Essay Abstract

While there is no evident reason for believing that the world was anyhow made of discrete parts but overwhelming evidence for a rather continuous evolution, signal processing is superior if based on discrete values. This seeming contradiction gave rise to an investigation on how analog and digital approaches relate to each other and to reality. Three interrelated mathematical pillars of physics were found to suffer from unjustified generalization: Points instead of endpoints, once and twice redundant equivalences. A realistic interpretation of abstraction-made ambiguity sheds new light on quantization and apparent symmetries.

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See http://www.fqxi.org/community/forum/topic/369

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Dear Eckard,

That was a really enjoyable essay, clearly combining your vocation and your avocation.

"Complex calculus is so excitingly superior...". Yes, isn't it. In "The Road to Reality" Penrose speaks thirteen times (I counted) about "complex number magic". I believe that I mentioned to you in the past, but if not, I'd like to recommend now Paul Nahin's "An Imaginary Tale, The Story of SQRT(-1)". I think you'll love it.

You mention Heisenberg's imaginary difference and Minkowski's ict. I would be interested in your opinions or insights into this aspect of reality.

Thanks for a refreshing essay.

Edwin Eugene Klingman

    • [deleted]

    As a non-physicist physician, having read a multitude of your posts on this website, I am embarrassed to admit that I cannot tell whether you think that we need the physics to prove the math, the math to prove the physics, or neither of the above.

    To me, as a lay person, the ideas of zero, i, and the infinities are mathematical, whereas the issue of divisibility is physical. The problems seem to arise when we use physical constructs for the former (singularities and the like) and mathematical precision for the latter (space, time, etc.)

    Thank you for a very interesting essay.

    John S. Minkowski, M.D.

      Dear medical Doctor,

      Thank you for your mature comment. I fear, your essay is a bit too parsimonious as to really hurt anybody. You wrote: "This representation is where the money is, and so therefore, it must be the truth." Maybe, Jan M. (1916-1991) relates to the Hermann M., and you relate to the former. Otherwise, I would suspect behind your sarcastic utterance an almost antisemitic attitude.

      Perhaps you know that Christian Betsch won 1,000,000.00 Mark which was a fortune in 1925, i.e. after inflation in Germany. Why? His price-winning book "Fiktionen in der Mathematik" dealt with Hans Faihinger's theory of the as if, and the point is, he pleased even Fraenkel by letting the question whether Cantor's set theory is correct or wrong undecided. So he did not hurt anybody.

      I agree with you: Notions like zero and i are mathematical while in reality we are doomed to find only finite pattern. I would however like to add that continuity of reality is also an untestable abstraction. And I tried to show with my Fig. 1 that both limitations are mathematically equivalent to each other via cosine transformation.

      I realized you writing "infinities" instead of infinity. Here we might disagree. I know that you are following the mainstream belief, and therefore I cannot expect you to easily swallow my arguments unless you are willing to read again and judge yourself without any prejudice what I wrote. In the latter case, I am willing to provide further support. Concerning this question my critics hurts every average mathematician. Sorry for that.

      Doesn't my Appendix 1 exemplary show that a more precisely reinstalled Euclidean notion of number avoids problems inside mathematics?

      Regards,

      Eckard

      Dear Edwin,

      I feel pleased by your kind interest, and I appreciate your hints. No, I did not yet read Penrose and Nahin. What about magics, I decided to quote [13] Peat's essay as to elucidate Pauli's mysticism and its influence on quantum theory.

      Having learned a solid step by step approach to complex calculus at TU Dresden from 1960 to '66 and subsequently taught fundamentals of EE for more than forty years at Otto v. Guericke University Magdeburg, I cannot confirm any mystery of i.

      Already Bombelli understood that a+ib is always appears together with its complex conjugate a-ib. There is a very simple but compelling argument against the belief that i may be interpreted in terms of reality: The chosen negative sign of the argument in exp(- i omega t) is not just arbitrary agreed on, but it even depends on which scientific discipline has been chosen. I was told: More than a hundred years ago, the EEs decided to prefer anticlockwise rotation by multiplication with i in order to get as little negative signs in complex power of propagating signals as possible.

