...a companion blog to "Math-Frolic," specifically for interviews, book reviews, weekly-linkfests, and longer posts or commentary than usually found at the Math-Frolic site.

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"Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." ---Bertrand Russell (1907) Rob Gluck

"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal." ---Bertrand Russell (1957)

******************************************************************** Rob Gluck

Tuesday, February 4, 2014

Should Mathematical Explanation Entail Mathematical Beauty?

                                          
                       Math
                         Logic
                          Beauty?

Is there any math writer/blogger left who hasn't written about the beauty of mathematics… on multiple occasions? It is an almost overwrought topic that audiences either 'get' by now… or likely will never comprehend. I have an acquaintance who, upon once being told that I was reading a book entitled "Love and Math," responded, "well, THAT'S an oxymoron; those two words should NEVER go together"… I presume she would say the same thing about the phrase 'math and beauty.' :-(

Nonetheless, "Mathbabe" blog has a wonderful new post up (from a guest-poster, not from mathbabe Cathy O'Neil herself) which is a great take on this matter; I recommend it to all even if you've read a plethora of these 'math and beauty' type posts before, not because it necessarily offers something new or profound, but just because it is so well-composed:

http://mathbabe.org/2014/02/04/guest-post-beauty-even-in-the-teaching-of-mathematics/
 

Further, the post notes that there will be a conference in Sweden, March 10-12, on this very connection of math and beauty, or as the author puts it, "Specifically, we will look at the question of whether mathematical beauty has anything to do with mathematical explanation. And if so, whether the two might have anything to do with visualization".

By coincidence, this post came along just as I was also reading some old 1960/70's essays on the history and nature of formal/symbolic logic, somewhat dividing that history into the pre- and post-Gödel periods. Anyway, this all got me to thinking how differently (I believe) logic is viewed from mathematics in this "beauty" regard... even though logic is very much at the foundation of mathematics.

Math enthusiasts easily perceive the beauty of their field (and wish everyone beheld it). But I suspect logicians don't perceive their subject in the same way (...but professional logicians please let me know if you do see beauty as an integral, prevalent feature of your field as well).
Formal logic seems to much more aptly fit the cold, dry, analytical (dare I say, boring) stereotype people so often apply to mathematics (not that there isn't any beauty to logic, but that you have to more deliberately look for it to see it, than in math).
The difference between logic and math seems somewhat akin to the difference between machine code (all 1s and 0s) versus the richer, higher level programming languages most people learn, even though the latter are ultimately based, in some sense, on the former. 

I guess I'm wondering if others (especially those more regularly working with academic logic) agree with my impression that logic is less pervaded by "beauty," and more befitting of the common stereotypical view many hold toward mathematics… or, alternatively, if taught and approached correctly, is logic also a matter of under-appreciated beauty?

ADDENDUM: I'll close this out with a concluding quotation from one of the essays I was reading (by Leon Henkin, 1962), which may bear some pertinence: 
"…perhaps of greater significance is the consensus of mathematicians that there is much more to their field than is indicated by such a reduction of mathematics to logic and set theory. The fact that certain concepts are selected for investigation, from among all logically possible notions definable in set theory, is of the essence. A true understanding of mathematics must involve an explanation of which set-theory notions have 'mathematical content,' and this question is manifestly not reducible to a problem of logic, however broadly conceived.
"Logic, rather than being all of mathematics, seems to be but one branch. But it is a vigorous and growing branch, and there is reason to hope that it may in time provide an element of unity to oppose the fragmentation which seems to beset contemporary mathematics -- and indeed every branch of scholarship."
 

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