How to Avoid Collapse of Modern Mathematics

Pythagoras struggling in vain to avoid collapse surrounded by a worried Society.

This is a continuation on a previous post about Norman Wildberger's mathematics education program Insights into Mathematics noting connections the Leibniz World of Mathematics and the BodySoul program. 

A common concern is the concept of real number and the set of real numbers $\mathcal{R}$ as the playground for most of modern mathematics. Wildberger takes a critical look on how these concepts are introduced in standard texts noting that basic difficulties are swept under the rug. View in particular this episode: Real numbers as Cauchy sequences does not work.

BodySoul takes a constructive approach viewing the natural numbers 1, 2, 3,..., to be constructed by repetition of the operation +1, the integers as solutions to equations $x+n=m$ with $n$ and $m$ natural numbers, the rational numbers as solutions to equations $q*x=p$ denoted $x=\frac{p}{q}$ with $p$ and $q\neq 0$ integers, while the real number $\sqrt{2}$ is defined as the positive solution to the equation $x^2=2$ or  $x*x=2$.

Recall that the Pythagorean society based on the concepts of natural and rational number, collapsed when it became public that $\sqrt{2}$ is not a rational number. Modern mathematics is based on the concept of  $\mathcal{R}$ as the set of all real numbers. Wildberger concludes that all attempts to bring rigour into the foundations of mathematics as the virtue of modern mathematics including Dedekind cuts, equivalence classes of Cauchy sequences and infinite sequences of decimal expansions, have failed. The trouble with all these attempts is the resort to infinities in different form. What will be the fate of the society of modern mathematics when this fact becomes public?

In the constructive approach of BodySoul there is no need to introduce infinities: In particular it is sufficient to work with rational numbers as finitely periodic decimal expansions or even more restrictive as finite decimal expansions, which makes perfect sense to anybody. But it requires making the notion of solution of an equation like $x*x=2$ precise, that is making precise the meaning of the equality sign $=$. 

We then have to make the distinction between exact equality or more precisely logical identity denoted $\equiv$ and numerical equality denoted by the usual equality $=$ as something different to be defined. We thus have $A\equiv A$ while writing $A=B$ would mean that $B$ is not identical to $A$ but equal in some restricted meaning to be defined. 

We then understand that $x\equiv\frac{1}{3}$ as exact solution to the equation $3*x=1$, while $x=0.333333333$ is a solution in a restricted meaning. We meet the same situation as concerns the solution to the equation $x*x=2$ with $x=1.414$ and $x=1.41421356$ as solutions in a restricted sense, or approximate solutions of different quality or accuracy. 

To measure the quality of a given approximate solution $x$ to the equation $x*x=2$, it is natural to evaluate the residual $res(x)=x*x-2$ and then from the value of $res(x)$ seek to evaluate the quality of $x$. This can be measured by the derivative $f^\prime (x)=2*x$ of the function $f(x)=x*x-2$, noting that a different approximate solution $\bar x$ is connected to $x$ by the mean-value theorem 

• $res(x)-res(\bar x) = f(x)-f(\bar x) = f^\prime (\hat x)*(x-\bar x)$     

where $\hat x$ lies between $x$ and $\bar x$. With knowledge that $x>1$ and $\bar x>$, we can conclude that $f^\prime (\hat x)>2$ and so

• $\vert x-\bar x\vert<\frac{1}{2}\vert res(x)-res(\bar x)\vert$

from which the quality of approximate solutions can be measured in terms of the residuals with $\frac{1}{2}$ as sensitivity factor. 

This analysis generalises to to approximate solution to equations $f(x)=0$ for general functions $f(x)$ with the derivate $\frac{1}{f^\prime (x)}$ expressing residual sensitivity. In particular we see that if $f^\prime (x)$ is small the sensitivity is large asking the residual to be very small to reach precision in $x$. 

But this argument is not central in modern mathematics where the notion of exact solution to an equation is viewed as the ideal. The exact/ideal solution to the equation $x*x=2$ would thus be viewed as a non-periodic infinite decimal expansion, which would require an infinite amount work to be determined, thus involving the infinities which Wildberger questions. The equality sign in this setting comes without quality measure in finite terms as an unattainable (Platonic) ideal. 

In the setting of the algebraic equation $x*x=2$ the notion of an ideal solution may not cause much confusion, but for more general equations such as partial differential equations it has generated a lot of confusion because the quality aspect of approximate solutions is missing. The quality of an ideal solution is infinite beyond measurement but also beyond construction.  

There is a notion in modern mathematical analysis of partial differential equations named well-posedness with connects to the sensitivity aspect of approximate solutions, but it has received little attention in quantitative terms.  

As a remedy, this is the central theme of the books Computational Turbulent Incompressible Flow and Computational Thermodynamics. There is much to say about mathematical equations and laws of physics with finite precision.

We may compare the Pythagoreans facing the equation $x*x=2$ with a notion of ideal solution, and modern mathematics hitting a wall confronted with the Clay Math Institute Millennium Problem on ideal solutions of  Navier Stokes equations. 

An opening in this wall is offered as Euler's Dream come true. 

PS Recall the famous Kronecker quote: "God made the integers, all the rest is the work of man". So the power of an almighty God was not enough to proceed and also make the real numbers. What are the prospects that man can succeed?