Essay Abstract

Do you believe that science is based on contradictions? Let me consider the common experience of seeing a dot of the pencil on a paper. If I call the dot ``zero length dimension'' or ``zero extension'', then certainly I have seen `nothing'. But, if I have seen `nothing', I wonder how I can refer to `nothing', let alone naming `nothing' as ``a point''. Therefore, the expression ``zero length dimension'' is a contradiction. Mathematical science, as of now, is based on this contradiction that results from the attitude of exactness, because exact ``zero'' is non-referable and inexpressible. Such attitude leads to incomplete statements like ``infinitesimal quantities'' which never mention ``with respect to'' what. Consequently, as I find, science becomes fraught with singularity and the definition of `field' seems either circular or incomplete in reasoning. I avoid this contradiction, by accepting my inability to do exact science. Therefore, I consider the dot as of ``negligible length dimension''. It is a practical statement rather than a sacrosanct axiom. The practicality serves the purpose of drawing geometry, that becomes impossible if I decide or choose to look at the dot through a magnifying glass. It then answers a different practical question, namely, what the dot is made up of. Certainly, reality of the dot depends on how I choose or decide to observe it. This is the essence of ``relational existence''. On the contrary, modern mathematical science is founded upon belief of ``independent existence''(invariant). My belief in inexact mathematical science and relational existence needs the introduction of an undecidable length unit to do arithmetic and leads to non-singular gravity. Further, the quest for justification of my choice or decision leads to my incompleteness -- ``I''-- the undecidable premise beyond science, the expression of which is a (useful) contradiction in itself as ``I'' is inexpressible.

Author Bio

I got a degree called ``Doctor of Philosophy'' in Science without understanding Philosophy of Science and used to chase a theory of ``everything'' without understanding the subtlety of a ``thing'' and the underlying contradictions of exact mathematical science. Now, I try to understand the value of philosophy and reasoning in the foundations of mathematical science. I am on my own and love to be on my own.

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This is an excellent essay, especially in terms of using a minimal number of generic antonyms like "never" (see below). Similar to neurosurgeon Christof Koch's writing style before his book "The Feeling of Life Itself", Abhishek's lengthy appeal sto biography are somewhat thermo-dramatic but "on point" here:

"However, now I find no shame in confessing that I have never really understood the definition of derivative in standard calculus [2], although I applied such tools, quite comfortably, in physics so as to build a career in a discipline of science, based on such acceptance and belief. "

When the essay dials in on "points" and "pointing out" it seems to favor classical number lines (for the sake of complexity?); I believe this could be edited by appealing to predictions and observations (like doctors without borders): "for the sake of non-fiction":

"I may emphasize that I am considering only positive numbers here for the sake of simplicity"

    Dear Abhishek Majhi,

    enjoyed reading your essay! I would, however, opt for a superposition of your father's and mother's advices as follows.

    Take, for example, Newton's

    F=ma

    On the left hand side we have a force, already known to club-swinging Neanderthal man, i.e. a phenomenon. At the same time F is a vector having magnitude and direction. On the right hand side we have two virtual or ideal quantities, mass and acelleration, not represented in natural language and thus acting as 'enablers' or 'jokers' warranting that the mathematical result of the equation does't CONTRADICT the associated phenomenon, with which it obviously cannot be identical. My point is this: math is very precise - provided it has been arrived at in the presence of phenomena. Whether F=ma holds near black holes or at the Planck scale - how could we know? The return path from math to the phenomena is blocked by the NOT-FALSE relation between math and the phenomena. Blindly following theory (here:earth sphericity) made already Colombo falsely believe he had arrived in India...

