In a recent post I said that one of the things that dissuades us from accepting the existence of a truly Platonic mathematical world, or believing in the timeless existence of its forms independent of human minds, is the habit we have of distinguishing ourselves from the rest of nature, despite all the evidence we’ve seen of nature’s tightly knit oneness. This is why Plato came to mind for me again, when I read an article about how fish can count.
The article describes some of the findings published in an animal cognition paper. Researchers found that angelfish can successfully distinguish between various sized angelfish groups, provided the groups differ by at least a 2:1 ratio. Data has also been collected that suggests that these fish can, more precisely, distinguish between the quantities 2 and 3. Since the 2:1 ratio doesn’t apply here, it is believed that this talent arises from a different neural mechanism. Researchers say that there is no reason the fish could not get beyond 3, but rather that there has been no particular advantage to them doing so.
A good deal of work has been done on the presence of some number sense in babies and other animals, often mammals. But these fish findings got my attention because fish, in our understanding of evolution, show up a few hundred million years earlier than mammals. And Robert Gerlai, who conducted much of the research in the article, played right into that observation when he said that given how widespread the basic math-related skills are throughout the animal kingdom, it’s possible that the action of counting, “could have arisen in one ancestral species from which all species with this ability evolved.”
What all of this suggests to me is a kind of biological underpinning of mathematics that can’t be disassociated from the world itself, or the universe for that matter. Everything neurological has its source in the signaling talent of individual cells, ultimately tied to the substances of stars. If counting grows out of this as does, breathing, digestion, movement and later memory and imagination, then it may be that Plato intuited that mathematics, as a thoughtful pursuit, was tapping into something very fundamental, something not quite ours, whose truth value was greater than the particulars to which we applied it. It seems inevitable that we will find that mental processes are as exploratory as the work of the senses.
I’ll close with a quote from Alain Connes:
One of the essential things a mathematician does is recognize the internal coherence and generative character belonging to certain concepts. It happens that very simple concepts can suggest all sorts of ideas or models. Investigating these, one truly has the impression of exploring a world step by step — and of connecting up the steps so well, so coherently, that one knows it has been entirely explored. How could one not feel that such a world has an independent existence?
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