Abstract
In this paper we critically evaluate the notion of the structure of the natural numbers with respect to the question how the internal structure of such a structure might be specified.
Research partially supported by the Portuguese Science Foundations FCT, projects Hilbert’s 24th Problem, PTDC/MHC-FIL/2583/2014, and Centro de Matemática e Aplicações, UID/MAT/00297/2013.
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Notes
- 1.
Historically, the example of the mutilated chessboard can be traced back to Max Black who posed it in 1946 as a problem in his book Critical Thinking (Black 1946, exercise 6, p. 142) (but starting off with the chess board, thus, leaving out the creative part of adding this structure as a first step). It is also reported that Emil Artin occasionally used this example in his lectures (see Zweistein 1963; Thiel 2006); it might well be that he took it from Black (or some other later source), but it was stressed in a obituary for Artin that he applied the idea of the solution within his mathematical activity, as he had “the very rare ability to detect, in seemingly highly complex issues, simple and transparent structures”. (“[Er hatte d]ie so seltene Gabe, in scheinbar hochkomplexen Sachverhalten einfache, durchsichtige Strukturen aufzuspüren” (Reich 2006, p. 39).)
- 2.
More exactly: the center of mass of the solar system, which consists of the Sun together with all objects of the solar system; but, clearly, the Sun dominates this mass in such a way that it is reasonable to identify the center of mass with the center of the Sun.
- 3.
We will shortly address abstract structures in Sect. 7.4. The distinction of these two forms of structures is ubiquitous for the working mathematician: the different character of natural numbers and groups is conceptionally self-evident. For the syntactic counterpart of structures we find it explicitly discussed in (Hermes and Markwald 1958, §§ 4.2 and 4.3) as heteronome and autonome Axiomensysteme. Although the authors presuppose an empirical base for the heteronomous axiom system (what we will not do for the concrete structures), they point to the fact that these axioms systems are chosen a posteriori (“nachträglich gewählt”). This is in accordance with our understanding of concrete structures relating to mathematical objects which are supposed to preexist.
- 4.
The semantic nature implies, in particular, that it is assumed that any (properly formulated) statement about this realm has a definite truth value.
- 5.
The distinction of the syntax and semantics of, let say, function symbols and functions themselves is, of course, fundamental in mathematical logic, see Kahle and Keller (2015). In the present context, however, we argue essentially entirely on the semantic side and will neglect the difference as it should not give rise for confusion.
- 6.
The (potential) infinity of \(\mathbb {N}\) could also be given by such a condition: by consecutively taking out elements from this bag, one will never completely empty the bag.
- 7.
Tactically, we use here 0 and S as syntactic entities to give names; if you think of their semantic interpretations, you would have always the corresponding concrete elements of the infinite set at hand; still, S would be supposed to be a semantic function telling you how to go from one element to the next one.
From another, constructive perspective one can also proceed the other way around: starting from 0, one constructs successively the elements of \(\mathbb {N}\) by applying the “successor function” S. In fact, it would be far better to call this “function” in this specific context “constructor”. This terminology is known from Computer Science, and the analogy is not accidental: the separation of the definition of a datatype, by constants and constructors, from the implementation of functions operating on this datatype corresponds to the distinction we have in mind here. The problem with the constructive perspective is that it cannot go beyond countable universes.
- 8.
Intentionally, we phrased this exercise as a trivial tautology: first-order definable functions and relations are, by definition, given by first-order logic formulas over addition and multiplication. The interesting question is, of course, which are these functions; it is just an empirical observation that they include essentially all number-theoretic functions used (and defined independently) in the history of mathematics; to show this inclusion is not a trivial exercise!
- 9.
The story was recorded by G. H. Hardy (1921, p. lvii f): “I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. ‘No,’ he replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.’ ” One may note, however, that Bernard Frénicle de Bessy had already published this fact in 1657.
- 10.
For instance, by defining ≺ as:
$$\displaystyle \begin{aligned} 1 \prec 2 \prec 3 \prec \dots \prec 0, \end{aligned}$$where 0 is supposed to be ≺-greater than every other natural number, it can be taken to represent ω (and any original n represents n − 1 of the ordinal world).
- 11.
Remember the dots in the definition of \(\mathfrak {N}\) at the beginning of Sect. 7.3.
- 12.
In fact, at a time where Skolem had already promoted first-order logic as the “one and only solution” for mathematics; but against the fierce opposition of Zermelo.
- 13.
This is in contrast to Hilbert who treated Arithmetic and Analysis equally in his second problem in his famous Paris problem list of 1900, as, by that time, he had no reason to restrict himself to first-order theories.
- 14.
“Brouwer and other constructivists were much more successful in their criticisms of classical mathematics than in their efforts to replace it with something better.” (Bishop 1967, p. ix).
- 15.
“Intuitionism was transmuted by Heyting from something which was anti-formal to something which is formal. When one speaks today of intuitionism, one is talking of all sorts of formal systems (studied by the logicians).” Bishop in (Bishop 1975, p. 515); for the valuable results see, for instance, (Kohlenbach 2008); for constructivity (Bishop 1967).
- 16.
Corry (2004, p. 259) describes this for Øystein Ore’s introduction of the term structure in the context of his concept of lattice in the following words: “The leading idea behind this attempt was that the key for understanding the essential properties of any given algebraic system lay in overlooking not only the specific nature, but even the very existence of any elements in it, and in focussing on the properties of the lattice of certain of its distinguished subsystems.”
- 17.
We may note that the way Bourbaki promoted the notion of abstract structure was critically evaluated by Leo Corry (2004) with the result that Bourbaki’s use lacks a satisfying specification of the notion of structure.
- 18.
- 19.
One will not be able, for instance, to impose the structure of the natural numbers on a finite set; or a field structure on a set with six elements.
- 20.
In the sense of Bernays’s “bezogene Existenz”, (Bernays 1950).
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Kahle, R. (2018). Structure and Structures. In: Piazza, M., Pulcini, G. (eds) Truth, Existence and Explanation. Boston Studies in the Philosophy and History of Science, vol 334. Springer, Cham. https://doi.org/10.1007/978-3-319-93342-9_7
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