fom
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Re: LOGIC & MATHEMATICS
Posted:
May 28, 2013 8:56 PM


On 5/28/2013 8:17 AM, Julio Di Egidio wrote: > "Zuhair" <zaljohar@gmail.com> wrote in message > news:f25c467c7fc94d2dbbaa9e421f4fda30@w5g2000vbd.googlegroups.com... >> On May 27, 6:51 pm, CharlieBoo <shymath...@gmail.com> wrote: >>> On May 26, 2:52 am, Zuhair <zaljo...@gmail.com> wrote: >>> >>> > Frege wanted to reduce mathematics to Logic >>> >>> What does it mean to "reduce mathematics to Logic"? >> >> It means that any mathematical discourse can be interpreted within a >> logical discourse. >> >> Logic is responsible for laying down a set of rules that results in >> generation of non contradictory statements in the most general manner. > > Logic is not just mathematical logic: case in point, you can make of > identity an axiom, but the notion of identity itself presupposes an > existential stance, so it is not a purely logical notion (in the sense > you have just stated). My take is that mathematics uses logic (as the > language of mathematics is as well logical) and logic uses mathematics > (as the logical calculus is mathematical), but neither can be reduced to > the other. Mathematics is the study and applications of "numbers", which > is not a purely logical endeavour, while logic is the study and > applications of "rational discourse", which is not an essentially > mathematical endeavour.
Frege concluded that his attempt had been an error. In "Numbers and Arithmetic" he retracted his logicism with the statement,
"The more I have thought the matter over, the more convinced I have become that arithmetic and geometry have developed on the same basis  a geometrical one in fact  so that mathematics in its entirety is really geometry"
Frege
Although generally ignored, Kant invoked a strict demarcation between logic and mathematics. Frege's logicism is essentially the idea that one can have "logical objects" and this had been one of the things that Kant had rejected. There is a nice discussion of Kant and Frege in sections 4 and 5 of John MacFarlane's dissertaion,
http://johnmacfarlane.net/dissertation.pdf
Of course, there have been significant efforts at recovering Frege's program. I can give you links if you are interested, but MacFarlane's paper gives an analysis similar to your own statements. So, I thought you might find it interesting.
> > I'd think the only way to operate the "unification" you have in mind is > by reducing *everything* to just its operational side, the calculus: > then there is not even any difference left between logic and > mathematics, indeed you'd have destroyed the very nature of both. >
This is exactly how I feel when I run into views that reject model theory. I am at a loss of how a "syntax only" view of mathematics is coherent. But, that is one of the developments in the history of mathematics as different "number systems" evolved.
Among classical writers, De Morgan seems to have seen this as an outcome of adopting these "new" arithmetical systems as part of mathematics:
"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
In other words, each of us interprets the language of mathematics individually according to our experiences with it.

