Tag Archives: math

The Joy of Circles

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If ever there was a book that perfectly summed up the case for why I love books, [The Archimedes Codex] How a Medieval Prayer Book is Revealing the True Genius of Antiquity’s Greatest Scientist by Reviel Netz and William Noel would be exhibit A. The reasons why, as points covered in this wonderfully entertaining read for bibliophiles and lovers of multidisciplinary fields in action, include, but are not limited to, the following:

  1. The book as a material object
  2. The book as a historical record
  3. The book as a conveyor of information
  4. The book as a technology
  5. The book as an advancer of technology

All this and more comes together in one. Noel and Netz take turns in the telling according to their areas of expertise. Noel, as Curator of Manuscripts at the Walters Museum in Baltimore was given the opportunity by the anonymous owner of the codex to steward the study of this famous palimpsest. Netz’s specialty is in ancient science, and so between the two we get a very through understanding and deconstruction (literally) of book provenance, structure (with forays into paper, ink, and binding), forgeries, conservation, and cutting-edge methods of reading the unreadable, as well as a brief history of Archimedes, his impact on the whole history of math and science, the differences between how math was approached in ancient Greece compared to our own age, and quite a bit of the actual math involved. For me, it was a thrilling read. History, science, math, literature, and book studies all in a single object—the most ubiquitous and under-rated technological wonder of them all: a humble book.

A palimpsest, for those not familiar with the term, is a document (in this case a codex, which is a book in our familiar form as opposed to a book in scroll form, say) which has been erased (in this case, scraped away off the parchment, as opposed to erased off of paper) and written over again. What looked like a simple prayer book, was actually written over several books of Archimedes. Of those Method survives in the palimpsest alone. No where else! What may seem to be an act of unforgivable folly—using Archimedes text as scrap paper! is the very thing that allowed its improbable survival. And so we are grateful.

The process of reading the Archimedes text underneath the prayer book (and to add extra fun to the challenge, a modern-day forgery of illuminated illustrations), is difficult difficult lemon difficult* not to mention painstaking. I will admit that I have at least a passing interest in rare books and book conservation, so the technical aspects of the work of uncovering the text was fascinating to read. But, I would think it interesting to any reader if for no other reason than to gain a better understanding and measure of respect for a book’s structure and material evolution (or de-evolution as is sometimes the case—I’m looking at you, acidic paper!)

But, fascinating too were the passages dedicated to Archimedes, his way of thinking, and enormous impact on science, in fact, some of the most sophisticated technology employed in the effort to read his text would not have been possible without his proofs and methods.

The revelations of Archimedes true intent in regard to the Stomachion, for instance, read like a mystery novel. The Archimedes Palimpsest, incredibly, has pushed back the historic timeline of when combinatorics were first thought to be robustly considered and developed. Combinatorics, I might add, had no practical use to Archimedes, and yet, without that particular field of mathematics, computers would not be possible and you would be sadly deprived of learning about this book from me. Full circle. Is there anything more satisfying?

*to randomly quote, as I am wont to do, the very funny film In the Loop

**Illustration from p 45 of [The Archimedes Codex] How a Medieval Prayer Book is Revealing the True Genius of Antiquity’s Greatest Scientist

The Nectar of Mathematics

It is better to do the right problem the wrong way than to do the wrong problem the right way.
Richard Hamming quoted, Julian Havil, Impossible: Surprising Solutions to Counterintuitive Conundrums (50)

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My kind of geometry: The Doughnut

I was deep into my morning walk a few weeks ago when a powerful craving for doughnuts caught up with me. But proper doughnuts require a little time and a small crowd to partake in the pleasure, so I waited until the right moment.

For every complex problem, there is a solution that is simple, neat, and wrong (H.L. Mencken quoted, 82).

I find that I tend to read a math book or two every year. I’m not sure what it is in me that compels me to plow through the complex equations that I have little to no real understanding of, but I do it anyway. I like the ideas that the math symbolizes, I suppose. I take a strange pleasure in relating events in my life to mathematical equations.

