One Sunday morning, following the visiting speaker’s sermon that dealt in some way with the beautiful, the service leader asked members of the congregation to put forth the things they found beautiful. Finally, a retired professor of physics stood and asked whether anyone found mathematics beautiful. No one admitted to it.
As someone who finds beauty in a lot of different things - music, art, nature – I was made to ask myself whether I had ever found beauty in any mathematical formula and decided that it had never happened. It turns out that for mathematicians there are beautiful formulas and ugly formulas, and a beautiful formula causes the same area of the brain to be stimulated as listening to Bach or viewing a Van Gogh does in many of the rest of us.
Mystified, I went in search of explanations of mathematical beauty, and this article in BBC News by James Gallagher explains it as well as it can be made understandable to those of us who are mathematically inert.
On an intellectual basis I get it. Certain seemingly unrelated mathematical concepts like, pi, imaginary numbers and prime numbers may be shown to be related in mathematical equations, and the more concise that demonstration the more beautiful. Thus Euler’s equation,
reveals an unexpected relationship between all of these conceptual numbers. The value e, like pi, is an irrational number that is approximately 2.71828, and like pi it is a mathematical constant. The value I is the unit imaginary number and is equal to the square root of (-1). As one mathematician explained the beauty of Euler’s equation, it in the most concise manner explains the relationship between the constants.
Another mathematician when discussing the equation that expresses the fact that any prime number divisible by four with a remainder of one is the sum of two squares was beautiful, but not especially for the equation. In his words that equation is like the final chord in a symphony. The derivation is the symphony.
I understand that on an intellectual basis, but not on the emotional basis that occurs with art or music.
Last evening we were watching television and Lynn paused the recording while she answered a phone call. The image on the screen was of a beautiful, rocky coastline, and it reminded me that when I first read, in the mid-1970s, about fractal geometry and Mandelbrot Sets they were beautiful, not for the equations, which just look like equations to my eye, but for what they represent when expressed visually.
The equations of fractal geometry - which were impossible to perform before computers because they involve continuous equations in which the result of the first equation is entered into the second, the result of which is entered into the third out to infinity - are beautiful because they express the real world.
“"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."
Benoit Mandelbrot
Certainly for graph-able functions, I assume that a mathematician can see the graph without doing the plotting. If so, I certainly could see how those equations would be beautiful.
tabg, thanks for commenting. I use math in much the way that you do with ceramics. a^2 + b^2 = c^2 to see if your picture frame is square, and a lot of ways, but I don't look at some elegant mathematical proof - abstract math - and say that is beautiful.
In that way I feel like a tone deaf musician, that knows which strings to pluck, but not without someone else writing the music. I think real mathematicians are alternately wired. They see the answer in advance of the proof.
JMac, I like that image. I read chaos theory back in the 90s and tried to explain to my partner how it applied to everything from weather patterns to the stock market, left a book on his desk and picked it up after it lay in the same place for six months. He was a trader and I think it bothered him to think that the movements of the market over time were basically increasingly unpredictable.
"i work my way in reverse order"
Sort of like visualize the David and then take away everything that is not him?
I came to love the elegance within mathematics as a child discovering the mysteries of the number 9. Later, in the 8th grade, algebra fascinated me, and the next year, geometry had me in its grip.
As a musician, I readily understood and admired the number relations within time and tempo and the structure of tones. It still serves me well when I want to transpose a tune from one key to another, or figure out the chord patterns or harmonies behind a melody line.
It was well after most of my formal schooling was done that I found even more to love. In preparation for transferring my New York State teaching certs to Washington State, I found I needed 6 college credits of mathematics above and beyond the 6 credits I had when I got my Art degree. The options were either to enroll in two expensive and time-consuming college courses, or try to teach myself and take a qualifying exam. I chose the latter, and had a ball learning it. Trigonometry and more algebra! Set and number theory! Probability! Whoooo! Gloriously logical and so beautifully constructed.
So, yep. mathematics can be beautiful.
I get it in music easier than math. I like the beauty of logic and concise expression.
Rosi, thanks for bringing logic and probability into the discussion. Those two, to me, are beautiful. Although statistics was not intuitive, I used it all of the time when I worked and loved feeding in all of the data and waiting to see how things worked out.
And, tabg, geometry was the math course that I loved. It was the only branch of math that from the very beginning was intuitively easy for me. When asked for a proof I knew something was, or was not, true. It was only a matter of going through the motions..
True story; my partner when I was still working and I decided to get a cup of coffee at lunch. We each picked up one of the flattened insulators to slip on our cups and he said, "Hard to guess that that becomes a cylinder." And I said, "That's because it's a truncated cone." He was a chemical engineer before going to medical school and I was a biology major. We laughed and he said, "That's the difference in an algebra mind and a geometry mind." I think that was the difference; we just see the world in equally valid but different ways.
Still, the mathematicians who see beauty in math have the same reactions in their brains when they see an elegant formula that others have on viewing one of the impressionist masters' paintings.
I once walked into a room at the Chicago Art Museum unprepared for the fact that I was going to be coming face to face with one of Van Gogh's Sunflowers paintings and almost went to my knees. It was like a religious experience.
I felt bad for Ed, the physics professor, who must have felt alone as the only one in the room who found mathematics beautiful.
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