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Author Topic:   Is mathematics a science?
Brad McFall
Member (Idle past 5032 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 10 of 48 (239089)
08-31-2005 3:47 PM
Reply to: Message 1 by nwr
08-28-2005 4:24 PM


let discovery begin!
quote:
Appendix B
Ars Combinatoria
(some German)
1. Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind — as Eddington put it — the colorful tale of the subjective storyteller mind. In Appendix A we have discussed the structure of mathematics as such. The present appendix deals with some of thesimplest structures imaginable, the combinatorics of aggregates and complexes. It is gratifying that this primitive piece of symbolic mathematics is so closely related to philosophically important problems of individuation and probability, and that it accounts for some of the most fundamental phenomena in inorganic and organic nature. The same structural viewpoint will govern our account of the foundations of quantum mechanics in Appendix C. In a widely different field J. von Neumann’s and O. Morgenstern’s recent attempt to found economics on a theory of games is characteristic of this trend. The network of nerves joining the brain with the sense organs is a subject that by is very nature invites combinatorial investigation. Modern computing machines translate our insight into the combinatorial structure of mathematics into practice by mechanical and electronic devices.
It is in view of this general situation that we are now going to insert a few auxiliary combinatorial considerations of an elementary nature concerning aggregates of individuals. The reader should be warned beforehand that in their application to genetics the lines are drawn somewhat more sharply than the circumstances warrant. In the progress of science such elementary structures as roughly correspond to obvious facts are often later recognized as founded on structures of a deeper level, and in this reduction the limits of their validity are revealed. This hierarchy of structures will be illustrated in Appendix D by the theory of chemical valence.
page 237-8 in PHILOSOPHY OF MATHEMATICS AND NATURAL SCIENCE by Hermann Weyl Princeton Uni Press 1949
If one thinks one can understand the reduction of said sharp genetic line(s) in terms of a hierarchics of valences (contingent nonetheless) then the math that bridges this text of Weyl could be considered science. I often think I am permanently in such possession. But that is such (relative to this thread head) IF the computer science structure Weyl passed on can be empirically such as to reduce the same structure. I do not think that it does. Hence math is not science. A philosophical position from Wolfram’s new kind of science however might have one think (thought previously) that it is (so irreducibly complex) but again philosophy is not math or science now is it? Theology is not teleology.
quote:
By breeding experimens one has succeeded in dissolving the genetic consitution in to an aggregate of individual genes or 'points', much like as chemistry dissolves a molecule into an aggregate of atoms
op.cit. p240
Viscosity was not the word but the math could cover it.
Empirical mathematics and breeding empirical mathematicians are two different combinations. Experimental space both contain.
There is a metaphor mathematically where Weyl used a simile.

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Brad McFall
Member (Idle past 5032 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 14 of 48 (239406)
09-01-2005 7:03 AM
Reply to: Message 13 by Dr Jack
09-01-2005 6:02 AM


Re: Math as a language
Nice.
Jack,
I could not have said it better myself!

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Brad McFall
Member (Idle past 5032 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 19 of 48 (239668)
09-01-2005 3:29 PM
Reply to: Message 16 by crashfrog
09-01-2005 10:56 AM


Re: Math as a language
There are other "grammers" that do not exist and yet are purely cognizable FROM Boole, x=x^3 and higher powers GIVEN THE PROSE in his{rules of thought} work. It remains to be seen if these "lingos" will be called "math" of today or "organons" anon. Claiming the plurification of Boole categorically will never occurr is more dangerous than asserting the small probability that God exists not existing.
This message has been edited by Brad McFall, 09-01-2005 03:31 PM

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Brad McFall
Member (Idle past 5032 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 43 of 48 (240221)
09-03-2005 8:26 AM
Reply to: Message 34 by Chiroptera
09-02-2005 11:39 AM


"/&/ "as genes? math or biology? normal rxns??
Some day I expect the shapes of the symbols in rings will affect biological signing but still it is the multiple "///" tones that matter in e/c. I really dont know that this must be not math. It seems like math when a notion of theoretical biology is sustained. I dont think that that the "////" can be blinking lights as Stu Kaufmann was motivated to think (that he could not "do" Kant etc).
Now that I know yous gls are really good math people I might try to figure out how the 70s' advances in group theory might be science via math in biogeography. Later.
Weyl wrote
quote:
Another point of debate is the question whether the numbers are independent objects or whether arithmetic is concerned merely with the concrete numerical symbols, "whose shape is recognizable by us with certainty independently of place and time, of the particular conditions of their manufacture, and of trifling differences in their execution" (Hilbert). Thus e.g. Helmholtz(Zahlen und Messen, loc. cit., p 359):"I consider arithmetic, or the theory of pure numbers, as a method built upon purely psychological facts, by which the consistent application of a system of symbols of unlimited extent and unlimited possibility of refinement is taught. In particular, arithmetic investigates what different modes of combination of these symbols (numerical operations) lead to the same result." Only recently Hilbert carried this point of view consistently into effect (compare Section 10), in a manner unassailable even by the crticism directed against it by Frege (Grundgesetze der Arithmetik, 1893). A succession of strokes ('ones') offers itself as a suitable symbol. If I hear a sequence of tones, I put down a stroke upon hearing each one, placing one stroke after another: ////. A second time I proceed similarly, again obtaining a symbol consisting of a succession of strokes. If I were immediately able to judge the equality or disparity of the 'shape' of the two symbols, a numerical comparison would be accomplished. Here the representation of data by strokes has the function of putting the data into a 'normal' form of such a kind that a difference in shape at once indicates a difference in number.
I always considered Mandelbrot's discussion of the word "fractal" to point to this problem of what FORM other than 'normal' such sounds heard in science might be maths for.
quote:
(For a directly given whole, number is meant to describe a relation between the whole and parts of it as are considered as units...)
both quotes of Weyl, page 35 in PHILOSOPHY OF MATHEMATICS AND NATURAL SCIENCE
I am torn between determining transfinte e-numbers or reflecting on groups further for the permutable subtraction I think evolution remands biologically from n! no matter the divisions. I really do think that whatever that "math" is it is pure in itself no matter how it sorts difference of natural and artifical selection in the statistical normal distribution approximation.

