On 5/28/2013 2:17 PM, Charlie-Boo wrote: > 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.
*I* am doing nothing.
I merely answered your statement as it is presented in the literature historically.
> 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, > > MATH > > ARITHMETIC > > Natural Numbers > > | | > v > > Any particular infinite Set > > Classes > > What about (a) reducing negative integers? Arithmetic of real > numbers? Other branches of mathematics e.g. trigonometry or number > theory?
The most concise statement of this can be found in the introductory chapter of Kleene "Introduction to Metamathematics".
It is in my remarks above as "issues associated with geometry motivated a general arithmetization of mathematics".
And, if you want to know one way of doing it, you may revisit my impenetrable post,
> > 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?
Once again, I answered your question according to the historical meaning of that phrase. You are applying certain modern views that would be compatible with Hilbert's formalist views, although the symbolic representations you are taking as possible models have a particular constructive origin.
Please do not hold me responsible for defending that history since I do not ascribe to logicism in that sense.
> > 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? >
Define "mathematics in general"
Define "logic in general"
And, before you speak of "equating a small part of mathematics to an arbitrary r.e. set", please read Frege's "Foundations of Arithmetic".
Frege and Hilbert had different views on the nature of mathematics. Hilbert, for the most part, won. So, your questions do not make sense with respect to the context.
>>> 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.
There is so much here, I cannot even touch this.
I do not reject your *opinion*. But I think it would be interesting to see you justify it with respect to the context of the philosophy of mathematics.