Dear Sir,

You have hit the Bull's eye with the naming of your essay. It could not have been described better. The Renaissance mathematicians did not 'liberate' mathematics; they led it astray 'by means of clever restricted constructs' 'to maximally generalize models and to interpret results immediately in an artificial mathematical domain as if they were automatically valid in reality too'. We fully agree with your introductory tenets. In our essay here, we have shown how Russell's paradox of set theory contradicts relativity.

Ancient Indian texts use zero, infinity and negative numbers as interpreted by you. According to them, a number divided by zero remains unchanged (not become infinite). We have discussed this with justification elsewhere. However, imaginary numbers are not mathematics as shown in our essay here. Dr. Schneider has written a beautiful essay here, where he has discussed complex numbers and its effect on relativity. The concept of minus infinity via 0 to positive infinity is a totally wrong concept as shown in our essay. Numbers are a property of all substances by which we differentiate between similars. If there are no similars, it is one. If there are similars, it is many; which can be 2,3....n depending upon the sequence of perception of 'one's. Thus, mathematics cannot be divested from objective reality. Your notion of time is identical to that in our essay.

Newton and Leibniz evolved calculus from charts prepared from the power series, based on the binomial expansion. The binomial expansion is supposed to be an infinite series expansion of a complex differential that approached zero. But this involved the problems of the tangent to the curve and the area of the quadrature. In Lemma VII in Principia, Newton states that at the limit (when the interval between two points goes to zero), the arc, the chord and the tangent are all equal. But if this is true, then both his versine must be zero. In that case, he is talking about a line so that neither the versine equation nor the Pythagorean Theorem applies. Hence it cannot be used in calculus for summing up an area with spatial dimensions.

Newton and Leibniz found the solution to the calculus while studying the "chityuttara" principle of ancient India, which is now called Pascal's differential triangle. To solve the problem of the tangent, this triangle must be made smaller and smaller. We must move from x to ﾎ"x. But can it be mathematically represented? No point on any possible graph can stand for a point in space or an instant in time. A point on a graph stands for two distances from the origin on the two axes. To graph a straight line in space, only one axes is needed - the connecting line. For a point in space, zero axes are needed. Either you perceive it directly without reference to any origin or it is non-existent. Only during measurement, some reference is needed.

In many ways, our essays here complement each other.

Regards,

basudeba

    Dear basudeba,

    You are blaming not just me wrong but also some experts in history of mathematics. I maintain: Renaissance, enlightenment predominantly in non-Catholic countries of Europe, and both application and upcoming scientific exchange LIBERATED mathematics by ignoring the ancient restriction to the forward countable elements of reality. Let me give a perhaps most simple example:

    It is of course possible to describe sound in terms of absolute pressure of air. However, acoustics benefits a lot from what you called "going astray", a mathematical trick. Absolute pressure can be considered as two fictitious components, a constant one and a superimposed one that alternates between positive and negative values.

    Thank you for pointing me to the somewhat strange ancient Indian mathematics and to the essay by Schneider. I will read his and yours.

    Regards,

    Eckard

    Dear Ken Hon Seto,

    My references include meanwhile heretical papers by Euclid and Galileo. Unfortunately you didn't refer to scientific work. Therefore I am not in position to judge your Model Mechanics. My difficulties begin with your claim for absolute time. What do you mean with it?

    I consider the currently elapsed time the natural reference.

    Eckard Blumschein

    Akinbo,

    While I answered your question "Why do you include volume among unphysical mathematical notions" I didn't yet answer the following numbered questions:

    1) Can the Universe perish (e.g. in a Big Crunch) or can the Universe be created from nothing (e.g. in a Big bang)?

    2) If you answer is Yes, does that mean that the Peirce-continuum along with its points also perish?

    My answer is: The universe is by definition just a mental container of anything physical, the whole of space including all the stars, galaxies, and possible even multiverses.

    I abstain from any comment on belonging speculations unless they look simply like obviously nonsensical artifacts. For instance, even H.-D. Zeh distrusted white holes ascribed to a Schwarzschild solution. A God is not trustworthy because he is to similar to hoe humans look.

    When I attributed the definition of a genuine continuum to Peirce, I was aware that it was not a new insight by himself but just formulated the infinite divisibility. Why felt C.S. Peirce overwhelmed by set theory? Stanford Encyclopedia omits a lot. If I recall correctly, Peirce was proud of having found Dedekind's definition of infinity before Dedekind.

