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.
This mean that for a system to be logical it must be PRODUCTIVE, i.e. have a methodology by which one can produce a lot of string of symbols (statements), and it must have a truth labeling of those statements in such a manner that no statement and its negation are both labeled as True. And above all the system must do that in the most general manner, so its rules must not be limited to a special sector of concepts, in other word it must be topic free.
Mathematics do not have the same 'general' motivation logic has, so arguably logic is more general than mathematics. Mathematics mainly pivots around strong concepts of structure, construction, succession, etc.. many of which are not so general as logical concepts are, the later involve itself with general concepts of Truth labeling, Predication, Identity, Proof, etc.. Frege added also to those the concept of "extending predicates". Those are very general concepts that can work virtually in any field of knowledge. The above system is also motivated along the same general lines logic has and it use the same tools and styles. Seeing that a particularly motivated system like second order arithmetic is interpenetrate in such a general kind of discourse about logical concepts really reduces mathematics to logic, we no longer need those particular motivations about constructions and structures nor about space and time that customarily mathematics are easily seen to negotiate, those are proved to be so STRONG, we don't need them to have mathematical systems, actually very weak motivations about general kinds of consistent discourse (i.e. logic) are just enough to produce mathematical systems. This reduce mathematics down to the level of general analytic machinery, rather than it being about a particular high content realm.
In other words mathematics is just an offshoot of symbolic logic.