Atiyah's Jellyfish

Mario Livio wrote

MICHAEL ATIYAH, one of the greatest mathematicians of the 20th century, has presented an elegant thought experiment that reveals just how perception colors which mathematical concepts we embrace-- even ones as seemingly fundamental as numbers. German mathematician Leopold Kronecker famously declared, “God created the natural numbers, all else is the work of man.” But imagine if the intelligence in our world resided not with humankind but rather with a singular, isolated jellyfish, floating deep in the Pacific Ocean. Everything in its experience would be continuous, from the flow of the surrounding water to its fluctuating temperature and pressure. In such an environment, lacking individual objects or indeed anything discrete, would the concept of number arise? If there were nothing to count, would numbers exist?

He was quoting part of a Times Higher Education review. The relevant passage is

But let us imagine that intelligence had resided, not in mankind, but in some vast solitary and isolated jellyfish, deep in the depths of the Pacific. It would have no experience of individual objects, only with the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count.

Even more fundamentally, in a purely static universe without the notion of time, causality would disappear and with it that of logical implication and of mathematical proof. Connes actually alludes to this philosophical dilemma in the context of relativistic cosmology.

This is an argument that mathematics is human invention, not discovery. Need we take it seriously?

Let's dismiss the static universe first. In a static universe, thought as we know it could not exist. There would be nothing that could move from a state of ignorance to a state of knowledge, because there could be no change. So there could be no invention and no discovery. But still, in such a universe, if there were a ball in one place and a ball in another place, there would be at least two balls. Mathematics could not be done in such a universe, but it could still be true about things in it.

Now let's take the jellyfish. Have you ever watched a real jellyfish? If not, try this short video of a deep sea jellyfish. Does a jellyfish have something discrete to count? Yes indeed, its own pulsations. So if a jellyfish were able to think, it would be able to discover counting.

But would either of these thought experiments, if sound, establish anything to the purpose? No, both of them are fully consistent with the view that mathematics is discovered, not invented. In the static universe, mathematics would not be in the thoughts of any agents, because there could be no activity, no agents, and no thoughts. So nothing could be said to know mathematics. That does not mean that mathematics would not exist! The imaginary jellyfish could not be said to know mathematics either. Again, that does not mean mathematics would not exist, only that the jellyfish wouldn't know about it. Only if you start from the presupposition that mathematics cannot exist unless it is invented would these examples be relevant at all.

Let me offer a different example: a game player (possibly a human, possibly a program) wandering through a maze/dungeon. There are coins, potions, scrolls, and an abundance of deadly enemies scattered throughout the dungeon.

I suggest that it is precisely the same with mathematics and science generally. There is a human element to mathematics; there is a social element to science. The questions we ask are influenced by our nature, environment, society, and desires. The answers we eventually get are not. Whether a culture is interested in counting is a contingent fact about that culture. What base they count in, if they are interested, is a contingent fact about that culture. How far their number names go is a contingent fact about that culture. What laws of arithmetic they are aware of is a contingent fact about that culture. Whether they can prove anything about them from axioms by sound rules of inference, or even have the concept of proof, is a contingent fact about that culture. But 3*3+4*4=5*5 is a brute fact, not contingent on any culture.

As I understand it, no Platonist would dream of denying that our interest in counting or the ways we express it or our awareness of any specific law of arithmetic or whether we have the idea of proof or what we are able to prove are contingent on our nature, experience, and history. The Platonic claim is that once you are able to ask whether 3*3+4*4=5*5 you find that it is, and that that fact has nothing to do with us.

But what about geometry, where for millennia people thought Euclidean geometry was the only geometry, then non-Euclidean geometries were discovered, and then one of them was found to fit the world of Physics better than Euclidean geometry did? Or what about set theory, where it turned out that the Axiom of Choice is independent of the other axioms?

A Platonist would say that in both cases, people discovered something and thought it was one thing. They turned out to be mistaken. There is nothing surprising about people being mistaken. If mathematics were a human invention, people could just define the problem away by saying that only Euclidean geometry is real and it would be so. But if people discovered geometry, nothing would be more natural than its discovers being mistaken about it. For example, there is a place in New Zealand called “Banks Peninsula”. When James Cook's expedition saw it in 1770, he called it “Banks Island”, and the mistake was not realised until 1809. Now we know there is more than one geometry and more than one set theory. So? Just what you would expect if mathematics is discovered, not invented.