Continuation Causes Superior but Unrealistic Ambiguity by Eckard Blumschein

Dear Israel Omar Perez,

You wrote in your thread:

- "I have read your essay. I can see we have some points in common, particularly, what you mentioned in a previous post about the idea of adimensional points. As I told sometimes these things become a prejudice quite hard to get rid of it."

My reply_

I have to apologize for my poor memory. Could you please explain to me your position? Do you consider Euclid's definition of point a prejudice of mine?

- "I also agree with the idea of the infinite quantities, this is a problem that dates back to Aristotle and it has not been appropriately solved."

My reply:

Well, Aristotle's meant: Infinity actu non datur (There is no actual infinity).

I consider this still correct if seen from the perspective of counting, i.e. inside the rational numbers: While there is already general agreement on that irrational numbers, e.g. sqrt(2) correspond to a never-ending rational representation, I am arguing that rational or even natural numbers are likewise unimaginable if seen belonging to a Peirce continuum of truly real numbers. Infinite accuracy is an unreal purely mathematical fiction. I agree with you that the mathematics of Cauchy, Weierstrass, Dedekind, Cantor, Hilbert, etc. solved such problems by arbitrary definitions rather than appropriately. I am arguing that there is no reasonable option but to understand the real numbers in the sense of Peirce's true continuum. When Peirce distinguished himself from "pseudo-continuum" he felt perhaps not strong enough as to clearly add the for the sake of logics necessary distinction between the two mutually excluding and complementing ideal worlds of continuity R and discreteness Q, which is of course irrelevant in physical measurement.

- "I think your work emphasizes theses issues that I am sure it could shed light on the nature of numbers, points, etc."

My reply:

I do not see a reasonable alternative to Euclid's definition of a point and also Peirce's definition of a continuum. Physics and mathematics itself will benefit from more precisely reinstated Euclidean definition of a number as a measure.

- "In fact, I have been studying the problem of infinite quantities, you may be interested in seeing the surreal numbers, that apparently solve the seven indeterminacies and say something about the division by zero."

My reply:

While I did not deal with surreal numbers, I looked at Robinson's hyperreals, and I did not find any tangible justification, neither in logics nor in practice. Aren't they just a fabrication that is based on Cantor's questionable naivety?

What about division by zero, engineers like me do not shy back from equating anything/zero with infinity and anything/infinity with zero. I do not see progress in these questions since Bernoulli. I agree with Terhardt: Fake rigor is an unnecessary obstacle.

Let me add:

The main reason for me to deal thoroughly with fundamentals of mathematics was the rejection of R "for mathematical reasons". Practice has been benefiting from cosine transformation for many years.

Regards,

Eckard

Dear Steve,

Well, I am trying to purify science from mysticism. It seems to be a hard job.

You wrote: "The reals are the reals".

Really? I quoted the book Numbers by Ebbinghaus et al. as to show what is a widespread weakness among mathematicians. The have problems to understand in what the reals are different from the rational numbers p/q. The reals are thought the union of irrational and rational numbers. Weyl called the rationals bones and the irrationals a sauce embedding the latter. I do not share this view.

When Ebbinghaus himself admitted problems of understanding, he did it cautiously. Even more cautiously encrypted he uttered his attitude towards Cantor's transfinite numbers by quoting Lessing at the begin of the belonging chapter.

Is there just one number y=f(x) because x is a unique number? I am saying no.

Best whishes to you,

Eckard

Dear Steve,

All three of my appendices are anything than harmless. In contrast to Ray Monroe, I am in a comfortable position: I can blame you for uttering guesswork when you claimed having found applications tor aleph_2 and aleph_3. You would be the first one worldwide since more than one hundred years.

My appendix B explains what generations of mathematicians were either unable to find out or perhaps unwilling to admit: How did Georg Cantor manage cheating not just himself?

I am hopefully not the only one who understands: While there is only one ideal property infinity, the same infinity has two aspects depending on the side from which we look at it.

