Thursday, June 03, 2021

Famous Math Problems 19: The most fundamental and important problem in mathematics (English)

The rational number line and irrationalities

In this video we present a basic and profound solution to the most important and fundamental problem in mathematics (which is: How to model the continuum?) This is the rational number line. Our presentation is geometric, assuming a prior theory of affine geometry: this balances the far more familiar arithmetic approach.

We briefly outline a view of a more algebraic and logical orientation towards mathematics which rests on the rational number line as the basic continuum. Here is a real alternative to the current set-up of modern mathematics---very briefly sketched.

We then discuss a projective view of the rational line, with constructions going back to 19th century mathematicians von Staudt and Hilbert. Basic properties of the rationals are described. Then we discuss the introduction of `irrationalities' via Euclidean geometry and the attempts at solving polynomial equations.
There is more below.

Ниже есть продолжение.

Stevin numbers, infinitesimals and complex numbers

Here we discuss important developments re the mathematics of the continuum arising from modern European developments, beginning with the Flemish engineer Simon Stevin who in 1585 introduced essentially the modern idea of a decimal number and arithmetic with them, and then moving on to further issues brought about by the calculus.

Stevin's introduction of decimals was very practically oriented, he proposed that decimal number arithmetic was a simpler alternative to arithmetic with fractions. This is a reasonable position, but Stevin's arithmetic was in some important senses more limited than fraction arithmetic-- a point that is still relevant today as we proceed to examine the modern difficulties with `real numbers'.
These include the role of infinitesimals: do they really exist, and what are they? as well as the role of complex numbers: do they really exist, and what are they? We meet the idea of an algebraic extension of the rational number line, and also finite variants.
Dedekind cuts and computational difficulties with real numbers

In this final video on the most fundamental and important problem in mathematics [which happens to be: How to model the continuum?] we tackle the seriously unfortunate developments leading to the current misunderstandings about the so-called 'real numbers'. Of course this name is a complete misnomer: they are not 'real' at all; rather they constitute a desperate attempt to enforce the existence in mathematics of objects which are actually unattainable without resorting to an infinite number of computational steps (whatever that might actually mean!)

In this video we give a bird's eye view of the various misguided attempts at establishing 'real numbers' and sketch some of the logical and technical difficulties that students are usually shielded from. The basic construction arises from Stevin's decimal numbers extended, using a dollop of wishful thinking, to arbitrary infinite decimals, not just the repeating decimals encoded by rational numbers. Understanding the difficulties with this approach is not that hard, and in essence the same kinds of problems resurface in the various variants which we also discuss: infinite sequences of nested intervals of rational numbers, monotonic and bounded sequences of rationals, Cauchy sequences of rationals, equivalence classes of Cauchy sequences, and finally the icing on the cake of irrationality: Dedekind cuts.

Students of mathematics! Listen carefully: none of these approaches work. This is the reason why not one of these 'theories' are properly laid out in front of you when you begin work in calculus or even analysis. To those who would try to convince you otherwise, via appeals to authority or numbers, name-calling, or by special pleading on behalf of all those lovely 'results' that supposedly follow from the required beliefs: ask rather for explicit examples and concrete computations.

These are the true coin in the realm of mathematics, and will not lead you astray.

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