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


Re: For the readers of WM: Resources on empirical time to consider as a foundation for mathematics
Posted:
Mar 21, 2013 1:52 PM


On 3/21/2013 8:26 AM, AMeiwes wrote: > "fom" <fomJUNK@nyms.net> wrote in message > news:1ZudnajAj7TF09fMnZ2dnUVZ_vydnZ2d@giganews.com... >> On 3/20/2013 4:36 PM, AMeiwes wrote: >>> "fom" <fomJUNK@nyms.net> wrote in message >>> news:wIydnYEnbJNcdTMnZ2dnUVZ_sqdnZ2d@giganews.com... >>>> >>>> In the post, >>>> >>>> news://news.giganews.com:119/ce55c74c8c9244d28e9686e487e1549e@y9g2000vbb.googlegroups.com >>>> >>>> WM has made the remarks, >>>> >>>> "This is deplorable because mathematic >>>> like no other part of "arts"depends on >>>> physics." >>> >>> wrong => math is not art. physics depends on math. >>> >> >> In fairness, the reference to "art" here involves the >> distinction between the convocation of Masters of Arts >> degrees and Masters of Science degrees in the system >> of higher education. That, however, is irrelevant to >> what is meant in relation to the relative logical >> priority between mathematics and physics. >> > > ..... so what does "relevant relative logical priority" actually mean?
So, for example, consider the end of the nineteenth century. The situation in mathematics had been such that the multiplicity of geometries and the arithmetical methods being used to approximate solutions in the calculus created the conditions for "logicism".
Frege's purpose for defining the natural numbers in terms of concept extension had been to demonstrate that mathematics was a branch of logic.
At the time, the sense of things would have been that geometry was the 'science of form' and arithmetic was the 'science of number'. As the foundations of geometry had been shaken, the focus of attention had shifted to 'the arithmetization of mathematics' so that "all" of mathematics was, in fact, arithmetical.
Because of the use of classes in syllogism,
All men are mortal Socrates is a man  Socrates is mortal
Mathematics would not have been considered primarily as 'the science of classes'. That would have rested with the study of logic.
The deductive calculus we refer to today when speaking of 'mathematical logic' is largely based on Frege's analysis of negation in relation to sentences used in syllogisms. But, what had also been involved was the introduction of quantifiers in the modern sense of mathematical logic.
For Frege, these new ideas about logic provided a means to define natural numbers in terms of classes. At the core of it is the definition of the number zero as the class of objects satisfying selfcontradictory statements.
This provides a unique, grounded class upon which to formulate a definition of succession based on classes and their extension. The sense of it is that 'the number of a concept is like the direction of a line'. There can be a multiplicity of parallel lines. But, they have the same direction. Recalling that Frege had also been dealing with the interpretation of definite articles (the 'the' in the analogy he used) the multiplicity of objects  that is the class of objects  satisfying a predicate is given the same definite sense as the multiplicity lines in a parallel class.
So, if one accepts the Fregean doctrine (one he retracted at the end of his career), then one has that since all of mathematics can be arithmetized and since arithmetic can be represented by classes, then mathematics is a part of logic.
This is what is meant when speaking of 'logical priority'.
Obviously, WM's use of 'arts' in relation to the convocation of degrees is irrelevant. His statements concerning the relationship of mathematics and physics are.
At the end of the nineteenth century physics had had such success that other sciences had been trying to ground their own claims in relation to physics. I have read accounts that Bertrand Russell made an observation on this phenomenon. While the empirical sciences were rushing to ground themselves upon the explanation for reality given by physicists, the physicists themselves had been rushing to ground their own field on "objects" that could only be defined mathematically.
So, you decide:
biology depends on chemistry
chemistry depends on physics
physics depends on mathematics
mathematics depends on logic
Do the 'sciences' organize themselves into a logical hierarchy?
And, if you were to personally argue against WM's foundational claims, how would you do it? What ground would you choose for mathematics?