On one hand, according to Archimedes insight, there is no limit to the process of counting. And there is also no limit to the mathematical process of splitting. Weierstrass introduced rigor into the mathematics of rational numbers when he formalized Cauchy's method of limit. Cantor correctly mentioned that this so called potential infinity is not yet actually infinite.

On the other hand, a Cauchy sequence approximates its limit as good as we like. While an irrational numbers like sqrt(2) cannot be found within the rational numbers of any finite precision, it has nonetheless its unique place on the Peirce-continuous line of logically correct understood real numbers.

The infinite precision of a real number is as unphysical as are less mystified notions as for instance one, line, point, sin(omega t) and zero. In this sense, the absolute infinity is not special at all. Adding any number to infinity yields infinity again as multiplication of any number with zero yields zero.

Cantor's infinitum creatum sive Transfinitum and his transfinite alephs have proven useless phantasm. Their only alleged basis was the second diagonal argument which can be easily rebutted as indicated in my appendix B.

Eckard

Dear Edwin,

Perhaps you are correct in that we personally may loose the battle. However the battle will certainly go on after my dead.

I have already a lot of ammunition, and I do not expect finding any tangible counterargument in the book by Nahin. Nonetheless I will try to get it as soon as someone has given it back to our library.

You are definitely not the only one who has no logical argument with my contention.

What about Schroedinger, his papers in 1926 were the first ones I looked into as to find out where he might be wrong. In the fourth one, I found the most crucial guess as an almost inevitable logical consequence of the usual superficial use of complex calculus. This explains why the team Born, Kramers, Heisenberg, Jordan, and Pauli independently arrived at the same mistake as Schroedinger and Weyl and also Dirac and others.

They altogether rejected the possibility of negative wavelength, frequency and energy while it was quite natural to them that time can be positive as well as negative. In brief, they were unable to really understand that Heaviside's approach is based on tacit preconditions, which were no longer valid.

Regards,

Eckard

Dear John,

Isn't money in general based on counting and therefore discrete? Nonetheless, I learned the expressing "how much does it cost", not how many.

The reason for some physicists to look for smallest particles seems to be nonetheless promising after Guericke's likewise expensive search for empty space has led to the first industrial revolution and also to electricity.

Regards,

Eckard

Dear Steve,

You are among the few who do not feel hurt by my perhaps difficult to rebut arguments. Before I will answer any reasonable question, let me comment on the what has been solicited:

• How can a spacetime or other continuum--with continuous symmetries--emerge from a 'digital' description?

- I am arguing that the symmetry of Schwarzschild solution is most likely an artifact rather than a description of reality. Nonetheless, differential equations are certainly best suited to describe dependencies on spatial and temporal distances if there are no relevant discrete structures to be considered. In other words: A superimposed Heaviside function is often the only relevant discrete structure. It does not emerge. It has to be obeyed.

• What is the nature of space? How would a discrete universe expand without the discreteness becoming evident? Or, does it become evident?

- This question might bother those who intend to see spatial and temporal distance subject to some quantization.

• What are the implications of a minimal length, time, or energy, and how could we observe them now? Or, is this the wrong way to view fundamental discreteness?

- Planck mass could be measured. Counting peas does usually not make much sense.

• Is a universe that is infinite in various ways incompatible with a digital description?

- I prefer guessing that reality might be potentially infinite, and I do not see this lack of limitation an obstacle that could hinder digital approximations.

• How is a digital description consistent with a 'flow' of time? How does causality work?

- In practice, the potentially infinite rational numbers are always sufficient. Theoretically, I am arguing that Peirce was correct, and the fictitious entity of all real numbers should be seen as a true continuum. I see causality the most indispensable and irrefutable hypothesis of science. Of course, it is strictly speaking often impossible to exactly define single countable objects. Panta rhei. Natura non facet saltus. Causality must not be confused with determinism.

• Can the World be modeled as (or even be) a digital computation? Where does this picture lead us?

- Back to determinism a la La Mettrie: Man a machine.

• Are simple discrete models like cellular automata, etc., effective approaches to physics?

