collection of three primes a, b, c does not exhaust the entire collection of prime numbers as initially assumed. So, the prime numbers are more than this collection, whose existence we assumed in advance, and hence we can conclude that prime numbers are more than any assigned multitude of prime numbers.
Euclid’s proof—consistently (and rightly) acclaimed as exemplary in the history of mathematics—is perfectly acceptable from the point of view of today’s standards, except for one detail of style, which today we would possibly write in a different way. What I mean is the statement that the collection of prime numbers a, b, c legitimately represents the idea of “any assigned collection.” Today we would prefer to refer to “any assigned collection of prime numbers” as a1 , a2 , a3 , . . . , an . In this way, we are able to symbolically represent the idea of a completely general collection of natural numbers with the help of the combined use of a general index n and the three dots that clearly indicate an indefinite amount of numbers in the collection. Euclid indicated that he meant “any collection,” but in fact he took a rather specific one, namely, one comprising three numbers. Logically speaking, doubt might arise as to the general validity of Euclid’s argument. Perhaps, one might insist, the proof would not work with any arbitrary collection, because the inner logic of the proof presented by Euclid may contain a step that depends on the fact that the collection has exactly three numbers, and no more than that. Of course, this is not really the case with this specific proof of Euclid, but I stress this seemingly trivial point in order to emphasize once again just how much the existence of the right notation and symbolism may become crucial to our ability not only to express but sometimes even to think correctly about the mathematical situation involved in a particular question. The numbers involved in Euclid’s proofs have proper names a, b, c that help us referring to them, but notice that Euclid did not operate on these letters (with operations such as addition, multiplication, or division) in the same way we can with our modern algebraic notation for the numbers a1 , a2 , a3 , . . . an . Moreover, in Euclid’s text, the number d is not defined as the “product” of the three, but rather as “the lowest number among those which are simultaneously measured by these three numbers” (we call this the “lowest common multiple”). Euclid’s letters, then, are only labels rather than algebraic symbols that allow for formal manipulation. This is a crucial difference that will be overcome only as a consequence of a long and complex process described throughout the various chapters of this book. Another interesting proposition appearing in the arithmetical books of Euclid is VII.30. It states that if a prime number p divides a product of two numbers a·b, then either p divides a or p divides b. For example, the prime number 7 divides 42, which is a product 42 = 2·21, and hence it must divide either 2 or 21 (which indeed it does: 21 =3·7). But 42 can also be written as 42 = 14·3, and again 7 must divide one of the factors, in this case 14. With time, this general property became so intimately identified with the concept of prime number itself that it was sometimes perceived as a possible alternative definition. However, as the concept of number continued to evolve, new abstract and more general systems were defined in the nineteenth century where the two meanings of a prime number (namely, on the one hand its being divisible only by itself, and on the other hand the property embodied in Proposition VII.30) were no longer coextensive. It was only then, as we will see, that these two properties were clearly separated (see Section 10.2). ARITHMETIC BOOKS OF THE ELEMENTS
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