Ramble, Part 71: Where Do We Begin?
May. 24th, 2009 03:08 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe Formalist program required the selection of a list of fundamental axioms from which all of mathematics could be (in principle) constructed. More broadly, each branch of mathematics should have its own list of axioms (presumably deriving from the fundamental list), subject to the same constraints as mentioned in the last Ramble.
David Hilbert himself was one of the first to successfully reconstruct Euclidean geometry. As mentioned before, Euclid's formulation had flaws, most notably in connection with the existence of intersections of circles with lines and other circles. Hilbert's reformulation corrected this problem, but he found it necessary to include an axiomatic description of the real numbers and, in particular, a "completeness" axiom. (This axiom is equivalent to the assertion that every Cauchy sequence has a limit, or that every Dedekind cut marks a real number.)
Cauchy and Dedekind had shown that the existence and properties of the real numbers could be derived from that of the rational numbers, and it is not hard to show that the rationals can be derived from the integers and thence from the natural numbers - the usual counting numbers. It was natural, therefore, to ask for a minimal set of axioms for the natural numbers. Dedekind had made some progress in that direction, but it was Giuseppe Peano who came up with a simple formulation, which has come to be accepted as fundamental. Peano's version involves two concepts, "number" and "successor", subject to five axioms, as follows.
That's it. All the fundamental operations and theorems of arithmetic can be constructed on that basis.
For example, we can define the sum of any two natural numbers m and n, thus. For any natural number m, m+0 is, by definition, m. If, for two natural numbers m and n, m+n has been defined, then we define m+S(n) to be S(m+n). 1 is, by definition, the successor of 0; so m+1 is simply the successor of m. 2 is the successor of 1, so m+2 = S(m+1) = S(S(m)); and so on. Addition is thus defined as repeated succession, or the repeated addition of 1.
What about multiplication? Multiplication is simply repeated addition, so its definition should come as no surprise: m x 0 = 0, m x S(n) = m x n + m.
What is interesting about Peano's formulation is that mathematical induction, which had been first enunciated by Pascal more than two centuries earlier, moves front and center: to discuss the natural numbers, we need induction, and if we can do that much, we can do it all - all of mathematics becomes accessible.
Nothing's ever that easy.
Ramble Contents
David Hilbert himself was one of the first to successfully reconstruct Euclidean geometry. As mentioned before, Euclid's formulation had flaws, most notably in connection with the existence of intersections of circles with lines and other circles. Hilbert's reformulation corrected this problem, but he found it necessary to include an axiomatic description of the real numbers and, in particular, a "completeness" axiom. (This axiom is equivalent to the assertion that every Cauchy sequence has a limit, or that every Dedekind cut marks a real number.)
Cauchy and Dedekind had shown that the existence and properties of the real numbers could be derived from that of the rational numbers, and it is not hard to show that the rationals can be derived from the integers and thence from the natural numbers - the usual counting numbers. It was natural, therefore, to ask for a minimal set of axioms for the natural numbers. Dedekind had made some progress in that direction, but it was Giuseppe Peano who came up with a simple formulation, which has come to be accepted as fundamental. Peano's version involves two concepts, "number" and "successor", subject to five axioms, as follows.
- 0 is a natural number.
- Every natural number n has a successor S(n), which is also a natural number.
- 0 is not the successor of any natural number.
- If two natural numbers are not equal, their successors are not equal.
- If a set K contains 0 and, whenever it contains a natural number n, it also contains the successor of n, then it contains all natural numbers.
That's it. All the fundamental operations and theorems of arithmetic can be constructed on that basis.
For example, we can define the sum of any two natural numbers m and n, thus. For any natural number m, m+0 is, by definition, m. If, for two natural numbers m and n, m+n has been defined, then we define m+S(n) to be S(m+n). 1 is, by definition, the successor of 0; so m+1 is simply the successor of m. 2 is the successor of 1, so m+2 = S(m+1) = S(S(m)); and so on. Addition is thus defined as repeated succession, or the repeated addition of 1.
What about multiplication? Multiplication is simply repeated addition, so its definition should come as no surprise: m x 0 = 0, m x S(n) = m x n + m.
What is interesting about Peano's formulation is that mathematical induction, which had been first enunciated by Pascal more than two centuries earlier, moves front and center: to discuss the natural numbers, we need induction, and if we can do that much, we can do it all - all of mathematics becomes accessible.
Nothing's ever that easy.
Ramble Contents
