the mathematics after Descartes?

In the short article "The Book First of Descartes’s Geometry" by André Warusfel :

It should be noted in passing that his technique of taking a length as a unit, which is now so commonplace, was at the time an innovation of unheard-of abstraction!

This is, it's there some good text where is possible to find a comprehensive comparison of mathematical treatments with and without equivalence classes (perhaps specifically, with equivalence classes applied to rational/fractions, not necessarily in equivalences related to integers).

The question borns to possibly distinguish style of mathematics before, even, the implicit introduction of equivalences into arithmetics. It is not a requirement, but, it would be nice if such a text has geometrical illustrations.

The question born due to a YouTube user called John Gabriel. He has some information but, speaking to him is like 'walking in eggshels'. Anyway, my intention here is certainly not a critique beacuse the guy has been expelled of several forums due to his belligerent ways of communication, but rather the knowledge.

What attracts my attention is how remarkable statements in one 'arithmetic' may be non remarkable in the other. Is there some other authors that can surf the two worlds, so to speak ?

  • $\begingroup$ Sources with discussion of post-Cartesian math include: Badiou: A Subject to Truth; The Mathematical Soul; Beyond Cartesian limits: Leibniz's passage from algebraic to “transcendental” mathematics sciencedirect.com/science/article/pii/S0315086004000199 $\endgroup$
    – DJohnson
    Aug 3, 2023 at 12:25
  • $\begingroup$ It's look meaty the reference :P $\endgroup$ Aug 4, 2023 at 21:59
  • $\begingroup$ COuld you expand a little on the entire paragraph that mentions "The logarithmic curve provided the general solution to this problem." $\endgroup$ Aug 4, 2023 at 22:04
  • $\begingroup$ Also in the past was certain about that the definition of commensurable is in the present what was in the past. The modern definition makes sense to me. Is the usage of natural numbers, somehow, implicitly invoking equivalences classes, in the context of the definition of commensurability. Following the style of Landau's Foundations of Analysis, he starts with rationals, and after, he proceeds to include negatives. Some author also structure 'number' more like a vector "separating" the 'sign' from the 'quantity'. Is it the mere introduction of natural numbesr also introducing equivalences ? $\endgroup$ Aug 4, 2023 at 22:20
  • $\begingroup$ As I undertand, in a time before Descartes, probably nearer the greeks' time, the "geometrical" operations with line segments, were closed. It is very tempting to introduce natural numbers in definitions, because they are primitive. $\endgroup$ Aug 4, 2023 at 22:24

1 Answer 1


Descartes is credited with introducing coordinates to geometry and it is often called analytic geometry compared with the 'synthetic' geometry of Euclid. However, Weyl denounced this as an "act of violence" and was part of a movement to return to a more synthetic form of geometry. As this was pioneered by Euclid, we really ought to say that post-Descartian geometry and so also post-non-Euclidean geometry of Riemann manifolds and so on (but not of the absolute geometry of Bolyai which is already synthetic) is a return to Euclid and so Euclidean geometry but not in the sense normally used for this term but as a synonym for synthetic geometry.

  • $\begingroup$ Is there some comprehensive reference of what means by synthetic ? through the ages the word synthetic denote the same ? In the past I thought that 'synthetic' meant the same now or 2000 years ago, but right now I has a air of haze regarding this digression $\endgroup$ Aug 4, 2023 at 22:09
  • $\begingroup$ @IgnacioBotayaVera: I don't think I've seen a formal definition of the term. Its seems to be more folklore usage. I expect its use comes in part from being the opposite of analytic. $\endgroup$ Aug 4, 2023 at 22:34

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