Beholding Mathematics...

“Beauty is the first test: there is no permanent place in this world for ugly mathematics.” -- G.H. Hardy


Just a little Sunday rumination on mathematical beauty today….

Math bloggers come in a variety of flavors, sometimes with few specific overlapping interests, but what they often hold in common is an overarching desire to share math's beauty and wonder with others.
"The beauty of mathematics" is a phrase most math bloggers have probably used at some point… it rolls off our tongue (or keyboard), so obvious is the "beauty" of mathematical thought and application we see, despite its frequent lack of resonance with other lay folks.

This article about young Princeton University professor Manjul Bhargava well captures his enthrallment with the "beauty" of math:

http://www.livescience.com/28508-numbers-manjul-bhargava-nsf-bts.html

In it Bhargava says: "When you discover things about numbers, it's very beautiful. When mathematicians are thinking about their problems, we're not thinking about their various applications, but rather are pursuing beauty. That's how pure mathematicians think."

Bhargava is of Indian origin... same as the remarkable Ramanujan who also found incredible, and uncanny, beauty in mathematics, which he felt was communicated to him directly in visions from an Indian goddess. 

And most anyone who reads science on the Web has probably seen this Richard Feynman clip which goes semi-viral from time to time. In it he describes, touchingly, how he, as a scientist, is capable of perceiving beauty, just as much as one of his artist friends can do… perhaps even more-so. As he essentially says toward the end, 'How does science detract from the ability to perceive beauty… it only adds, it only ADDS!!' …and so too mathematics, one could say:



So why do people universally comprehend the notion of "beauty" when applied to art or music or fashion or even food, but not when it comes to mathematics?
Usually beauty is beheld directly through the senses, vision, hearing, taste, touch… but math is more abstract, beheld at a different cerebral level where we discern patterns or connections mentally, whether or not they are physically evident.
Certainly not all math is beautiful, and it is indeed interesting how some theorems or proofs can be viewed as elegant or beautiful, while others simply are not (EVEN though they may be equally true or valid). As Paul Erdös was fond of saying some proofs are "from the Book" (God's creation manual).

An article from Smithsonian Magazine notes "...two of the essential requirements for mathematical beauty. First, it is surprising... Second, it is simple."

Read more:
 http://www.smithsonianmag.com/science-nature/the-natural-beauty-of-math-174842751.html#ixzz2KS3JBicz

Meanwhile S.Lang, in "The Beauty of Doing Mathematics," addressing those who view mathematics as dry and dull, writes "Last time, I asked: 'What does mathematics mean to you?' And some people answered: 'The manipulation of numbers, the manipulation of structures.' And if I had asked what music means to you, would you have answered: 'The manipulation of notes?' "

Robert Krulwich, Marcus du Sautoy, and Simon Singh explore the topic in this short clip from the World Science Festival:



And below a longer piece from the same Festival (including, at the 76-minute point, comments on Platonism vs. non-Platonism in math):

http://worldsciencefestival.com/videos/mysteries_of_the_mathematical_universe

There the debate continues over whether mathematics is discovered or invented; whether it exists in some outer 'real' world or only within the processes of the human brain.
And one may reflect on and on and on about mathematical 'beauty,' but surely it at least (like all beauty), ultimately exists in the… mind… of the beholder (be that beholder Man or God).