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  • Post #143

CHRIS ISHAM'S VERY INTERESTING TOPOS-THEORY MATHEMATIZATION OF PHYSICS MAY HAVE ORIGINATED WHEN HE DROVE EDWARD SIEGEL[J. NONCRYSTALINE SOLIDS 40, 453(1980)], ONCE AT QUEEN MARY COLLEGE/UNIVERSITY OF LONDON AND INPE/CNPq, SAO JOSE DOS CAMPOS, BRAZIL, FROM IMPERIAL COLLEGE TO A CONFERENCE AT OXFORD CIRCA 1990 AND SEEMS TO FOLLOW ON EDWARD MACKINNON'S CALL FOR AND JACK AND IAN STEWART'S[THE COLLAPSE OF CHAOS: DISCOVERING SIMPLICITY IN A "COMPLEX" WORLD, PENGUIN(1994)] CALL FOR WITHOUT IMPLIMENTATION OF "COMPLI-CITY"(BOTTOM UP MANY TO FEW INDUCTION) AND "SIMPLE-XITY"(TOP-DOWM FEW TO MANY DEDUCTION) BOTH SIMULTANEOUSLY AUTOMATICALLY VIA EDWARD SIEGEL'S[SYMPOSIUM ON FRACTALS..., MRS FALL MEETING, BOSTON(1989)-5-SEMINAL-PAPERS!!!-ATTACHED HEREIN] FUZZYICS=CATEGORYICS=PRAGMATYICS("Son of 'TRIZ'")/CATEGORY-SEMANTICS COGNITION(AKA SYNERGETICS PARADIGM AND DICHOTOMY) ANALYTICS DISCOVERY, IN AND UNIFYING PHYSICS TO "PURE"-MATHEMATICS, OF ARISTOTLE'S "SQUARE-OF-OPPOSITION" WITH PLATO'S "FORMS" AND HEGEL'S "BECOMING" BY A SYNTHESIS OF WILLIAM LAWVERE CATEGORY-THEORY/TOPOI WITH KARL MENGER[DIMENSIONTHEORIE, TEUBNER (1929] WITH WIERZBICKA-LANGACKER-LAKOFF COGNITIVE-SEMANTICS WITH PAWLAK-ZIARKO-SLOWINSKI ROUGH-SETS THEORY.

SIEGEL'S RESULT IS A (2+1)-DIMENSIONAL MATRIX(LOOKING LIKE A TIC-TAC-TOE DIAGRAM) OF SYNONYMS/FUNCTORS VERSUS ANTONYMS/MORPHISMS VERSUS ANALOGY/METAPHOR.

AMAZINGLY NEUROIMAGING SCIENTIST JAN WEDEEN(HARVARD MEDICAL SCHOOL) RECENTLY DISCOVERED THIS EXACT STRUCTURE IN HUMAN BRAIN IMAGING!!!

SIEGEL'S FUZZYICS=CATEGORYICS=PRAGMATYICS("Son of 'TRIZ'")/CATEGORY-SEMANTICS COGNITION(AKA SYNERGETICS PARADIGM AND DICHOTOMY)ANALYTICS, BY PARSIMONY ELIMINATION VIA DISAMBIGUATION OF JARGONIAL-OBFUSCATION("LOTSS FANCY SHMANCY LINGO TO SNOW THE RUBES")THUS PROGRESSES UP THE STIKELEATHER "HIERARCHY-OF-THINKING"(HoT)WHERE EACH LOGIC-LEVEL IS META VIA "MINING" THE NEXT-LOWEST LOGIC-LEVEL:

DATA -> INFORMATION -> KNOWLEDGE -> UNDERSTANDING -> MEANING -> WISDOM -> INSPIRATION -> "SPIRITUALITY" -> UNIVERSALITY -> SIMPLICITY ->??? EVOLVING TO VIA PHILOSOPHY OF PARSIMONY TO "K.I.S.S.(KEEP IT SIMPLE STUPIDS)!!!Attachment #1: 2_FULL_PAPER_COMPLEX_QUANTUM-STATISTICS_IN_FRACTAL-DIMENSIONS.pdfAttachment #2: STRATIFIED_FUZZYICS__HoT_HIERARCHY-of-THINKING_from_ASKING_ORTHOGONAL_QUESTIONS_WHAT_WHEN__WHERE__WHY.doc

9 months later

Tom,

Do you agree with Feynman? I know your great love for the mathematical...

