Frege took himself to have completed that part of the project that deals with the natural numbers (i.e. the finite cardinal numbers 0, 1, 2, 3, and so on). That is to say, he provides here the promised purely-logical proofs of highly-analyzed versions of a handful of fundamental truths about the arithmetic of the natural numbers. The idea, once again, was that these truths would in turn suffice for the proof of the whole of the arithmetic of the natural numbers. The completed part of the Grundgesetze project also includes the beginnings of a theory of real numbers.
A critical part of the formal system of Grundgesetze is its inclusion of a term-forming operator which, when attached to a predicate-phrase, gives a name of the value-range (Wertverlauf) of the associated function. Value-ranges are the modernized, formalized version of the extensions of Grundlagen, and obey a similar extensionality principle. Specifically, the principle is that for functions F and G, the value-range of F = the value-range of G iff (∀x)(Fx = Gx). Where F and G are concepts (first-level functions of one argument whose value is always a truth-value), the identity-conditions of the associated value-ranges are exactly those of the earlier extensions. Value-ranges play in Grundgesetze a role similar to the central role played by extensions in Grundlagen: numbers are now, in Grundgesetze, understood as value-ranges of concepts, and the extensionality-principle governing these items (Basic Law V) is essential to the most-important results regarding numbers.
Frege’s work on the project was stopped short by the letter from Bertrand Russell in 1902 noted in section 11.1.1, in which Russell points out the inconsistency in Frege’s system.11 The inconsistency arises as follows. Consider the predicate R, a predicate satisfied by exactly those objects o such that: o is the value-range of some concept C such that ~C(o). Now we consider the value-range of R, which we’ll call “r.” We ask whether that object r falls under the concept R, that is, whether R(r). If R(r), then (given the definition of R) r is the value-range of some concept under which it, r, does not fall. From this it follows (given extensionality) that r does not fall under any concept of which it is a value-range. But this contradicts our supposition that R(r), since r is of course the value-range of R. Thus far, we have shown that ~R(r). But this too leads to contradiction: if ~R(r), then r is the value-range of a concept (namely, R) under which it does not fall. So, by definition, R(r). At this point, we have deduced both that R(r), and that ~R(r), which is a contradiction. Since we did this using just Frege’s fundamental principles about value-ranges, we see that something is very badly wrong with those principles.
The problem this raises for Frege’s logicist project is twofold. First of all, it means that the purportedly-logical axioms on which Frege meant to found arithmetic are not in fact truths of logic. That this is a serious difficulty follows from the fact that there is no straightforward way to replace the problematic parts of the system by principles that are both purely logical and strong enough to develop arithmetic as Frege understands arithmetic. The second difficulty raised by the contradiction stems from the fact that Frege’s all-important analyses of arithmetical truths turn crucially on the notion of value-ranges, that notion that is shown by Russell’s contradiction to be incoherent. Both the analytic stage and the proof-theoretic stage of Frege’s logicism are therefore undermined by the difficulty over value-ranges.
Frege’s immediate reaction to the paradox was to attempt to modify his principles regarding value-ranges so as to maintain the general structure of his proofs while regaining consistency.12 He eventually came to realize, however, that no such “fix” would work, and that no principles governing value-ranges could count both as ‘purely logical’ by his own lights, and sufficiently rich to ground the existence of an infinite collection of objects, that is, the numbers. His conclusion was that the paradox, and the consequent failure of his fundamental principles, shows that there is no way to give a purely-logical grounding of arithmetic.
