God, Santa, and The Tooth Fairy

[Belacqua]

There are no numbers without a brain, computer, abacus, piece of paper, or the like to be imprinted on or processed within. All of those things take up space. Numbers and ideas and concepts can't be demonstrated to exist without matter and energy in space and time.

Well, sure. Numbers exist in the brain. In the brain, they occur as electrochemical events among synapses. These have extension, but it's pretty small. There are thousands of synapses in every voxel of an fMRI scan.

So OK: numbers are synaptic events.

Now if we look at a piece of paper, if there are numbers on that paper, there must be synaptic events on the paper, since we've established that numbers are synaptic events. But strictly speaking, paper doesn't have synapses. The numbers on the paper are something else. Some of them are only a couple of millimeters tall, but I've seen billboards with numbers two meters tall. 

So now we have two types of numbers which are completely ontologically different. Synaptic events and printed symbols. And even the symbols can differ, since two can be shown as 2 or 二 or 弐 depending on culture and context. 

And then we have the abacus, which has neither synaptic events nor printed symbols. So it appears that there are all kinds of numbers in the world. And each of these has materiality and extension. 

Frankly, I think none of these things we're talking about are numbers. They are symbols which refer to numbers. We have developed a variety of useful ways to symbolize and refer to numbers. 

Now if all the numbers we use in daily life are symbols or referents, to what exactly are they referring? Perhaps the REAL number two is kept in a vault in Zurich. What is it made of? Is it heavy? Could Lupin III steal it?

No, I'm sorry, I think that's all silly. We use material symbols for convenience when talking about numbers, but numbers themselves have no materiality, no extension, and no location. They are immaterially real.