Sums of the Harmonic Series

 We shall prove the following:



Let

$$m=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$$

Then  $m\notin \mathbf{N}$.


We will use the following simple lemma:


Let $n$ be a positive integer and $k$ the maximum number for which $2^k\le n$. Then for all $m\le n$ then exponent of $2$ in the prime factorization is either strictly less than $k$ or else $m=2^k$.


This is because if we had $m=2^k{p}_1^{s_1}...p_t^{s_t}$  then $n\ge m>2^k{2}^{s_1}..{.2}^{s_1}=2^l$ with $l>k$ which is a contradiction.
 
Consider then

$$m=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$$

and multiply both sides by $n!$ to obtain:


$$n!m=n!+1\times 3\times ...\times n+..1\times 2..(n-2)\times n+(n-1)!$$

Let $k$ be the maximum exponent such that  $2^k\le n$. Consider  $1\times 2\times ...\times n$.
In the prime factorization of each of the terms $1,2,.,k,...,n$ there will certainly be for the exponents of $2$  all the numbers from $1$ to $k$ and each at least once and possibly several times except for $k$ which can occur maximum once. There are three possibilities for $n$. Either it is $2^k$, or has a prime factorization with exponent of $2$ a $c<k$ or it has no factor $2$. In the first case it is easy to see that the exponent $p$ of $2$ of $(n-1)!$ is strictly less than the corresponding exponent of each of the other  terms of
$$n!m=n!+1\times 3\times ...\times n+..1\times 2..(n-2)\times n+(n-1)!$$

So we can divide the equation by $2^p$ and obtain an odd number equal to an even and thus arrive at a contradiction.

In the second and third cases we have that there is a $t=2^k<n$ and there will be exactly one term $1\times 2...\times (t-1)\times (t+1)\times ...n$ which will have an exponent  $p$ of $2$ strictly less than that of all the other terms (because $k$ as an exponent can occur only once by the lemma) and we can divide the equation in the same way to obtain an odd number equal to an even number and thus a contradiction.

Quest for true mathematics

Another thing to be investigated: the perversion of mathematics through abstract algebra in the 20th-century (field theory and abstract ring theory and algebraic geometry based on them).  Our moto; back to Galois, Gauss, Dedekind, Riemann, Weierstrass, Poincaré ! Bring back the study of Euclid and Archimedes.  Several complex variables and 'elementary' number theory. The more concrete analytic study of (partial and ordinary) differential equations rooted in geometric intuition and the reality of 'real' physics (Kepler, Newton, Leibniz, Cauchy, Lagrange, Laplace, Hamilton, etc.).  And study the philosophical light brought by formal logic and recursion theory before it became perverted by logical positivism and ZFC set theory and model theory.

Mathematics though difficult and demanding dedication yet ultimately must be a self-illumination of the intelligible structure of the soul. Once such an illumination is obtained (which involves the development and purification of intuition) then the result should be permanent. If this is not the case then we are in the presence of a 'sick' and 'deviant' mathematics that, besides lacking philosophical (and even logical) value, does spiritual and psychology harm: this appears to be the case of much of abstract algebra and abstract analysis together with ZFC set theory and model theory. Perhaps there is more true mathematics or philosophy in hard classical physics problems than generally realized.

The fountains of authentic mathematics remains: mathematical analysis in ${\mathbb{R}}^n$ including PDEs (and its its extensions and abstractions in complex analysis, measure theory and function spaces) and finite mathematics.

Update: the history of abstract algebra is of utmost interest. We must distinguish between natural and good abstraction and faulty abstraction. Thus Grassmann's linear, exterior and multilinear algebra is a brilliant example of good and natural abstraction.  So too is the theory of groups, specially the theory of finite groups (cf. the famous textbook of Burnside). Likewise Boolean algebras and other structures considered by Whitehead in his Universal Algebra.   But there are other structures in which further abstraction is not a good idea. Thus finite fields, algebraic number fields and quotients of polynomial rings over a common field - these are fine as they are. A more abstract perspective (such as Noetherian rings, etc.) is misleading and wrong.  In fact so much of the theory of finite fields and algebraic number theory can be carried out in a wonderful way in the context of elementary number theory.  The same goes for category theory. The general theory has no interest compared to Abelian categories but this is just a faulty abstraction of categories of modules.   So too is general topos theory basically sterile.  Only toposes of sheaves are really interesting - though there are also some interesting alternative structures.

The quest for true geometry

There are many problems and paradoxes with standard geometry and topology (and differential equations) based on the classical notions of differentiability and smoothness which fall short of the conceptual perfection of analyticity or its extensions. Indeed the classical view of the real line is problematic.  The work of Lawvere in synthetic differential geometry is of outstanding importance. And yet from a higher philosophical and scientific point of view, what is the true geometry ? Perhaps it lies within the inner mind of the classical painter, sculptor and writer.

But let us understand better the qualitative topological revolution of Poincaré.  Qualitative and topological knowledge only is meaningful in the context of a strictly quantitative situation (a given equation, a given problem, a given theory). On the other hand quantitative analysis is meaningless without being able to draw some qualitative conclusions, all intelligible descriptions of the results of such an analysis must involve or implicitly posit qualitative notions.

What being is in its essence determines its qualitative characteristics and correlates them with certain definite quantitative changes by the fundamental law of its own nature. The magnitude does not determine the quality, nor does the quality determine the magnitude, but the roots, both of the quantitative and qualitative elements in being, lie deeply concealed in the fundamental essence.  - J.G. Hibben, Hegel's Logic, An essay in interpretation.

We should investigate the fundamental mathematical, scientific and philosophical problem of the strange separation between the concepts of smooth (infinitely differentiable) maps and analytic maps (real and complex). This is closely connected to locality and determinism. 

Analytic functions are mysterious objects. We are told that analytic functions form a sheaf. For disjoint open sets if we take constant functions taking different values on each of them, then the (trivial) 'gluing' exists but if we take the complement of the domain of holomorphy of this  analytic function on the union, then it must topologically separate the two open sets ! This is all trivial, but is shows how connectivity or simple connectivity are bound up with the nature of analytic functions.

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.