Hopefully I could find out why the notion infinity has a record of causing confusion. Just a few examples: Bernhard Bolzano (1781-1848) who created the notion Menge (= set) wrote "Die Paradoxien des Unendlichen" Reclam Leipzig 1851

Cantor: On the different points of view with respect to the actual infinity (in German) Halle 1890

Hilbert's hotel is famous, also his paper "Über das Unendliche" in Math. Annalen 95(1925)165

Lavine „Understanding the infinite".

I see the latter inappropriate to someone who tries to benefit from "understanding".

I am already not sure whether to criticize Wolfram's definition of infinity as "an unbounded quantity that is greater than every real number or just smile because the real numbers are conceptualized as endless. To me, the property of being infinite (= unbounded = endless) is an absolute quality, not a quantity.

Relative endlessness means something very useful while logically quite contradictory. Being endless with respect to something as introduced by Bernoulli and Leibniz tamed the nonmathematical quality, made it a usable quantity as if it still was a quality.

There are consequences: Singularity functions like e.g. sign(x) do strictly speaking not fit into what I am calling a mathematics of continuity (anything flows) or with other words Rolle's mean value postulate.

Accordingly, the definition sign(0)=0 is unfounded, etc.

While Laurent Schwartz felt guided by his desire to be rigorous and logical, his basic idea effectively was just the opposite: Abandoning Euclid's indivisibility of a point for the sake of the analogy with electrical dipoles, quadrupoles, and so on.

The theory of distributions followed Rolle when introducing so called testfunctions D, i.e., smooth functions which have compact support.

Already the notations and conventions were "correcting" plausible mathematics:

N0:={0}в€ЄN

вЉ† means subset, вЉ‚ will not be used

Etc.

What about claimed applications cf. "Some applications of the theory of distributions" Lectures on Modern Mathematics, vol. I, New York: Wiley, 1963, p. 23-58

Seen from the perspective of engineers and physicists, the intriguing possibility of endless repetition of the operation differentiation is rather an unnecessary and distracting support of e.g. Hausdorff.

Well, one may calculate to some extent within the theory of distributions and adopt Dirac impulses. However, be cautious and don't ignore arguments against unjustified consequences of belonging topology. If singularities are justified at all, then certainly not inside smooth functions. Don't get me wrong, I am a fan of ideal (not smooth) singularity functions like ramp, step, pulse and doublet. See my earlier essays.

Point charges, line currents and the like are ideals that should not be extended by force to inappropriate models. It is just often best to calculate as if a dot was a point, provided one understands what one does.

If you pretend to calculate as if you were blindly following a pied piper who attracts crusaders (nowadays by means of seemingly compelling videos) then be careful: You need not giving rise in public for getting burned as was Giordano Bruno. You are not obliged to become one more crank among thousand who failed to find and point out what actually went wrong with putatively compelling mathematical evidence.

For instance, G. Cantor's transfinite numbers seem to contradict common sense. Their defenders called them counterintuitive as if common sense was inappropriate. Actually, it was Cantor who was led by his naïve intuition: "einfaches Hinüberzählen, mehr als unendlich viele Zahlen". The distinction between Cantor's aleph zero and aleph one is plausible to me. Anything else including the diagonal arguments should be judged from the perspective of the (so far missing) application in science. While a critical study of Fraenkel's 1923 textbook reveals the cardinal logical mistake, the deeper reason for the acceptance of the rather strange transfinite set theory may be found in a desire of mainstream mathematicians to cope with inconsistencies affecting the notion number: Are they anything one may calculate with? Is this a good question? The natural numbers combine an idealization (the number one) with equality based repetition (addition). In this sense, mathematics is based on logics but not Hilbert's other way round.

At Quora, Cristinel Stoica recently reminded of his FQXi essay 1357. So far I didn't find a current essay by him. May I hope for himself or someone else taking issue concerning what I am calling mistakes?

Eckard Blumschein

Reply to Eckard Blumschein's comments:

Could you accept the idea that the use of the math characteristics produces problematical physics and should be indicate a model that needs a redo (my thesis)?

