Every day we use numbers - they form the fundamental language of science, which has allowed us to gain a tremendous amount of practical insight into our world and how to exploit it to improve our lives. But is it possible to use them to not only understand the relationships between things, but what things actually are as well?
As per one of my other blog posts, I have a very simple line of questioning here to illustrate the problem I've been thinking about.
Q. I am currently holding an apple in my hand. For all practical purposes, I know it's an apple, but what is it, actually?
A. The apple's made up of cells that were formed by the tree from which it came.
Q. Okay, but what are those cells?
A. The cells are made up of molecules, which are in turn made up of atoms.
Q. But what are those atoms?
A. Atoms are made up of sub-atomic particles like protons, neutrons and electrons. Those are then made up of quarks.
Q. So what are those quarks?
A. Well, we have some theories about what quarks are made of...
Q. So what is that apple?
A. Like I said, we have some theories...
So what is it that's preventing us from saying what something is?
Mathematics, I think, is a language just like English or German or French. Think about the number one, for example. Where in the real world does the number one actually exist? Nowhere - it's a concept in our heads. What I think we're saying when we speak of one apple is the following: "what I am holding in my hand is stable enough to be considered distinct from its surroundings".
What happens when you try to hold water in your hand? It spills out all over the place. With regard to the little bit that you have left in your hand: do you say that you have "one water" in your hand? Of course not! Without a container, we cannot consider water to be a relatively stable, distinct entity from its surroundings. If you place a certain amount of water into a jug perhaps you would be able to call it one litre of water, but what you're saying there is that the jug is of a standard, relatively stable size such that it can contain a certain amount of water. The jug, then, is stable enough to be considered distinct from its surroundings.
What's the point of this then? Well, what this means to me is that our concept of one is derived from a visceral experience. The concept two is both a logical extension of this experience and can also be a visceral experience itself (I can hold one apple in each hand, thus holding two relatively distinct clumps of matter). The number 10 billion, however, is quite a different thing. Have you got a visceral understanding of what 10 billion is? Have you ever been able to count 10 billion of something, such that you could recognise 10 billion of that thing when you see it? I hope not - I would imagine that to be a tremendous waste of time and energy.
So, numbers are derived from visceral experience - they are logical extensions of the concept of one. In a similar fashion, fractions are extensions of this visceral experience of one. We slice one apple up into two pieces, and thus we (approximately) have two halves of an apple, and so on.
[It's important to note here that I'm not discussing the historical roots of the concept of "one" - I am sure that other people have researched this topic extensively.]
So, just like colour, or weight, or smell, numbers are derived from observable properties of physical phenomena - they are not the phenomena themselves.
Let me clarify that: when you say you have one apple, you are saying that the clump of matter you are holding in your hand is relatively distinct from its surroundings - you are describing properties of that clump of matter. The concept one defines how well the matter sticks together in a relatively stable form. In no way does it actually tell you what the apple is.
It's a well-known fact that mathematics is the language of science. For example, according to Newton's laws of motion, the displacement that an object undergoes (s) is related to its initial velocity (u), its acceleration (a), and the number of seconds for which it travels (t) by the following equation:
s = ut + (1/2) a t2
This is a mathematical equation that tends to hold true at relatively low speeds for simple linear motion. Note how this equation says nothing of the object in question - it only defines the relationships between various observable properties of the object.
And why is it that it feels as though mathematics has an existence separate to our own, as if it lives in a different world? (Sir Roger Penrose, in The Road to Reality, calls this the "Platonic mathematical world"). I think it's exactly because of the fact that numbers and formulae deal exclusively with properties of things, and not with the things themselves.
All languages fail in their ability to describe what things are - mainly because, in simple terms, the word apple is not the apple itself. This applies equally to mathematics.
As per this article which I wrote last year on the work of David Bohm, we have some evidence to show that "things" might not be as separate as they seem to be. What if, for example, everything (the entire universe) were an unbroken whole? Bohm calls this the Holomovement, and Pirsig calls it Dynamic Quality. What would this do to the validity of mathematics if one apple wasn't really distinct from its surroundings on a deeper level?
For all practical purposes, being an engineer myself, I think mathematics is incredibly useful in helping us exploit our surroundings to our benefit. But as for unlocking the deepest mysteries of the universe, I think it is only as useful as any other language: in other words, not very useful at all. Towards understanding reality then, I think that we need to start to relate to reality in other (newer and older) ways in addition to languages (mathematical and otherwise).
[Along this line of thinking, I would highly recommend reading "Zen and the Art of Motorcycle Maintenance" and "Lila" by Robert Pirsig, as well as "Wholeness and the Implicate Order" by David Bohm.]
So what hope is there then of science, in its present form which is heavily reliant on the language of mathematics, being able to tell us what things are? In my opinion: none.
And what's so important about knowing what things are anyways? Why don't we just carry on with our lives, making the best of them with the understanding that we've gained from science? That, I suppose is the more pertinent question that arises from this essay. I personally think that to know what everything is will tell us something very deep and fundamental, if not why we're here then at least in which direction we should be heading. For all practical purposes.
That's one thing science has never been able to tell me: which direction to choose.