- Perhaps they are not ingenious enough.

• Is there a deep, foundational reason why reality must be purely analog, or why it must be digital?

- Something should be called "foundational" or "deep" if it obeys logic without paradoxes and if it has proven fertile in applications. I gave examples for pseudo-foundational speculations and for foundational pragmatism. Instead of a direct answer I offer a touchstone. Judge yourself.

Eckard

Steve,

You asked me: "Do you believe in God?"

Earl Bertrand Russell wrote: There are many religions but at best a single one can be the true one.

I already confessed that I consider causality in the sense of rejecting mysticism the only indispensable hypothesis of science.

Accordingly I would feel in principle unbiased concerning any further hypothesis including genesis alias big bang, spacetime and antiworlds. They might be entirely or partially correct or wrong. However, there are reasons to be skeptical:

Don't big bang, primordial elements and inflation remind of old religions?

Is there at all any reliable way and any need to see beyond the logical horizon?

Isn't spacetime a construct closely linked with the twin paradox due to Lorentz transformation? Isn't it bewildering to explain the reportedly observed flatness of universe as an illusion due to inflation?

Arn't antiworlds and physical singularities possibly artifacts of unjustified interpretation of mathematics? Will quantum computing become feasible? Will Higgs be found?

For such questions I am happy to have found one more independent touchstone. If it puts me inside the mainstream, then the better. I am prepared to accept any convincing argument.

Eckard

Dear John M.,

You wrote "digital reality costs money, although much less than before." Perhaps you meant digital virtual reality and compared it with perceptually equivalent not yet digitalized alternatives.

I rather felt reminded of the cost for equipment like LHC that is designed as to search for a suspected digital structure of time and space. I see these efforts similar to the intention of medieval alchemists to make gold. Tschirnhaus and Boettger made white gold: china. Isn't the www a spin-off of Swiss physicists?

"Despite Aristotle"? Isn't his view "infinitum actu not datur" still correct? By the way, there were already ancient symbols for zero. Most likely ancient and medieval mathematicians hesitated using it because they understood the two the first number after what we are calling the neutral element of multiplication: the one. Not by chance, Leibniz invented the dual numbers and also the infinitesimals. I see an overlooked problem not in infinity and not even in zero but in a strictly ideal understanding of the basic measure one.

You complained: "I fear the unintended side-effects for my children." If I guess correctly, you do not like your children excessively consuming digital pleasures like gameboys, headphones, TV, internet, etc. Well, as I wrote: "Signal processing is superior if based on digital values." I even partially ascribe the present turmoil in Egypt and elsewhere to the desire for participating in such tempting pleasures. However such questions are off-topic. At least my essay deals with deeper issues.

Since I am retired, I do not fear side-effects that put my employment at risk when I frankly utter what is definitely highly unwelcome to many experts.

Regards, Eckard

Dear community,

While I am still waiting for responses by several contestants including Cristinel Stoica, I highly appreciate the private response to my essay I got from David Joyce. He is a professor of mathematics, an expert in history of mathematics. I will ask him again for his permission to put his reply on this public location.

In the meantime I would like to try and clarify to him and perhaps to you all which two old notion I would like to suggest for reinstating more precisely:

i) Euclid's notion of number as related to the ideal notion of unity

ii) a continuum every part of which has parts as still stated by Peirce.

Both old notions were accepted until the 19th century. While the notion unity is not identical with any object of consideration like length, area, a countable item, etc., it can nonetheless be attributed to it, and it shares its extension.

When the notion of number relates to the notion of unity, and a true continuum cannot be composed of any finite amount of extension-less points, which do not have parts, then there are no singular numbers. Brouwer's intuitionism comes close to it.

Accordingly, we should not declare Aristotle, Galilei, and Leibniz outdated and Peirce just an often drunk stupid American. Even if Peirce was an outsider, I consider him more intelligent than the whole folks including Dedekind, Cantor, Frege, ...

Let's judge the foundational consequences for physics. Shouldn't we consider numbers and continuum quasi orthogonal to each other?

Eckard