14 DAY TRIAL // Roger Penrose described having various encounters with people who seemed to be utterly incapable of understanding the concept of "canceling" ? why can one "simplify" a fraction in the way it is usually done. He attempts to clarify the issue in the preface of his book The Road to Reality but I think he creates even more confusion. And I also think there is a profound reason for why this otherwise brilliant mathematician doesn't manage to properly explain this uncomplicated mathematical concept.
He tries to explain that a fractional number is an infinite set of equivalent pairs of integer numbers. For example 1/2 is equivalent to 2/4 and to 3/6 and to 4/8 and so on. One can represent the number 0.5 in all these different fractional forms (there is an infinite number of equivalent fractional representations of the number 0.5). But what some people don't understand is why one can go from one fractional representation to the other via that specific process of "canceling" ? why is canceling the correct method? It seems like some sort of "magic" to them.
In his attempt to explain this he writes:
"It is better to think of 3/8 as being an entity with some kind of (Platonic) existence of its own, and that the infinite collection of pairs is merely one way of our coming to terms with the consistency of this type of entity."
Probably this is what many mathematicians think about numbers ? they think of them as some kind of Platonic "entities" ? and I think this is precisely why normal people are incapable of following their explanations. The reference to Platonic "entities" simply increases the "magic" instead of clarifying anything.
Moreover, as I'll explain now, I think this is also ultimately wrong. Numbers are not "entities" ? they are relations. (This idea isn't original of course, I have it from the very beginning of V.I. Smirnov's classic series A course of higher mathematics.)
But before explaining that, let me make the problem of "canceling" more explicit. You can easily prove that 1/2 is the same number as 3/6 because if you actually divide 1 by 2 and then 3 by 6 (use Windows' Calculator) you get the same result in both cases: 0.5. But the issue is to get from 3/6 to 1/2 directly. This is what canceling does. But there's a problem with canceling: it cannot take you from e.g. 3/6 to 2/4. This seems a downfall because we are told that 1/2, 2/4, 3/6 and so on, are equivalent. So why does canceling skip 2/4? The "magic" of canceling seems to have its shortcomings.
The fact that numbers are relations cannot be clearer than in the case of fractional numbers. In this case we literally have one "entity" presented to us as actually being a relation between two other "entities". For example the number 0.5 can be presented to us as the relationship between 1 and 2 ? or between 2 and 4 (and so on). But probably this doesn't sound very revealing yet: what are "1" and "2"?
What are numbers?
To make the long story short, the issue is: The numbers are relations between what? The answer is: Any number is the result of a measurement. When one measures something one has to have a measurement unit. So, any number is a relation between the thing being measured and the measurement unit.
In case of measuring lengths you take the measuring unit and see how many times you have to repeat the unit to obtain the length you are measuring. The result of this "how many times" is a number that represents the measured length in relation to your unit of choice.
But suppose now you wonder what the length of the measuring unit is. You can express this length in relation to itself and then its length is 1 (any length enters inside itself 1 time and 1 time only). But you can express it in relation to some other (again arbitrary) measuring unit.
Let's denote the original length with M (the thing being measured), the first measurement unit with U and the second measurement unit with V. The length of M relative to U is M/U ? the result of this fraction tells you how many times you have to multiply U for obtaining M (this is an actual physical process of multiplying and counting). If M is longer than U than M/U is a number greater than 1, otherwise it's smaller (for example if U is twice as long as M the length of M relative to U is 1/2).
If by multiplying U a certain number of times we obtain M exactly, then we say that the length of M relative to U is a natural number (or a positive integer).
But what happens when we don't get M exactly? Well, let's choose another unit of measurement, V, and let's choose V in such a way that both M and U can be expressed as natural numbers relative to V. We can multiply V and obtain U exactly and furthermore we can also obtain M exactly. So both M and U are natural numbers relative to V. In this case the length of M relative to U is a fractional number (also called a rational number).
For example if U is twice as long as M we can choose V to be equal to M. Then the length of M relative to V is 1, and the length of U relative to V is 2. Thus, the length of M relative to U is 1/2. But, remember, the choice of V is arbitrary. We can choose some other unit of measurement, W, which is e.g. half of V. Then M is 2 relative to W and U is 4 relative to W ? thus the relation between M and U, expressed with the help of W, is 2/4.
Now it should be clear why 1/2 and 2/4 are the same number ? they express the relationship between the same two things (M and U) with the help of various arbitrary units (V, W etc.). And what canceling does is to change the unit from W to V in such a way as to obtain the largest possible such unit. Moreover, canceling involves only integer changes from units such as W to V (one takes V to be twice or three times etc. the length of W, one never takes V to be, say, one and a half the length of W) ? that's why canceling skips some fractional representations of the same number.
From what I've said so far one can also understand that fractional numbers are not fundamentally different from natural numbers, it's only that in their case the measurement unit isn?t "properly" chosen. If M is a fractional number relative to U, you can always represent M relative to some other unit U' instead of U and then M becomes a natural number.
Irrational numbers
But the story doesn't end here. There are other kinds of numbers. In some cases, one cannot find any unit V to fit exactly inside both M and U: for example this happens if one tries to measure the length of the diagonal of a square taking as unit of measure the side of the square ? this number, the diagonal of the square relative to its side, is called the "square root of 2" (or sqrt(2)).
It is said that when one of Pythagoras' students, Hippasus, demonstrated that sqrt(2) cannot be represented as a fraction of two integers, Pythagoras was so pissed off by such "irrationality" that he ordered his drowning. He never accepted the reality of irrational numbers ? most of the later mathematicians did, but the name stuck to this day.
