The Infinite and the Infinitesimal
What is the blown-hair sword?
Each branch of coral holds up the moon.
– Blue Cliff Record, Case 100
It’s true that the most dazzling of mathematical constructs can be reduced, in some sense, to the integers: 1, 2, 3, … . Forget the small fry, like the paltry number needed for counting the books in the Library of Babel (see LoB in Counting To One, I); instead, focus on the dot dot dot business at the end. We use the ellipsis, or …, to indicate that these integers just keep marching off in an orderly, never-ending fashion. It seems a little strange that we can’t actually count all the counting numbers. And the ellipsis, or the lemniscate (∞), or “infinity,” does not really represent that which lies beyond wherever our counting gets to. These symbols don’t accurately describe neverendingness in the same precise (yet incomprehensible) manner that 301,312,000 describes the LoB book count. Rather, they point to the mathematical fact that “here be dragons.”
There are ways to accurately represent certain types of infinity, and they don’t demand any mathematical sophistication. They’re drawings. Here’s one of my favorites:
While this circle may not seem boundlessly large, the ability of the infinite to nestle within the finite is one of infinity’s unnerving qualities. It’s not required that one zoom off to the outer reaches of mental space to encounter the infinite; it lives in the neighborhood. In the case of a circle, the infinite naturally springs up in our attempt to precisely describe geometrical reality with number. I’ll flesh out how our circle embodies the infinite, but let me first put it in context.
Linking geometry to mathematics goes back at least several thousand years, as does the notion that mathematical thinking is as close as we can come to truth (or Truth, if you prefer an aura upgrade). Folks who have so thought, and think, are no slouches. The Math = Truth club includes: Pythagoras, Plato, Newton, Einstein, Russel, Godel and Hawking—to name just a few.1 One might call it the Certainty Club. It has a lot of appeal, and if one thinks that we live in an age of math/science, then we live in an age where the Certainty Club holds cultural sway. The certainty of mathematics rests on its precision, which mathematicians use to build structures of logic. The blown hair sword of reason is mathematics.
Since geometry arose from perception, its intimate connection with number has been a central area of mathematical exploration since Pythagoras.2 There have always been marvelous problems in attempting to map the continuum of geometry with discrete numbers. The Ancient Greek philosopher Zeno was a master at devising situations that revealed the enlivening gap between that map and its territory. His story of a race between Achilles and the tortoise nicely illustrates the dilemma.
The tortoise is given a head start, and Zeno uses that initial lead to prove that Achilles can never catch the tortoise. His technique is to analyze Achilles’s motion in terms of an infinite number of ever smaller motions. Basically, every time Achilles reaches the position where the tortoise was in some previous interval of time, the tortoise has moved to a new position beyond where he previously was. Here’s what that looks like: say the tortoise is 10 meters ahead of Achilles as the race starts, and that Achilles runs 100 times faster than the tortoise. By the time Achilles reaches the tortoise’s original position, the tortoise has moved 1/10 of a meter from that position; when Achilles reaches that point, the tortoise has moved 1/10 of a centimeter; when Achilles reaches there, the tortoise has moved 1/100 of a millimeter, and so on ( . . . ).
Zeno’s point is that it’s mathematically possible to slice up the race into an infinite number of parts. Since nothing can actually proceed through an infinite number of steps, the perceived motion violates logic, and therefore does not really occur. He concludes that motion is an illusion, or a delusion.3
Zeno’s logical sword cuts the weak link: the infinitesimal or infinitely small (1/∞). While we experience physical motion as smooth, not a series of discrete steps, whenever we try to mathematically describe this continuum of movement with numbers, the infinitesimal emerges. The abstraction of the infinitesimal is just as problematic as the abstraction to the infinite. In terms of counting, one can’t even count to one, if one counts by ever smaller fractions:
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …
And there’s the ellipsis, dark star of analysis. The infinite is not in the hinterlands, well beyond numbers like the LoB count; it lurks between zero and one.
