(This article is continued in More about math and neurons).
Sometimes we have a feeling of déjà vu in a situation where we know we have never been before. I have had two very strong occurrences of that in my life. One was when I saw St Cuthbert’s Church in Wells in England, and the other was the first time I saw St Martin’s in the Fields in London. Now my ancestors are mostly from England, some even from the south of England (the English ancestors of most white southern Americans are from the north of England, as are some of mine). Was this ancestral memory? Was it memories from a Previous Life? Well, I didn’t believe that, but the feeling was remarkably strong.
Many years later I discovered the reasons for the feelings in both cases. Adelbert Stone Chapel on the Case Western Reserve University campus in Cleveland (where I taught for 35 years) is an exact copy (on the outside) of St Cuthbert’s Church. Independent Presbyterian Church in Savannah is a three quarters size copy of St Martin’s in the Fields, and when I lived in Savannah as a teenager I frequently rode past that church on the bus.
There is presumably a neuron assembly (or something like that) in the brain devoted to recognizing things “I have seen before”. No doubt this can be triggered in the brain by mistake. Being triggered does not have to mean you had a previous life, it may mean a mistake in the recognition devices in your brain. The fact that I eventually understood my two experiences is in fact irrelevant. If you have the feeling of déjà vu and know you haven’t been there before and you never are able to explain it it still doesn’t prove you had a previous life or anything else supernatural. The feeling means only that a certain part of your brain was triggered and you don't know why.
When I deal with mathematical objects such as numbers, spaces, or groups I tend to think of them as “things” that are “out there”. Every time I investigate the number 42, it is even. Every time I investigate the alternating group on 6 letters it is simple. If I prove a new theorem it feels as if I have discovered the theorem.
There is also presumably another neuron assembly that recognizes that something is “out there” when I have repeatable and consistent experiences with it. Every time I push the button on my car door the door will open, except sometimes and then I consistently discover that it is locked and can be unlocked with my key. Every time I experiment with the number 111 it turns out to be 3 times 37. If some math calculation does not give the same answer the second time I frequently find that I made a mistake. I know this feeling of consistent “out there” behavior does not prove that numbers and other math objects are physical objects. The feeling originates in a brain arranged to detect consistent behavior. The feeling is not evidence that math objects exist in some ideal space.