      I collected a lot of seemingly mysterious oddities and asked myself for reasonable explanations. I am a bit ashamed because I not immediately understood that the oddities with use of complex calculus go back to something that proved at least also superior as compared to ancient mathematics as now is complex calculus. I am speaking of what deserves the name first mathematical revolution: introduction of differential calculus.

      You asked for opinions and insights concerning ih and ict. Well, you got aware of as I am claiming most important consequences affecting 20th century theories in physics. Thank you for addressing this. If you are ready to read my essay carefully and open for accepting your own conclusions, you will hopefully not shy back from what seems to be unbelievable. I am waiting for those who will try to rebut may basic argument: No matter how useful negative as well as imaginary numbers are, reality can be expressed without them. A lot of apparent symmetries can be attributed to inappropriate interpretation of per se correct mathematics.

      In my abstract I mentioned three affected interrelated mathematical pillars of physics. Did you understand from my essay how they are interrelated?

      Regards,

      Eckard

      • [deleted]

      Dear Eckard,

      Jan M. Minkowski was my father, and Hermann Minkowski's grandfather is a common ancestor.

      Regarding "This representation is where the money is...": When first read it sounds sarcastic (as you correctly identified), but actually it can also be read as a tautology in that the vast majority of the money of the world is represented digitally, as I alluded to earlier in the essay.

      Thanks for the reply and please no offense intended.

      John M.

        • [deleted]

        Dear Eckard,

        Congratulations for this work full of rationality.

        The reals are the reals.

        Best and good luck

        Steve

          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

            Eckard,

            While I am completely with you on the attempt to remove mysticism from physics [we seem to be losing this battle, by the way] I also agree with Penrose that complex calculus is almost magical in it's beauty.

            But do not read Penrose first (if at all). Paul Nahin is an Electrical Engineer and I really think you would love his book.

            I have no real argument with your contention that "No matter how useful negative as well as imaginary numbers are, reality can be expressed without them." In the bottom figure on page 2 of my essay, the 'arrow' goes through the 'surface'. If I introduce the complex 'i' as a means of keeping these two directions orthogonal, I can derive Schrodinger's equation almost immediately. I'm trying to write this up in a convincing manner.

            As for the last question, I will re-read your essay and come back to you. I've read many essays in the past few days, and the details get fuzzy fast.

            Edwin Eugene Klingman

            • [deleted]

            Dear Eckard,

            Well indeed it's necessary and you make a very good Occam Razzor if I can say,it's sad that people doesn't take these rationalities with more pragmatism,

            I beleive you focus a little too much on history of sciences.And you lack a little of universality thus a little of generality.

            But you are skilling and rational, it's interesting,rare and important.

            In fact you must think by yourself about numbers and sometimes you mix the ideas of others for a description of some words as here the infinity.But be sure I like read your posts.A very skilling scientist.And in the same time we learn this history of physics.

            Well the most important is to uinderstand the finite physicality, and the infinity due to our young age at the universal scale.

            It's totally different.

            With your little f(x)=......there you make the same than the people with their decoherences.......

            we can calculate our finite systems....we can appraoch them......we can add or multiplicate......but the finite system rests as they are.The infinity appears only in our multiplications of these finite systems.....RELEVANT VERY RELEVANT AS HABIT YOUR ESSAY.

            Best Regards and wishes

            Steve

            • [deleted]

            All that is important in fact.

            Let's take the binar paradox about the diagonal of Cantor and the infinity.Always a different appears.......surprising infinity +1.

            The different orders appear.

            What do you think dear Eckard about the rationals and the reals.Of course they are more numerous these rationals due to a fractionalization.This aleph 0 with N and Q.We can calculate this serie of the entires.It's a numerable system.