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Replies to this message:
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Brad McFall
Member (Idle past 5032 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 45 of 48 (240521)
09-05-2005 8:23 AM
Reply to: Message 44 by crashfrog
09-03-2005 7:46 PM


to speak a langugage or not, is that a question?
Hey froggy,
I am not tempted to think of it "a priori" as a language or lingo.
What would happen if some one other than me (rhetorically speaking (I am not spitting venom here) figured out what Brown's letters "S" and "T" mean in terms of my annotation on his questioning? The three strokes I attached in my side comments WOULD NOT BE LEIXICALLY a language or even a part of it but only a pure mathematical thought that would have implementations differently than is currently being taught in biology because of evolutioanry influence I contenD. I met and spoke with Brown on a few occassions and even talked with him about set theory once. Compared with other mathematicians at Cornell (some coming from Harvard etc except one who left for Berkely) he is really an OK kind of guy.
The back cover of the book
contains my notes on how to relate group operations to biology but this is done in terms of human teleology. Weyl clearly indicated(I speaking for the entire book PHILOSOPHY OF MATHEMATICS AND NATURAL SCIENCE, not a part or trap of it inversly) that teleology could connect into math DIFFERENTLY than evolution by force as it was DISCUSSED in the late 40s. I suppose I will only lurch you if I show how the "business cycle" of teaching evolution is only the alternative "phase" of matter that Wright explicitly did not include & show that macroevolution is not within a collective aggregate of that supply. Oh, well......I suppose I will have to do that by enlaring this "thumbnail" above but that is not math as science but science as itself. If Theus ever decides to respond I'll pick that up with 'him'.
The "apriority" of which you write seems to me, if I understand you well enough, to be the result of probabalisms (no matter how reflected on) not any DETERMINATION made from a prior mathmatical mind whether only revealing a statstical regularity or regularity as law etc (in nature by nuture etc etc). So,,, I have to disagree with you and agree with some others in this thread.

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Replies to this message:
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Brad McFall
Member (Idle past 5032 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 48 of 48 (240930)
09-06-2005 8:20 PM
Reply to: Message 46 by Chiroptera
09-05-2005 6:41 PM


Re: "/&/ "as genes? math or biology? normal rxns??
Here is the "mental concept" am concerned about.
Can Weyl be mistaken that "quantity" is not much longer an issue but instead that the whole scholarship he attempts to close, should be reopened, as the " quantity of genes" is adumbrated. Is there not something to Mr.Jack's query as to if there is something else a foot?? I for one was completely struck to sit in a graduate seminar in ecology and evolution@CUin 86 to hear a new professor Will Provine used contra Johnson in 96 ask formerly without response from other profs, as to what a "gene" was. How we "quantify" them seems to invert the relative importance of metric and geometry in Weyls' thought in this context but I should rather speak of yours or mine. Quatification can have purposes in mind, not pure- granted. Thanks so much for your clear response. You and others in this thread have gained real respect from me time around.
quote:
On the character of mathematical cognition
From time immemorial mathematics has been looked upon as the science of quantity, or of space and number. (Though we also find this definition with Leibniz, the mathesis thus delineated is to him but a part of the more comprehensive ars combinatoria.) Today this view appears much too narrow in consideration of such fields as projective geometry or group theory. Consequently we need not worry particularly over an exact determination of what is meant by quantitative. In fact, the development of mathematics itself rasies doubts as to whether quantity is a well-determined and philosophically important category. Geometry, inasmuch as it is concerned with real space, is no longer considered a part of pure mathematics; like mechanics and physics, it belongs among the applications of mathematics. Under the influence of the general arithmetic of hypercomplex numbers and later of the axiomatic investigations, of set theory and symbolic logic, the distinction between mathematics and logic is gradually obliterated.
page 62 Weyl Philosophy of Mathematics and Natural Science.
I think we are tempted to think of bioinformatics as an implementation applicable to Weyl's perspective but I think this is mistaken as it entails TOO much Greek Society reference which just does not exist in post-modern culture. I thought Weyl failed to follow through the Katian LOGICAL horizon organonically. Yes, I need to justify that last sentence biologically but there is no distortion in my so thus thought such applied albeit it be.

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