A recipe is like a math equation: n( x + y) (s/t/r) + nfº = Ne (That’s n ingredients, multiplied by speed and time of rotation, plus n degrees fahrenheit, equals the nectar of mathematics: in this case: Apple-cider doughnuts.). Of course we ran into some problems.

Now that we have complex numbers properly placed and our mind receptive to lurking difficulty, we will consider what should be a simple computation for a calculator (44).

Ah yes, the lurking difficulty. Well, that is something one must always be prepared for. I had my heart set on apple cider doughnuts. My children and I were all visiting friends who had kindly procured all the necessary ingredients. I only needed 1/2 cup of apple cider (which I would reduce to 2T) and my friend wondered what to do with rest as they didn’t care for cider. I told her not to worry, my boys would take care of that. The next morning, I awoke, ready to prepare the dough when I realized our error. I neglected to tell the boys that there had been a reason, other than their enjoyment and ever-lurking thirst, for the purchase of the cider. They had made quick work of it. Good communication is important. In math, baking and life—that holds true.

Put succinctly, to increase the chances of success the team must adopt the somewhat counterintuitive strategy of being wrong together, not correct together (53).

Something strange that I love about math, as it feeds some sort of philosophical truth I seek, is that not only can there be multiple ways to reach a solution, but there are multiple solutions to a problem. It just depends on what system, matrix, or units of measurement and/or data you are using. There is not as much firm ground as we like to think. There are just abstract ideas and evolving methods of problem-solving.

Of course making apple cider doughnuts is not that complex of a problem. I solved the equation, in fact, by a simple adjustment of words. Rather than making Apple-cider Doughnuts I replaced the 2T reduced apple cider with milk and renamed the solution: Plain Doughnuts.

*title from p 128: “Certainly, [the proof] is more secure and in looking at it we can taste the nectar of 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)

this + this = this + 2

Rabbit doesn’t want to tell him anything. The more he tells, the more he loses.
– John Updike,  Rabbit, Run

DSCI0013The question I am always interested in, and often even ask is, “What are you reading?” But what I really want to know is, “Why?” I worry that there might be a taint of accusation or judgment in the asking. But I mean it quite straightforwardly – how does one come to a book?

Lately, my reading has been influenced by others, combined with the order of the library of congress system. When I enter the red zone of the stacks in my library job to return a random book,  I can hear the catcalls of the surrounding volumes- “Hey Baby, check me out…” And I do. I hate to be rude.

Rabbit, Run was one such book. I have read a lot about John Updike, but never actually read any of his books. I always suspected I would not like them.

He sees in the dark she is frightened; her big black shape has that pocket in it, that his instinct feels like a tongue probing a pulled tooth. The air tells him he must be motionless; for no reason he wants to laugh. Her fear and his inner knowledge are so incongruous; he knows there is no harm in him. (75)

Initially I didn’t think I would relate to Rabbit, I am long past the particular stupidity of youth, that ridiculous time of life when you are suppose to make life-long decisions without having any concept of the time frame. Now more experienced, the stupidities at least take on the profundity of mortality. Still,  something in Rabbit’s brutal moments of truth and sweetness kept me reading. And then there were passages of writer-ly genius from Updike that  did for me what I always hope reading will – he made me stop reading, and think. It is that translation from the order of all the facts of the story into meaning that I really enjoy.

I have been watching a lecture by Buckminster Fuller. I love his ideas. He describes the comprehensive metaphysical universe as “the aggregate of all of humanity  consciously apprehended in communicative experience.” That’s a sentence that gives some serious pause. He is also very charming, adding an irrepressible hehe on the end of nearly all of his sentences. Framing his ideas in math and physics, the synergy of all of his “generalized principles” is wonderful.

The “this + this = this + 2” is what makes us human. It is the meaning that our limitless minds quite miraculously glean. As Rabbit flees from one experience to another, from one girl to another, the depressing physics is exposed. His immaturity and internal disconnect leaves his love in the behavioral stage. He doesn’t know what his love is. We only know how it behaves. It mysteriously comes and goes, he’s barely pierced by it when it flees. The physics of our lives, as Buckminster explains, is the path of least resistance. Rabbit thoughtlessly runs towards his missing center by whatever path is easiest at any given moment.