    Having quoted: "Peirce says that if a line is cut into two portions, the point at which the cut takes place actually becomes two points..." you are asking:

    "Does this mean that 'the point at which the cut takes place' has two parts? If so, this contradicts the geometric definition."

    It looks as if you are aware of the calamity which I ascribe to Dedekind's notion of number as a pebble.

    Regards,

    Eckard

    Eckard,

    You said you have trust only in the definition of a point as having no extension. In my opinion, this may be one example of "Your Physics Suffers from Unwarranted Interpretations", the title of your essay.

    In my 2013 Essay, I briefly discussed how your definition came about. Let me copy and paste for your convenience the relevant portion, with bold and italics to highlight the important things to note:

    "Divisibility played a key role in apprehending the objects of geometry and arriving at a best definition of them. If all that has magnitude must be divisible, then only a zero magnitude cannot be divided and this will be the fundamental unit of geometry. One other argument attributed to Simplicius, goes as follows. If a body, having 3-dimensions is one dimension away from surfaces, then surfaces will have 2-dimensions. If a line is one dimension away from surfaces, then lines will have 1-dimension. By this logic then, the point would be one dimension away from a line and thus be of 0-dimension [1], p.157.

    Right from the early era the definitions have been contentious. The Platonic view was that things in the physical world are imperfect replicas of things in a perfect realm and should be taken as such. On this basis, it meant nothing for a line to have no breadth, even though nobody had seen such a line in reality... The Aristotelian view on the other hand appeared more inclined to getting descriptions as practical as possible to the reality we can actually behold. Like the Pythagoreans who were of the view that points were extended objects and called them monads to differentiate them from the dimensionless object of the Platonic school, Proclus and Aristotle also felt that points must really exist and had the attribute of position, but they were unclear at what magnitude the point could then be defined as the limit of divisibility.

    Although commonly portrayed as the arrowhead of the view that points were of zero dimension, Plato himself is quoted as somewhat disclaiming this. In Metaphysics, book I, part 9, paragraph 14, Aristotle tells us, "...Plato even used to object to this class of things as being a geometrical fiction". Instead, Plato is said to prefer that points be referred to as the 'beginning of lines' or as 'indivisible lines'. This defense was however denied him as Aristotle counters that if that were so, then the same argument and logic that makes lines exist must then equally prove that points also exist.

    For a fuller account of these arguments, other ancient definitions, criticism by commentators and modern views, see [1], pp.155-157 and [2], pp.76-80. Both also quote Aristotle's Metaphysics and Physics frequently. In summary, the fact as to whether a point is an extended object or a zero dimensional idea has had to be postulated, i.e. has had to be an act of faith and not from evidence. In our thinking, to accommodate the contending views, Euclid restricts himself to a least contentious, middle-of-the-road definition, but not as explicit in its detail, i.e. all are agreed that the fundamental unit of geometry would have no part. A point having no magnitude and of zero dimension would have no part. Likewise, an infinitesimal magnitude not further divisible into parts of itself would also satisfy Euclid's definition 'that of which there is no part'. This ambiguity may however have implications for the foundation of our physical theories, space being all pervasive".

    References

    [1] Heath, T.L., The Thirteen Books of Euclid's Elements. (3 vols.). New York: Dover Publications (1956).

    [2] Proclus, A Commentary on the First Book of Euclid's Elements, translated by Morrow, G.R., Princeton: Princeton University Press (1992).

    So Eckard, since you are following the Platonic interpretation may I ask:

    1. Is your point a geometric fiction or is it real?

    2. If it is real, can something that is of zero dimension exist? If so, how and where? Can it occupy a position?

    3. If it is not real, can something that is not real be a part of what is real? A line is real, at least in 1D (even though my line is 3D). Can an unreal point form a part of a real line which Euclid calls the extremity?

    4. When you divide a line into segments, each of those segments must by Euclid's definition have points as their extremity. From this does your point constitute lines?

    5. Finally, you talk of the continuum. In most cases, the linear continuum consists of an infinity of points. In your exchange with Edwin above, you said "Akinbo Ojo suggests splitting pebble-like non-zero dimensional numbers". No, I didn't say so. A non-zero dimensional point cannot be split. It satisfies Euclid's definition and cannot be split into parts. It is the smallest indivisible extension. That is why, Dedekind, Peirce and others have been worrying: "how then can we cut a line, if it consists of points, finite or infinite, since at every incidence for cutting a pint is located therein. My hypothesis suggests a solution. I never knew of Peirce till I read your essay. Thanks. I will read Giovanni Prisinzano views after this.