This pessimistic conclusion was not universally shared in the light of Russell’s paradox. In section 11.3.1, we outline the central ways in which logicist programs are still being pursued, ways which share a good deal with Frege’s original program, but depart from it in some significant ways. Frege’s own conclusion was that the combination he had envisioned, a conception of arithmetic in which arithmetic is about distinctive objects and yet grounded in pure logic, is untenable. Judging by some late manuscript notes, he appears to have held at the end of his life that the most promising account of arithmetic would maintain the view of arithmetic as concerned with distinctive objects, but would ground the truths about those objects in facts given by pure intuition.13
11.2.4 Later Work Outside of Grundgesetze
To return to Frege’s chronology: in the final few years of the nineteenth century, and into the twentieth, the mathematical world in Germany and much of Europe saw an explosion of interest in logic, and in the nature of mathematical theories. An important part of this development involved increasing sophistication regarding the nature of axioms, and in the understanding of the nature of, and of proof-techniques regarding, the relations of dependence and independence that obtain between parts of theories. In 1900, David Hilbert published a monograph entitled Foundations of Geometry, in which the newly-emerging techniques for demonstrating consistency- and independence-results, and the new, modern conception of axioms, are presented.14 In the years 1899–1900, Frege engaged in a correspondence with Hilbert, in which Frege presents an opposing view of the nature of axioms, and of the fundamental logical relations of consistency and independence.15 Frege continues to develop his position in two series of essays, each entitled “On the Foundations of Geometry,” published in 1903 and 1906.16 At the heart of Frege’s argumentative strategy is a defense of his view that mathematical theories, and hence their axioms, must be understood as collections of thoughts. This view is in direct conflict with the view defended by Hilbert, in accordance with which the axioms of a theory ought to be understood essentially as partially-interpreted sentences. On Frege’s view, each mathematical theory is a set of truths about a determinate subject-matter, while on Hilbert’s view, a theory is a set of general structural requirements, applicable to a wide variety of different domains. Hilbert’s conception of theories is geared towards the investigation of the structural properties of those various domains, and of demonstrating such results as the consistency and independence of sets of axioms, the categoricity of defining conditions, and so on. Frege’s conception, on the other hand, is geared toward the investigation of the fundamental truths of a given subject-matter, as these concern a specific collection of objects, functions, and relations. In the debate between the two logicians, and in Frege’s follow-up essays, we find a rich articulation of a fundamental cleavage between two ways of conceiving of the nature of logic and of its role in mathematics. The issue remains alive today, with both Frege’s and Hilbert’s sides defended in various ways as part of ongoing investigations into the nature of mathematical and logical knowledge.
In the years 1918–25, Frege returns to the philosophy of language, writing a series of essays on sense and reference that expand upon the conception of language articulated in the period 1891–2.17 In these later years, he treats, for example, the problem of “indexicals,” that is, words like “I” and “yesterday” whose reference depends in part on the context in which they are used. Frege’s fundamental idea here is that the context of use is part of what determines the sense expressed by the use of a sentence, so that, for example, two utterances of “It’s raining today” will express different senses if they occur on different days. Frege also pursues in more detail than previously the nature of thoughts (i.e. as sketched already, the senses of sentences). As against the worry that thoughts are somehow too ephemeral to be real, Frege replies that thoughts are indeed real, and that (in keeping with his general anti-psychologism) they are to be distinguished from anything subjective, like ideas. Thoughts exist in what he calls in 1918 a “third realm,” a realm of objects that differ from m
aterial objects in not being concrete, but that differ from ideas in not being subjective.18
Frege died in Bad Kleinen in 1925, and is buried in the Friedhof at Wismar, the city of his birth. Though he judged his logicist project to have been a failure, he seems to have recognized nevertheless the importance of his logical investigations. Six months before his death, Frege expressed to his son Alfred, regarding a number of unpublished essays, a sentiment that applies as well to the published work:
Even if not all is gold, there is gold in them. I believe there are things here which will one day be prized much more highly than they are now.19
11.3 AFTERMATH: FREGE’S LEGACY
11.3.1 Logic and Mathematics
Frege’s idea that logic can be pursued via the use of formal systems is so commonplace now as to be worth hardly a second thought, but it was an entirely new idea at the time, one that Frege introduced and pressed into extraordinarily fruitful service. Though our means of employing formal systems now incorporates some elements that Frege would have rejected, as he explains in his controversy with Hilbert, nevertheless the fundamental idea of encoding logical inferences via syntactic transformations remains at the heart of modern logic. Frege’s introduction of the quantifier, similarly, brought logic into the modern era.20 His analyses of fundamental arithmetical concepts in terms of simpler logical ones came out of fruitful interaction with the mathematics of his day and cannot in all respects claim originality, but Frege’s single-minded and rigorous employment of those analyses in pursuit of an epistemologically-significant goal has left a permanent mark on the philosophy of mathematics.
