Physics Suffers from Unwarranted Interpretations by Eckard Blumschein

Dear Michel,

Grothendieck's "Esquisse d'un programme" relates to point-set topology, something that my essay puts in question because General topology cannot even perform a symmetrical cut. I am arguing that the distinction between open and closed intervals of real numbers is only justified in the paradise by Cantor and Dedekind. Maybe, Grothendieck disappeared when he got aware of related inconsistencies.

Already Cantor got insane. While I see other possible victims too, I as a pedestrian didn't get aware of tangible practical results of point-set theory so far.

Best,

Eckard

Best,

Eckard

Dear Michel Planat, dear Tim Maudlin,

I hope to manage making my essay a bit more understandable to experts:

What happened to mathematics? Part 1

Georg Cantor's naГЇve transfinite cardinalities have not by chance proved useless despite of evidence that was more or less accepted in 1873 and 1890.

Let's understand the ideal continuum as a directed measure every part of which has parts while a point has no parts, and any amount of points, no matter how "dense" can therefore not constitute a continuum. For the sake of simplicity we may restrict to one continuous dimension: Length means the distance from a point zero of reference to something that refers to a chosen basic measure called one. This agrees with Euclid's definition of number as a measure and with Salviati alias Galileo Galilei who already used the method of bijection when he arrived at the reasonable conclusion that the relations smaller than, equal to, and larger than don't apply for infinite quantities but only to finite quantities: There are not more natural numbers than squares of them. The converse is not true. With infinite quantities Galileo meant infinite series. The Latin word quantus asks for a measure: how large? The distinction between finite and infinite doesn't ask for a quantity but for a quality of a chosen model.

In point set theory the expression "subset of a set" is used instead of "part of continuum", and a pair of points could be interpreted as the measure, a piece of continuum. However, G. Cantor's point sets are thought to be set together of distinct, i.e. zero dimensional, points, not of pairs of points. Although Fraenkel admitted in 1923 that Cantor's naГЇve definition of an infinite point set is logically untenable [ ], there is still no corrected one. The naГЇve set theory has been replaced by bundles of axioms, e.g. ZFC. Nobody so far put the due attention to a seeming trifle: Not just Cantor but also Dedekind gave preference to the archaic pebble-like notion of number instead of Euclid's measure.

A yardstick may illustrate the fundamental difference between both notions of number, see Fig. 1. Modern mathematics follows those who attribute the notion number to pebble-like marked points on it instead of a sum of spaces between points. Imagine the yardstick chopped into single pieces being still immediately adjacent to each other. Then each cut alias point is located exactly in between two pieces.

In modern point set topology a point a is considered embedded into the surroundings a-пЃҐ and a+пЃҐ. This causes trouble in case of a=0 within IR+ and gave rise to tricks like (-1,0)U(0,1)=(-1,1)\{0}, a so called deleted neighborhood.

With the traditional Euclidean notion of number as a measure, a point is something that has no parts and accordingly no measure. Here one cannot perform the numerical distinction of a single point that is embedded within the continuum of real measures. Singular points can then only be addressed in the discontinuous rational measures. Likewise there is no chance to distinguish between open and closed REAL measures, and only non-zero measures can be numerically excluded or added.

Obviously, topology strongly depends on the choice of the notion number.

Best,

Eckard

James,

Thank you for pointing me to isodual mathematics. I will have a look at it.

If you are unbiased looking for more pearls you might find out what is behind the essay of Phipps, and how Renaissance, propagation of ideas, and the experimental method made Europe rapidly superior to Asia.

Eckard

Corrections to part 1:

? in a-? and a? should read epsilon

1873 and 1890 should read 1874 and 1891, respectively

[ ] should read Fraenkel, A: "Einleitung in die Mengenlehre" Springer, Berlin, 1923

Fig. 1 can be described as follows:

0----|----|----|----|----|----|----| yardstick

1 1 1 1 1 1 1 measures between subsequent points

0 1 2 3 4 5 6 7 numbers attributed to subsequent points

|...................6..................| number 6 attributed to six measures

|.........3.......||.......3........| twice three measures

There is no distance between ||

Eckard

Part 2

At the time of Cauchy and Gauss in the middle of 19th century, the plurality of those who were teaching mathematics grew rapidly. Why did Abel see mathematics in a mess? Abel meant its lacking rigorous foundation. While the irrational numbers were well known to be different from the rational ones, it was common practice to ignore this trifle. When Riemann suggested his surfaces already in 1851, he referred to points, not to measures. After Cauchy had defined irrational numbers as limit values attributed to convergent series, it was and it is still tempting to interpret limit-values as limit-points instead of limit-measures and furthermore to logically equate the never-ending procedure that defines a series of quantities with the limit attributed to it.

Among a lot of pre-thinkers of Dedekind and Cantor were not just rather self-taught ones like Bolzano and Dirichlet but also Heine and the utterly influential teacher Weierstrass. Even Cantor's strongest opponent Kronecker shared their intention to make the continuum algebraic.

Dedekind declared the real numbers an extension of the rational ones. Actually we have to interpret the transition from rational to real as an essential extension, as the transition from discrete to continuous, i.e. from quantity to quality. In other words, in contrast to algebraic numbers, real numbers are not numerically distinguished from each other. For instance, 3.14... is a rational number, no matter how many subsequent decimals. Its limit-measure pi is qualitatively different.

