What are Numbers?
This is deep.
Numbers seem to include and infinitude of infinities.
For most of us, our first encounter with number is with integers: 1, 2, 3, 4, 5, …..
For a few years integers (whole numbers) is where it’s at for most of us.
Then we start to get into fractions. This can happen quite early if we have siblings – halves, thirds, quarters, etc.
It seems that there is no end to the possible number of either integers or fractions. We can always add one to even the largest integer we can come up with. This idea is usually our first encounter with infinity.
Then we find that there is an infinity of fractions between any two integers (an infinity of infinities).
It is worth noting at this point, that our physical universe seems to be finite – large, and finite. Thus there is a limit to how big or small things can get in our physical universe, but as puny human beings, we are a long way from any of those limits.
If we continue exploring numbers, we find all sorts of interesting things.
We find irrational numbers, like Pi (the ratio of the diameter to the circumference of a circle), which cannot be expressed as a fraction in any integer mathematical base. There is actually a mathematical proof that the number of irrational numbers must be larger than the number of fractions – it is a greater infinity (yet more infinities of infinities).
If one starts playing with numeric bases, one can have a lot of fun. Things get really strange when you start using irrational numbers as numeric bases.
Then there are logarithms – more fun!
Then there are ideas like imaginary numbers – which are multiples of the square root of negative 1. As any negative number multiplied by itself gives a positive number, it is not possible to have the square root of negative one, but because the equation can be written, the idea exists.
Then there is the whole world of Turing machines, and control codes for such things.
Then there is the world of mathematical theorems, and the work of people like Stephen Wolfram and the team working with him, doing things like enumerating all the possible theorems in mathematics.
There is games theory, with ideas like multiple stable state equilibria.
There is multidimensional mathematics, with tensor calculus.
There is Hilbert spaces, and worlds of probability vectors.
Then there is the strange strange world of Quantum Mechanics.
It seems that the logical spaces opened by explorations of mathematics is infinitely variable.
At least once a day I take an apple from the bowl on the bench, take a knife from the knife block, and on a chopping board, place the apple, and make a cut through the centre. If I don’t hold it firmly, the two halves fall away.
If I do hold it firmly enough, I shift my grip and spread my fingers a little, and make a second cut through the middle, but 1/6 of the distance around the circumference from the first. Now I have two sixths and two thirds, and if my grip was OK, they are all still together in one piece. Then I dissect the final two thirds, and I have 6 pieces of apple, which I then take back to my chair, and consume.
As I eat them, I first notice the texture, then the taste. Then the slight elevation of mood. The pleasant filling sensation as my belly fills with the apple pulp and saliva. Then comes a slight warming of the body, as the metabolic heat from the breakdown of the sugars in the apple spreads out through my body.
Those 6 sixths of an apple give me definite pleasure. They are not merely relational concepts, they are real portions.
And this illustrates the fundamental distinction. Numbers are all conceptual. Reality simply is what it is. Each of those sixths of an apple contains around 10^24 molecules, in many similar repeating cellular arrangements; giving it texture, colour, flavour, and all the metabolic properties that they have.
Numbers are human inventions, labels for concepts.
Trying to understand reality without number is a lost cause. Numbers seem to be fundamental to how reality works; but not, I’m afraid, the sort of nonsense that the Pythagoreans dreamt up. There are not four elements – there are over 100 elements, and many forms of energy.
The actual numbers that are important in the real science of reality seem far more random. For the most part they are not whole numbers. For the most part, many of them don’t seem to have any particular mathematical significance other than the fact that they are the particular constants that one must plug into the particular sets of equations in order to understand what is going on at the level of the very small that seems to underly all of this experience of being that we label reality.
So number is very important. Becoming numerate is lifelong exercise. Every day I make a practice of calculating at least one thing I have never previously calculated, and most days I do it many times. After Les’s funeral my daughter looked at me looking at the casket and said “you’re doing that number thing you do aren’t you – stop it!” – I didn’t stop it. I calculated weight distribution and moments of inertia, to be certain I could stabilise the coffin as we carried it out if one of the other bearers tripped.
Driving home yesterday at one point I calculated the average life of vehicle cost of fuel vs capital cost for a few different vehicles, and for most vehicles the lifetime cost of fuel is about the same as the capital cost; and they can be most economically purchased at about 70% of their design life and run until they stop. And there are lots of variables in there that work well for averages, but say little or nothing about any particular vehicle.
It is much slower calculating in my head than using spreadsheets. I have thousands of spreadsheets I have created around particular and general classes of problems. I have a calculator I wrote which allows me to change between any numeric base, and displays any numeric base from 2 to 36.
Numbers allow us to define shape and colour, and intensity and any number of dimension we care to define and enumerate – if you doubt that – go watch a digital 3D movie – all done with numbers – zeros and ones – lots of them – the colours, the sounds, all of it!
Everything on this page – just rather simple sets of zeros and ones in a sense, yet so fantastically unique in another sense.
If you allow for all the characters normally found on a US keyboard (93 of them), then If you had had a machine typing out all possible combinations of characters on new lines, at 10 characters per second, when the universe began, then the combinations it would be producing today would only be 9 characters long – about one line per second.
To get the particular arrangement of characters I have written here at random is such a fantastically improbable event. Yet it seems to be what the application of a few simple filters through evolution by natural selection at successively more complex levels, can and does deliver.
I love the way that different folks get different (and seemingly valid) implications from Goedel’s work.
I love Goedel’s use of logic to show that all systems descriptions are essentially incomplete, even if a system follows strict rules, and even if all the starting conditions are known.
So I guess it depends very much on what is meant by “mystical”.
I have long ago accepted that all knowledge is probabilistic, and that however much I know, I will be surprised from time to time.
In that sense, I reject Rand’s starting assumptions.