On 5/27/2013 10:51 AM, Charlie-Boo 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"?
Historically, mathematics had been seen as treating the science of number and the science of form. Classes had been considered the subject of logic. As mathematics developed in the 19th century, issues associated with geometry motivated a general arithmetization of mathematics. The Fregean program of logicism involved establishing the foundations of mathematics by defining arithmetic in terms of classes.
A more modern author who makes a simple statement of such is Quine in "Methods of Logic" if I recall correctly.
> The comments I > see after this first post seem to debate what that means, as well. If > (since) you are going to give (giving) a formal answer, then what is > the formal problem? Trigonometry is part of Mathematics. How would > we "reduce trig to Logic"? Or start with a simple case: What is the > criteria for something said to reduce number theory to logic? > > Computers process only zeros and ones. Anything you do on paper can > be done with a computer. If 0 is replaced by FALSE and 1 is replaced > by TRUE, does a computer reduce mathematics to logic? >
Actually, Boole's idea had been to address issues in logic more mathematically. So, your example reflects replacing the traditional semantical notions of logic with the Boolean arithmetical representation.
This is opposite to what you ask.
Sometimes one sees reference to Boole as being associated with an algebraic approach to logic (a Boolean algebra is a logical algebra, right?) in contrast to the symbolic approach to logic associated with philosophical treatments.
I would probably classify your reference to what can be done "on paper" along the lines of a symbolic approach, and, the Russian school of constructive mathematics is explicit in their treatment of number along such lines.