This continues to be confused Sung. "Arbitrariness" can certainly be applied to axioms, but not to mathematics itself. Effectiveness is in the nature of mathematics. Mathematics, according to Benjamin Peirce, echoed by Charles, is the science that draws necessary conclusions (from premises of any kind). The measure of such conclusions is "effectiveness" - Wgner's surprise is that this effectiveness is scale invariant. That is, for example, that expressions of the large apply equally to the small.
Steven On Friday, July 4, 2014, Sungchul Ji <[email protected]> wrote: > (For an undistorted figure, see the attached.) > > Dear Ben, > > I appreciate your detailed comments which I find very informative and > thought-provoking. > > The following is my response. > > > Sung, list, > > > > Your syllogism fails not only because of its equivocation with the word > > 'sign' > > All signs, by definition, are equivocal, unless their meanings are further > constrained by associated context. Let us remember the first principle of > the Taoist philosophy which reads in Korean “Doh Gah Doh, Be Sahng Doh”, > which, as I understand it (Perhaps some Taoist philosopher on this list > can enlighten us here.) , can be translated as > > “Principles/concepts once articulated/represented (070414-1) > are no longer permanent principles/concepts.” > > I think some of Peirce’s writings may be in harmony with this principle. > > The equivocality of the term “mathematics” appearing in my UAM > (Unreasonable Arbitrariness of Mathematics” syllogism would be removed if > the specific examples of the mathematical functions that I provided in > Table 1 attached to my original emails are taken into account. > > >but also its equivocation on the regard in which arbitrariness is > > involved. > > Again, the word “arbitrariness”, of course, is equivocal, since it is a > word. The arbitrariness of this word would be removed in the context of > the UAM discussion if (and only if) you take into account the specific > examples that I provided by way of illustrating what I exactly mean by > “arbitrariness”, i.e., a given set of data being able to be fitted by two > very different mathematical equations. I agree with your argument below > that there can be numerous meanings of the term “arbitrariness”, so what I > mean by “arbitrariness of mathematics” is not the same as what Sassureans > mean by this term in the expression “arbitrariness of signs”, for example. > But the main point of the UAM thesis is that my use of the term > “arbitrariness” coincides that used by the Saussureans, which I know you > do not agree with. > > > The arbitrariness of which among various synonyms one uses to > > denote an object - the arbitrariness of the Saussurean sign and the > > Peircean linguistic symbol - does not involve the arbitrariness which > > you are imputing to mathematics, > > What is your evidence for this conclusion ? > > > an arbitrariness of what conception, > > meaning, intension one uses to denote an object, except to the extent > > that the synonyms are not quite perfectly synonymous and instead differ > > in meaning or, more generally if one is to speak of icons as well as > > symbols, differ in what they represent as the object's character, either > > by resemblance or symbolically by imputation. > > > > Now you've found two mathematical objects whose graphed value sets are > > approximate icons of each other and, to a lesser degree, of some > > positive phenomenon's set of graphed measured values. > > Yes. > > > Since the two > > mathematical objects do not have quite the same set of values, further > > testing of the positive phenomenon may show that one of those objects > > tends to be a better fit than the other to the phenomenon > > This is a distinct possibility. Another intriguing possibility, suggested > by a first-year student, a young lady who is performing research with me > supported by a Rutgers Arresty Research Assistantship this summer, is that > a combination of the two functions may actually fit the phenomenon better > than any one of the individual functions. > > But I still cannot rule out the possibility that, no matter how accurately > the phenomenon under investigation is measured, the long-tailed > distributions that result will be able to be fit into the two functions we > have already found and to still others yet to be discovered. But the > arbitrariness will be removed if it can be proven that only one of these > equivocal functions can be DERIVED mathematically based on first > principles, as M. Planck (1858-1947) had done with his blackbody radiation > equation. The importance of DERIVATION, when first pointed out by Howard > some time ago, I did not fully understand but I think I do now, because of > Figure 1 below, according to which mathematical models of nature can be > both deterministic/effective and arbitrary. This idea may be expressed > diagrammatically as shown in Figure 1. > > a b > Phenomenon --------> Data -------> Mathematical Model > | ^ > | | > |________________________________________| > c > > Figure 1. The irreducible triadic relation among phenomenon, data, and > mathematical models. a = measurement which is deterministic and follows > the laws of physics; b = mathematical modeling which is arbitrary (i.e., > more than one models can fit the same data); and c = mathematical > derivation of the models based on first principles and hence is a > deterministic step. > > > and, if such > > isn't the result, then you have two mathematical objects neither of > > which quite fits the phenomenon although both resemble it. It is hard to > > see how this corroborates the idea of an unreasonable arbitrariness of > > mathematics. > > > > I think your argument is irrefutable. One possible solution to this > dilemma may be that both UEM (Unreasonable Effectiveness of Mathematics) > and UAM (Unreasonable Arbitrariness of Mathematics) are implicated in the > complex, irreducibly triadic relation among phenomenon, data, and > mathematical models as postulated in Figure 1. > > > Your other example is of the quantum mechanics of de Broglie / Bohm > > versus the Copenhagen interpretation and the like. It's hard for this > > example to be of general use to philosophers and semioticians most of > > whom don't have the mathematical and physical background to follow the > > argument and hope to bet on the right horse. For example, one reads > > physicists arguing with each other about whether de Broglie / Bohm > > mechanics does or does not succeed in predicting all the phenomena that > > 'standard' quantum mechanics does. In other words, most of us are not in > > a position to assess whether these two systems of quantum mechanics work > > equally well and whether the choice between them is _/arbitrary/_, as > > per your view, as far as physics is concerned, although many of us have > > philosophical preferences and maybe a kind of sporting interest in the > > subject. Anyway, there are always more things to say about views of > > quantum mechanics, and plenty of people to say them, so I'll leave it > > there. > > I recently asked Sheldon Goldstein, a Rutgers professor who is leading a > group of physicists researching on the Bohmian mechanics (BM), whether BM > can account for all the well-established quantum mechanical phenomena as > well as the non-deterministic wave mechanics of the Copenhagen > interpretation (CI), his answer was an unequivocal YES. > > Now, if we can apply Figure 1 to the debate between BM and CI, it may be > predicted that whichever mathematical model of quantum mechanics (QM) can > be successfully derived from first principles will emerge as the TRUE > model of nature. If this ever happens (or this might already have > happened but I am ignorant of it), then we may be justified to conclude > that the arbitrariness of the mathematical models of QM is only > temporary and the ultimate mathematical model will turn out to be > deterministic and unreasonably effective, consistent with the Winger’s > UEM thesis. > > With all the best. > > Sung > __________________________________________________ > Sungchul Ji, Ph.D. > Associate Professor of Pharmacology and Toxicology > Department of Pharmacology and Toxicology > Ernest Mario School of Pharmacy > Rutgers University > Piscataway, N.J. 08855 > 732-445-4701 > > www.conformon.net > >
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