A game of 5x5 Go

(;GM[1]FF[4]PB[You]PW[COSUMI]SZ[5]KM[0]RE[B+4] ;B[cc];W[cd];B[bd];W[dc];B[bc];W[de];B[cb];W[db];B[da];W[ed]
;B[be];W[ce];B[ca];W[eb];B[ea];W[];B[])

Euclid Book V

David Fowler and C. Zeeman have made many important contributions to understanding this book of Euclid's which is assumed to be reorganization of the work of Eudoxus.

The theory is quite abstract and can be framed in terms of an action of the natural numbers $\mathbb{N}$ on any linearly ordered structure $\mathcal{A}$. We write $a:b=c:d$ for $a,b,c,d\in \mathcal{A}$ iff the following three conditions are satisfied:$$\mathrm{\forall }n,m\in \mathbb{N}((n.a=m.b)\iff (n.c=m.d))$$$$\mathrm{\forall }n,m\in \mathbb{N}((n.a>m.b)\iff (n.c>m.d))$$$$\mathrm{\forall }n,m\in \mathbb{N}((n.a<m.b)\iff (n.c<m.d))$$ 

In the case of $\mathbb{Q}$ and the action is the usual multiplication the first condition is sufficient. It just says that $\mathrm{\forall }q\in \mathbb{Q}(\frac{a}{b}=q\iff \frac{c}{d}=q)$.  What is amazing is that Eudoxus' definition works for the real numbers too ! Consider the particular case in which $b=d=1$. Then $a:1=c:1$ is just equality in $\mathbb{R}$. Eudoxus' definition is saying in this case that $a$ and $b$ determine the same Dedekind cut.

Borcherds' lectures on Elementary Number Theory

 https://www.youtube.com/playlist?list=PL8yHsr3EFj52Qf7lc3HHvHRdIysxEcj1H

 https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8

 

Foundations of Elementary Pure Mathematics

The foundations of pure mathematics are, for arithmetic,  the elementary theory of numbers,  field (or Galois) theory and some basic finite group theory and for geometry,  complex analysis and basic algebraic topology  - to which we add advanced linear algebra and some combinatorics.

Pure mathematics took a wrong turn with the development of algebraic geometry in the 20th century.  We mean abstract algebraic geometry based on commutative algebra (after Weil's attempt using field theory) via scheme theory (to allegedly bridge the gap with algebraic number theory). The correct mathematica communis was later revealed to be not commutative algebra but category theory and specially topos theory (more generally from the perspective of homotopy theory and higher category theory). It is absurd  and sterile to study algebraic varieties  merely from the point of view of commutative algebra; rather they must be studied in the context of the interdisciplinary richness of complex analytic geometry (Hodge theory, D-modules, Kähler and Symplectic structures...); also semi- and subanalytic and real algebraic geometry must not be neglected. Thus to approach pure mathematics you should do either number theory (which is heavily grounded on field theory) or pure geometry i.e. complex analytic geometry - though the two are connected in interesting ways (modular forms, Riemann zeta function). Indeed infinite series arise naturally in both arithmetic (as Dirichlet series, etc.) and complex geometry (meromorphic functions).   The most important Lie groups are analytic...

We will discuss later the necessity of generalizing analytic functions to be suitable for physics and applied mathematics (such as Sato's theory of hyperfunctions), to be able to furnish an alternative to smooth functions or the standard classes of Lebesgue measurable functions and distributions.

String theory is not science and not physics (and it does not even address what we feel to be the main philosophical and mathematical problems of quantum theory) although it uses abstract versions of some physical concepts. Rather it is an alternative (experimental) form of pure mathematics, perhaps somewhat in the spirit of Ramanujan.  String theory is actually a beautiful form of alternative pure mathematics which uses some transposed concepts of physics, to obtain surprising results or insights.