"Ross A. Finlayson" <email@example.com> wrote in message news:firstname.lastname@example.org... > On May 28, 6:17 am, "Julio Di Egidio" <ju...@diegidio.name> wrote: >> "Zuhair" <zaljo...@gmail.com> wrote in message >> news:email@example.com... >> > On May 27, 6:51 pm, Charlie-Boo <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. >> >> 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. >> >> Just my 2c, feedback welcome. > > Or, both as of objects of plain reason (then with as well geometry): > the philosophical, logical roots are of the nature of the objects of a > system with truth.
But logic is not bothered by matters of true, logic is all and only about validity, i.e. non self-contradiction on logical forms. In particular, logic is that which abstracts from the specific nature of objects or any other contingencies. As you may expect, on the other hand, I do not see how geometry is not just a mathematics.
Then one might retort that the logical true of abstract form constitutes the object of logic, but that's just a paralogism: logic is purely self-referential, not just purely abstract as mathematics is.
> That the or a same fundamental principle can be > see to bestow upon the objects of logic, numbers, and geometry their > nature is rather compelling where the same criterion applies to each > object. > > Basically seeing points, numbers, and collections or distinctions as > primary (in geometry, number theory for mathematics, and set theory > for proper "logic"), then, how does an axiomless system of natural > deduction see the inference and deduction of and about them? > > Points and their properties of being any different than another point > can see built a geometry with less axioms than Euclid's, here courtesy > their being objects, ditto numbers and Peano/Dedekind, ditto sets and > Zermelo.