neck; there were verses from the Qur’an he was to recite every day. The boy he had been sat at the edge of the worn velvet seat and trembled; after two weeks of treatment, when his mother asked him about the farishte, he had said:
“They’re gone.”
That was a lie.
My theory stands as firm as a rock; every arrow directed against it will quickly return to the archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.
—Georg Cantor, German mathematician (1845–1918)
In a finite world, Abdul Karim ponders infinity. He has met infinities of various kinds in mathematics. If mathematics is the language of Nature, then it follows that there are infinities in the physical world around us as well. They confound us because we are such limited things. Our lives, our science, our religions are all smaller than the cosmos. Is the cosmos infinite? Perhaps. As far as we are concerned, it might as well be.
In mathematics there is the sequence of natural numbers, walking like small, determined soldiers into infinity. But there are less obvious infinities as well, as Abdul Karim knows. Draw a straight line, mark zero on one end and the number one at the other. How many numbers between zero and one? If you start counting now, you’ll still be counting when the universe ends, and you’ll be nowhere near one. In your journey from one end to the other you’ll encounter the rational numbers and the irrational numbers, most notably the transcendentals. The transcendental numbers are the most intriguing—you can’t generate them from integers by division, or by solving simple equations. Yet in the simple number line there are nearly impenetrable thickets of them; they are the densest, most numerous of all numbers. It is only when you take certain ratios like the circumference of a circle to its diameter, or add an infinite number of terms in a series, or negotiate the countless steps of infinite continued fractions, do these transcendental numbers emerge. The most famous of these is, of course, pi, 3.14159…, where there is an infinity of non-repeating numbers after the decimal point. The transcendentals! Theirs is a universe richer in infinities than we can imagine.
In finiteness—in that little stick of a number line—there is infinity. What a deep and beautiful concept, thinks Abdul Karim! Perhaps there are infinities in us too, universes of them.
The prime numbers are another category that capture his imagination. The atoms of integer arithmetic, the select few that generate all other integers, as the letters of an alphabet generate all words. There are an infinite number of primes, as befits what he thinks of as God’s alphabet…
How ineffably mysterious the primes are! They seem to occur at random in the sequence of numbers: 2, 3, 5, 7, 11…There is no way to predict the next number in the sequence without actually testing it. No formula that generates all the primes. And yet, there is a mysterious regularity in these numbers that has eluded the greatest mathematicians of the world. Glimpsed by Riemann, but as yet unproven, there are hints of order so deep, so profound, that it is as yet beyond us.
To look for infinity in an apparently finite world—what nobler occupation for a human being, and one like Abdul Karim, in particular?
As a child he questioned the elders at the mosque: What does it mean to say that Allah is simultaneously one, and infinite? When he was older he read the philosophies of Al Kindi and Al Ghazali, Ibn Sina and Iqbal, but his restless mind found no answers. For much of his life he has been convinced that mathematics, not the quarrels of philosophers, is the key to the deepest mysteries.
He wonders whether the farishte that have kept him company all his life know the answer to what he seeks. Sometimes, when he sees one at the edge of his vision, he asks a question into the silence. Without turning around.
Is the Riemann Hypothesis true?
Silence.
Are prime numbers the key to understanding infinity?
Silence.
Is there a connection between transcendental numbers and the primes?
There has never been an answer.
But sometimes, a hint, a whisper of a voice that speaks in his mind. Abdul Karim does not know whether his mind is playing tricks upon him or