## Mathematical Realities

## Part C: Validity

### Real Maths

We can distinguish between different types of maths depending on what kind of things they apply to.

Statistical or probabilistic maths is concerned with the average behaviour of multiple entities that are roughly the same, where individual items can't be tracked or don't have to be. Examples range from the odds of throwing a double six in dice, to statistical mechanics such as heat and gas laws.

At the other end of the scale is deterministic maths, in which the behaviour of individual entities can be described and predicted more or less exactly by general equations. Examples are Newton's laws of motion and the equations of general relativity.

The recent TV series "Numb3rs" also provides some interesting "real world" examples of both kinds of maths.

Despite appearing at the opposite ends of a spectrum, these are united by what links them to reality – what makes them *actually work*. Each depends on the fact of reality which underpins the validity of concepts: multiple entities share qualities which can unite them. Those common denominators both allow us to form valid concepts that subsume multiple entities – and allow abstract mathematical equations to describe and predict the behaviour of individual or collective items of the same type.

Let us examine both these cases in more depth.

### Taking Chances

One can derive equations based on the abstraction of perfect randomness, just as one can derive an equation for the circumference of an abstract "perfect circle", without either having to exist in reality – yet with both applicable to those entities in reality that approximate them, to whatever degree of precision is involved.

For example, consider a bag of black and white balls, all alike except for their colour. If you mix them in a cement mixer, the position of any individual ball rapidly becomes impossible to predict. Even though the path taken and final position of each ball is deterministically caused by its nature and that of everything it interacts with, the system is chaotic. That is, its next state is sensitive to tiny variations in its current state, so any attempted calculation rapidly breaks down due to exponential amplification of unmeasurable uncertainties in position and velocity.

The result is an assortment of balls which is effectively random. Because the balls all share salient qualities *to the required degree* and the system is *effectively* unpredictable, the abstraction of "randomness" (whose essence is "completely the same and completely unpredictable") applies. Hence, one can accurately predict things such as the odds of picking out 5 black balls in a row.

Of course, the balls are *not* identical: there will always be subtle variations in qualities that affect their paths, such as mass and shape. However in the context we are talking about, the resulting non-randomness in distribution is undetectable, being either too subtle or taking too long to become measurable.

Notice how this corresponds to the nature of concepts. While the entities we group into concepts are not identical (usually), all that matters is that their differences *don't matter in the context of the use of the concept*.

These considerations apply to any system in which the distribution of entities approximates randomness, not only simple physical systems such as tossing coins or rolling dice, but complex biological systems such as the distribution and spread of genes in populations or even the average behaviour of human beings. Note again how this links back to the nature of concepts: we can ignore extraneous details such as the physical nature of the system in order to focus (validly) on the relevant factor they share in common, namely "random" behaviour.

Of course, in some cases the natures of the existents will cause observable deviations from randomness. This in itself tells us something more about that nature. Indeed, it is a foundation of statistics, whose equations compare actual to random outcomes: calculating how likely it is that any differences are themselves merely random fluctuations, or due to causation in a particular direction.

### Being Determined

Why can one derive equations that accurately describe the free trajectory of any object, from a brick to a boulder, from glass to steel? Because the nature of matter is to attract other matter according to specific rules: and therefore, that is also the nature of all things made of matter.

Thus again, we see that maths works for the same reason that concepts work: the differences in size, shape and composition are irrelevant to gravitational attraction and therefore to free trajectories.

Again, the variations from the equations are instructive. Continuing with the example of trajectories, air resistance obviously is important. But again, this follows from the nature of the existents: in this case, all matter obeys Newton's laws of motion. As a result, we can calculate the effects of density and shape on how the trajectory will be altered by air resistance. If we look closely enough, we can even find chaotic effects, due to things like fluctuations in air pressure, wind direction and even absorption, reflection and radiation of energy – though these are negligible in most contexts.

More interesting are cases where behaviour deviates from equations in the *absence* of external influences. Then, as with deviations from randomness in our earlier example, we learn something new about the entities" nature, which differs in some way from the assumptions of the maths. But for the same reasons, that too will be amenable to yet other mathematical formulae.

### The Law of Mathematical Validity

From the above we can distil what I'll call the *Law of Mathematical Validity:*

Mathematics is an abstraction which is valid and effective for the same reasons that all other abstractions are valid and effective.

That is, concepts work because they are valid, and they are valid because things in reality* do, in fact*, share common qualities that allow them to be grouped naturally and thought about – in relevant contexts – as a single mental unit. And the same is true of mathematics, replacing "thought about as a single mental unit" with "described by a single set of equations."

To view this from another perspective: existents affect each other according to laws of cause and effect, which simply means according to their respective natures, and being able to describe that mathematically is a natural consequence. That is, concepts are valid because entities *do* share the fundamental qualities by which they are grouped into concepts. Therefore, the instances of concepts *will* behave the same way in response to the same causes, and numerical regularities and relationships will be observed in nature accordingly.

We look more deeply into this in Part D.