On Tuesday, July 30, 2013 2:16:56 PM UTC-7, fom wrote: > On 7/30/2013 8:29 AM, Peter Percival wrote: > > > From time to time a definition of what mathematics is is offered or > > > requested--I don't know why. Here is one due to Donald Monk: > > > > > > As a tentative definition of mathematics we may say it is an a > > > priori, exact, abstract, absolute, applicable and symbolic scientific > > > discipline. > > > > > > [Foot of page 1 of J. Donald Monk, Mathematical logic, GTM 37] > > > > > > It might be amusing to identify seven things which aren't mathematics > > > that satisfy all but one (a different one in each of the seven cases) of > > > - a priori, > > > - exact, > > > - abstract, > > > - absolute, > > > - applicable, > > > - symbolic, > > > - scientific; > > > thereby demonstrating their independence. Actually, the definition of > > > scientific is just as controversial. On page 2 op. cit. Monk explains > > > that by absolute he means 'not revisable on the basis of experience.' > > > > > > (The book arrived today, and it's going to be a tough read. If I need > > > any help I will, of course, seek it in one or both of the scis dot logic > > > and dot math.) > > > > > > My second thought (the first related to 'scientific') was that > > > 'symbolic' is nugatory: every discipline uses written language, and the > > > special symbols of mathematics are just taking the place of words in > > > natural language ((i)for reasons of brevity and (ii) for > > > understandability among people who use different natural languages, I > > > suppose). > > > > Other disciplines have not found themselves > > confronted by semantic indeterminacy in the > > manner of mathematics. From a foundational, > > logical point of view, Bolzano argued for > > undefined language primitives. I do not > > see that as relevant here. However, there is > > a particular quote from De Morgan that I > > think applies. It is what I associate with > > the algebraic point of view. He writes: > > > > > > "As soon as the idea of acquiring > > symbols and laws of combination, > > without given meaning, has become > > familiar, the student has the notion > > of what I will call a symbolic > > calculus; which, with certain symbols > > and certain laws of combination, is > > symbolic algebra: an art, not a > > science; and an apparently useless > > art, except as it may afterwards > > furnish the grammar of a science. > > The proficient in a symbolic calculus > > would naturally demand a supply > > of meaning. Suppose him left without > > the power of obtaining it from > > without: his teacher is dead, and he > > must invent meanings for himself. > > His problem is: Given symbols and > > laws of combination, required meanings > > for the symbols of which the right > > to make those combinations shall be > > a logical consequence. He tries, > > and succeeds; he invents a set of > > meanings which satisfy the conditions. > > Has he then supplied what his teacher > > would have given, if he had lived? > > In one particular, certainly: he has > > turned his symbolic calculus into a > > significant one. But it does not > > follow that he has done it in a way > > which his teacher would have taught > > if he had lived. It is possible > > that many different sets of meanings > > may, when attached to the symbols, > > make the rules necessary consequences." > > > > Augustus De Morgan > > > > > > > > Well, perhaps this excerpt is included > > in the sense of your remarks. Perhaps > > not. > > > > Another abstraction that would be more > > universal than "mathematics" would be > > Quine's double standard for radical > > translation (or something like that) > > derived from his exposition in Word > > and Object. > > > > http://en.wikipedia.org/wiki/Intentionality#Intentionality_poses_no_problem_for_science > > > > http://en.wikipedia.org/wiki/Indeterminacy_of_translation > > > > I think it might apply to your remarks > > concerning people who use different natural > > languages interpreting the same mathematical > > symbols. > > > > Anyway, just a little you may not have > > run across concerning "symbolic" (although > > I doubt it -- you seem extremely well-read > > on theses matters).
So I browse to sci.logic and find it veritably infested. (The front page is full of spam.) Then to find some of the regulars true to its purpose, I wanted to compliment you for carrying on an at least semi-urbane conversation on the matters.
As to definitions in mathematics and definition in mathemations, what strikes me is the very roots of the word: "de-" as the unbounding of here "-finite", that which is bounded. In a sense it is as of the abstraction, of things, then, as to the very results of abstration of things (where here the regulars might know then I ascribe an axiomless system of natural deduction as theory).
Few points: geometry is analytic (vis-a-vis a space-filling natural continuum as a geometry for points and spaces, corresponding to the above), one might expect that any abstraction of abstraction could arrive at the same conclusions (Platonism), semantics are synthetic (then as to that plainly the synthetic is analytic), again in the utterly fundamental qualititative and quantitative are same. These are concerns for the nascent in mathematics, then that as to the applied in the mesoscale, these distinctions then so arise then then into the macroscale they are again so erased, that they again so erase.
Then as to not writing in some time, largely it is that many of the points of my program for mathematics are made, what I hope is then over time to assemble and organize said statements of theory and theoretical import as to then a mathematics (in foundations of collections, numbers, and geometry) with results: that mathematics is the communication of results of logic and their means, and reasoning on reason.