Dear Heinz Luediger,

Thank you for your comment. You are an EE as I was too,and you were perhaps not forced to deal with Hegel's dialectics while I didn't like dialectic materialism which was obligatory to me as a precondition of promotion. Didn't Heg,l say "Die Atomisten denken zu materiell"? Donn't get me wrong.I consider thesis, antithesis,and synthesis valuable. However, if I recall correctly, it was Marx who criticized "abstruse Hegelei". What about physics I would like to rather trust in Kant who wrote in Kritik der reinen Vernunft: "Die Eigenschaft von Größen, nach welcher an ihnen kein Theil der kleinstmögliche ist heißt Continuität. Raum und Zeit sind quanta continua". Notice his word Eigenschaften. Both being finite and being infinite are properties, in other words qualities, not quantities. Please read $ 2 of my essay carefully.

I didn't use the word metaphor exactly as perhaps do you. To me, a metaphor is an imaginary way of describing something by saying that it is something else which has the qualities that I am trying to describe.

Because you seem to be a prolific essay writer, I would like to ask you for any hint to some overlooked imperfection in basics of mathematics and/or physics.

Of course, I would also like you to take issuw concerning my views on the addressed in my essay positions by Fourier, Hilbert, Hausdorff, Einstein, and Feynman.

Regards

Eckard Blumschein

Bravo Eckard,

and I can add little to Ed's accolades. You say you are no mathematician but you have always had me fooled, and I am only just understanding enough of what mathematics IS to appreciate it.

Your essay should be awarded the full 1/3 weight of scoring for relevance by the judges, for if not then none other should be. I've bookmarked it on my reader as a favorite, there is such a wealth of information and accompanying reference.

I must do more reads, especially to get my head around your stating that you have discovered an error that profoundly impacts Fourier. If you would elaborate a bit that would be instructive. I recall some years ago dialogues between you and Rob McEachern which I at the time thought was simply a difference in preference for either cosine, or Fourier transforms. I have thought in my own simple modeling that cosine transforms were ontologically most representative of physical realism in energy concentration in a 3D+T soliton waveform, while Fourier Analysis more applicable to distinguishing specific frequencies among polychromatic emissions.

I viewed an interesting science program recently and was surprised to discover that whales are a relatively young species. What was in particular pertinent to the program was about the whale's ear bones which are remarkably dense, and which enable whales to locate direction of sound through the water. It corroborates your contention that the ear distinguishes frequency from interpreting backwards from 'end of signal'. Where humans under water hear sound from all directions, it is because human tissues are nearly the same density as water so the sound wave penetrates through the head and reaches both inner ears at so close to the same time that the regions of the brain interpreting direction from a lag time between the ears, cannot differentiate. The sound wave warping around the head in air provides a significant time difference, which the high bone density of the whale provides as a direction finder.

Your essay lays out the distinction between physics and math, and addresses the Topic of the Essay Contest in a most fair, informative and respectful manner. Thank-you jrc

Dear Eckart,

I don't think we really disagree on the notion metaphor, rather we seem to look at it from different perspectives. Maybe my forthcoming essay can clear up matters a little bit...

Generally my reasoning rests on the impression that neither physics nor mathematics are monolithic (coherent) blocks of knowledge, but sets of Absolutely non-contradictory (orthogonal) theories, theorems, etc. So, everything seems to be connected by being Absolutely disconnected in quasi-space. If so, logical positivism tried to logically (the math is just heuristic) connect things which share no common measure and thus created loads of conundrums. Logical positivism is dead - long live logical positivism!

Heinz Luediger

Eckard,

page 6 [15]

physically, if we pass the pole of a magnet past a conductor, a rise and fall of induced voltage will occur, but the direction of the induced current will not reverse bias. The rate of change of flux density in proximity to the conductor is proportional to the value of induced voltage, so it makes clear sense that only positive reals are necessary and while the sinusoidal shape of rate of change would approach a nil value of voltage tangent to either the upper or lower limits of a continuous sinus, there would exist a real positive voltage level in the same direction of potential in the conductor.