    Best regards and thanks for the conversation

    Akinbo

    Eckard,

    I just read Giovanni Prisinzano's essay that you recommended in a comment. I made some comments but unfortunately Giovanni may not be able to reply yet due to some problems.

    Regards,

    Akinbo

    Akinbo,

    I am sure; instead of '"Your Physics Suffers' you meant 'your "Physics Suffers'. Yes, I consider your interpretation of Euclid's definition (a point is something that has no parts) logically unwarranted although it can indeed bee seen as compromising with those who objected to Anaxagoras and suggested a-tomos parts.

    Leukippos, Demokrit, Epikur, and Lukrez did not yet distinguish between reality and mathematical models. They were correct concerning the limited divisibility of real objects but wrong concerning mathematical atoms. Any introduction of a smallest or a largest number contradicts the axiom of infinity which expresses irrefutable insights of Archimedes and Aristotle. Mathematics doesn't obey reality; it isn't an empirical science. Therefore there is no experimental evidence for the attribution of zero-dimensionality to a point.

    The Pythagoreans were proved wrong already in ancient time:

    There are definitely incommensurable ratios, and points cannot reasonably be thought as extended objects. They are dimensionless Platonic objects.

    Your blown up point reminds me of the extended reals that are thought to include infinity.

    To me, point, line, area, and volume altogether are ideal mathematical, not physically existing objects, and a n-dimensional continuum is not at all composed of (n-1)-dimensional elements. I see this not a matter of faith but of reasoning. In mathematics, the distinction between existing and non-existing might be difficult and futile. Comprehensive self-consistency is more important. This might answer your questions 1-3.

    4. Infinitesimals between two borders each are not zero-dimensional and can therefore constitute a larger continuum. They can however not be split into points.

    5: "A non-zero dimensional point cannot be split.It satisfies Euclid's definition and cannot be split into parts."

    This is your strange dogma.

    "It is the smallest indivisible extension."

    No. It has no extension at all.

    ""how then can we cut a line, if it consists of points, finite or infinite, since at every incidence for cutting a pint is located therein. My hypothesis suggests a solution."

    I don't understand what you meant with a pint (of beer?), perhaps a point. Anyway, I strongly disagree. Are there really finite or infinite points?

    Points cannot be cut. My solution to the conundrum isn't based on Dedekind's pebble-like numbers but on measures instead. In other words, I too don't use 0-dimensional and 1-dimensional perspectives at a time. Nonetheless, I consider points zero-dimensional.

    Kind regards,

    Eckard

    Your outcome sourcing fines definite improving notions which attracted conventional sources of measuring.

    With best regards,

    Miss. Sujatha Jagannathan

      Hi Eckard,

      Excellent, in-depth analytical essays in the spirit of hard Cartesian doubt. You talk about "unwarranted interpretations". Romanovskaya T.B. in [link:www.philosophy2.ru/iphras/library/physics.html#73 ] Sovremennaya fizika i sovremennoye iskusstvo - paralleli stilya// Fizika v sisteme kultury [Modern physics and contemporary art - parallels of style // Physics in the culture system][/link]. The author speaks about "crisis of interpretation and representation" in fundamental physics. Morris Cline says that "mathematics loss of certainty". The problem of the foundation of mathematics (better - justification or basification) for over a hundred years ... What are your ideas on a single foundation of "fundamental knowledge"? What interpretations "warranted"? What is "justified" basis of physics?...

      "Truth should be drawn ..." A.Zenkin "SCIENTIFIC COUNTER-REVOLUTION IN MATHEMATICS".

      Kind regards,

      Vladimir

        Miss. Sujatha Jagannathan,

        No matter how hard I am trying to understand your sentence, I have no clue what you meant. I am just a German. My dictionary tells me:

        If you are fined, you are punished by being ordered to pay a fine.

        A source is something from which something emerges. What is sourcing?

        An outcome is the result of a process or an action.

        What are definite notions?

        Eckard Blumschein

        Hi Vladimir,

        Dedekind and Georg Cantor were friends, at least for a while. I suspect a possible reason to Hilbert for popularizing Cantor instead of Dedekind to be found in Hilbert's admiration for stunning ideas. More than infinite, what a wonderful silly idea!

        I quoted two authors besides Zenkin who shed light into the matter: Mückenheim and Spalt.

