A Short Note on Kronercker’s “God created the natural numbers, the rest is the work of man.”
Any student of mathematics today would be able to call upon standard sets of numbers such as N (the set of natural numbers), Z (the set of integers), Q (the set of rational numbers) and so on. However, providing clear definitions for these sets was a real challenge to foundational mathematicians. Such deep anxiety was in part expressed by Leopold Kronecker in his famous comment: “God created the natural numbers, and the rest is the work of man.” In my view, this quotation contains two significant ideas insofar as the philosophy of math is concerned. The first idea is that given the set of natural numbers, we can ‘arbitrarily’ construct all other sets of numbers. The second being that the starting point, namely the set of natural numbers, is somehow provided. This essay first deals with the first idea, showing how one might “construct” other sets of numbers such as Z from merely natural numbers and minimal conditions. Then, I will comment on the second idea by drawing upon the theories of ordinals and show that we can provide natural numbers in a near ex nihilo fashion with just basic notions of least element 0 and “successor”. Finally, the essay concludes briefly on a more philosophical note, commenting on why Kronecker might dismiss some’s refutation of the ideas.
Assuming we start only with the set of natural numbers N, we can indeed set-theoretically construct “the rest”. Since it would be too much to show the construction of every standard set of numbers, I would focus on the case for integers. To do so, we first need to introduce notions of relations on a set, equivalence relation and equivalence class. A relation between sets X and Y is defined as a subset R of the Cartesian product X×Y. In particular, a relation R on X is called an equivalence relation if it satisfies all three of the following criteria:
R is reflexive, meaning that (x, x) ∈ R for all x ∈ X
R is symmetric, meaning that (x, y) ∈ R implies (y, x) ∈ R
R is transitive, meaning that if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R
Further, we define:
Given any x ∈ X, the R-class of element x is the set R(x) of all the elements that are in the equivalence relation R with x. In notations, R(x) is defined as {y ∈ X : (x, y) ∈ R
With these notions in place, we can start building the set Z from N, assuming that N is “given” together with its arithmetic operations of addition and multiplication, and the usual “≤” order. Within the set of natural numbers, we represent the solution x to the equation b + x = a (a,b ∈ N) with the ordered pair (a, b), where x is given as (a-b).
It is clear that for any x, the values a and b could take are not unique. Hence, x is represented by an equivalent class of infinitely many pairs (a, b), where the equivalence relation (a, b) ∼ (c, d) is defined by
“(a, b) ∼ (c, d) represent the same x if and only if a + d = b + c” …….(1)
As a result, the set of integers Z is essentially the equivalence classes of pairs (a, b) with a,b ∈ N with respect to the equivalence relations defined by (1). Hence, we have “constructed” Z as (N×N)/∼, in which every integer can be identified with [a,b], which denotes the equivalence class determined by the pair (a, b).
With integer numbers now constructed, we still need to define the three “rules” we have taken from N in Z, namely addition, multiplication and the “≤” order. By using the [a,b] notation, this is quite simply done:
1.Addition rule: define [a,b]+[c,d]=[a+c,b+d]
2. Multiplication Rule: define [a,b]∙[c,d]=[ac,bd]
3. Less-than-or-equal order: define [a, b] ≤ [c, d] ↔ a + d ≤ b + c
Finally, we just need to define a function E: N→ N×N/~ = Z As such, N is being embedded into Z in such a way that the arithmetic operations in N are carried over: E(n) is defined as [(n, 0)].
We have shown how we could construct the set of integers in terms of equivalence classes of pairs of elements from N. Not only have we obtained integer numbers, but we have also managed to maintain the arithmetic and order features of natural numbers. Similar procedures can be carried out to obtain Q and ℝ . Once we have done that, we would have gotten a sense of what Kronecker meant by “the rest is the work of man”, as we can simply set-theoretically construct those standard sets with N as a foundation.
However, some may question whether we indeed have such a foundation – “what if God did not give us N?” I believe this would not be too much of a problem either. With Dedekind’s (1988) ingenious view on ordinals, God only needs to give us “nothingness”, and we can then construct N ex nihilo by viewing it as an ordinal (Bostock, 2009). To do this, we first want to formalise our view on natural numbers as sets. In particular, we only need to let 0 be the empty set ∅ (hence “nothingness”) and treat 0 as the “first element” in N. Then, we can obtain the second element as a successor of 0, where a successor of X in this context is defined as the union of X and the set containing X itself as an element. Further, every element in X is defined to be smaller than the new element {X}. As such, by simply using the notion of “0 as the empty set”, and “successors”, one can iterate the process to obtain all natural numbers. In more concise terms, N can therefore be described as having 0 as a member, and also has a member the successor of each of its members, and it is the smallest set satisfying these conditions. Thus, even the second idea is “strong”, that we indeed can accept the existence of N because we can create N “out of nothing”.
Although we have explained how one could set-theoretically construct from natural numbers, Z, Q, and ℝ, we have not been able to construct the set of irrational numbers from in a finite manner. Some could argue that maybe this shows that not everything is “work of man”.
However, the way Kronecker chose to phrase the quote also has deep ideological significance. It perhaps reveals a deeper belief of his in intuitionism, which rejects mathematical objects that cannot be constructed. Thus, Kronecker might simply respond to these critics by saying that given precisely this non-constructive nature of irrational numbers, they do not exist and hence is not considered to fall under “the rest”. On this point, Kronecker has another equally famous quote to Lindemann -- “of what use is your beautiful investigation regarding π? Why study these problems when irrational numbers do not exist?” (Reid, 2012)
References
Bostock, D. (2009). Philosophy of Mathematics: An Introduction. Wiley-Blackwell.
Kreisel, Georg. 1983, “Hilbert’s programme”, in Philosophy of Mathematics, Paul Benacerraf and Hilary Putnam, eds., Cambridge: Cambridge University Press, 207–238, 2nd ed.
Reid, C. (2012). Hilbert. Springer Science & Business Media.