One thing I treat lightly (lack of space) was the place of error analysis/statistics in misleading and inadequate for physical models. This point was explored in Nielsen, Guffanti & Sarkar arxiv:1506.01354 "Marginal evidence for cosmic acceleration from Type Ia supernovae". This point was further explored in Sabine Hossenfelder's recent interview of S. Sarkar https://www.youtube.com/watch?v=B1mwYxkhMe8&list=PLwgQsqtH9H5fe4B5YCF3vcZgIkMMULS7z

Let me add a bit on a previous comment on your essay, The truncated Fourier analysis results in the next term after truncation is the Uncertainty (Heisenberg's Uncertainty?). The photoelectric experiment where the slope of the energy vs frequency line has a slope of h (Plank's constant). So, the best a measurement can do is within 1 h . So the Fourier series in representing a observed value is a truncated Fourier series where Heisenberg's Uncertainty is the next term. The idea of conjugate pairs is an artifact of assuming particles are infinite waves.

Would you comment on the idea that all the added dimensions, imaginary numbers, and things like Fourier constants do not improve Understanding or physics. They merely mask better physics.

I suggest that nature does "calculate" and does arrive at answers. So, Godel's and Turing's theorems do not apply to physics but to math. A common factor is that they both use ordinal numbers as an important part of the proof. This is what triggered the suggestion that ordinal numbers do not belong in physics. As you see, I hold only cardinal numbers as useful with irrational and transcendental function as contributing to the error between observation and math. This suggests the "natural numbers" includes a man-made part - the ordinal numbers.

Now consider what the cardinal numbers are counting - the standards of measurement. Physics starts with assumptions/postulates about what the standards of measurement are. Advances in physics is primarily about redefining the more reduction standards. For example, Newton defined gravity as a assumed measure in the "The Principia". Gravity was limited to action between masses. In Newton's "Opticks" 1730 edition (careful - different editions have different Query numbering)

Qu. 17 -22 the gravity was caused by an aether which had additional experiment observations of directing corpuscles in diffraction. That is, the aether characteristics are causeless and all other effects emerge such as gravity and diffraction of light observations. Further, the STOE's plenum includes the aether concept and adds the explanation for astronomical observations such as rotation curves (dark matter) and Planet 9.

I understand the natural number's interest is describing an extension of a point to a line to want to include such numbers. But I reject imaginary numbers as being an unnecessary crutch. The distinction between a dot and a point is that there is no distinction. In physics, the cardinal numbers include the zero to signify the beginning of a standard of measure.

You noted "the map is not the territory". Mathematical transforms (maps) as a procedure to solve difficult equations is helpful. However, interpretation of the map should not imply physical effects or measures without the inverse transform. For example, General Relativity field equation has the real measures (mass, distance, time) on the Right Hand Side (RHS) and the transform on the Left Hand Side (LHS). The LHS parameters are NOT physical space or time. Singularities or infinities are not physical and are indicative of incorrect calculation. The speed of gravity is frequently and falsely measured with LHS parameters as equal to the speed of light (the maximum possible speed by assumption).

The STOE developed a Universal Equation with real quantities on both sides of the equation. It started with the Quasi Steady State Cosmology (QSSC) of A Source at the center of Spiral Galaxies. This equation was applied to astronomical problem observations and to light interference.

Current morals (not ethics) are distinctly NOT suited for a significant cooling period. There have been many such periods in history. If a society has grown such that the population level requires the warm period, the following cool period means less food. In high population, humanity morals care for the sick and old and infirm young. Indeed, this is so ingrained that even primitive societies overextend (see Tainter "The Collapse of Complex Societies"). However, there are societies that practices exposure of sickly or unwanted (read unable to support) infants. The Bible chronicles Moses was such a child. A further lesson in the life of Moses is where he is unable to contribute to society and unable to cross the river (read a major test). A similar practice was chronicled by Bronowski "The assent of man" episode 2 of the Bakhtiari nomad peoples' response to their harsh life. We see a lesson for humanity that is ignored. The last cold spell in the early 1800s was solved by technology (invention of fertilizer). Will technology do it again?

The STOE suggest principles of life and for physics should be the same (see https://fqxi.org/community/forum/topic/3032 "Fundamental principles criteria"). For example, life functions with feedback loops (Sarengetti Rules). The STOE suggest feedback is a fundamental method which results in fine tuning of parameters. For example, the CMB temperature is determined by such a loop.

Hodge

    Dear John Hodge,

    I appreciate the opportunity you gave me to hopefully better explain what I meant with "calculate as if the model was identical with the basic reality behind it".

    While I agree: "Singularities or infinities are not physical", I don't see them "indicative of incorrect calculation" but rather of careless assumptions and/or interpretations.