"Physics is not mathematics, and mathematics is not physics. One helps the other. But in physics you have to have an understanding of the connection of words with the real world. It is necessary at the end to translate what you have figured out into English, into the world, into the blocks of copper and glass that you are going to do the experiments with. Only in that way can you find out whether the consequences are true. This is a problem which is not a problem of mathematics at all." (Feynman, 1965, p. 49).

"Mathematicians are only dealing with the structure of reasoning and they do not

really care what they are talking about. They do not even need to know what they

are talking about, or, as they themselves say, whether what they say is true." (Id.)

"In other words, mathematicians prepare abstract reasoning ready to be used if

you have a set of axioms about the real world. But the physicist has meaning to

all his phrases." (Id.)

"Mathematicians like to make their reasoning as general as possible,...[whereas]

the physicist is always interested in the special case." (Feynman, 1950, p. 50)

"[T]he poor mathematician translates [the special case] into equations, and as the symbols do not mean anything to him he has no guide but precise mathematical

rigour and care in the argument." (Id.)

Therefore, it is up to the physicist to narrow the scope of the problem and define what is required of the mathematician in rather specific terms.

Strangely enough, Feynman concluded his discussions on the relation of mathematics to physics with the following observation: "the mathematical rigour of great precision is not very useful in physics." (Id., pp. 50 - 51) The reason is that great precision can dampen or limit the intuition and creative imagination of the physicist, which he needs in order to modify his original ideas or guess at new solutions. (Id., p.51) An approximate mathematical conclusion is often more helpful.

Regards,

Akinbo

    Hi Akinbo,

    Sure, I agree. I have said repeatedly that unless it can be shown that a mathematical model is independent of its physical result (i.e., that meaning is independent of language), there can be no demonstrated correspondence between the theory and the physics, and hence no rational -- no truly objective -- theory.

    The main way in which special and general relativity differs from quantum mechanics is that relativity is written in mathematical language that allows exact solutions to the equations that incorporate the theory (this is true of all classical physics). The reason that quantum theory is subject to the many interpretations that we hear, is that it is not based in equations that incorporate the theory, it is based in the phenomenology of quantum mechanics -- as Feynman said, it's all about explaining "the experiment with the two holes."

    Quantum theorists from Bohr, Feynman, Bell and to the present day, have struggled to develop from scratch a quantum foundations theory independent of the experiment; they have not been successful. The tendency among theorists now is to simply accept that nature does not obey a strict mathematical structure, that events are random at foundation.

    Just to add, the reason that I don't often respond any more to Pentcho or to Peter J, is that they have a conviction that relativity can -- like quantum mechanics -- be "interpreted" starting with physical results rather than from what the theory says. Nothing could be further from the truth, and such discussions go nowhere. Rational theories are true by correspondence, not by interpretation.

    Best,

    Tom

    Thanks Tom. In your short reply 'truth' appears at least thrice. Now are there two truths, one mathematical and one physical? If there are two types of truth, which one are you looking for?

    Mathematically, if something arrives later than it used to under a given condition, it can be interpreted that clocks run slower or time is dilated OR the speed of propagation is reduced/ affected. All are mathematically equivalent.

    But in physics, they may not be all equivalent. Some of the mathematically correct choice of interpretation may lead to riddles, paradoxes and absurdities. In that case, the philosopher is likely to prefer the interpretation free of these. What is mathematically false, cannot be physically correct. What is physically false, cannot be mathematically correct.

    Your point is however important. As someone said, you serve as a Quality Controller.

    Regards,

    Akinbo

    *Regarding my statement in italics, a line having length and of zero width is physically impossible, same with a surface of zero thickness. Mathematicians should therefore take a look at this conflict.

