Part 1) Maths in practice
Point: Maths needs definitions to work, but it's people who make those definitions.
I want to make clear that regardless of what I'm about to say I have never been a relativist about mathematical truth. Within Euclidean Geometry Pythagoras' theorem is absolutely true in an indisputable and special way and no amount of grim reality can change that. That being said: mathematicians present an image to the public, this image suggest that mathematician's sole job is the prove things. We start from basic assumptions and from there prove Pythagoras' theorem, that any map may be coloured with at most 4 colours or some other mind-bending thing. We have, from Euclid to Russell, projected an image that we work from basic unshakable assumptions that can never be disputed, “a point is that which has no part” or some other such high minded thing.
But, I want to suggest that this image only gives part of the story. Compare it to football. You could say that the point of football players is to kick the ball into the net, but this misses a large part of the story. You can no more play football only ever shooting at the goal than you can build a house from the roof down. And there is the same phenomenon in maths. You can say that what we do is take definitions and see what they imply and hence prove theorems. But this idea misses something. It misses where those definitions come from.
If we are to study what infinity is and how it acts, but if we dont define it we have nothing to talk about. We must consider what we mean by infinity. And this is not a trivial matter. There are very different ways that you might like to define something that give very different results. A good example of this is given by geometry. At school everyone gets taught about Euclidean geometry, that is, shapes on a flat piece of paper. And as Every Schoolboy Knows, in this system we find out that triangles have angles that sum up to 180 degrees or two right angles. That is a proven fact that holds true for all triangles. But now consider a thought experiment:
Suppose that I start out facing west in the Atlantic at the point where the equator and the Greenwich meridian meet, and I then walk (for the purposes of this example I am Jesus) in a straight line West, across the Atlantic, across south America, until I get to the Galapagos Islands which are on the Equator, but now 90 degrees west of Greenwich. I now turn through 90 degrees to face due north, I walk in a straight line to the north pole. I get to the north pole and now turn through 90 degrees again so I am facing along the Greenwich Meridian, and walk down the UK and Africa to get back to where I started and turn through a right angle to face west again. … Hang on. Something strange has happened here. I've walked along 3 straight lines joined at 3 angles, so I must have walked along a triangle. But the angles I turned through were three lots of 90 degrees. Three right angles not two, is what Every Schoolboy Knows wrong?
This example shows us that it is important to define what we mean clearly. It is true that 3 lines joined by 3 points on a flat piece of paper will always have angles that add up to two right angles. But this is not true for 3 lines joined by 3 points on the surface of a sphere. (Homework: find a saddle or a Pringle or something else of that shape, and draw as big a triangle as you can fit on the surface of it, what do you notice about the angles?) What is important is that we have a clear idea of what we are talking about before we try and understand it.
We dont find that definitions and assumptions are carved in stone somewhere, we cant find them with a microscope, we dont get told them by some almighty god of maths. We have to invent them ourselves. The interesting and remarkable thing about maths is that not all definitions and assumptions will do. We can, if we want, define all numbers to be equal to 2, but we can clearly see that this wouldn't be very helpful in actually understanding what 2 was. Likewise we can say that the number 3 is Julies Caesar, but this doesn't feel natural. It doesn't match our intuition.
Mathematics is very intuitive. If I find some problem and I cant feel involved in what's happening, if I cant see the structure of the problem, then I get bored. This is a problem with a lot of school maths, you cant relate to it, it's not easy to get inside and see what's happening, and that makes it boring. We want to make mathematical structures and ideas that we can get a good understanding of and which fit together neatly. The idea of a square is natural, it covers a lot of things that all have the same sorts of properties, (they all have right angles etc). If I was to define a new word, bibbly, to mean “a shape with either 5 or 6 sides”, that wouldn't feel right. It's a perfectly good concept, I can tell you if a given thing is a bibbly or not. But there aren't many properties that all bibblys have. Things that have 5 sides are very different to things that have 6 sides, and to try and force them together is unnatural. It's like pork in jelly. It's perfectly edible, it's good food sure. But it's just not right.
We then have an interesting philosophical question, why is it that only some definitions seem to “work”. We can think about this in a few ways. The main camps are:
- Platonic, there are only a few valid definitions because there are ideal objects in some reality outside this one. We find that “square” is a good idea because there is an ideal square that exists and is a perfect timeless thing. Plato argues that we know about squares because we come into direct contact with the ideal form of squareness before we are born.
- Intuitionist, our brains are wired up to be happy with some concepts for various evolutionary and historical reasons. In this theory squares are easy to understand because we have evolved to be able to understand square things.
- Empiric, maths exists to solve physics problems, we think about squares because squares help us to understand the world around us. They are a useful approximation of something that exists in the physical world.
- Logical, we can deduce all definitions from the basic rules of pure logic. We understand that 1+1=2 because proposition 110.643, Volume II, 1st edition, page 86 of the pricipia mathematica by Russell and Whitehead proves it from logic.
- Aesthetic, we choose those definitions that give the most visually and intellectually pleasing equations. If we want to define what means, we choose that definition that produces the result because this looks nice on a piece of paper and is elegant as an idea.
I dont propose to pick a side in this debate. I would defiantly agree with the idea that mathematics has an “unreasonable effectiveness in the natural sciences”, as Wigner put it. Though I would not say that this suggests that we create maths with physics in mind, often the most useful ideas are created decades or centuries before their application in physics becomes apparent. And I would tend to dismiss logic as a basis here, because of the many people who have tried to put all our definitions on one basis, all have run into paradoxes and contradictions and have been forced to use assumptions that are not truly logical. Intuitively the platonic theory is most like how maths feels in practice, but I dont pretend that this must reflect a true philosophical reality. I'm far too much of a materialist to pretend that this is evidence. Sorry to end this first part of the discussion on such a woolly note, but I dont want to pretend to have a final answer to this question, anyone have any thoughts on this?
What we realise then is that definitions are tricky. If we define a triangle as 3 lines joined at 3 points and then prove things based on the assumption that they are all in a flat plane we will run into error if we try and thing about triangles on spheres. Likewise if we have a bad definition of infinity we will end up with proofs that are nothing to do with the idea we're thinking about. We need to be very careful not to make blind assumptions or else we will end up in all sorts of paradoxes.
Paradoxes of infinity
If we are to understand infinity we must be able to deal with the flaws in reasoning that lead to such paradoxes as:
- Zeno's arrow. I fire an arrow at someone, before it hits them it must get to the half way mark, before it gets there it must get to the quarter way mark, before it gets there to the 8th, 16th, 32nd etc. So before the arrow can hit its target it must do an infinite number of things, going between any two of these marks takes some amount of time, even if that time is very small. So it must take some small amount of time and infinite number of times. This will take forever. So the arrow will never hit the target.
- Considering the arrow again. The arrow passes through each point in its trajectory only once and only for an instant. In that instant where the arrow is a some fixed point it cannot be moving, because movement within one instant of time is impossible. So then how is it that the arrow is moving if at every instant it is stationary?
- Thompson's lamp. I have a lamp with a very fast switch that I can turn on and off as much as I like. I switch it on, then half a minuet later switch it off, then quarter of a minute later switch it on, and so on. Is it on or off exactly one minute after I started? What about later times?
Summary: If your definitions aren't sorted out you can end up confusing and contradicting yourself. We dont find definitions set in stone, we have to make them to suit our needs.
Next time: defining infinity.