    In sharp contrast to Newton (randomly picked example):

    |ОЁSвџ© = c[{sj}]|{sj}вџ©

    Does it speak of the efficiency of angels' wings in vacuum?

    good luck for essay,

    Heinz

      Thank you Daniel for reading my essay in the first place and for the compliments. However, I would like to clarify that a vital message of this essay is the inexactness of practical expressions of the ideal thoughts of pure mathematics. Please note that ``number line'' is the geometric representation of the thought of real numbers. While the thought is ideal and arbitrarily accurate, the expression is not. When you draw a line with your pencil, your expression is only accurate up to the extension of your pencil tip. But, only with that inaccuracy of the object of expression that you can express. Same is the case for measurement or comparison e.g. if you do not see the cross-wire of the eye piece you can not make measurements in a physics lab, although the thickness of the cross-wire, that lets you see, itself serves as the basic irremovable error -- the immeasurable lets you measure. If you do not accept this limitation of the accuracy in the expression (or measurement), you can not express (or measure). The same is the case for language as well -- you and me can communicate only up to the limitation of our knowledge in English -- if you use a word that I do not understand, then communication fails. And, without expression you can not use your thought in for practical purpose e.g. if I have not written this essay to express my thoughts, you would not be reading this. This, interestingly leads to the other crucial issue i.e. relational existence. This essay does not exist if you do not read it, it does not have a value unless you find a value in it. Even if you find value in it, it is subject to the way you interpret it or relate to my expressions of thought, where the premises are English language, mathematics and symbols.

      So, I may write that, if I have favored some viewpoint in this essay, it is practical reasoning that incorporates inaccuracy of expressions and inexact measurement. This is why I have talked about `practical' numbers. Have you thought what or how you count when you say `I have five fingers'. I see immense confusion in this statement if I want give formal reasoning and try to do exact mathematics, because each one of my fingers are different from each other (e.g. by virtue lengths, breadths, fingerprints, wrinkles on the skin, etc.). It seems to me that I am adding like

      [math] $1_a+1_b+1_c+1_d+1_e.$ [/math]

      But I do not know how to make sense of this. For me, in practice, I use the fact that the fingers are both identical and different (contradiction!) according to need. I perceive of the fingers as different, but express their counting as being identical so as to do arithmetic (sum) by disregarding information on purpose e.g. something like

      [math]$1+\delta_a+1+\delta_b+1+\delta_c+1+\delta_d+1+\delta_e=5+\delta~\ni \delta=\sum_{i=a}^e\delta_i$[/math]

      where $delta$ is simply ignored for practical purpose. Looking into those different $delta$-s serves a different practical purpose i.e. if one chooses or decides to ask what does a finger look like and starts investigating what is the meaning of a finger.

      In a nutshell, the aim of this essay is to convey the role of contradictions in practice and in mathematical science (e.g. how the issue of seeing a point leads to some interesting consequences from standard geometric calculus that is plagued with incomplete statements made by Cauchy).

      Dear Heinz,

      I am glad that you have enjoyed my essay. However, I am not capable of understanding in what way you superposed my parents' views. Rather, it seems that it is my misfortune or incapability that I have not been able to convey my message through my writing. I think so because of the following reasons.

      While you write F=ma, that involves a definition of acceleration in terms of standard derivative. I shall be glad to know if you have enjoyed reading my critiques of calculus. Because, I have discussed in what way standard calculus, as applied in physics (where units are involved), is based on incomplete statements. The word ``vector'' that you have used is explained on the basis of such incomplete statements. The word ``black hole'' that you have used is also explained with a theory (differential geometry) that is based on such calculus.

      By the way, if you have enjoyed reading the later section on non-singular gravity, then you should have seen that the so called singularity problem has been resolved by showing that gravity is asymptotically safe. The math is simple -- just series expansions. The picture that was snapped last year, was that of the asymptotic safe region where motion could not be realized giving the appearance of black.

      Thanks a lot for reading my essay.

