On May 27, 12:43 pm, fom <fomJ...@nyms.net> wrote: > 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.
You say you are reducing mathematics to logic by equating arithmetic with certain classes. But (a) you are equating only arithmetic, and only arithmetic of the natural numbers, to (b) the set of classes that can be represented by finite expressions that you choose to represent certain classes. That is,your
MATH | | V Logic
is really just,
| | v
Any particular infinite Set
What about (a) reducing negative integers? Arithmetic of real numbers? Other branches of mathematics e.g. trigonometry or number theory?
And (b) doesn?t this apply to any infinite set? For example, instead of the expressions for representing certain classes, can?t we equate the natural numbers to any r.e. set we use in mathematics e.g the set of wffs or strings in general or matching parentheses (), (()), ()(), (()()) etc?
So you are just equating a small part of mathematics to an arbitrary r.e. set. What does that have to do with mathematics in general or logic in general?
> > 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. > > So, to decide the issue, what is a number?
There is no such thing as number. People throw in anything that is needed to solve an equation that is often solved by the natural numbers. They throw in negative integers, then rationals, then irrationals, then real numbers, imaginary number, complex number, transcendental numbers etc. It is a moving target with no single definition. In general, it is what mathematics have decided to include when they are trying to solve an equation and what they have defined to be a "number" does not provide any solution, so they invent a "new kind of number".
The natural numbers are simply delimiters that divide everything into things. But that started a whole series of equations and the addition of whatever they need to solve their equations.