A light-ish interlude.

OK. The last entry was possibly a step and a half. And the final assertion that "nothing that exists is 'outside' science" is something of a brainful. So this post is a gentle meander around the idea until I'm happier with it.

Bud has made an interesting comment:

"... modeling anything with science is, in itself, a leap of faith"

This is worth dwelling on. It's an important part of understanding the difference between the scientific method and a scientific model, and between a model and reality itself.

A model is a leap of faith. You are being asked to accept that what the model tells you is the same as what you would experience in Real Life. Take the seemingly simple example of numbers and arithmetic. As a young child, you are taught to accept that the sum "5 + 3" is comparable to adding five apples to three bananas. You are asked to believe that every single time you bring together five apples and three bananas they will always combine to make eight pieces of fruit.

This is a leap of faith or an act of belief. See the previous posts on belief in science and a belief in belief.

Two things turns this act of belief into a scientific model. Firstly, the model is expressed in a way that is comprehensible and useful. The vast majority of human beings can be taught basic numeracy and can apply it in extremely useful ways to Real Life. Secondly, the model is expressed in such a way that it is testable. We can compare the results that the model of numbers and basic arithmetic give us, with the results that we observe in Real Life. We can go out and buy five bananas and three apples and we can put them together and count them and (fortunately for us) every single time we will count eight pieces of fruit.

Pure Mathematics

While we're meandering and talking about arithmetic, let us take a brief stroll through the scented meadows of Pure Mathematics. Some people don't even label pure maths as a science. Even the term 'pure' has a strange, un-sciency feel to it. Here's what wikipedia has to say about it:

"Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application."

Which is more a definition of what it isn't than what it is. Essentially, pure maths takes the 'useful' part of our definition of a scientific model and puts it to one side. It still takes a model and expresses it in comprehensible terms (comprehensible to other pure mathematicians, that is), but it does not care whether the results are applicable to the Real World.

Let's take the schoolboy favourite concept of an infinite number. There is a finite amount of 'stuff' in the universe. So the concept of an infinite amount of something is, to all intents and purposes, entirely useless. At the same time, it is an extremely cool idea (well ... to pure mathematicians and schoolboys at least). Pure maths is built upon piles and piles of extremely cool but generally useless things.

There are of course exceptions. The simple ends of pure mathematics are useful: numbers and geometry for example. Many other abstract and 'pure' mathematical models have turned out to have application in the Real World, although typically they become useful decades or centuries or even millennia after they are first investigated 'for fun'.

Intuition

Intuition is great. The human brain has the capacity to observe the world and make lightning fast decisions about what is happening and what is likely to happen. We can throw and catch a ball without performing any complicated calculus to work out its trajectory. We can distinguish between faces and voices from objectively tiny differences. We can store and recall and relate information about things with unlikely speed.

But intuition has a very definite limit. The human brain works on a human scale. If you go very far outside that scale, either very much bigger or very much smaller, then intuition falls down. Things become quite literally 'strange'.

Take symmetry as an example. Symmetry is an intuitive concept. Imagine you take a square of plain white paper and place it in front of you. You can turn it through a quarter, a half, or three quarters of a turn (90°, 180°, 270°) and it will appear the same. You can reflect it in a mirror and it will still appear the same. It has some symmetry.

Now, I used the word "imagine" there for a reason. You don't actually have to take a square of paper. You know these things from experience and intuition. You've seen a square before. You know how squares work. You can imagine the same process with a rectangle or a triangle or any number of familiar shapes and you would understand how their symmetry worked.

There is a simple model of symmetry in a branch of pure maths called Group Theory (no wiki link as it would confuse more than anything else). It provides a formal model for what we understand intuitively about symmetry. But, because it is pure maths and not bounded by a need to relate to the Real World, it goes some steps further and provides a formal model for aspects of 'symmetry' that are utterly outside our intuitive understanding. It can, for example, model the idea of an object that needs to be turned around twice (720°) before it looks the same. Clearly, that makes no sense at all for our intuitive understanding of the physical world.

Except (either beautifully or irritatingly, depending on your perspective), there is an application for this bizarre and unintuitive model in Quantum Mechanics. It is counter intuitive because it is not on a human scale, but it is still useful. And I'm definitely not linking to the wikipedia article here since the page acknowledges itself that "All or part of this article may be confusing or unclear."

So:

• a scientific model is a leap of faith: the scientific method requires you to TEST the model
• pure maths is only science-ish since it investigates models for their own sake
• a model does not need to be intuitive to be useful or scientific