The descriptive use of logic in mathematics (Part II of 'Axiomatizations of arithmetic...')

(This is the second part of the series of posts with sections of the paper on axiomatizations of arithmetic and the first-order/second-order divide that I am working on at the moment. Part I is here, and I pick up immediately where I left yesterday.)
========================================

As mentioned above, what Hintikka presents as the descriptive use of logic in mathematics was the predominant approach in the early days of formal axiomatics, in the second half of the 19^th century (the notable exception being Frege, more on whom shortly). And indeed, Dedekind’s famous letter to Keferstein (published in (Van Heijenoort 1967)) is quite possibly one of the most vivid illustrations of the descriptive approach in action; because this approach was still something of a novelty at the time, Dedekind carefully explains his procedure when formulating an axiomatization for arithmetic (basically what is now known, by a twist of fate, as Peano Arithmetic).

How did my essay come to be written? Certainly not in one day; rather, it is a synthesis constructed after protracted labor, based upon a prior analysis of the sequence of natural numbers just as it presents itself, in experience, so to speak, for our consideration. What are the mutually independent fundamental properties of the sequence N, that is, those properties that are not derivable from one another but from which all others follow? (Letter to Keferstein, p. 99/100)

Thus, a pre-existing structure, the sequence of natural numbers, presents itself ‘for our consideration’, so that we can attempt to determine what its basic properties are. Dedekind then lists what appear to be the key properties of this structure, such as that it is composed of individuals called numbers, which in turn stand in a very special kind of relation to one another (the successor relation). One might think that, by offering an (apparently) exhaustive list of properties which, taken together, seem to describe the basic facts about this structure, the axiomatization would be complete also in the sense of picking out an unique referent, namely the intended structure, the sequence of the natural numbers. But Dedekind quickly adds that this is (unfortunately) not the case:

I have shown in my reply, however, that these facts are still far from being adequate for completely characterising the nature of the number sequence N. All these facts would hold also for every system S that, besides the number sequence N, contained a system T, of arbitrary additional elements t [satisfying certain conditions previously stated]. But such a system S is obviously something quite different from our number sequence N, and I could so choose it that scarcely a single theorem of arithmetic would be preserved in it. What, then, must we add to the facts above in order to cleanse our system S again of such alien intruders t as disturb every vestige of order and to restrict it to N? (Letter to Keferstein, p. 100)

In other words: while the properties he had just listed are all indeed present in the sequence N, they do not seem to exhaust the relevant properties of this structure, because they are equally true of other structures which are demonstrably very different from N. So an axiomatization guided only by these properties is not categorical because it does not uniquely refer to the intended sequence N; indeed, it also refers to structures strictly containing N but also containing additional, disruptive elements. How doe we get rid of these alien intruders?

For our purposes, it is important to notice that the properties listed by Dedekind up to this point have in common the fact that they can all be expressed by means of (what we now refer to as) purely first-order terminology. To be sure, that there may be an important distinction between first- and higher-order logics is something that became widely acknowledged only in the 1930s. Until then, first-order logic was not recognized as a privileged, particularly stable fragment of the logical system developed for the logicist project of Russell and Whitehead. So Dedekind had no reason to notice this peculiarity about these properties, or to make an effort to exclude higher-order terminology.

Dedekind then notices that solving the issue of the alien intruders t was the hardest part of his enterprise, as the problem is:

How can I, without presupposing any arithmetic knowledge, give an unambiguous conceptual foundation to the distinction between the elements n [the legitimate numbers] and the elements t [the alien intruders]? (Letter to Keferstein, p. 101)

The solution to this conundrum is offered by the technical notion of chains, which he had introduced in previous work. He explains this crucial notion in the following terms:

… an element n of S belongs to the sequence N if and only if n is an element of every part K of S that possesses the following two properties: (i) the element 1 belongs to K and (ii) the image φ(K) is a part of K. (Letter to Keferstein, p. 101, emphasis in the original)

(The function φ had been introduced previously.) Dedekind’s notion of chain can be glossed in modern terminology as “the minimal closure of a set A in a set B containing A under a function f on B (where being “minimal” is conceived of in terms of the general notion of intersection).” (Reck 2011, section 2.2) What matters for our purposes is that the notion of chain involves quantification over sets of elements n, and thus cannot be expressed solely with first-order terminology.

Dedekind correctly claims that the notion of chain offers a satisfactory solution to the problem of the alien intruders t. But as he had no reason to be parsimonious in his use of higher-order terminology, one might think that there might perhaps be other solutions to the problem of alien intruders not requiring the move to second-order quantification, which he did not consider. However, it is now known that first-order axiomatizations of arithmetic are inherently non-categorical (as per the usual Löwenheim-Skolen considerations). Second-order terminology is indeed required for an axiomatization to describe only the ‘intended structures’ – the sequence of the natural numbers and structures isomorphic to it – and not what are known as the non-standard models of arithmetic.

The need for second-order terminology to achieve categoricity in the case of axiomatizations of arithmetic is an illustration of the general point that the descriptive use of logic for mathematics, as defined by Hintikka, will generally require quite expressive logical languages. Arguably, first-order languages will systematically fail to deliver the expressive power required for the precise description of non-trivial mathematical structures, and this may be one of the causes for the (purported) inadequacy of first-order logic to account for ‘ordinary’ mathematical practice (Shapiro 1985).

TO BE CONTINUED...


Part III

Part IV