Other examples of simple irrational numbers are: the diagonal of a cube relative to its side (a number called sqrt(3)), the length of a circle relative to its diameter (which is called pi) and so on. Such irrational numbers are fundamentally different from natural (or fractional) numbers. The set including both the rational and the irrational numbers is called the set of "real" numbers. Then, in a similar way as one has fractional numbers as pairs of natural numbers, one can have some other numbers, called "complex" numbers, which are pairs of real numbers. As fractional numbers are not fundamentally different from natural numbers, neither complex numbers are fundamentally different from real numbers (the manifest border exists only between rational and irrational numbers).
Equilibrium and summation
There is another way of measuring things besides the idea of repeating over and over the same measurement unit and fitting this sequence inside the thing being measured. One can measure certain properties by using the idea of equilibrium. For instance one can measure weights with a balance. Two groups of things have the same weight if when they are placed each on one scale, the scales are in equilibrium. By repeating a measurement unit one gets the mathematical concept of multiplication, while the idea of measuring using equilibrium leads to the concept of summation. The sum of two weights is equal to another weight if the group of two weights is in equilibrium with the other weight.
Obviously one can also introduce the idea of measuring unit here: we can first see that two bodies have the same weight and then use them together to equilibrate a third weight ? the third weight is then twice as heavy as each of the two weights taken separately.
In the same way as the concept of multiplication creates some unusual numbers (the fractional numbers), the concept of summation also brings to the party its "unnatural" numbers: the negative numbers.
Suppose that you have two weights, A and B, which together are equal to another weight, C. When we remove B the scales will no longer be in equilibrium. But then we can pull C's scale upwards with a certain force until the balance gets balanced. The weights push the scales downwards, our force pulls upwards. The upward force on one scale is equivalent with a downward force on the other scale. In other words our upward force is a negative weight. Instead of having A + B = C, we now have A = C + (?B), where ?B is the upward force. If we have nothing on one scale and both the weight B and the force ?B on the other scale, the balance is in equilibrium: B + (?B) = zero. (As you can see, all algebraic equations, "something = something", represent this idea that the two somethings are in equilibrium.)
So we have two different methods of measuring things and, in consequence, the two fundamental algebraic operations, multiplication and summation. It is worth mentioning that they are equally fundamental ? it is often said that multiplication is repeated summation, it's one thing added to itself for a certain number of times. This obviously isn't a proper definition of multiplication because it uses the very idea of repeating something (i.e. of multiplication) in order to define what repetition means. One often sees the following "definition": n*a = a + a + ... + a, and beneath the dots there is an accolade saying "n times". This is downright silly.
I must say I haven't met anyone incapable of understanding the idea of canceling a fraction, but I have met various people who couldn't grasp the idea of distributivity. Distributivity tells you how summation and multiplication are connected. The law of distributivity is this: x*(y + z) = (x*y) + (x*z). But why is this law true?
This is easy to understand if you think about weights in balance. What does x*(y + z) mean? It means that you have a number of x groups of y and z. The numbers y and z represent the magnitudes of the weights (y + z is a group of two weights one pushing downwards with the force y and the other one pushing with the force z) and x tells you how many such groups there are. On the other hand, what does (x*y) + (x*z) mean? It means that you have a number of x weights having the magnitude y and x weights having the magnitude z. It's pretty obvious that you have the exact same number and types of weights on both scales, they are just arranged (or grouped together) differently ? on the left scale the weights are grouped in x pairs, while on the right scale all the y weights are grouped together (there are x of them) and all the z weights are also grouped together (and there are x of them).
One can think about all the other properties of summation and multiplication in similar ways. There are only two possible things that can happen in algebra: either the measurement unit is changed or the things are grouped in different ways. And the main point one has to keep in mind is always that a number is not a "thing" (an abstract "entity") but a relation between two (very concrete) things. There is no such thing as a "pure" number.
Is mathematics really that "abstract"?
One doesn't have to be intimidated by the fact that mathematicians almost never mention the concrete things and delude themselves they are being "abstract" and "formal". You can come up with the concrete things yourself ? any you shouldn't feel apologetic about it. Here is a nice story told by Richard Feynman (Surely You're Joking, Mr. Feynman!, A Different Box Of Tools):
"Topology was not at all obvious to the mathematicians. There were all kinds of weird possibilities that were 'counterintuitive.' Then I got an idea. I challenged them: 'I bet there isn't a single theorem that you can tell me -- what the assumptions are and what the theorem is in terms I can understand -- where I can't tell you right away whether it's true or false.'
It often went like this: They would explain to me, 'You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?'
'No holes?'
'No holes.'
'Impossible! There ain't no such a thing.'
'Ha! We got him! Everybody gather around! It's So-and-so's theorem of immeasurable measure!'
Just when they think they've got me, I remind them, 'But you said an orange! You can't cut the orange peel any thinner than the atoms.'
'But we have the condition of continuity: We can keep on cutting!'
'No, you said an orange, so I assumed that you meant a real orange.'
So I always won. If I guessed it right, great. If I guessed it wrong, there was always something I could find in their simplification that they left out.
Actually, there was a certain amount of genuine quality to my guesses. I had a scheme, which I still use today when somebody is explaining something that I'm trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they're all excited. As they're telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) -- disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn't true for my hairy green ball thing, so I say, 'False!'
If it's true, they get all excited, and I let them go on for a while. Then I point out my counterexample.
'Oh. We forgot to tell you that it's Class 2 Hausdorff homomorphic.'
'Well, then,' I say, 'It's trivial! It's trivial!' By that time I know which way it goes, even though I don't know what Hausdorff homomorphic means.
I guessed right most of the time because although the mathematicians thought their topology theorems were counterintuitive, they weren't really as difficult as they looked. You can get used to the funny properties of this ultra-fine cutting business and do a pretty good job of guessing how it will come out."