“All things are number.” — Pythagoras (582 BCE - 496 BCE)
There was a time when I thought about the world in the following way:
But this view left me with two questions that troubled me for a long time:
Today, however, I have come to think differently. I now regard the world itself as a mathematical fact, comparable to truths such as these:
The largest of those 26 sporadic simple groups is known as the Monster group.
The Riemann Hypothesis has not yet been proven. However, Hardy in 1914, and Hardy and Littlewood in 1921, showed that infinitely many zeros lie on the line Re(s) = 1/2. In that sense, the statement that infinitely many zeros line up on a single straight line is already a mathematical fact.
In 1984, the physicist Steven Weinberg made the following remark:
The Universe is an enormous direct product of representations of symmetry groups. It's hard to say it any more strongly than that.
I suspect that the universe may, quite literally, be a direct product of representations of the 26 sporadic simple groups.
Mathematical facts exist independently of time. Asking when they were born does not really make sense.
Mathematical facts are also independent of location. There is no need to ask where they exist.
Imagine a state with no time, no space, and no matter: a state before the birth of the cosmos. The only things that could exist in such a state would be mathematical facts.
Among those facts, there might even be sentient beings who understand that their own world is itself a mathematical fact. This is a striking kind of self-reference.
To close, here is a passage written in 1983 by the physicist Freeman Dyson:
I have a sneaking hope, a hope unsupported by any facts or any evidence, that sometime in the twenty first century physicists will stumble upon the Monster group, built in some unsuspected way into the structure of the universe.
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