I have long thought that such a scenario as this, rather than the typical plot with a 'zero' baseline and positive and negative deflections commonly interpreted as the signature of a transverse wave; might be the realistic signature of a soliton of linearly projected electro-magnetic emission. Perhaps both Newton's corpuscle of light, as well as the wave model, are at least half correct. jrc

Dear John,

"... your stating that you have discovered an error that profoundly impacts Fourier. If you would elaborate a bit that would be instructive."

I promise I will do my best in an even more understandable manner. When I just supported Tim Palmer's criticism of Hilbert space, I was nonetheless aware of the importance to avoid terminology like "infinity-adic" which is definitely not appealing to those like Lorraine Ford and Rob McEachern. In this respect I am trying to learn from Feynman although he was sometimes also forced to admit that he couldn't explain something like half-spin without resorting to a sophisticated theory (spinors).

Let me begin for this morning with telling you why I didn't shy back from critically reading the many pages of Fourier's "Théorie Analytique ...". Reacting to FQXi attempts to question causality I decided to look for the reason behind non-causalities from which I was befitting for fife decades. Admittedly, I had collected insight over the decades in which I dealt with complex calculus, dual circuits, power electronics, arc welding, auditory frequency analysis, and even a bit of neurophysiology. The fact that Fourier transform implies an arbitrary and therefore redundant phase was not new to me. My surprising revelation was that the mistake arose from what I quoted in my essay: A function of time(arbitrarily) given by a mathematician is not identical with how nature works. Mathematics is too wide. Now I have to make a break.

If you like me to continue,I will later refer to details with respect to Neumann and Dirichlet boundary conditions.

Eckard

Dear John,

I don't have [15] at hand, and I am not yet aware of your essay. Perhaps, it will take me time to check your idea althoug I am not unfamiliar with hearing and waves.

Eckard

Thank you, Eckard.

I am not writing at all presently, nor have an essay to submit. I was only using a physical example of how positive reals would suffice. And it has long seemed to me that many times, the practices used in graphing actually cause a non-physical interpretation.

I agree that time as treated mathematically, and in physical theory, is not what it is in nature. It is a problem that I think arises in the analytical need to isolate a particular aspect of time, and then treat that as the whole measure of time itself. Yet we have our metronome counting time in our laboratory in unit seconds, while we calculate a light velocity proportional difference across electrical and magnetic field strengths, so we have at least two kinds of time in practice at the same time. At each of any instant in that one second duration, that proportional light second is existential. And that definitely does NOT mean that past, present and future are the same thing. But we are actually confronting two different aspects of time.

And, take your time, you have brought a lifetime of accumulated knowledge to bear on the Topic, most of which I have only casual acquaintance with but at least enough to find your essay illuminating in your presentation. I very much liked how you laid out the problems of how a continuum can or must be treated, and whether we can even say that a continuum is possible. Yet if it were not conceivable intellectually, we wouldn't bother with discussion. Best Wishes jrc

Dear all,

Rethinking the question

"Are there, for example, real consequences for physics -- including quantum mechanics -- of undecidability and non-computability?"

I would like to once again support T. Palmer: Back to basics!

Past and future are not undecidable from each other, except for theories that are used as putative fundament of physics.

I promised to John Cox to elaborate, and this will take me some time.

Just one detail:

Having checked my essay for decisive loopholes in my reasoning I realized that I did perhps not yet clearly enough distinguish between Fourier's undecided with respect to the choice of t=0 and therefore redundant complex transformation on one hand and Heaviside's fictitious split into an even and an odd component on the other hand. Heaviside introduced the reference t=0. Fourier or Heaviside, on which fundament should QM be based? For convenience, one may calculate as if Fourier was right.

Eckard Blumschein

    Heinz,

    While I in priciple appreciate your dealing with categorization errors, I will perhaps have to read your essay more carefully in order to learn something. Why did you not even give references and page numbers? From the very beginning of your essay I feel challenged to disagree. To me the chicken-egg problem and Zeno's paradoxes are just stupid confusions while the question whether the universe is finite or infinite is most likely undecidable.

