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Topic: For the readers of WM: Resources on empirical time to consider as
a foundation for mathematics

Replies: 10   Last Post: Mar 21, 2013 11:34 PM

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fom

Posts: 1,968
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
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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:wIydnYEnbJNc-dTMnZ2dnUVZ_sqdnZ2d@giganews.com...

>>>>
>>>> In the post,
>>>>
>>>> news://news.giganews.com:119/ce55c74c-8c92-44d2-8e96-86e487e1549e@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 self-contradictory 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?









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