fom
Posts:
1,969
Registered:
12/4/12


Re: LOGIC & MATHEMATICS
Posted:
Jun 4, 2013 8:43 PM


On 6/4/2013 9:19 AM, CharlieBoo wrote: > On May 28, 8:13 pm, fom <fomJ...@nyms.net> wrote: >> On 5/28/2013 2:17 PM, CharlieBoo wrote: >> >> >> >> >> >> >> >> >> >>> On May 27, 12:43 pm, fom <fomJ...@nyms.net> wrote: >>>> On 5/27/2013 10:51 AM, CharlieBoo 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. > > I am asking for a formal definition of "reducing mathematics to > logic" (RMTL). In other words, a condition under which something can > be said to be RMTL if it meets that condition. But you simply refer > to some system and claim that it does RMTL without ever defining what > constitutes RMTL. References to philosophy and other people's words > do not constitute a statement of a formal definition of what you are > accomplishing. It simply gives you a (fake) excuse for saying > whatever you want and making claims about it, without ever giving a > precise definition of what you supposedly accomplished. > > Here's a discourse of mathematics: "To bisect an angle, place the > point of a compass on its vertex, draw 2 arcs that intersect within > the angle, use a straightedge to draw a line connecting the vertex to > that intersection, and that line bisects the angle." Isn't that > mathematics? But you can't right here reduce that to logic, can you? > > CB
Make me laugh some more. You are a good clown.
If you insist on acting stupid and childish, then you must define 'mathematics' and 'logic'.
Then I shall tell you if anything I might try to say could convince you of anything. I do doubt that by virtue of your response.

In Chapter 9 of "Vectors and Tensors" by Bowen and Wang you can find the definition of a Euclidean point space. This is how the usual topology of real spaces is attached to the algebra of inner product spaces.
Bowen and Wang do a wonderful job of formulating all of the algebra up to that point (Perhaps you should review "Universal Algebra" by Graetzer to understand how to proceed in the most formal manner that does not presume geometricallybased grouptheoretic forms).
What may perhaps be required as prior to some of these constructions is the classbased construction described by Kleene in "Introduction to Metamathematics" which reviews the intent of the work done by Dedekind and Cantor.
For what this is worth, your beliefs as an individual are irrelevant to the meaning of "reducing mathematics to logic". The phrase reflects historical events. If you choose ignorance in these matters, that is what you choose.
The general