A similar infinity awaits us in that simplest of geometrical shapes: the circle. If its diameter is the simplest of all numbers, one, then what is its circumference? You may recall from math class that such a circle has a circumference equal to “pi.” The number that “pi” represents cannot be written down, or computed by any computer – even one as large as the LoB. We can calculate it approximately:
Pi = 3.1415926535897932384626433832795028841971693993751058…
Please note the ellipsis. Whether this circumference is expressed to a forty digits or 301,312,000 digits makes no difference, either expression remains infinitely far from a full expression of the ratio of circle’s circumference to its diameter. The digits that unfurl are without pattern, i.e. completely random. The next integer in the series can never be predicted, but must always be calculated. Since this “number” has no pattern it is incompressible (unlike the number of LoB books, which is simple to write). There is no way to shorten its expression and still entirely capture it. Like life, neither predictable nor compressible, one needs to show up to see what occurs next, step by step.4
There’s more. It turns out that one can prove that the number of numbers like pi—numbers that require an infinite number of digits to express—far outnumber the number of “regular’” numbers—numbers that only require a finite number of digits to express. That is, virtually all numbers are of the type whose expression requires an endless series of digits, with virtually none of the numbers requiring only a “few” digits—like 301,312,000 digits.
While there are an infinite number of numbers whose expression requires only a finite number of digits, like 13 or 1/33 or books in LoB or 1/(books in LoB), there are a higher order infinite number of numbers whose expression requires an infinite number of digits, like pi or 1/pi. To get a sense of how many more of the pi-type numbers there are then the ‘regular’-type numbers, recall the progression of fractions found in Zeno’s paradox: 1/2, 1/4, 1/8, 1/16, 1/32, …1/8192, 1/16384, …, where the ‘space’ between adjacent fractions diminishes by a factor of two as the progression proceeds. Clearly there are an infinite number of such ever-smaller fractions packaged between 1/2 and 0, an example of the regular-number-type infinity.
One can mathematically verify that no matter how weensy the “space” between adjacent fractions become, one will find an infinite number of fractions like 1/pi—fractions requiring an infinite number of digits to express—sandwiched between any two neighboring regular-number fractions. An infinity of pi-type fractions lies within the gap between adjacent regular-type fractions found in Zeno’s progression. This remains the case even in the limit, as the gap between neighboring fractions is sliced to zero. That’s a measure of the infinity of pi-type numbers compared to the infinity of regular-type numbers.
The fact that one can use mathematics to demonstrate the logical necessity of this result speaks to the keenness of its edge. The mathematical blade is continually sharpened so that it may cleanly separate what had previously appeared muddled. But then each cut, while honing that logical edge on the resolution of the current paradox, at the same time produces new and more profound structure and paradox. Precise pruning makes for luxuriant profusion. Less makes more. If mathematics is the free creation of mind, then this process is neither dry nor barren, but mysterious and alive, teeming with surprising and incalculable creatures, rather like an underwater coral garden illuminated by moonlight. Mind swims—a penetrating brightness—into its own astonishment.
- Juzhi was once asked whether he thought that mathematics was as close as humans could come to the truth. His response of raising a single finger was sharp enough to draw blood.
- The truly classical example of this is the much-loved Pythagorian formula relating the lengths of the sides of a right triangle: A2 + B2 = C2.
- Zeno was no fool, neither was his teacher Parmenides. He totally got that swift Achilles could race past the tortoise. Since this experience violated logic, it clearly showed the senses were not to be trusted, if one were interested in knowing the truth.
- Wislawa Szymborska’s poem “Pi” gives a feel of this endlessness. Here’s the beginning:
The admirable number pi:
three point one four one.
All the following digits are also initial,
five nine two because it never ends.
It can’t be comprehended six five three five at a glance,
eight nine by calculation,
seven nine or imagination,
not even three two three eight by wit, that is, by comparison
four six to anything else
two six four three in the world.
The longest snake on earth calls it quits at about forty feet.
Likewise, snakes of myth and legend, though they may hold out a bit longer.
From Poems New and Collected, 1957-1997(Mariner Books)