Tag Archives: mathematics

Simple and Elegant Do Not Mean Easy

Gone are the two theories, gone their troubles and delicious reflections in one another, their furtive caresses, their inexplicable quarrels; alas, we have but one theory, whose majestic beauty can no longer excite us. Nothing is more fertile than these illicit liaisons…
André Weil quoted in Edward Frenkel’s Love and Math (103)

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Simple and elegant Swedish bread twists with almonds and cardamom: an essential component of higher math comprehension

Math is passion. And like passion, it has its dark side. I have written before about the identity crisis that maths seem to provoke ( here and  here), I admit I may have some slight obsession with the subject. In my own life I have, like others, found deep peace and contentment in the objective exactitude of math, but, also like others, when math seems to veer off the course of what we have understood as the applied rules, it is deeply unsettling.

In math, the problem is always well defined, and there is no ambiguity about what solving it means, you either solve it or you don’t (56).

After reading Edward Frenkel’s very fine book Love and Math, I feel I may have come to glimpse the nature of my fascination with math theory. I don’t believe it is our fault that our world-views are thrown by concepts such as imaginary numbers. The trick is, I think,  one must push through the black hole of comprehension that engulfs us, otherwise it is very easy to lose our psychological footing. I believe it all comes down to subjective versus objective truths.

Mathematics is separate from both the physical world and the mental world (234)

Let me first say that I do not in any way want to present myself as somebody who read Love and Math with anything close to full comprehension of the complex and creative math that Frenkel heroically tried to bring within my intellectual reach. But that is not the point. It’s not why I seemed unable to put the book down, nor is it, I think, why he wrote it.

Indeed, the square of any real number must be positive or 0, so it cannot be equal to -1. So unlike √2 and -√2, the numbers √-1 and -√-1 are not real numbers. But so what? (101).

BUT SO WHAT????!!! SO WHAT!? That is the very heart and soul of the identity crisis of myself and many others!? Not so what!? Math is objective. What is the meaning of truth? Where are we then? Who am I? What is real? Why do I matter, oh god, what is the meaning of life? But wait….hang on…a light, a sliver of understanding…while Frankel described how it was in fact true that 2+2=1, I had a Eureka! moment. Yes. I see it! It is true 2+2 does equal 1. The truth is not altered. The truth is objective, it is only the means by which I got there, the translation I used, that altered. The solution is “created” but that creation has nothing to do with the solution other than its ability to allow us to perceive what is already there: the truth. That’s objectivity on an entirely different order. Wow. What a moment. It’s true, it’s like falling in love.

The deeper I delved into math, the more my fascination grew, the more I wanted to know. This is what happens when you fall in love (28).

I believe that our subjectivity is absolute. Inescapable. The only measure by which to ground ourselves in our subjectivity, however, is the purely objective language of math. It’s pure objectivity profoundly orients us. It is the discrete objectivity of math that connects. What a marvelous completeness the totality of subjective and objective truths gives us.

In truth, the process of creating new mathematics is a passionate pursuit, a deeply personal experience, just like creating art and music. It requires love and dedication, a struggle with the unknown and with oneself, which elicits strong emotions (233).

Frenkel’s book is wonderful on multiple fronts, his personal history growing up towards the end of Communist Russia, describing his struggles to overcome the systemic anti-semitism that pervaded the culture, is riveting. His charming delight connecting math to all aspects of life culminating in his 2010 film, Rites of Love and Math, is inspiring and beautiful. He draws on every aspect of life to help bring understanding to the complex math he is explaining, for example he refers to his mother’s borscht recipe to explain particle content of quantum field theory. This , however, brings me to a very serious breakdown in my comprehension, to which I must bring Frenkel to task:

For example, let’s look at this recipe of the Russian soup borscht, a perennial favorite in my home country. My mom makes the best one (of course!). […] Obviously, I have to keep my mom’s recipe secret. But here’s a recipe I found online (196).

My dear Mr. Frenkel, I am afraid that that is not at all “obvious” to me. Please explain, or send recipe.