    I used the word truth only once, Akinbo, and not in the context you ascribe to it. My usage of "truly" means "really," and the use of "true statement" means "logically coherent." In the final case, my use of the word truth means "facts." I am not concerned with truth as you use it -- my concern is for rational, objective knowledge.

    You write: "Mathematically, if something arrives later than it used to under a given condition, it can be interpreted that clocks run slower or time is dilated OR the speed of propagation is reduced/ affected. All are mathematically equivalent."

    No. You are assuming an ideal clock that doesn't exist.

    "What is mathematically false, cannot be physically correct. What is physically false, cannot be mathematically correct."

    Nothing is ever physically false. There are only physically false interpretations of phenomena. That is why a mathematically complete theory that corresponds with the physical phenomena it incorporates and describes, is not open to interpretation.

    "*Regarding my statement in italics, a line having length and of zero width is physically impossible, same with a surface of zero thickness. Mathematicians should therefore take a look at this conflict."

    There is no conflict. The ideal line is a metric of 1 dimension, and the ideal surface is a plane of 2 dimensions. Your confusion results from thinking that because we live in 3 dimensions we cannot describe existence in any other terms. Language itself, however, is dimensionless.

    Tom, the title of this blog is Why Mathematics is an Illusion instead of Why Mathematics is NOT an Illusion as you may prefer. I understand your sentiments but you hide much with the use of words. Should I say semantics.

    Your confusion results from thinking that because we live in 3 dimensions we cannot describe existence in any other terms.

    Of course, we can. Jason has been describing existence with ghosts, spirits, etc. While you and others may enjoy looking for truth and describing how you can exist in a 2- or 10- dimensional world, many who are not confused believe we can apprehend truth in the 3-dimensional world which in your own words you admit we live in. So when you say, "The ideal line is a metric of 1 dimension, and the ideal surface is a plane of 2 dimensions", what does 'ideal' mean? Does ideal exist? Can you show me a physical surface of zero thickness? Can multiple surfaces of zero thickness constitute a body of 3-dimensions? Where did the thickness of the body come from, or can multiple of zero thickness give rise to a real measurable thickness? Let's talk about the REAL world.

    Regards,

    Akinbo

    The real world is described by mathematics, which is independent of the real world.

    In principle, all mathematics can be translated into natural language. Therefore, there is no difference (except in quality and precision) between one describing the real world in the English we are using, and describing it in mathematical language.

    Tom,

    "The real world is described by mathematics, which is independent of the real world". What does this sentence mean?

    Perhaps, you meant to say: The ideal (platonic realm) world is described by mathematics, which is independent of the real world.

    Regards,

    Akinbo

    Tom,

    Are quality and precision not rather important in physics? I suggest also that mathematics often gives us no insight into the physical mechanisms of nature. That means it can hopelessly mislead those who believe it does. It is then a false god. The one most in QM presently pray to.

    Best wishes

    Peter

    "Perhaps, you meant to say: The ideal (platonic realm) world is described by mathematics, which is independent of the real world."

    I meant what I have said many times: that meaning is independent of language. The correspondence between physical events and their mathematical description tells us whether or not the language describes anything physical.

    The reason that multiple "interpretations" exist in QM, is rather different than Tom's statement that:

    "The reason that quantum theory is subject to the many interpretations that we hear, is that it is not based in equations that incorporate the theory, it is based in the phenomenology of quantum mechanics ... Rational theories are true by correspondence, not by interpretation."

    The reason is that mathematical identities, may imply different physical manifestations. If the only thing that can be observed, is the end result, R, then it is not possible, even in principle, to decide which manifestation "corresponds" to reality.

    a(b+c) = R = ab + ac, is an identity.

    But one side of the identity implies the existence of one multiplier while the other implies the existence of two. If the only thing that can ever be known, is R, then one can never know whether one, or two, multipliers, exist in the "black box" of reality.

