Philosophy of Mathematics

by Øystein Linnebo

This is precisely what Frege thought. He pioneered the influential view known as logicism, which holds that at least all truths about numbers—the naturals up through the reals—are reducible to logic and thus analytic. The following table summarizes our discussion so far. analytic synthetic a priori Frege Plato (pre-Copernican) Kant, Brouwer (Copernican) a posteriori ✘ Mill, Quine As mentioned, some views fall outside of this table because they deny that mathematical theorems are true.

In short, Frege’s aim of gap-free proofs led him to formulate what is now known as a formal system, which is an artificial language with clearly formulated axioms and inference rules, all described with mathematical precision.

First-order quantifiers generalize into the position occupied by a singular term or proper name, while the more controversial second-order quantifiers generalize into the position occupied by a predicate. Consider, for example, the claim that Socrates is mortal, symbolized as Ms. While first-order logic allows us to infer that there is someone who is mortal, ∃x Mx, second-order logic allows us to infer that there is some “concept” (as Frege put it) under which Socrates falls, ∃F Fs.3

If the proof relies only on “general logical laws and definitions,” then the proposition is analytic. But if the proof requires assumptions that “belong instead to the domain of a particular science, then the proposition is synthetic”

When I assert “These are five trees,” for example, I am saying of the first-level concept TREE that it is quintuply instantiated. Frege is particularly pleased that “[t]he extensive applicability of number can now be explained” as follows. Numbers are ascribed to concepts, which can apply to absolutely all kinds of object, “the physical and mental, the spatial and temporal, the non-spatial and non-temporal”

One natural continuation would be to regard numbers as second-level concepts that ascribe cardinality properties to first-level concepts. For instance, “These are five trees” might be analyzed as “5x(TREE(x)),” much like “There is a tree” is analyzed as “∃x(TREE(x)).” Frege subjects this analysis to a barrage of objections. The most serious is that “[e]very individual number is an independent object” and thus not a second-level concept. As evidence, he points to the fact that we speak of “the number 1,” where the definite article serves to class it as an object. In arithmetic this self-subsistence comes out at every turn, as for example in the identity 1 + 1 = 2. (Ibid., §57) This argument has attracted much scholarly attention because of its unusual mix of linguistic and metaphysical considerations.5 The premises involve linguistic considerations about how we talk about numbers. But the desired conclusion is metaphysical, namely that numbers are objects, not second-level concepts. That is, the conclusion is an instance of what we earlier called “object realism.”

To spell things out further, we need to talk about semantics, which is the branch of linguistics concerned with meaning, reference, and truth—in short, with the relation between language and the world. The Fregean argument, as I shall call it, relies on the following two premises: Classical Semantics. The singular terms of the language of mathematics are supposed to refer to mathematical objects, and its first-order quantifiers, to range over such objects. Mathematical Truth. Most sentences accepted as mathematical theorems are true. We now reason as follows. Consider sentences that are accepted as mathematical theorems and that contain one or more mathematical singular terms. By Mathematical Truth, most of these sentences are true. Let S be one such sentence. By Classical Semantics, the truth of S requires that its singular terms succeed in referring to mathematical objects. Hence there are mathematical objects.

“How, then, are numbers to be given to us, if we cannot have any ideas or intuitions of them?” (§62). This is a version of what we have called “the integration challenge” (cf. §1.5).6 Since numbers cannot be perceived or tracked by means of instruments, how do we manage to refer to them, let alone gain knowledge of them? Frege’s response to this challenge takes us right to the heart of his philosophy: Only in the context of a sentence do words have meaning. We must, therefore, define the sense of a sentence in which a number-word occurs.

Frege applies the context principle to the question of how the numbers are “given to us.” This is a semantic question about how number terms come to refer to numbers. By the context principle, this question cannot be answered in isolation, outside of any sentential context. The question must instead be answered by explaining how sentences containing number terms come to be meaningful. In this way, Frege hopes, the context principle will transform the question about our semantic and epistemic “access” to numbers into an easier question about the meanings of arithmetical sentences.