Rabbit leaves his old home depressed, with a feeling of his heart having slumped off center. (229)

But then. I stop reading and think. Is it possible that this idea explains the meaning behind all of our decisions and actions? I have made some pretty painful decisions in the last couple of years, but if I look at them in terms of the path of least resistance…well yes. What may look like a path of ridiculous hurdles and mountains of Everest proportions, is a water slide compared to the alternatives.

The path of least resistance either exposes the depth of feeling that makes it the easier route, or it reveals the shallowness of feeling: proving the path unworthy of being blazed. I have discovered that, for me, any path is fundamentally easier than one that is littered with the hidden landmines  of qualifiers and suppression. We can’t really know what anyone else finds more or less resistfull, but if we understand that that is the force that guides us, then we can begin to ask the why- why is this less resisting?

Rabbit never asks why it is easier for him to run. The results are tragic.

This childish mystery – the mystery of “any place,” prelude to the ultimate, “Why am I me?” – starts panic in his heart. (283)

The truth is there is nothing childish about it.  Just stop, and think. See what your chosen path reveals.

I find that our whole education system around the world is organized on the basis of the little child being ignorant. Assuming the little child is born, is going to have to be taught, in a sense is empty waiting for information to be given by the grown ups. And so, the little child demonstrates time and again an interest in the whole universe. The child is very enthusiastic about the planetarium. The little child asks the most beautiful questions abut the total universe, continually embarrassing the grown-ups who have become very specialized and can’t answer the great comprehensive questions. We find the child then, with his propensity to comprehend totality- willing to be synergetic. Yet our education is to say – never mind about that universe, come in here and I’m going to give you and A and a B and a C…  – Buckminster Fuller

2i – 1

Another kind of Mathematics

We know that one times one is one,
but an unicorn times a pear
have no idea what it is.
We know that five minus four is one
but a cloud minus a sailboat
have no idea what it is.
We know that eight
divided by eight is one,
but a mountain divided by a goat
have no idea what it is.
We know that one plus one is two,
but me and you, oh,
we have no idea what it is.

Oh, but a comforter
times a rabbit
is a red-headed one of course,
a cabbage divided by a flag
is a pig,
a horse minus a street-car
is an angel,
a cauliflower plus an egg
is an astragalus.

Only you and me
multiplied and divided
added and substracted
remain the same…

Vanish from my mind!
Come back in my heart!

Nichita Stãnescu,  translated from Romanian by George Mustea

Truth of Ho

"Eve"- oil clay by Victoria Accardi

Sometimes I come across a group or order of words that strike me as particularly funny or apt. The “truth of Ho” was a little gem from my statistics textbook (for anyone whose memory is being jogged, yes- Ho and Ha hypothesizes).

What is the truth of Ho? As I see it, it is the constant battle we are all engaged in with regard to the pressure to sell ourselves. Some sell their bodies but most sell out our inner selves to serve some relative societal expectation. It’s a really rude awakening  when the schism between whom you present and who you truly are is exposed. When you have to deaden yourself to get through the day, why, it can make one feel something like a whore.

I watched The French Lieutenant’s Woman the other night and although I do love Meryl Streep when it came to the quasi eponymous line: “I am the French Lieutenant’s…WHORE!” I laughed out loud. Oh for God’s sake – what does that even mean? I suppose that’s the point of the book/film. The female is accused of being a whore when she is clearly not, while the male protagonist is busy selling his soul and future to his bitchy nitwit of a fiancé and a bourgeois ideal.

I find myself amused to distraction at how language is used in my statistics class. For instance- the null hypothesis: it is always assumed to be true. So, if the null hypothesis is not true, then the alternative hypothesis must be true. Still with me? Good, because here is the fun part – you can reject the null hypothesis or  fail to reject said hypothesis. Who said mathematicians are not philosophers? That takes some deep thinking: you don’t simply choose to accept the alternative hypothesis, rather, the choice is to reject or fail to reject. If that doesn’t sum up the majority of decisions made in a lifetime…

Eve -Victoria Accardi (bottom back view)