    Wave-functions are like multipliers; they can only be observed "within" the "mathematical identities" of QM, not the final result. Hence all the interpretations pertaining to the nature of wave-functions etc.

    Rob McEachern

    Peter, the problem with the quantum mathematical model is that it is not demonstrably based in first principles. It is "lawless" so to speak. Classical theories like Newton's and Einstein's can be shown complete -- i.e., one can simply imagine a mathematical structure, without any knowledge of physical phenomena, that behaves in a certain way according to a certain set of variables, and then test that described behavior against real world effects to determine whether the mathematical description is identical to the way the physical world actually behaves.

    This is why Newton's 'hypotheses non fingo' is so important to the philosophy of physics, and to science in general. The story of Newton's apple is probably apocryphal -- and misleading -- because it assumes no prior knowledge of how things move in space. Newton's actual thinking is abstract, based on the earlier work of Galileo and Kepler, in which we know both mathematically and experimentally that objects fall at the same rate in a gravitational field, and that planetary orbits are elliptical. Newton reasoned, and invented a calculus to describe, the unity of motion that applies to projectiles under the influence of a gravitational field both near and extended, both horizontal to the plane and around a curved manifold. Thus, there is no difference in principle between an apple falling perpendicular to the plane of the Earth, and the moon falling around the curvature of the Earth or the Earth around the curvature of the sun. All that energy of mass and inertial motion is precisely accounted for in the mathematical model. That is why I asked you that if you think Laithwaite's gyroscopic experiment describes more than Newton's laws account for, to calculate that discrepancy -- for if it were true, the laws of motion would be violated and anti-gravity would be a fact. We could not trust energy to be conserved.

    The mathematics of quantum mechanics in the microscopic domain does not proceed from the classical laws of motion -- sometimes the apple falls to Earth and sometimes it doesn't. If we want a unitary physics governed by the same laws of motion at every scale, we won't get there by simply assuming that the classical laws are "somehow" projected from the quantum phenomena. We have to know where and how the quantum effect becomes calculably transformed to the classical laws -- that's where models incorporating an extra degree of freedom come in, allowing a continuous measurement function to describe the transformation, without using arbitrary boundary conditions.

    Best,

    Tom

    "The reason is that mathematical identities, may imply different physical manifestations."

    Not in a complete theory made of closed logical judgments. The physical results are dependent only on the value of the variables. In E = mc^2, e.g., the proportional constant c describes the limit of the changing identities in the relation between mass and energy.

    I agree, if "complete" refers to "Physically" complete, rather than merely "Mathematically" complete.

    In other words, Mathematical completeness is not sufficient, to guarantee any relevance to Physics. It is necessary, that there also be a demonstrable, one-to-one correspondence, between "every" physical entity/property, and each mathematical symbol, relevant to the physical phenomenon being described.

    Demonstrating such a one-to-one correspondence is where the rubber meets the rubber; it is what distinguishes math from physics.

    In the case of QM, the only such correspondence is with the amplitude of the Fourier Transforms, not the Fourier Transforms themselves, or their "internals".

    Rob McEachern

    Should have said "where the rubber meets the road"

    It would be nice if one could edit.

    Rob McEachern

    We agree, Rob.

    Mathematical completeness is necessary, though not sufficient.

    As you say, "It is necessary, that there also be a demonstrable, one-to-one correspondence, between 'every' physical entity/property, and each mathematical symbol, relevant to the physical phenomenon being described."

    A mathematically complete theory, then, is one that guarantees measured correspondence between symbol and physical result.

    "Demonstrating such a one-to-one correspondence is where the rubber meets the rubber; it is what distinguishes math from physics."

    Absolutely. The language (mathematics) and the meaning (physics) are independent of one another.

    "In the case of QM, the only such correspondence is with the amplitude of the Fourier Transforms, not the Fourier Transforms themselves, or their 'internals'."

    I agree that the Fourier transform does not provide the degrees of freedom required to completely describe the physical phenomenon.

    Having just dropped 600 bucks on a new set of Firestone tires, I appreciate that more rubber is reaching the road than before. :-)