My interest in mathematics is more personal than it is academic. I learned what I know formally, in the usual sequence of undergraduate and graduate math courses. But it has penetrated my personal life, and I have come to see mathematics as deeply rooted in a fundamental human drive to live more, or to live more fully. It is this conviction that has motivated me to search for mathematics in sensory mechanisms, in the ways we learn, or the ways we make art and music. Some of mathematics’ historical development has fascinated me because there one can see how the combined effect of intuition and rigor reveal the power of a precise examination of thought. Other math enthusiasts share my desire to search out its ubiquitous presence in nature, happy to find some of our imagined possibilities in, for example, the navigational processes of insects. But it is only in Daniel Tammet’s latest book, Thinking in Numbers, that I find the unexpected glimpses of mathematics that can tell us more about how it lives in us.
I wrote about Tammet’s book last week. It is interesting to me that, while Tammet’s experience bears little resemblance to mine, his view of mathematics has a good deal in common with mine. Mathematics is woven into Tammets experience in a very personal way – in his sensations – making it all much more immediate for him. For me, it was by taking classes that I learned the power of its precision, got a look at its highly imaginative structure, and was able to enjoy its often surprising results. These had an emotional impact on my own search for meaning. For Tammet and for me, mathematics is an aspect of life itself. So today I decided to highlight two of his observations that I found provocative (and perhaps unexpected). They each focus on infinities, the source of mathematics’ great richness as well as its many paradoxes. The first of these is Tammet’s discussion of how the calculus informed Tolstoy. The other is his view of the infinite stories contained in dreams. (Tammet also tells his story about Tolstoy in this short piece published by The Guardian this past August).
Tammet quotes from War and Peace in both The Guardian piece and the book:
“The movement of humanity, arising as it does from innumerable arbitrary human wills, is continuous,” he writes. “To understand the laws of this continuous movement is the aim of history … only by taking infinitesimal units for observation … and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.”
And he adds:
Mathematics, Tolstoy understood, is like literature: a way in which the world expresses itself. Words and numbers: both allow us to entertain pure possibilities, immune from prior experience or expectation. Perhaps that is why some of Count Leo’s closest friends were mathematicians.
I particularly like the phrase, “a way in which the world expresses itself.” Tolstoy used the language of calculus to criticize historians for what he considered erroneous simplifications and for their failure to grasp the significance of change brought about by a multitude of infinitely small actions. Change itself, Tammet points out, is generally misperceived.
Change appears to us mysterious because it is invisible. It is impossible to see a tree grow tall or a man grow old, except with the precarious imagination of hindsight. A tree is small and later it is tall. A man is young, and later he is old. A people are at peace, and later they are at war. In each case, the intermediate states are at once infinitely many and infinitely complex, which is why they exceed our finite perceptions.
Mathematics often reveals the way our finite perceptions are misleading and it corrects them. In The Guardian piece, Tammet provides a link to a paper written by Stephen Ahearn for the Mathematical Association of America on the same topic in 2005.
In another essay entitled Book of Books, Tammet reflects on sleep.
Our dreams contain the infinite. Uninhibited by wakefulness, words and pictures and emotions circulate and combine freely inside our head….Dreams defy our finite scrutiny; too often they evaporate in the narrow light of day…Like a book, like a life, where does the explanation start? A dream has no beginning, and therefore no middle and no end.
This reflection is connected to the, often made but little understood, observation – that the unconscious mind can solve problems and write stories. According to Tammet, “the Unconscious mind has authored some of the greatest works in literature: Goethe and Coleridge are only two of its pseudonyms.” And in this context Tammet talks about how Vladimir Nobokov wrote his famous Lolita. He explains that the novel
began life on a long series of here-by-five index cards. He sketched out the story’s closing scenes first. On subsequent cards Nabokov jotted down not only paragraphs of text but also plot ideas and other bits of information…Every so often Nobokov would rearrange his index cards, searching for the most promising combination of scenes.
Tammet imagines some of the potential versions of Lolita, ones that would be viable alternatives, and suggests that there are more than a million of them. This imagined array of possibilities, based essentially on the possible permutations of the index cards that make up its three hundred fifty plus pages of text, is particularly interesting to me. It is about meaning-making as well as permutations, which is what the mind does all of the time, including what happens in our dreams. But Tammet is placing his attention on the meaning-making itself, the possibilities for other stories, other ways, other lives. And this admonition about the presence of these alternatives is one that mathematics has always provided me
Nabokov himself makes a math analogy:
Reality is a very subjective affair…You can get nearer and nearer, so to speak, to reality; but you never get near enough because reality is an infinite succession of steps, levels of perception, false bottoms, and hence unquenchable, unattainable.
The question that comes up in my mind is how is it that we come to explore these intuitions precisely. It is the precision of math ideas that will make mathematics uninteresting or irritating to the student who happens to be required to learn them. But it is precisely this precision that permits us access to what we cannot easily see about ourselves. So how did we come to intuit the enormous value in formalizing what seem to be living processes? This is a big question, and I won’t attempt to answer it, but feel free to let me know what you think.
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