        Meanwhile I am sure: The belonging key question relates already to Dedekind's changed notion of number. I see it unwarranted to abandon Euclid's definitions. My arguments arose from obvious inability of a professor of mathematics to explain logical inconsistencies. He pointed me to Weierstrass.

        My primary concern was a strict separation between past and future. Read Phipps' essay in order to see from a quite different side that spacetime is indeed not warranted but IR+ is valuable.

        When I dealt with IR+, I came the the history of negative and imaginary numbers.

        My message is: Interpretations directly in complex plane is not necessarily warranted.

        Kind regards,

        Eckard

        Thank you very much, Eckard! My highest appraisal and another "eternal question" which I ask all mathematics and physics. John Archibald Wheeler left to physicists and mathematicians a good philosophical precept: "Philosophy is too important to be left to the philosophers".When physicists and mathematicians speak about the structure and the laws of Universum for some reason they forget about lyricists that the majority on Mother Earth. I believe that the scientific picture of the world should be the same rich senses of the "LifeWorld» (E.Husserl), as a picture of the world lyricists , poets and philosophers:

        We do not see the world in detail,

        Everything is insignificant and fractional ...

        Sadness takes me from all this.( Alexander Vvedensky,1930)

        It is by a mathematical point only that we are wise,

        as the sailor or the fugitive slave keeps the polestar in his eye;

        but that is sufficient guidance for all our life.

        We may not arrive at our port within a calculable period,

        but we would preserve the true course. (Henry David Thoreau, 1854)

        Do you agree with Henry David Thoreau?

        Kind regards,

        Vladimir

        Yo put the same question to Tom Phipps. I largely agree with his answer with a few caveats.

        Quote: "I do not have any particular religion of my own. That would partly close my mind, which I prefer to keep open. I do, however, have a sort of frankly irrational suspicion -- which is akin to faith -- that when we understand the fundamental ways in which nature works we shall be far more stunned, shocked, amazed than even the lyricists, poets, etc. have it in their power to imagine." Unquote.

        Reasoning tells me that irrational analogy between e.g. Nemzov and Jean Jaurès is hopefully unwarranted.

        Does nature "work" at all in the same sense as do humans? I would rather postulate causality.

        At least I don't feel stunned, shocked or amazed by getting aware that primitive amimals/humans/religions tend to behave irrational as do rabbits, as if unlimited growth was feasible.

        When Alfred Nobel did love poetry, this attitude helped him to envision the only rational road to peace.

        Because, I did never before hear of Little Jack Horner and Henry David Thoreaut, I prefer more directly understandable arguments in a scientific discussion among unspecialized participants.

        Kind regards, Eckard

        Dear Eckard,

        I ask this question, because a modern physical picture of the world very poor in meanings, it semantic incomplete and without ontologic justification. The solution of fundamental problems of physics and mathematics and need of new heuristics demand deep judgment of the philosophical foundations of these two fundamental sign systems. I hope that you will read also my essay to conduct subject discussion on the philosophical bases of physics and mathematics.

        Yours faithfully,

        Vladimir

          Dear Vladimir,

          Your essay guided me to an "Ontology of Mathematical Discourse" by G Gunter in Russian language. Chapter 2 deals with "Interpretation of Existence in Mathematics" and lists concepts by G. Cantor, by Brauer, and by Hilbert. Cantor mentioned the TND. As usual, I cannot accept his ideas, and I prefer reading at best his original papers in German which is difficult enough for me.

          I looked in vain for Brauer in the bibliography. Perhaps Richard Brauer (1901-1977) is meant. Can you please confirm this?

          Yours faithfully,

          Eckard

          Dear Eckard,

          At the bottom of the article three links Brouwer L.E.J.

          65. Brouwer L.E.J. On the foundations of Mathematics // Collected Works. V.1. Philosophy and Foundations of Mathematics. Amsterdam - Oxford - New York, 1975, p.11-101

          66. Brouwer L.E.J. Guidelines of Intuitionistic Mathematics // Ibid., P. 477-507

          67. Brouwer L.E.J. Historical Background, Principles and Methods of Intuitionism // Ibid., P.508-515

          Kind regards,

          Vladimir

          Dear Vladimir,

          Thank you for revealing my mistake. When I read "Brauer" I was mislead. Brauer is a frequent German name and means brewer (of beer). Brouwer means the same in Dutch and is indeed pronounced almost like Brauer. If I didn't skip reading the text then I should quickly have realized that Brauer meant Brouwer. While I wondered how Richard Brauer relates to the matter, Brouwer makes more sense.