            My question is this one, is it continu for you.

            The aleph 1 and the reals is continu at the exp.....if we tend to the aleph 2 and the curves.....the aleph 3 is probably the sphere and the aleph infinity is the same .....EUREKA for the distribution of numbers hihihi

            Hypothese of the continuity is solved because all is ......in fact all is a question of domains , limits and utilized methods...that depends of what we want describe.....for the fractal....finite serie.....for the binar .....for this ....

            Sincerely

            Steve

              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

              • [deleted]

              hihihih it's full of relevance dear Eckard, I was right, you are the most skilling here on FQXi.

              But here is my question.

              How could you differenciate those two infinities, the infinity inside the physicality and the infinity behind our walls, these physical limits, quantics and cosmological?

              lET'S TAKE the number of particles in our universe, the number of cosmological spheres and you multiplicate them .......what is this number finite or infinite....?

              Regards

              Steve

              • [deleted]

              Do you think the equipotence and the omnipotence show the road of a real explaination for the link between the absolute , the physical reality and its fractals of maths.......

              The spheres are better than points of course.if a real abstraction of the reality is really harmonized.The infinity doesn't need to be calculated, it is just a tool of calculation for some series. That permits the real definition of domains ....and codes also for some binar systems for example.

              In all honesty dear Eckard, do you beleive in God ?

              Me I am frank, I am a pure universalist, I beleive in something of infinite above our perceptions.It's not a religion or other, no it's just that it's not possible to have a so beautiful sphere in evolution and all its codes without a kind of absolute.Now of course we live in the physicality and if we can't see this infinity, there is a reason I think.The informations continue their road dear Eckard.It's an evidence.

              The human inventions are totally different than our pure universality.There you shall understand the difference ,I am persuaded, between this said universality and on the other side some religious explainations of humans.When I read the sacred books,the bibble, the talmud, thje coran, the hindouism,the boudhism.......a simple conclusion is everywhere...the love of all creations simply.We are all linked, uniques and precious,.......the real universality is the total respect of all creations and the pure complementarity of harmonization.It's better than chaos.

              The aleph is the first letter in hebraic writing......the 1 is the 1 dear Eckards and 1x0.......has no sense!!!!

              Best Regards for your maturity also .

              EQUIPOTENCE+OMNIPOTENCE=SPHERIPOTENCE.

              Steve

              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

              • [deleted]

              You wrote:

              "I believe you focus a little too much on history of sciences."

              Let me try and explain some of my reasons:

              - When I got aware of inconsistencies, I looked for possible mistakes, and eventually I found crucial decisions mainly in the German mathematics of the 19th century and in the German physics of the 20th century. Because German is my mother tongue, I had ideal possibilities to read the pertaining original papers, and I found all my suspicions confirmed.

              - Before retirement, I was with Otto von Guericke University, and our faculty was named after Werner von Siemens. Therefore I dealt with historical facts. I got in particular interested in two questions:

              On which basis did the utterly fertile differential calculus and belonging technical and industrial progress arise after Galilei, Kepler, Descartes, Guericke, Leibniz, Newton, Watt, etc.?

              How did arise the application of Fourier's, Maxwell's and Heaviside's theories and complex calculus? Why did Cantor manage causing so much unnecessary quarrel?

              - I consulted mathematicians as to understand some arbitrary definitions, and they pointed me to literature that guided me back to Euclid, Bombelli, Dedekind, and others.

              - I collected valuable insights from public discussions and from literature including a booklet by the outsider Mückenheim.

              - Having experienced very different political systems, I do not trust in any seemingly overwhelming propaganda.

              - Having participated in FQXi contests, I got aware that physicists seems to be still unable to reach full agreement on just one theory without paradoxes. Perhaps John Merryman is not the only one who is highly skeptical concerning modern theories.

              - Even a lifetime would presumably be too short as to thoroughly deal with all branches of speculative physics. My life is almost over.

              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