ریاضیدان و فیزیکدان مشهور، راجر پنروز در کتاب The road to reality به موردی مشابه اشاره میکند. متن کمی طولانی ولی فوقالعاده جالب است (در این متن منظور از 3/8 کسر سه هشتم است):
One of my mother’s closest friends, when she was a young girl, was among those who could not grasp fractions. This lady once told me so herself after she had retired from a successful career as a ballet dancer. I was still young, not yet fully launched in my activities as a mathematician, but was recognized as someone who enjoyed working in that subject. ‘It’s all that cancelling’, she said to me, ‘I could just never get the hang of cancelling.’ She was an elegant and highly intelligent woman, and there is no doubt in my mind that the mental qualities that are required in comprehending the sophisticated choreography that is central to ballet are in no way inferior to those which must be brought to bear on a mathematical problem. So, grossly overestimating my expositional abilities, I attempted, as others had done before, to explain to her the simplicity and logical nature of the procedure of ‘cancelling’.
I believe that my efforts were as unsuccessful as were those of others. (Incidentally, her father had been a prominent scientist, and a Fellow of the Royal Society, so she must have had a background adequate for the comprehension of scientific matters. Perhaps the ‘stern face’ could have been a factor here, I do not know.) But on reflection, I now wonder whether she, and many others like her, did not have a more rational hang-up—one that with all my mathematical glibness I had not noticed.
There is, indeed, a profound issue that one comes up against again and again in mathematics and in mathematical physics, which one first encounters in the seemingly innocent operation of cancelling a common factor from the numerator and denominator of an ordinary numerical fraction.
Those for whom the action of cancelling has become second nature, because of repeated familiarity with such operations, may find themselves insensitive to a difficulty that actually lurks behind this seemingly simple procedure. Perhaps many of those who find cancelling mysterious are seeing a certain profound issue more deeply than those of us who press onwards in a cavalier way, seeming to ignore it. What issue is this? It concerns the very way in which mathematicians can provide an existence to their mathematical entities and how such entities may relate to physical reality.
I recall that when at school, at the age of about 11, I was somewhat taken aback when the teacher asked the class what a fraction (such as 3/8) actually is! Various suggestions came forth concerning the dividing up of pieces of pie and the like, but these were rejected by the teacher on the (valid) grounds that they merely referred to imprecise physical situations to which the precise mathematical notion of a fraction was to be applied; they did not tell us what that clear-cut mathematical notion actually is.
Other suggestions came forward, such as 3/8 is ‘something with a 3 at the top and an 8 at the bottom with a horizontal line in between’ and I was distinctly surprised to find that the teacher seemed to be taking these suggestions seriously! I do not clearly recall how the matter was finally resolved, but with the hindsight gained from my much later experiences as a mathematics undergraduate, I guess my schoolteacher was making a brave attempt at telling us the definition of a fraction in terms of the ubiquitous mathematical notion of an equivalence class.
What is this notion? How can it be applied in the case of a fraction and tell us what a fraction actually is? Let us start with my classmate’s ‘something with a 3 at the top and an 8 on the bottom’. Basically, this is suggesting to us that a fraction is specified by an ordered pair of whole numbers, in this case the numbers 3 and 8. But we clearly cannot regard the fraction as being such an ordered pair because, for example, the fraction 6/16 is the same number as the fraction 3/8 whereas the pair (6, 16) is certainly not the same as the pair (3, 8). This is only an issue of cancelling; for we can write 6/16 as 3x2/8x2 and then cancel the 2 from the top and the bottom to get 3/8.
Why are we allowed to do this and thereby, in some sense, ‘equate’ the pair (6, 16) with the pair (3, 8)? The mathematician’s answer—which may well sound like a copout—has the cancelling rule just built in to the definition of a fraction: a pair of whole numbers (a x n, b x n) is deemed to represent the same fraction as the pair (a, b) whenever n is any non-zero whole number (and where we should not allow b to be zero either). But even this does not tell us what a fraction is; it merely tells us something about the way in which we represent fractions. What is a fraction, then? According to the mathematician’s ‘‘equivalence class’’ notion, the fraction 3/8, for example, simply is the infinite collection of all pairs
(3, 8), ( -3, -8), (6, 16), ( -6, -16), (9, 24), ( -9, -24), (12, 32), . . . ,
where each pair can be obtained from each of the other pairs in the list by repeated application of the above cancellation rule. We also need definitions telling us how to add, subtract, and multiply such infinite collections of pairs of whole numbers, where the normal rules of algebra hold, and how to identify the whole numbers themselves as particular types of fraction.
This definition covers all that we mathematically need of fractions (such as ½ being a number that, when added to itself, gives the number 1, etc.), and the operation of cancelling is, as we have seen, built into the definition. Yet it seems all very formal and we may indeed wonder whether it really captures the intuitive notion of what a fraction is. Although this ubiquitous equivalence class procedure, of which the above illustration is just a particular instance, is very powerful as a pure-mathematical tool for establishing consistency and mathematical existence, it can provide us with very top-heavy-looking entities. It hardly conveys to us the intuitive notion of what 3/8 is, for example! No wonder my mother’s friend was confused.