          Kind regards,

          Eckard

          5 days later

          Dear Mr. Blumschein

          I read with interest your essay and like a few remarks such as "Moreover, they use Heaviside's trick which tempts to unwarrantedly interpret results of complex calculations". This is right. He, not Maxwell coined the "Maxwell Equations" with a wrong Ampere's law and a nonexistent "displacement current"

          Another good part is "Leibniz and Newton merely agreed on that acceleration is an absolute quality. Let's show Newton's mistake with the metaphor of an unlimited to both sides box [14]. Only if there is a preferred point of reference, it is possible to attribute a position to it. In space, such point is usually missing." However, I think you are helping the relativists defending their ideology. Newton was perfect in insisting on absolute velocities with reference to space. All astronomers are measuring peculiar motions of stars and galaxies. And we know that we are travelling through space with an absolute velocity of 371000 m/s towards the Virgo cluster. With the CMB zero this discussion is finally closed and relativity is dead.

          You also warn about the mindless use of singularities in math. But you fail to mention that these singularities created by illegal divide by zero operations in Levy-Civita's tensor math have finally led to monstrosities like the big bang and black holes. These are purely mathematical constructs and misled physicists and a wide public to believe in such singular objects. They even claim to be able to imagine such singularities in space and time. Here you mathematicians have strong duty to warn urgently. Nature hates singularities; beware of them!

          All the best for your future work

          Lutz

            Dear Professor Kayser,

            While I appreciate confirming comments like yours, I am even more grateful for frankly uttered criticism. Maybe, you mistook me "helping the relativists defending their ideology." I neither intended to do so nor do I agree with your lecturing: "Newton was perfect in insisting on absolute velocities with reference to space."

            I am distinguishing by capitalizing the ideology Relativity from relativity as understood by Galileo. The latter did also not refer to absolute space. Given Virgo cluster and CMBR were absolutely at rest, do you imagine velocities of light toward and away from this reference differing accordingly?

            In endnotes of an earlier essay of mine I explained my interpretation of constant velocity: Distance between the location of emission at the moment of emission and the location of arrival at the moment of arrival divided by the time from emission to arrival. Even if the locations may change, this distance does neither depend on the movement of light with respect to a hypothetical medium attributed to space nor on the movement of its emitter or receiver wrt a reference location. Wasn't Newton wrong?

            I didn't say that the use of singularities in mathematics is mindless. I merely criticize unwarranted interpretations in terms of reality. And yes, I omitted hints to further belonging implications. Average experts have already to swallow a lot and may then get proud of revealing some implications for their own.

            We certainly agree with Thomas Phipps on that the wrong expectation and interpretation of Michelson's Potsdam/Cleveland experiment led away from the correct Maxwell-Hertz equations to the corrected (in the sense of spoiled) Maxwell's equations. Perhaps you are right in that Heaviside is to blame for the "correction". I just vaguely recall that Gibbs arrived at the same equations.

            When I referred to Heaviside's trick I dealt with a different matter which I consider to have most serious consequences too.

            Not even you seem to accept what I meant when I criticized Dedekind's mutilation of the notion number and belonging "not even wrong" interpretations.

            All the best,

            Eckard

            Dear Lutz,

            You asked: "Who has invented symmetrical relativity the first time? Was it Poincare?"

            In my previous essay "Peace via Discoveries and Inventions", I expressed between the lines that I see both Henry and Raymond Poincaré tragic historic figures. Science and mankind, respectively, payed high prices for their successes.

            Henry P. admired Lorentz and did not just coin the term Lorentz transformation. He also introduced the notion relativity and suggested x^2+y^2+z^2+(ict)^2 already in 1904, i.e. before Einstein's Relativity and Minkowski's spacetime.

            We agree at least with Phipps on that unwarranted expectation and interpretation of Michelson's 1881/87 experiment implied Heaviside's variant of Maxwell's equations. The same misinterpretation caused FitzGerald and Lorentz to fabricate length contraction.

            Looking for origins of the abstruse symmetry between past and future time, I delved into some much earlier made mistakes concerning how mathematics was founded and interpreted. So far I am not in position to at least skim through all essays. I merely got aware of Akinbo Ojo and Giovanni Prisinzano who dealt with related enigma.

            Thank you for the request.

            Best,

            Eckard