Last week’s post was going to be about the number two, but I got diverted into talking about existence/non-existence instead. Existence/non-existence is only one of the many attributes that comes in only two possible varieties or types. Up and down, left and right, in and out, positive and negative.
These attributes might be associated with another attribute representing a magnitude, such as distance, weight or other attribute. So we may say 20 metres to the left, thus locating the object or event in relation to the datum or origin. Both attributes are required in such circumstances, since the directional attribute (left/right) does not completely locate whatever it is, event or object. Neither does distance, by itself, locate the event or object.
Relative to datum, in a three dimensional world, any three axes define direction and the datum itself divides the direction into two opposite parts. If you include the fourth dimension of time, the datum, now, still divides the direction into two parts, before and after. This of course can be extended to as many dimensions as you may choose to conjecture.
One interesting two-ism is the two-ism of a mirror. When you look in the mirror you see an image of yourself. When you move your left hand, the image appears to move its right hand, and the image’s hair parting appears to be on the opposite side to yours. This is a mind trick, since if you see a person raise the hand on their right as you look at them, your mind says that it is their left hand that has been raised. If they have a parting on the left as you look at them, your mind tells you that their parting is on their right.
This illusion is so strong that people misunderstand the reason why words appear reversed in the mirror, and why it is hard to trim your moustache, or pluck hairs in the mirror.
Many people are puzzled because a mirror appears to reverse things left-to-right but not up-to-down. It doesn’t – your left hand is still on the left, and your right hand is still on the right, your head is still at the top and your feet are at the bottom.
The trick is that your nose is closer to the mirror than the back of your head and the same is true of the image. The image’s nose is closer to the mirror than the back of the image’s head. If you draw a map of yourself, the mirror and the image, you will see that the mirror reverses the axis between the original and the image. The front/back axis. Once you see that, it is obvious, and it is hard to see how you could have thought otherwise. It doesn’t help your coordination when you part your hair though!
When we consider the number two, it is an interesting integer, the second of the natural numbers. Interestingly we use the second ordinal number to describe the second natural number, and we use the second ordinal number in that definition too. I’m sure that the circular nature of this description is apparent.
I’m a fan of the axiomatic approach to number theory. An axiomatic system consists of a set of axioms that are used as the basis of reasoning. A theorem in such a system is a set of steps leading from a premise to a conclusion. A premise should be the conclusion of a previous theorem.
Skipping a lot of details, one axiomatic approach is to define a function S, the successor function. S(x) then refers to the successor of x, where x is a natural number. So S(7) is 8, S(1,000,000) is 1,000,001. S(1) is 2, and we have a non-circular definition of the number 2. Erm, almost. The number and its successor form a pair and a pair has how many members? Two. There’s still a whiff of circularity there, to my mind.
Two is an even number and the first of them. An even number is a number which can be split into two in such a way that the two parts are the same number. To put it another way, if you take an even number of stones and put them alternately into two piles, you will be left with two piles each with the same number of stones. If you take an odd number of stones, and perform this test, you will find that the two piles have a different number of stones.
If you consider the set of even number and the set of all natural numbers you might conclude that there will be less even numbers than natural numbers. Paradoxically, there are as many even numbers as there are natural numbers.
It is possible to demonstrate this by a process of mapping the even numbers to the natural numbers. 1 then maps to 2, 2 maps to 4, 3 maps to 6 and so on. This mapping process is also called ‘counting’. For each and every natural number there is a corresponding even number and for each and every even number there is a natural number. The two sets of numbers map one to one. If two sets map one to one, it is said that their cardinality is the same, or in common language, they are the same size.
We are more used to finite sets of things (like the set consisting of a pack of cards) than infinite sets of things (like the set of even numbers or the set of natural numbers). If you take half the members of a finite set away, you have a smaller set of things. For example if you take all the black cards out of a set consisting of a pack of cards, the resulting set is smaller, but for infinite sets of things like the natural numbers this is just not true. If you take the odd numbers from the set of natural numbers, the resulting set of even numbers is the same size as the original set, not smaller.
Much of the above is far from rigorous, and I’m aware of that. However, the main thrust of the arguments is still, I believe, valid. Numbers are fascinating things, with each one having unique properties, and a whole lifetime could be spent considering just one number.