    I don't at all reject imaginary numbers as being unnecessary.

    Is "the distinction between a dot and a point is that there is no distinction"? Well, I blame the mathematicians for ignoring Euclid's indivisible point. They are instead operating with something for which I don't have a better denotation in English language than the word dot. Words like dot-set topology theory and dot-charge sound strange. They make aware of denied mistakes. Also, we should strictly speaking distinguish between absolute endlessness and the admittedly very reasonable relative infinity.

    Again: Calculate as if but be careful.

    Carefulness demands too that I will explain to you later in what I see you wrong.

    Best,

    Eckard

    Best,

    Eckard

    Wonderful. I appreciate you comments where you suggest I'm wrong. It would certainly help the development of my STOE. I would appreciate you views if they are supported by observations/experiments and not be rejected by observations/experiments.

    You may also note that messages may be sent to my RG page where the many papers of the STOE are displayed.

    Hodge

    Stay calm, John Hodge,

    I did not yet at all deal with your STOE, and I see you my ally concerning birth control.

    Nonetheless I will today begin with evidence against your opinion that "the idea of conjugate pairs is an artifact of assuming particles are infinite waves."

    In Fig. 1 of my third FQXi essay I showed frequency vs. elapsed time plots which were calculated by means of COSINE transformation, not the complex-valued Fourier transfotm. The f vs.t hyperbolas exhibit the same uncertainty relation as found by Heisenberg for P vs. q/h. Notice, Planck's constant is just required as to get an dimension-less argument. Other conjugate pairs are omega and t as well as k and r.

    Can you provide a source for your wrong claim or is it your own guess?

    Eckard

    John Hodge,

    In his sentence "The map is not the territory" Alfred Korzybski used the metaphor map as do I according to the meaning in in common sense like in German Landkarte, in French carte géographique, and in Russian geograficeskaja karta: a drawing of an area as it would appear if you saw it from above. Maps show an area's main features or give special information about it.

    When you wrote "Mathematical transforms (maps) you presumably meant something quite different because you suggests that "nature does 'calculate'". By the way, I already briefly reminded in my essay of the huge number of mathematical constructs that are typically fabricated as modifications of Fourier's transform. Most of them seem to be rarely used and rather useless to me. Some of them do indeed not even have inverse or are no transforms at all, e.g. Hilbert transform. Cosine transform in R plus+ is an exception. I am calling CT and R plus the mothers, not the daughters (not just special cases) of R=R plus + R minus and complex Fourier (real + imaginary) transform.

    You called singularities unphysical. I agree. I do however not blame them as "indicative of incorrect calculation". Well, several maps (theories) are most likely just incorrect or at least incomplete. Maybe, there is no theory of everything at all because some basic assumptions contradict to each other. Having just recently near to the end of my life pinpointed a cardinal mistake by Fourier I feel the revelation of very basic inconsistencies hard work.

    Eckard

    Eckard

    I am impressed by your theoretical knowledge. So, i have much to learn from the article. It is a very broad article.

    Thank you for writing on my page also.

    Since you mention Euclid I havre question for you (if you are interested you can answer on my page):

    Is it correct to define parallellism by a point that does not exist?

    With best regards from _________________ John-Erik

      John-Erik,

      First of all, I have to apologize for an embarrassing typo in my essay: Factionalism should read fictionalism.

      I gave a link to David Joyce from whom I got my admittedly very limited knowledge of Euclid's Elements.

      The question you are alluding to is more directly addressed for instance in https://www.jstor.org/stable/2973238?seq=1#page_scan_tab_contents

      I dislike unnecessary academic disputes concerning mathematical existence, Platonism, finitism, etc.

      Instead I suggest calculating with careful critical interpretation as if our models and laws (maps) were the nature (the conjectured real territories). Accordingly:

      - Nature doesn't model (calculate) itself.

      - Bolzano was wrong when he meant that a line of doubled length contains twice as many points.

      - Common sense tells us that Euclid's points are something ideal, quite different from dots.

      - Experience tells us that one must not deny causality even if this is not obvious in laws of nature.

      - Being infinite is also just an ideal feature. Singularities and closed loops are unphysical.

      Eckard

      RE: "Can you provide a source for your wrong claim or is it your own guess?"