Dedekind made two unwarranted claims:

- He postulated that a given irrational number can be subject to the distinction between equal to, larger, or smaller than a given rational number. He admitted; nobody can provide evidence for that.

- He imagined the real line composed of just enough densely located points. While this contradicts to the definition of a point as having no parts and the continuum of real numbers as endlessly divisible, this idea of Dedekind can be rescued by replacing points with infinitesimal measures.

G. Cantor started with the naive idea that there must be more real than rational numbers and spoke of more than countable many (überabzählbar viele) alias transfinite numbers. As did already Galileo he used the method of bijection in order to demonstrate that the infinite amount of natural numbers is sufficient as to count any rational or algebraic irrational constructs. The latter was a surprise that paved the way for publishing his otherwise silly idea despite of rejection by Kronecker already in 1874. While Galileo had plausibly concluded that there are not more natural numbers that their squares because a quantitative comparison is not reasonable for infinite series, Cantor and Dedekind followed the intuition that only counting means (well) ordering from smaller to larger, a view that denies actual infinite non-linear divisibility. Cantor's proofs rested on the assumption that any number must be smaller than, equal to, or larger than a given rational one. Ignoring the possibility of incomparability Cantor's evidence is logically cyclic. Up to now, mathematicians tend to fail understanding what makes real numbers different from rational ones, cf. an admission by Ebbinghaus in his textbook "Numbers".

Fraenkel 1923 reveals the decisive trick in Cantor's 1891 diagonal argument: Cantor assumed that all of infinitely many digits are fixed. Likewise one could fix all of the natural numbers and then show that by adding something that there are evidently more than infinitely many natural numbers: "Infinity plus anything is larger than infinity" ?

Already the word "many" instead of "much of" is misleading: In common sense, an endless amount is not countable. Cantor's "mathematical" notion of being countable contradicts to the traditional understanding. The entity of all natural numbers is not countable. As Archimedes understood, numbers are endless. They are just tools. The allegedly difficult to understand Cantor's Set Theory is easily understand as a useless intentional denial of sound Galilean reasoning.

What was the intention? Why declared the Bourbakis Set Theory the fundamental of mathematics? See above.

Eckard

Aldo Filomeno replied on Apr. 30, 2015 @ 18:28 GMT

... Your essay looks interesting and original... I'm curious to see how you argue that (A), i.e. that the world displays spatiotemporal patterns, is hardly indisputable. How could it be disputed? I took it as a premise of my argument ... (By the way I don't understand your last observation!)

In parentheses he referred to my claim:

"I should add that I maintain that I consider it unjustified to integrate in case of frequency analysis over available past and not yet written future data. Cosine transformation is evidently as good as Fourier transformation except for it omits an arbitrarily chosen zero."

I guess, Aldo accepted that future data are not available in advance. Most likely, he merely objected to my last sentence because real-valued cosine transformation is a particular case of complex Fourier transformation.

While this objection is mathematically correct, the application of Fourier transformation on functions of time requires Heaviside's trick that uses the mirrored past instead of the unknown future in order to make an integration from -oo to +oo feasible.

In other words, Heaviside's trick introduced redundancy and led to seemingly symmetrical results. That's why the complex representation does not at all convey more information except for the arbitrarily chosen phase reference.

So far, nobody could otherwise explain why the usual spectrogram is non-causal, why in practice MPEG coding works well with cosine transformation, how the rectification by OHCs might be possible in complex domain, etc., etc.

Fourier transformation of a function of time or space and before him John Wallis (1616-1703) and after him Einstein just adopted the Parmenidean (fatalistic) philosophy. Fourier was not even wrong because he dealt with a closed heat conduction ring. However, treating time or space as a loop is certainly not warranted.

If Heraclit and Popper are correct then only the past can be described for sure by a spatio-temporal pattern. The future did not yet happen. It is undecided. I share this view although it deviates from the philosophy behind the majority of the essays in this contest including Aldo's up to average fqxi members. Who is correct?

Eckard

Part 3

A mathematics that is relevant for physics should be free of arbitrariness. When I first came across to the criticism that modern mathematics was made rigorous by means of an iron bar, I felt inspired to ask what this meant. Among many other other examples, the definition sign(0)=0 proved unwarranted and in practice misleading.

It is commonly accepted that the division by zero is not defined and therefore forbidden. Natural mathematics does not define and forbid. At least I am not aware of any reason to forbid the conclusion that division by zero results in an infinite result. There is merely no logical justification for subsequently calculating with the property oo as if it was a number. This follows from basic and obvious to everybody axiom by Archimedes: To every number n there is a larger one n+1.

G. Cantor's largest finite number oo does not by chance look like omega. His "überabzählbare" (more than countable) numbers contradict to this axiom. That's why his set theory was obviously naive. As Fraenkel admitted, axiomatic set theories replaced this flawed belief by postulating in the last axiom of ZFC the property of endlessness to a set: "There is an endless set". Already the first axiom of ZFC used the notion set in algebraic, i.e., finite sense. The maneuver managed to hide the logical split.

Eckard