This makes identities of the form “the number of Fs = the number of Gs”—or in symbols, “#xFx = #xGx”—especially important.7 Putting things together, the original problem of explaining how the numbers are “given to us” is thus transformed into the problem of explaining the meanings of identities of the form “#xFx = #xGx.” The meanings of other sentences involving number terms can be explained later. Frege proposes a brilliant analysis of the mentioned identities. He proposes that the meaning of “#xFx = #xGx” be “recarved” as the claim that the Fs and the Gs can be one-to-one correlated. What it means for the number of knives on a table to be identical with the number of forks, for example, is that each knife is correlated with a unique fork and vice versa. This analysis is attractive. It seems to avoid all use of the number terms whose meaning we are trying to explain. All that the analysis requires are the resources to express that the Fs and the Gs can be one-to-one correlated. And these resources are available in pure second-order logic, where a second-order quantifier can be used to state that there is a relation that one-to-one correlates the Fs and the Gs. Thus, Frege’s proposed analysis seems both noncircular and purely logical. The proposed analysis is encapsulated in the principle where ‘F ≈ G’ abbreviates the mentioned claim about one-to-one correlations, or, as we shall put it, that the Fs and the Gs are equinumerous. Because Frege claims to be taking a cue from ...

object realism is the view that there exist mathematical objects. Another form of realism is concerned, not with the question of what there is, but with objectivity: Truth-value realism. Every well-formed mathematical statement has a unique and objective truth-value which is independent of whether it can be known by us or proved from our current mathematical theories. Frege accepts both forms of realism. As we have seen, he compares mathematicians with geographers, who discover new continents whose existence and characteristics are independent of us. Is there any relation between object realism and truth-value realism? In sciences other than mathematics, there certainly is.

Platonism, we recall, holds that abstract mathematical objects are just as real as ordinary physical objects and play analogous roles in their respective sciences (cf. §1.4). This suggests the same order of explanation as above, namely that mathematical objectivity is underpinned by the existence of mathematical objects, which secure and explain the objective truth-values of statements concerned with such objects. Frege rejects this order of explanation. He takes questions about the meaning of complete sentences to be explanatorily prior to questions about the reference of singular terms. On this view, the existence of mathematical objects is to be explained in terms of the objective truth-conditions of statements concerned with such objects rather than the other way round.

In sum, while Frege endorses both object realism and truth-value realism about mathematics, he stops short of full-fledged platonism. He reverses the platonist’s view of the relative explanatory priority of mathematical objects and mathematical objectivity.12

What exactly is the arithmetical theory that Frege wished to reduce to pure logic? The target theory is what is now known as second-order Dedekind-Peano arithmetic. This is an arithmetical theory that is formulated in a second-order language with three (for now) nonlogical symbols: a constant ‘0’ for the number zero and predicates ‘Pxy’ and ‘ℕx,’ stating that x immediately precedes y and that x is a natural number, respectively. The theory has the following axioms:13 (1) ℕ0 (2) ℕx ∧ Pxy → Ny (3) ℕx ∧ Pxy ∧ Pxy′ → y = y′ (4) ℕy ∧ Pxy ∧ Px′y → x = x′ (5) ℕx → ∃y Pxy (6) ∀F (F 0 ∧ ∀x∀y(ℕx ∧ Fx ∧ Pxy → Fy) → ∀x(ℕx → Fx)) To reduce this theory to pure logic, Frege’s first task is to provide a logical analysis of the three symbols still left unanalyzed. He begins by observing that zero is the number that applies to concepts under which no objects fall. This suggests that ‘0’ be defined as ‘#x(x ≠ x),’ that is, as the number of objects that are distinct from themselves. Next, what about the relation of one number immediately preceding another? This is simple enough to explain to a young child. Suppose you have some objects F and remove one of them, say a. Then the number of things left immediately precedes the number of things with which you started. This means that ‘Pxy’ can be defined as Finally, there is the concept of being a natural number. Informally, we often say that x is a natural number just in case x = 0, or x = 1, or …. But clearly, the ellipsis ‘…’ would not ha...

