On May 28, 8:33 am, Zuhair <zaljo...@gmail.com> wrote: > 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.
"reduce" is a verb so to "reduce mathematics to logic" means to do some action,
You are making a statement that at best is true or false. That statement is not doing something.
And how is "logical discourse" quantified? Is the statement, "For all mathematical discourses, there is a logical discourse in which it can be interpreted."?
What have you done that constitutes the action of "reducing"?
> 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. > > Zuhair