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Fregean Logicism

The following is a tutorial essay for Frege’s Grundlagen with Fabian Pregel. In effect, it is a quick agenda for an hour-long discussion on a guiding question, which for this week was ‘Assess Frege’s arguments for preferring the logicist strategy he eventually pursued in Grundlagen to the ones he rejected.’. A free-standing and much more readable version of some of the material may eventually go up on my Substack.

I. Introduction

In his Grundlagen, Frege takes the number of a concept F to be the set of concepts G such that there is a bijection between the instances of F and the instances of G (§72). He takes the natural number 0 to be the number of a contradictory concept (§74), and shows how to generate the other natural numbers: given a sequence of successive natural numbers (0, . . ., n), appending n + 1—that is, the number of the concept of being a natural number in that sequence—yields another sequence of successive natural numbers (§82). In this way, Frege constructs the natural numbers in something like a purely logical way, and is able to show that they behave as expected. This constitutes the strongest of Frege’s arguments for his approach: namely, that it works. “Definitions show their worth by proving fruitful,” he writes; indeed, “we can derive from our definition…any of the well-known properties of numbers.”

In selecting this strategy, though, Frege rejects two others as unsatisfactory. The first strategy—call it the quantificational strategy—simply specifies the conditions under which some number belongs to some concept, but declines to specify that numbers are objects (§55). The second strategy—call it the algebraic strategy—specifies the conditions under which the number of some concept is identical to the number of some other concept, but declines to specify which objects numbers are (§62-§65).

§II assesses Frege’s arguments against the adjectival strategy (§56); §III assesses Frege’s arguments against the implicit strategy (§56). In each case, such an assessment is aided by comparison to alternative arguments and countervailing considerations. To anticipate, the conclusion will be that Frege’s objections are relatively uncompelling; the alternative strategies are tenable enough that the best strategy must be decided by comparing which proves the most fruitful.

II. The quantificational strategy

Let us first sketch the quantificational strategy in a bit more detail. There are (at least) three Fs just in case ∃x1∃x2∃x3(Fx1 ∧ Fx2 ∧ Fx3 ∧ x1 ≠ x2 ∧ x2 ≠ x3 ∧ x3 ≠ x1). More generally, there are (at least) n Fs just in case ∃x1, . . ., xn(⋀iFxi ∧i≠j xi ≠ xj). Pragmatically, though, saying that there are n Fs usually communicates that there are exactly n Fs, which is to say also that there are not m Fs for m > n. If I have three cats, then I have two cats; but saying that I have two cats would be misleading, because you expect me to be a sufficiently cooperative conversation partner such that, if I had three cats, I would tell that to you instead of only telling to you the weaker fact that I have two cats.

If we want a unified treatment of 0, then ‘there are zero cats’ is trivial, but sounds false because it is incredibly misleading; but it’s plausible that 0 is special. That is, there are (exactly) zero cats just in case it’s not the case that there are cats. Notice that, effectively, the term ‘zero’ takes wide scope, such that the deep structure of the sentence is ‘zero(there are cats)’, just as the deep structure of ‘there are no cats’ is ‘no(there are cats)’. Compare, also, ‘there are some Fs’, which is true just in case ∃xx∀u(u ≺ xx → Fu). This gives some precedent for number terms like ‘three’ behaving similarly: the deep structure of ‘there are three cats’ may be ‘three(there are cats)’. In particular, number terms are higher-order concepts of type ⟨t,t⟩, which combine with ‘there are’ to form the quantifiers of type ⟨⟨e,t⟩, t⟩. Further, ‘the number of Fs’ is plausibly a function that returns n just in case there are exactly n Fs. (A similar treatment might be given for ‘the number n belongs to F’, but this is not natural English.)

Frege objects that the barest version of the quantificational strategy leaves open the possibility that ‘the number Julius Caesar belongs to F’ is true. But, on the above sketch, Julius Caesar is not a concept of the right sort and so cannot be a number. Of course, this response is a bit too quick, as Frege could equally well raise the ‘necessarily’ problem: why isn’t ‘the number necessarily belongs to F’ (or ‘there are necessarily Fs’) true? The more fundamental reason that Frege’s objection fails is that this strategy gives us an explicit catalogue of all the natural numbers, and neither Julius Caesar nor necessity are on it.

Frege objects further that the barest version of the quantificational strategy is far too weak: it cannot even prove the uniqueness of the number of a concept. But contemporary versions of this strategy (Goodsell & Yli-Vakkuri 2024) are sufficiently well developed to even prove all the Peano axioms—one of the core draws of Frege’s own strategy.

