[FOM] Mathematics ***is*** formalising of our thought and intuition

Kreinovich, Vladik vladik at utep.edu
Fri Jun 4 00:45:59 EDT 2010

May I rephrase Hendrik's opinion slightly, to emphasize the same point.

To a mathematician, when a political scientist derives some results from the axioms in a rigorous way, or a physicist derives some property from a known differential equation -- in other words, when they are proving theorems -- they are doing mathematics.

To a mathematician, any absolutely rigorous derivation is mathematics. Hilbert in his Foundations of Geometry emphasized that it does not matter what the arguments are about: they can be about numbers, about points, about beer mugs, about tables -- as soon as we formulate exactly the properties of the corresponding objects and make rigorous deductions, we are doing mathematics.

Yes, there is rigorous deduction in physics, in political science, in theoretical chemistry -- and to a mathematician, this make this part of physics also part of mathematics. When Steven Hawking and Penrose proved theorems about black holes and singularities in a rigorous way -- this is physics AND this is mathematics too. 

When mathematicians claim that mathematics is the most rigorous discipline they DO NOT mean to imply -- as Robert seem to have interpreted -- that other sciences are not rigorous. God forbid. These other sciences also have rigorous parts, and these parts are -- by definition of mathematics -- ALSO part of mathematics. Papers on mathematical physics, mathematical biology can be published in physics or biology journals -- but they can also be (and are) published in mathematical journals. Such papers regularly appear in journals published by American Mathematical Society, they  are routinely reviewed by Mathematical Reviews, the main review journal of the society, and many famous mathematicians have published such papers. 

There was a known case when Hilbert, a mathematician, published what is now called Einstein's equations of General relativity , I think, two weeks after Einstein (he did not have a detailed physical analysis). To Hilbert, it was a very mathematical work, to Einstein, a very physical one. 

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of hendrik at topoi.pooq.com
Sent: Thursday, June 03, 2010 6:01 PM
To: fom at cs.nyu.edu
Subject: Re: [FOM] Mathematics ***is*** formalising of our thought and intuition

On Tue, Jun 01, 2010 at 03:27:07PM -0700, Robert Lindauer wrote:
> That there are not currently as-good political theories as there are
> mathematical theories is not essential to (either) science.  Surely
> political scientists -could- carry on their discipline without error
> just as any mathematician might by simply being very conservative.
> Surely mathematicians have carried out poor calculations and/or
> proceeded under false assumptions.
> Surely it is the intention of every science to have error-free
> systematic understanding, this is not unique to mathematics.

Right.  But mathematics is the most important tool they use 
to achieve this when they do.  It seems to be a necessary tool, although 
it's not a sufficient one.

-- hendrik
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