      Regards

      Abhishek

      7 days later

      Dear Abhishek,

      A very interesting and deep essay in the spirit of the Cartesian doubt. It pushes thinking toward the centenary problem No. 1 for fundamental science and cognition as a whole - the justifcation (substantification, "foundations") of mathematics, which can more accurately be defined as the problem of the ontological basification of mathematics (knowledge). Overcoming the crisis of understanding in the philosophical basis of fundamental science (mathematics, physics, cosmology) is possible only on the basis of breakthrough ontological and dialectical ideas, first of all developing the ideas of Kuzansky - "coincidence of opposites", as well as ideas of the process metaphysics of Whitehead. One of the main reasons for "trouble with physics" (Lee Smolin "Trouble with Physics)", "loss of certainty" in mathematics (Morris Kline "Mathematics: Loss of Certainty") is the unsolved problem of the "foundations of mathematics" (justification + substantification = basification). As the doctor of physical and mathematical sciences, Alexander Zenkin, noted in SCIENTIFIC COUNTER-REVOLUTION IN MATHEMATICS: "the truth should be drawn ...". And I completely agree with him.

      Leopold Kronecker: "God made the integers, all else is the work of man" / Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk». ...If it is good to observe Nature, for example, how the vineyard sprouts from grape seed, then a slightly different conclusion can be made: God created the Universum only from the Absolute Forms of matter. Numbers are the work of man... Please also see my ideas on the problem of the philosophical basis of knowledge (framework, carcass, foundation).

      I wish you success, Vladimir

        Dear Vladimir,

        聽 聽 聽 聽Thank you very much for reading my essay and providing some valuable feedback. I am glad that you found the essay ``very interesting'' and ``deep''. However, the essence goes beyond ``Cartesian doubt''. A bit of reading Descartes gave me the impression that he used to consider `extensions' as the essence of existence. Although, I agree to some extent with Descartes, however, I do not know whether he considered ``relational existence'' and actually found a way to do mathematical science based on such philosophy (my view is akin to certain aspects of Indian philosophy, see end note).聽 Furthermore, unlike Descartes, I have not only raised ``doubts'', but I have actually shown how to clear those doubts in terms of mathematical expressions leading to fruitful results i.e. philosophy having a practical impact. I have never read the books of the people you have mentioned,聽but thanks for mentioning them anyways. However, the words聽 ``coincidence of the opposites'' looks very much like what I have called ``middle-way'' (borrowed from Nagarjuna, see end note). 聽 As far as the word ``truth'' is concerned, for me its ``relational'' and not ``absolute'' (Sunyata, see end note). As far as pure mathematical thoughts like numbers and geometry are concerned, I consider them to be ideal realizations of the mind which can only be used in practice with approximations.聽 Considering all these views, I may point out that this essay is a practical essay which does not stop at some philosophical discussion but shows how to apply philosophical thoughts through mathematics to solve, or at least lay the ground to solve, one of the hardest problems in physics called asymptotic safety, which I strongly believe is related to the problem of confinement. It is unfortunate for me that the simple (high school level) mathematics that I have shown towards the end of the essay, does not appear to the reader appreciable.聽聽As far as your ideas are concerned, I shall try to have a look at it if I can understand with my poor knowledge of mainstream philosophy.

        Regards

        Abhishek

        6 days later

        Dear Dr. Majhi,

        Thank you very much for the writing an essay on very foundational-level. It is very interesting that you bring the approximation in measurement (not "exactness") at very beginning which we do in practice. However, I have a different view regarding the use of the word "contradiction". If you consider the approximation at very beginning, you always might not get back the same results from considering "exactness".

        Good luck for essay,

        Gopal Sardar

          Abhishek Majhi re-uploaded the file Majhi_FQXIESS.pdf for the essay entitled "Contradictions, mathematical science and incompleteness" on 2020-04-24 05:20:08 UTC.

          Dear Gopal

          Thank you very much for making this comment. I fully agree with you, except that I want to replace ``always might not get back'' with ``never get back without making a fatal contradiction''.

          Thanks for the wishes.

          Regards

          Abhishek

          10 days later
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