Mathematics And Matter

In the posting "Matter In Space And Numbers", we saw that the numbers that we commonly use are actually a reflection of the density of matter in space. Aside from addresses and identification numbers the numbers that we use are almost always low, usually less than one hundred, and this is ultimately due to the extremely low concentration of matter in the space of the universe that we inhabit.

Today, let's have a look at how the mathematics which we use are a reflection of the reality that we live in by looking at what it might be like if that reality was different.

We develop mathematics to describe and measure the world around us. If there was nothing, no world and no universe, it would seem that there would be no mathematics. But yet, mathematics are actually the basic patterns of everything. Two plus two always equals four, whether it refers to cars or units of currency. A number itself has no real existence until it is manifested by something, but that does not mean that it does not exist. The number 72,814,932 still exists whether or not there is actually anything which manifests that number.

This can only mean that there is an entire universe of mathematics, of which that manifested by our universe and used by us is actually only a very limited or even an infinitesimal portion. Mathematics, at least as we know it, requires at least two entities with some type of boundary between the two. This most commonly is matter and space.

If we had space but no matter, there could be geometry but what would be the point? There would be nothing to use as reference points. Even to define a straight line, we need two reference points. Any point in the universe would be just as good as any other, there would be nothing to differentiate any place from any other place. With no matter, there would be nothing to count with numbers and no reference points to measure distance from or to.

We could say that we live in a universe in which space is the rule and matter the exception. Suppose that the reverse was true, with matter taking up most of the space. Mapping would be reversed with gaps between matter being charted, rather than the matter itself. Counting numbers would be less important, but angles and distances more important.

Suppose that space was different so that there were no fixed distances. This is related to what we saw in "Straight Lines And The Nature Of Space", on the cosmology blog. This would mean that matter may exist, but it would merge into other matter like clouds, rather than being fixed in location or form. Mathematics would be very different, with measuring rates of change in terms of space against time, predominating. Arithmetic would be of limited use.

If we lived in such a changing cloud-like reality in which matter was not mutually exclusive, the fixed geometry that we know would not make much sense without the ready reference points, but we might define the spaces between objects rather than the objects themselves. With change being much greater than it is in our universe, calculus being used to figure rates of change along with probabilities would be much more important.

Suppose that we had a reality similar to what we have now, but the number of spatial dimensions was different. More dimensions would mean the probability of more change, as well as greater challenges in navigation. It would be easier to get lost, making mapping more important. Reality would generally be more complex, with the basic forces such as gravity and electromagnetism having more dimensions to operate in.

What if all matter was just one "thing", with only one object in the universe? There would be distance measurement and counting, but only if the "thing" manifested some type of suitable reference points. If it was a flat sphere, with no reference points, then measurement and counting would be largely meaningless. Just like a universe of empty space, there would be no reason to map or measure the surface of the sphere if every point on it was the same as every other point. Distance from the sphere into space could be measured, but what purpose would that serve if there was nothing else in the universe to measure the distance to?

If there was no solid matter, but only the fundamental particles, the only mathematics that could be meaningful is rates of change. It would be much easier to measure the effects of the particles on one another, and the changes in these effects, than to count or otherwise measure the particles.

How about if there was no motion, but only stillness? There would have to at least be movement of electromagnetic waves or we could not see to measure anything. Distance measurement would be meaningless if we could not go anywhere. There could be two-dimensional geometry, but not three-dimensional because we could not go anywhere to get a perspective. Neither would there be any time measurement, because motion essentially is time.

Mathematics is a tool to help us in the reality that we inhabit. If we had greater powers, it makes sense that we would have less requirement for tools. If we could multiply things, or bring them into existence at will, we would have no reason to calculate how to do it. If we could move anywhere that we wanted to instantaneously, there would be no need to measure the distance to our destination.