The Edges of Reason
My younger brother is a musician and mathematician, an adventurer whose inquiries lead him to landscapes most people would never dream existed, much less find themselves exploring them. It isn’t enough to walk along planar geometries and simple proofs – that would be to walk in the Garden without admiring the flowers. The walk is well-trodden, and the flowers are the Mystery. So these flowers attract his attention, and once in a while we share some of their colors.
I dare say you would not be interested in the topics of our dialog; but then again, you might find some of the thoughts similar to considering the space between stars, or the far end of the Universe, should there be one, or its conception, if it had one. Ah, ha! You see, these days existentialism doesn’t score as well on the Nielsen Ratings as it used to.
Once you get a whiff of the Mystery, though, you begin to get this strangely comforting feeling that things are, after all, much larger than you projected them to be, and you (by comparison) are much smaller, a little bit less in control of your destiny than you imagined. It is a sweet gift; when your muscles aren’t bound up in such great effort to prove your permanence or grandeur, your relationships with people invariably improve. Not convinced? Then don’t read on, I’m not trying to convince you.
Infinite series. A number or number pattern which continues forever, without presenting a predictable sequence. How awfully dry, like chewing on stale crackers, don’t you think? Surely you remember learning about the number Pi: the number used to calculate properties of circles. Ok, I need to remember 3 or four digits of Pi to get close enough to the right answer.
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609...
In a human life there is either faith or its absence. In my case, while I maintain a certain distance from organized religion (praising the ethical and social basis they intend to create in this world), my faith in an ordered universe is boundless. I am shaken by human abuses, or a friend’s mortality, or my own, but it doesn’t change my belief in an underlying order in the world - at least it hasn’t yet; instead it breaks fragile assumptions, and forces me to look deeper for the structure that is inherent in everything.
Everything is in everything else.
So if some bit of this world we know keeps snaking out of our grasp, that is the most interesting place to investigate. Why, if there is such a thing as boundlessness in the world, might I not be boundless as well? I am of the same stuff as all of this I see around me. If the infinite exists, am I not part of it?
The number Pi has digits whose order does not repeat or present a pattern, for as far out as supercomputers have been able to calculate it. For thousands of years, we have known about the… sacredness… of this number, and manual calculation attained accuracy from two, to five, to a dozen digits. In the 19th century, a mathematician devoting a lifetime of calculation reached 707 digits (with one minor mistake at number 538, which invalidated the remaining 162); later attempts arrived at 800 and some, without errors. With modern tools we have stubbornly taken a run at it, digging deeper and deeper to get to the bottom of the thing… successfully without success. To one million digits and more, there is no bottom.
The same occurs when you run at subatomic particles, when you spin matter faster and faster and smash it together in nuclear acceleration, to break it into bits small enough to be called fundamental building blocks of creation… it all comes apart in your high-energy fingers, or becomes a reflection of your messing about, instead of the picture of a world independent of your messing about. Ah, the approximations are miraculous, they gave us the tunnel diode and so much engineering genius that the world – at least the material-rich industrialized world… at least, the wealthy of the material-rich industrialized world – live with what would some short years ago have been considered magic (or blasphemy, your pick).
This week, my brother the inventor and mathematician and musician brought the series of prime numbers to my attention: innocuous enough, not as silly as the number Pi with its perverse infinitude of decimal places, just simple integers: 1, 2, 3, 5, 7, 11, 13, 17… Problem is, the sequence of prime numbers has no pattern. Ever. Like the arrangement of stars in the sky. The space left between one prime number and the next has no pattern, like the space between the stars. Or thoughts and the spaces between thoughts. My brother was looking at the factors making up numbers as they grew in size, the integers which, multiplied together, would generate the number in question. He looked at the space as it grew larger and larger between successive primes. Is there not a handle to grasp, somewhere?
Big deal? It is a big deal, because you are smaller than you think you are, and a lot less permanent than you hoped you were, at least in the present packaging. So, sooner or later, the toys of occupation and entertainment you twiddle in your fingers become - sorry to say it - irrelevant. And what seemed to be completely useless bits of information become flashes of light emanating from a single incomprehensible Source.
The handle on transcendentals is beautifully beyond our grasp. But in that strange boot-strapping sleight of hand, we can create a handle for the Handle. They can be categorized because they live outside of our categories. They can be used if we dull them down so human hands can wield them; on their own they are the sharpest edges in the universe. They point to everything we have not yet understood, so looking at any one of them, as deeply as one can, is looking at every one of them, all these things at the margins of our command: pi, primes, big bang or oscillating universe, black holes, births and deaths: the edges of reason, where real reason lives.