      I notice you used wave type equations. Any use of Fourier is a wave use of particle. Bohm Interpretation is a particle can have a specific momentum and position - uncertainty is in the measurement. You Cosine function appears as a subset of Fourier which includes an infinity in its limits. So, any use of Fourier is suggesting an infinite wave.

      I use "map" as in transformation.

      I disagree with two guesses:

      1) "the idea of conjugate pairs is an artifact of assuming particles are infinite waves" and

      2) "there is no difference between point and dot".

      In both cases you are denying the reasonability of equivalent fictions. You are not alone. Already the existence monist Zeno exemplified such mistakes. Consider the half of an area of size one. You may endlessly divide each half into smaller parts (dots) and never arrive at size zero. There is, however, a fictitious end of the endless process, something that is assumed to have no parts: the ideal point as defined by Euclid.

      Although Fourier's analysis deals with waves, not with partial areas, the idea of endless division and calculation as if the fiction was reached is in principle the same. I asked:

      "Can you provide a source for your wrong claim or is it your own guess?"

      You replied: "I notice you used wave type equations. Any use of Fourier is a wave use of particle."

      Fourier analysis (and by the way CT too) decomposes a given function into a sum of partial waves.

      You added: " Bohm Interpretation is a particle can have a specific momentum and position - uncertainty is in the measurement. You Cosine function appears as a subset of Fourier which includes an infinity in its limits. So, any use of Fourier is suggesting an infinite wave."

      Engineers (and perhaps physicists too) may decide from case to case whether it is more appropriate calculating with electromagnetic radiation as if it resembles Huygen's waves or Newton's bullet-like particles and Einstein's mass-less but still body-like photons having measurable p and q.

      While my essay omits the various Bohm ideas, I maintain that Heisenberg's uncertainty is not restricted to p and q measurement but it affects all conjugate pairs including (elapsed) time and frequency.

      I found additionally out that such mathematical uncertainty is not restricted to Fourier transform but it does likewise affect cosine transform.

      Eckard

      I asked you:

      Is it correct to define parallelism by a point that does not exist?

      The link you gave did not work.

      With regards ____________ John-Erik

      John-Eric,

      I immediately found

      https://mathcs.clarku.edu/~djoyce/java/elements/elements.html

      This link is still listed on second place at wikipedia among "other external links". Because I didn't have contact to David Joyce for years, I can only hope he is well.

      What about the "existence" of the fictitious point at infinity, we should agree on that there are no tangible points and lines at all in reality. One may only calculate as if e.g. there were point charges, line currents, and other singularities.

      Albert von Sachsen (1316-1390) was close to this insight when he wrote: a Wooden bar of infinite length has exactly to many points as the whole 3d space.

      Eckard

      Eckard

      The new link did not work either

      In my opinion Euclid was wrong when he used a particle, assumed to be without existence, but nevertheless usable in a definition.

      John-Erik

      John-Eric,

      https://mathcs.clarku.edu/~djoyce/java/elements/elements.html

      does work. Try again or search for Euclid's Elements.

      https://mathcs.clarku.edu/~djoyce/java/elements/elements.html

      Did Euclid really use a fictitious particle for a definition? I rather understand his definition of an ideal point as something that has no parts like the logically consequent fictitious negation of the also fictitious endless divisibility. Atomos means indivisible.

      BTW, Albert von Sachsen was incorrect only in so far as Euclidean ideal points are uncountable. The 3d space has therefore not more points than the bar.

      Eckard

      Sorry,

      I wrote the link just on one line and just once. Somehow it was truncated and doubled. At Clark University, I didn't find Joyce anymore.

      The last link did not work either.

      Since I am very impressed by your article I take me the freedom of asking a very difficult question:

      Was it correct of Euclid to define Parallelism by a point that does not exist?

      Best regards from ________________ John-Erik

      Dear Eckard (if I may),

      thank you for a well-argued essay, there is so much food for thought here!

      It seems we have so many ideas in common, at least regarding the different uses of mathematical languages in physics. I would be thankful if you could take a moment to have a look at my essay, where I argue, among other things, that one could envision an indeterministic physics by rejecting the infinite precision of real numbers (a program of research that I carry out with Nicolas Gisin).

      More of a small historical precisation, the beginning of your essay states the common attribution of "shut up and calculate to Feynman", which however, seems not to be the case. David mermin has an eccellent story on this: https://physicstoday.scitation.org/doi/10.1063/1.1768652

      Anyways, good job, very high rate from my side!

      Best wishes,

      Flavio