*title from pg 201

 

 

Bastard Reasoning

What can be nothing one moment and something the next, yet disappears in the presence of anything? –  Robert Kaplan, The Nothing That Is (59)

IMG_0042 The other day, my fourteen year old son pointed out that life and death are not antonyms, “You can’t have death without life, therefore the opposite of life is not death, it’s nothing.” Nothing, as in, an absence- not even an absence- a void without context- Well, what is that? my other sons and I wondered… I had cause to think on this thought further as I was coincidentally reading The Nothing That Is: A Natural History of Zero by Robert Kaplan.

Zero is neither negative or positive, but the narrowest of no-man’s land between those two kingdoms. (190)

Kaplan takes the reader through the transition of numbers from mere adjectives to nouns in their own right, and then he hits us with the mystery and enigma that is zero. Either Kaplan is an extremely clear and gifted writer or my math skills are far more impressive than I ever knew. Alright, settle down, Jessica, that’s a bit of an overstatement. Maybe. It may be a simple thing for most people to get their heads around why any number to the power of zero is one, but I got through college algebra getting the answers right without knowing what it really meant, but which now, thanks to Mr. Kaplan, I do. The next day, I tried to get my eighteen year old son to appreciate how exciting it was that I actually understand the concept, but he wasn’t up for my enthusiasm at 8 am, if ever.

But I digress-  before Kaplan even gets into what zero is (or isn’t), he gives an account of the possible ways by which it came to be understood at all.

If you favor the explanation that the ‘O’ was devised by the Greeks without reference to their alphabet, its arbitrariness is lessened by noticing how often nature supplies us with circular hollows: from an open mouth to the faintly outlined dark of the moon; from craters to wounds. ‘Skulls and seeds and all good things are round,’ wrote Nabokov. (18)

One of my favorite images from this natural history is the method for computations that the ancient Hindus used: a board covered in sand to mark the numbers, subtractions, and additions as they went along. Kaplan tells us their word for “higher computations” is dhuli-kharma, ‘sand-work.’ But what is most intriguing to me  is the more metaphysical idea that the way in which they expressed zero (mostly as a place-value marker- which was a huge development) was by a simple finger impression, a dent of nothing formed by something…there is something perfectly beautiful in that…

Once zero was an official thing then things got a little complicated. Zero takes us out of the realm of  nouns into travels as a verb and things get a lot freaky. A number to the power of zero is one, but what of zero to the power of zero? How can that equal one and also, zero? What about division? Division is the first clue, in fact, that we have problems comprehending the magnitude and minuscule nature of the slippery zero.

‘Allez en avant et la foi vous viendra,’ said French mathematician d’Alembert: ‘Just go ahead and faith will follow.’ (157)

That math so elegantly sums up all the mystery and power of our universe is something I find fascinating. I love books such as this that make some of that wonder accessible to my rather limited mind.

After all, maybe, like zero, we are all indivisible in our center. Knowing we’re nothing, but only in the context of our absolute something. I kind of love that idea- we are one with zero. Kaplan draws from a gallimaufry of disciplines in  a poetic, profound and valiant attempt to describe zero, that “pure holding apart,” concept to which zero lends and points itself. The poetic justice of taking our psychologically linear perspective and wrapping it around into the perfect symbol- 0, stretches all boundaries of philosophy and meaning: circumference – everywhere; center – nowhere* …by the end of the book I felt as though my skirt had been caught up in the door of a moving car driving around in circles and I was just holding on for my life. When it stopped,  all I could do was smooth the tangents from my shirt, straighten the x-axis of my skirt and say: Kids,  I got nothing, I have no answers.  But whew, that was fun!

Opposites are an illusion of language. Something and nothing, you know, are equally false substantives. (218)

*Sphaera cuius centrum ubique, circumferentia nullibi

**“…space, which is everlasting, providing a situation for all things that come into Being, but itself apprehended without the senses by a sort of bastard reasoning…” Plato’s Timaeus quoted (63)