    You and Hegel cannot persuade me that being infinite is in any sense not eimply the lingistic as well as logic negation of being finite.

    You and Einstein will not make me disagreeing with Shannon on the distinction between past and future with no present (your reality filter)in between.

    What about the interpretation of QM, you should contact Rob McEachern.

    Eckard Blumschein

    Dear Eckard,

    already I'm in over my head. But in browsing to get a better idea of what (and how) Fourier distributions are and 'get back to basics', I ran across this very brief and concise paper on Fourier Transforms of a Heaviside Step Function: a Tragedy.

    cs.uaf.edu/-bueler/M611heaviside.pdf

    and have to wonder if Bueler is demonstrating the same, or similar, problem that you identify in the operations of F & H.

    It strikes me that if we take the energy signal as going to infinity, then that is really a mathematical convenience not an existential property. It simply allows an open ended process that would integrate any number of steps once a signal was switched on... that is, the duration of repeated pulsations is unknown and contingent on putting in by hand, an 'off' switch. Also, the 1/2pi term would naturally apply to the orthogonality of electromotive induction, or in a 3D+T wave model - the 90* rotation between the planes of direction of field strengths of the electrical and magnetic components of an EM signal. best jrc

    Dear John, dear all,

    "Fourier Transforms of a Heaviside Step Function: a Tragedy" ??

    I prefer using the word tragedy to human lives. Just a few excamples:

    Archimedes was killed: Don't disturb my circles

    Matrin Luther saved Stifel who had calculated and predicted the immediate doomsday

    Georg Cantor got insane, claimed having got the CH directly from God, and died in a madhouse

    Kronecker was mobbed and died

    Ritz and also Minkowski got suddenly ill and died

    Suicides by Boltzmann, Hausdorff, Turing

    Gödel's paranoia

    Grothendieck's disappearance

    In the case addressed by Buehler, there is a quite simple logical solution - admittedly one has to get free from traditional formalism but go back to basics as I tried to indicate in § 2 of my essay.

    My style of teaching is a bit different from Feynman's. I hope, you may find the solution yourself soon. I will give you just a hint. Feyman allegedly refused to explain half spin. Why didn't he just mention that a full circle (360°) of cos equals to 720° of cos squared?

    Once again, the solution is easily to find out for anybody with readiness to critical tinking. Don't shy back from questioning a very basis of mathematics. Admittedly I was inspired by a Professor Schwarz of South Africa whom I met in Milano in 1992, Dean Mückenheim provided me with many details, and I recall having read a lot of literature in German, e.g. Hans Gericke and Oskar Becker.

    Best,

    Eckard

    We should not neglect what John addressed: My suggestion "calculate as if there was no causality but be careful" does also relate to the artificial boundaries of the interval under consideration.

    However, I claimed having a simple logical solution to the Fourier transform of the Heaviside function. Here it is:

    H(t) can be split into two fictitious parts, the even one = ВЅ and the odd one = ВЅ sign(t).

    Notice: Frequency analysis of measured (i.e. past) data requires H(-t) and sign(-t).

    With cos(пЃ·t) + i sin(пЃ·t) as kernel of Fourier transform, integration from minus infinity to plus infinity yields the real part of H(пЃ·) = ВЅ and the imaginary part = 1/пЃ·i

    Bueler's example doesn't share the widespread mistake to define H(t) with t>=0 instead of t>0, and it illustrates that calculating as if setting t=0 in H(t) was correct may lead to wrong results. Use of distributions is not easy and perhaps unnecessary.

    Notice: Euclid's ideal point, something that has no parts, contradicts to a notion of number which is, as illustrated by Hausdorff, rather based on embedded dots.

    By the way, Heaviside hated geometrical evidences. Gauss criticized the desire for unnecessary acuity. Why? A point "at zero" cannot be split into a positive and a negative part. The only solutions are to calculate as if or to have a 0+ and a 0-. According to Salviati the relations larger than or smaller than are invalid for indefinitely large (as well as small) numbers. We may add: They are invalid for any truly real numbers, not only for infinity and zero.

    is

    Anyway, if there is no natural reference as with the t=0 of H(t) but not with Fourier, an arbitrary choice is unavoidable.