Frege also raises semantic objections; for instance, ‘the number of Fs’ must be an object. But this seems like a misleading surface form, analogous to Frege’s problem with the concept horse not being a concept. There is also the closely related issue of sentences like ‘the number of primes less than 10 is 4’. To make sense of such sentences, one might introduce higher-order numbers of the proper type (Goodsell & Yli-Vakkuri 2024). But this fails to solve closely related problems with sentences like ‘the number of primes less than 10 is identical to the number of cats I own’. The proponent of the quantificational strategy may have to employ rather nasty complicated type-shifting rules. This is a genuine problem, but should not count against the quantificational strategy because it is a problem for everyone who wants to accommodate sentences like ‘metaphysical modality is the broadest objective modality’—where a modality, like a number on the quantificational strategy, is expressed by a term (‘necessarily’) of type ⟨t, t⟩. So, Frege’s objections to the quantificational strategy seem rather weak when the strategy is developed even slightly more than he develops it.

III. The algebraic strategy

Frege develops the algebraic strategy—on which the number of Fs is the number of Gs just in case there is a bijection between the instances of F and the instances of G, and, further, numbers are objects—in a fair amount of depth, and defends it from some initial objections, before pressing one that he takes to be fatal to it. But analogues of his later development of his preferred strategy can be adapted to the algebraic strategy as well. In particular, the number of Fs is 0 just in case there is a bijection between the instances of F and the instances of a contradictory concept. Given a sequence of successive natural numbers (0, . . ., n), the number of Fs is n + 1 just in case there is a bijection between the instances of F and the natural numbers of that sequence. While we cannot say much about which object n + 1 is, we do know that it cannot be one of the previous natural numbers by Hume’s Law; we can append it to our sequence of successive natural numbers, and note that if the number of Fs is n + 1, then the instances of F also cannot be put into bijection with the new sequence. But we did not need to actually put the object n + 1 into our sequence as we built it; indeed, we did not need to start our sequence with the object 0. All that matters is the algebraic relations between the numbers; we decline to specify in independent terms which objects the numbers are.

The objection which Frege takes to be fatal to this strategy is again the Caesar objection. In particular, the worry is that the algebraic strategy fails to tell us the (supposed) fact that Julius Caesar is not a number. But the algebraic strategy, as outlined here, is perfectly happy to say that Julius Caesar could be a number after all (say, 5). Similarly, the algebraic conception of groups is perfectly happy to say that Caesar is a trivial group—perhaps most obviously the trivial group, but he and his toga could also be C2 if equipped with the requisite binary operation. Indeed, one draw of the algebraic conception is that more detailed specification leads to junk theorems. Frege’s strategy, for instance, seems to tell us that—assuming that all contradictory concepts of a given type are identical—the number zero does not change over time, but all other numbers do. In fact, if extensions satisfy something like the axiom of extensionality, then the number of people that I am is constantly changing: at one point, the number includes the concept of being a cat of mine, but at another, it does not. Further, the number 2 is of infinite cardinality. More contemporary versions of Frege’s strategy of specifying the natural numbers in independent terms also yield junk: for instance, any von Neumann ordinal n can be equipped with the topology n + 1 to form a T0 space.

Frege’s particular strategy of identifying the number of a concept with its equinumerosity class (and the direction of a line with its parallel class) generalizes disastrously to algebraic objects with important internal structure: for instance, the group up to isomorphism—say, C2—that some group—say, Caesar and his toga—constitutes certainly should not be identified with all groups isomorphic to it—after all, C2 has finitely many elements, and, more importantly, is a group. Even more worryingly, Frege’s particular strategy does not seem to clearly generalize to numbers other than the naturals. For instance, it is unclear how to generalize Hume’s law to irrational or even complex numbers, and it is unclear whether, for every irrational or complex number x, there is some property whose number of instances is x. But if there is no such property, then x would be identical to 0. So, Frege’s objection is again not very compelling. Indeed, if definitions are to be judged by their fruitfulness, the algebraic mindset—on which what objects numbers really are is more or less irrelevant—seems to be far more successful.

IV. Conclusion

Frege admits that certain strategies for logicism are very natural, but he quickly dismisses them in favor of an explicit definition of number. Unfortunately, his particular strategy seems unable to generalize to neighboring cases, like non-natural numbers or mathematical objects with important internal structure. His dismissals are also too quick. By developing the first alternative strategy—the quantificational strategy—slightly more than Frege himself does, we see that his Caesar objection, weakness objection, and logical objection are all uncompelling for various reasons. By developing the second alternative strategy—the algebraic strategy—slightly more than Frege does, we see that there is an attractive position that embraces the possibility that Caesar could be a number, and avoids the junk theorems that plague independent specifications of what numbers are.

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