Frege’s fix was to provide an explicit definition of the numbers. To do so, he followed the lead of contemporary mathematicians such as Dedekind, who had discovered how to handle abstraction by equivalence classes rather than abstraction principles. Consider the case of directions. Instead of accepting directions as a distinct kind of object, we may identify the direction of a line l1 with an equivalence class, namely the class of all lines l2 that are parallel to l1. It is straightforward to show that the principle (Dir) can now be derived and is thus not needed as an axiom. Frege suggested we apply the same strategy to numbers. There is no need to regard Hume’s Principle as an axiom governing numbers understood as a distinct kind of object. It suffices to define the numbers as equivalence classes. We may for example define #xFx as the class of concepts that are equinumerous with F: Hume’s Principle can now be derived and need not figure as an axiom.

For Frege’s revised strategy to work, however, we need a purely logical way to handle equivalence classes. Frege sought to achieve this by means of an abstraction principle for classes, namely his “Basic Law V,” which states that two concepts have the same extension just in case they are coextensive. Let us write {x | Fx} for the extension of the concept F. The law is then formalized as follows:15

Given any abstraction principle, we can define its abstracts as the appropriate equivalence classes and derive the principle from Basic Law V, using this definition. Basic Law V can thus be regarded as “the mother of all abstractions.” Given this single abstraction principle, all the others follow. But in 1902 disaster struck.

The “difficulty” is now known as Russell’s paradox. As Frege knew, a membership predicate “x ∈ y” can be defined as “∃F(y = {u | Fu} ∧ Fx).” That is, x is a member of y when y is the extension of a concept F under which x falls. Consider now the Russell class r, defined as {x | x ∉ x}. Is r a member of itself?

If we can legitimately and truthfully speak as if there are abstract objects such as directions, what more might be required for such objects to exist? There is no higher standard for the existence of objects of some sort than that which governs our discourse about such objects.

What is required for the existence of the directions is thus nothing over and above the existence of lines standing in appropriate relations of parallelism. There is no “metaphysical distance” between the former fact and the latter.

The problem arose, we recall, because Frege sought to reduce all forms of abstraction to a single form. When two concepts are coextensive, let us say that they have the same extension. This yields Frege’s infamous “Basic Law V,” which in contemporary notation can be written as As Russell discovered, this “law” gives rise to the paradox now bearing his name

STT distinguishes sharply between individuals, classes of individuals, classes of classes of individuals, etc. As Russell suggests, we may think of these as respectively individuals, families, clans of families, etc. In fact, the distinction between individuals and classes of the various levels is so sharp that each level has its own set of variables and constants, each with a superscript that indicates its level or type (as it is also called). For example, variables for individuals and for “clans” are of type 0 and 2, respectively. Furthermore, a membership claim s ∈ t is considered well-formed only if the type of t is exactly one higher than that of s. Thus, while it is permissible to say that an individual is a member of a family, or a family of a clan, it is impermissible to say that an individual is a member of a clan or of another individual. It follows that the condition meant to define the Russell set, namely ‘x ∉ x,’ is ill-formed. Russell’s paradox is thus blocked by a grammatical prohibition.

A cardinal number is supposed to be some form of property that two families share just in case they are equivalent in this way. So why not simply identify the cardinal number of a family with the clan consisting of all families equinumerous with this given family? This is mathematically elegant, and it ensures that two families have the same cardinal number just in case they are equinumerous. As a logicist account of mathematics, however, STT is now generally regarded as a failure. One problem is that the strict division into types is cumbersome and needlessly restrictive.3 A related problem is that its syntactic restrictions on membership claims block not only the paradoxes but also Frege’s celebrated bootstrapping argument, which shows that there are infinitely many numbers. The argument crucially turns on the fact that numbers too can be counted. Thus, if we have established the existence of numbers 0 through n, we have at least n + 1 objects available to count, which enables us to establish the existence of the number n + 1 as well. This argument is no longer available when numbers are construed in Russell and Whitehead’s way as clans of equinumerous families. As a result, we need a separate axiom of infinity, which states that there are infinitely many individuals. But it is doubtful that this axiom can be regarded as a purely logical principle. It certainly does not qualify as logical on our contemporary conception of logic, which requires a logical truth to be true ...