    Thanks Eckard,

    that is reasonable and easy enough to follow. The clarity rests with the distinction that "greater than, or equal to" zero; is fundamentally different than simply "greater than" zero. So Fourier requires the choice of calculating one or the other sign firstly, and separately. Not to become confused with corresponding terms of steps of the opposite sign. Some may say that casts the whole exercise in an arbitrary measure of observer dependence, but in physical fact the observation of any signal is fundamentally arbitrary. We set the criteria of what we want to observe in the first place. When we get a noise free result returned, then we have the information we want. So in that sense, yes, I can see how Heaviside's method would be 'noisy'. best jrc

    Yes, Eckard,

    on giving further consideration, I think with that clarification your essay makes the case quite well. It all holds together very nicely. Perhaps for the present we must content ourselves with the only thing definite about 'zero', is that it has an indefinite value. cheers - jrc

    Thanks for your comments on my page Eckard...

    This essay looks very interesting, judging by the abstract, and I shall look forward to reading it. For the record; Hilbert was not alone, and many people have put faith in the 'excluded middle' when in fact there was a middle ground, spawning what I call 'false dilemmas,' and much confusion of course.

    More later,

    Jonathan

    Dear Eckart,

    the more I read your essay and follow the discussion, the less I see the point you're going to make. Fourier states in J.B.J. Fourier, Theorie de la chaleur dans les solides, 1807:

    The integrals we have obtained are not only general expressions that satisfy the differential equations; they represent in a different way the natural effect, which is the object of the problem. This is the main condition that we have always had in view, and without which the results of the operations would appear useless transformations. [note the terms 'differential equations' and 'natural effect']

    What he says is, that mathematics is a desert with few oases called physics. My personal guess is that less than one percent of the totality of known math has correspondence in the 'world'. And this tiny part must be used - not questioned, for there is no ever knowable connection between these bits of math and the PHENOMENA.

    Best regards,

    Heinz

      Dear Heinz,

      Of course, we may follow not just Fourier and calculate as if there was no causality. Having quoted from page 7 of the English translation of Fourier's 1822 theory: "... mathematical analysis is as extensive as nature itself, it defines all perceptible relations, measures times, spaces, ..." I maintain my objection: Fourier was wrong in this decisive respect. My argument is quite compelling: Measured data which are available for mathematical analysis do definitely not extend from minus infinity to plus infinity but they only include the past. In other words Fourier was wrong because he uncritically adopted a widespread fatalistic philosophy that generalized too much (cf. the word general expressions in what you quoted).

      Did this better explain my point?

      Best regards, Eckard

      Dear Eckart,

      Fourier was a pre-modern, a man of classical physics, that's why I highlighted 'differential equations'. TIME to him was something totally different than for historians and logicians, i.e. the romantics. If my sources are correct, he wrote the variables on both sides of the FT as 'x' and 'u' (which are still used in Fourier-optics), not as 't' and 'f or omega'. So, we disagree on the concept of TIME with no chance of reconciliation. Nevertheless, good luck for the contest!

      best regards,

      Heinz

      P.S. Hegel abhorred of FICHTE'S Dreischritt of thesis-antithesis-synthesis, because for him dialects is not a process set in grammatical-logical-historical time but a principle of the mind. That's why for him 'evolution occurs at a single stroke'.

      Dear Heinz,

      At the time of Lagrange, Laplace, and Fourier, differential equations were not new, and heat conduction is not thinkable without time. Admittedly, I don't understand why you are not in position to accept compelling arguments and at least correctly write my name Eckard (neither Eckhard nor Eckart). Why do you disagree with my concept of time? I wonder if there is to engineers an acceptable alternative to time as something that includes past and future in common sense.

      Even if I am just a bit familiar with Fourier acoustics and not at all with Fourier optics, I am aware of wave numbers k, evanescent modes etc. Complex spatial frequencies correspond to ordinary complex frequencies as elapsed time corresponds to the likewise always positive quantity radius r, not to spatial coordinates x, y, z.

      Best regards,

      Eckard