## Mathematical Realities

## Part B: Abstractions

### Abstract Maths

Some people are puzzled that mathematics is so successful at describing reality. Others take its success to mean that fundamental reality might just be mathematical equations. Such conundrums vanish when we understand that all of mathematics is *abstraction*, and what that implies.

The concept of "number" is the fundamental underpinning of mathematics. But in reality there are just existents, each one distinct from all others. The very concept "number" is dependent upon conceptualisation. You cannot identify "two" of anything in the absence of the mental integration of multiple existents into one concept, based on their similarities. Thus, in order to count two oranges, you must first have identified "orange" as a particular kind of existent which you can distinguish from other existents; and the same is true of more general abstractions from "fruit" all the way to "thing".

As the process of forming concepts is a process of *abstraction*, it follows that all of mathematics, being founded upon abstraction at its lowest levels, is an abstraction. Indeed, it only gets more abstract from there, as is reflected in the history of number "types", characterised by increasing distance from the perceptual level.

The only numbers which are directly represented by concrete instances of concepts are the *positive integers*. You can have two oranges, 3 people or 100 ants – but there are no "negative existents" or "nothings". You will never see -1 people walking around.

Negative integers are a higher level of abstraction, dealing not with existents as such, but with actions done to existents (e.g., removal or destruction), and qualities of existents that cancel or oppose each other (e.g., directions or electrical charge). Referring to an absence, to a void, to literally nothing at all, is zero.

The most abstract numbers that are actualised in reality are the *rational numbers*, or *ratios*. We can directly refer to 3 out of 5 apples, or 99 out of 103 arrows.

Numbers more abstract than that have no *exact* representation in reality and thus are "purely abstract". This is all of the irrational numbers – those with infinite, non-repeating decimals, that cannot be expressed as ratios – which include surds like the square root of two and transcendental numbers like pi. The reason such numbers are not exactly represented in reality is that reality is not the infinitely smooth canvas of mathematical abstraction. For example, any real instance of a circle has a circumference line of nonzero width. For any such real circle, "its" value of pi is simply the ratio of its actual measured circumference to its actual measured diameter. Depending on how perfect the circle and how precise the measurements, this number will be equal or close to the value of the abstraction pi to a certain number of decimal places, and no more. There is no circle in the universe whose circumference divided by its diameter is exactly pi: it *has no* exact value, by its definition as a non-terminating non-repeating decimal.

A telling illustration is how surprisingly few digits of pi apply to any real circle. The number of relevant digits is only about the number of zeroes in the ratio of the diameter's length to its thickness (due to the difference in circumference between the "outside" and "inside" of the line). Imagine a perfect circle around our galaxy (diameter 100,000 light years, or 10^{21}m), "drawn" in a line only as thick as a hydrogen atom (10^{-10}m). That is a ratio of 10^{31}: so *only about 30 decimal places of pi are needed to calculate the upper and lower bounds of the circumference at that diameter plus or minus half that width!* And both increasing the diameter and decreasing the thickness by a factor of a billion each increases that by only another 18 digits! And this is ignoring the physical impossibility of actually *having* a perfect circle at such scales.

Even more abstract are the imaginary numbers, based on square roots of negative numbers. They don't even have approximate representatives in reality.

Note that such numbers are no more invalid or useless than any other abstraction which omits consideration of irrelevant measurements. A prime example of that is zero, which, while by definition referring to nothing that exists, is an extremely powerful part of mathematics. Similarly, the abstractions of pi, square roots and imaginary numbers give us tools for calculating dimensions in the real world to any required level of precision. In short, while we need to remember which numbers actually have exact representatives in the real world, purely abstract numbers can still be eminently useful for calculations on their approximate representatives, or for any other relevant application.

### To Infinity & Beyond

An unusual mathematical concept which is worth special mention is *infinity*. Philosophically, actual infinities cannot exist, because by the law of identity any entity must have a *specific* nature so cannot be "indefinite". One can have a "potential" infinity (e.g. of time, as in "the universe will never end"), but not an "actual" one (e.g. a universe that is infinite in volume).

As we've seen before, this is one of the few philosophical principles that would seem to have a direct implication for science: if there can be no infinities, then the universe must be finite, which restricts the possible cosmologies to finite ones (for example, a "closed universe" which is "finite but unbounded").

As with other mathematical abstractions, this is not to say that the concept (as a mathematical one) is invalid. Both infinity and its inverse, infinitesimals, have mathematical value. But just as infinitesimals are abstractions which have value for calculation but no referent in reality (because reality isn't infinitely divisible), so is infinity.

Given that, I do consider it meaningless to speak of different "sizes" of infinity, and I have yet to see a justification of that which did not reek of logical fallacies or sleight-of-definition. I believe that to attempt to extend the abstraction of infinity in such a way is to totally divorce oneself from relevance to reality in any form. Infinity is infinity, and one abstraction of endlessness can't exceed another.

### Mystic Maths

Tracing the conceptual hierarchy of mathematics in this way allows us to easily dispose of the notion sometimes encountered that as the universe is so well described by mathematics, perhaps reality is at base just mathematical equations.

Such a Platonic notion is a clear example of the fallacy of the "stolen concept." Mathematics describes the relationships between existents. To drop the existents yet attempt to retain mathematics is conceptually invalid because it is cut off from its own roots and validation. Mathematics is an abstraction: it has no independent existence outside of the concretes from which it is abstracted, any more than the concept "number" has an existence separate from *things* that can be counted, or the concept "apple" has an existence separate from the particular individual fruits it refers to. It is not that things exist because of mathematics: mathematics exists because of things.

The question of why maths works at all leads us down more interesting paths, and that's what we'll turn to next.