Mathematics was long considered to be an absolute body of knowledge. Its truths were held to be universal, eternal and absolute. Euclid’s geometry for example was believed to be the gold standard in human certainty. But over time mathematicians came to realize that the truth of their theorems is contingent on underlying axioms—and underlying axioms, even the seemingly ‘obvious’ ones were often seen to be false. As such, Einstein and Eddington were able to prove that Euclid’s parallel postulate did not hold for space, and therefore theorems based on the parallel postulate were not true of space.

The question then arose: What is mathematics? One school of thought (The Platonists) continued to believe that mathematics ‘discovers’ absolute truths about the universe. They held that the errors made by Euclid and others did not take-away from the essential quest of mathematics: absolute certainty. An opposing group of mathematicians (The Formalists) pointed to the fallibility of axioms and held that mathematics is a game, somewhat like chess, where symbols were manipulated according to rules legislated by axioms. Neither the theorems nor the axioms were true (or false) in any sense.

These schools—established in the early part of the twentieth century—hardened their positions and nothing much happened to resolve questions about the essential nature of mathematics. Then, a few years ago the mathematician-philosopher Hersh wrote that mathematics is a human activity, and the truths we discover about it are a function of our culture, values and biology. According to Hersh a different intelligence from ours would come up with radically different mathematics.

I understand Hersh’s argument and motivation. Large parts of mathematics seem to be sociologically driven, and he neatly avoids the Platonist-Formalist conundrum. But there is a natural question that arises: What about numbers? Are they a human invention or must any intelligence eventually find their way to 1, 2, 3…

It certainly seems that consciousness forces us to distinguish the ‘I’ from the other. As soon as an intelligence becomes aware of this separateness, they will need and invent numbers. But wait, Hersh might say—perhaps individual consciousness is a human peculiarity; perhaps there are intelligences that are collective in some sense and would find our notion of I-ness to be peculiar. Perhaps their bodies are intermingled with each other’s and they have a group-consciousness rather than a self-consciousness. For example, their intelligence could live in a gaseous cloud on some Jupiter like planet.

Hard to argue with that, but let us push a little. This gaseous, intelligent cloud, wouldn’t it observe things around it—wouldn’t it want to count stars for example? No, Hersh would argue. This gaseous cloud might live in a some homogenous, featureless cloudy soup where there were no discreteness of any kind.

But what about time? Wouldn’t any intelligence evolve to be aware of time? And as soon as they become aware of time, they will need to measure it, and once they need to measure it they will invent numbers. With numbers they would discover that 2 + 2 = 4, and with that they would get the richness of our number theory. Is it possible, Mr. Hersh, to visualize an intelligence that does not need time?

“Well,” he might say. “Why not?”

‘Why not’, indeed. There is no way to be sure either way. But surely one can see that only a barren intelligence—one unaware of time, distance, size and shape—could conceive of a mathematics that does not include numbers. Could they then even qualify as an intelligence?

*Check out http://www.edge.org/q2005/q05_7.html#sabbagh for a related post *--Gaurav Suri