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. > > Julio
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. 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.