(4 April, 2011)
“These are the kinds of mathematical puzzles that make An Introduction to Number Theory a treat for anyone who loves numbers.” (source)
“For anyone who loves numbers”?
Admittedly that statement is about number theory; but as I see it often applied to all of mathematics, I must vent a bit nevertheless.
I don't love numbers. I don't give a toss about numbers.
Now, their manipulation, making them the slaves for the building of the pyramids of mathematics… that's something! Who cares if the building blocks are numbers or essences of bovine geometry, if only they bend into groups where Hölder's inequality and Banach fixed point theorem hold!
Likewise, would you express your love of some very dear and special person as “a love of meat”? Or “a deep attachment to cells”? That would be a stupid, monstrous way to put it — it is not the individual pounds of flesh that you care for, but their combination, the structure which can smile and laugh and wink. What parts that person is made of does not matter much; your beloved could be made of silicon or virtual bits, as long as those parts could combine with the same thought in a smile, a laugh, and a wink.
Indeed, mathematics quickly and easily ascends into levels where you no longer speak of numbers — you manipulate objects to discover their interrelations, but you don't care at all what those objects are. This is better, naturally, because what you discover will hold true for any thingamajics that fulfill those few conditions you assume: you're not investigating and laying out just the structure of numbers, but of anything with enough rudimentary structure in it!
So with the examples I mentioned above: Hölder's inequality works for diverse objects. You might quibble and say you need numbers to have an inequality — very well then, but Hölder's is an inequality of the norms of two objects, and norms are a way of assigning a number (a “length” or a “size”) to each of a collection of objects. And what the inequality says, well, I can give you a nebulous definition right now: to have gotten so far, we must have some way of relating those objects to each other that we call “multiplication”: if we deal in cows, a way of looking at any two, and arriving at a third. And the inequality says that the norm (the “length”) of two objects so “multiplied” is not more than the norms (“lengths”) of those two taken separately and then multiplied as normal numbers are. If a and b are our objects and ||x|| stands for the norm of x, the inequality is this:\[ \|ab\| \leq \|a\| \|b\|. \]
What this hand-waving fails to convey is that such generalities are not drawn from numerous examples — no, the examples issue from this generality, which holds true for any specific case we might want to pursue. Mathematically proven truths stay true always — but the useful truths are not those of specific single cases. No, having found a truth in one area, we can look at the assumptions which the proof of that truth requires, and find out in which other realms assumptions like that hold. And thus, out of the realm of (say) mere numbers rises a Hölder's inequality for functions — for infinite sequences — for essences of cows if you were of a mind to define such — for so many cases that the profit of enumerating them pales in comparison with the pursuit of the general case; the jewels of each particular will follow from that much surer than a dawn follows the darkest night.
And in each particular case where those basic assumptions hold, the same structure sways into being without a second thought — the same chain of insights, of propositions, of lemmata, of theorems, of theories — in each of them, Hölder's theorem gives rise to Minkowski's inequality (the third side of a triangle is shorter than the other two together), which gives rise to Banach's theorem, and to innumerable results like a million towering crystal palaces of different hues in a million worlds of differing splendor, all after you've built them just once in the pale etheric realm of mathematical structure!
So no, I don't love numbers particularly much. Numbers are dull. The structure of which they are but one example is the pretty goddess which we people of mathematics worship and adore. She's the one at whose feet our love properly belongs.
(Or maybe this all is me quibbling over a choice of rhetoric; but in a pinch any pastime will do. For actual definitions, consult Wikipedia: Hölder's inequality, Minkowski's inequality, Banach fixed point theorem.)