The logicist project itself, despite its clear failure (indeed, to some extent because of that failure) has been an especially fruitful influence on subsequent investigations. The central reason for continued interest in Frege’s logicism stems from a continued interest in the nature of mathematical, specifically arithmetical, truth. In providing his analyses of arithmetical truths, Frege argues for a number of theses about arithmetic, including the claims that arithmetic is not based on anything psychological, that it is not grounded in intuition, and that arithmetic deals with a specifically arithmetical collection of objects, the numbers. The arguments Frege provides for these theses continue to play a significant role in the development of competing philosophical views about the nature of mathematical truth and mathematical knowledge. Because Frege’s views taken together require the truth of logicism (though not the success of his particular means of demonstrating it), any real engagement with Frege’s important arguments about the nature of mathematical truth and knowledge must come to grips with his logicism. Progress on these issues requires progress on the question of whether the logicism is, in the end, a viable thesis, and if not, then which parts of Frege’s edifice one must give up.
Bertrand Russell himself did not take the paradox he discovered in Frege’s system to be a reason to reject the logicist thesis. After the paradox, he pursues together with Alfred North Whitehead a revised version of logicism that is intended to be essentially in the spirit of Frege’s project.21 One difficulty with the Russell–Whitehead project, and a difficulty that would presumably have led Frege to view it as not quite “logicist,” is that the fundamental principles, the axioms, from which Russell and Whitehead attempt to derive much of mathematics include principles (especially the axiom of infinity and the axiom of reducibility) that Frege would not have regarded as purely logical. The difficulty here is that they lack the kind of self-evidence that Frege demanded of fundamental logical truths. On this question, subsequent scholars have primarily sided with Frege.
More problematic for the Russell–Whitehead version of logicism, together with that of Frege, are Kurt Gödel’s incompleteness theorems.22 These theorems were proven after Frege’s death, so did not directly affect Frege’s own research, but provide yet another blow to the kind of program envisioned both by Frege himself and by those, e.g. Russell and Whitehead, who worked on resuscitating a Frege-style logicist project in the aftermath of Russell’s paradox. Gödel’s first incompleteness theorem shows that, as long as the truths of arithmetic are expressible, as both Frege and Russell would have agreed, via a syntactically complete set of sentences of a formal language of arithmetic, then there is no way to axiomatize arithmetic. That is to say, Frege’s assumption that there is a manageable “core” of arithmetical truths from which the rest of arithmetic is provable is simply false. No decidable set, and certainly no finite set, of truths can serve as the proof-theoretic basis for all of arithmetic. This result by itself does not show that logicism is false, but it does undermine the straightforward Fregean way of attempting to demonstrate its truth, namely the strategy of proving the (purported) core arithmetical truths from truths of logic. In addition, it means that logicism can be true only if the truths of logic themselves form a quite unmanageable collection (specifically, one that’s not recursively enumerable).
The concept of value-range at the core of Frege’s difficulties with logicism is one of a handful of similarly compelling, and similarly problematic, concepts that have been at the heart of much mathematical work since the middle of the nineteenth century. The notions of the graph of a function, and more familiarly of a set of numbers or of functions, are of the same ilk as Frege’s value-ranges, and are similarly slippery: their role is sufficiently ubiquitous and foundational that it appears that, as Frege puts it, we “cannot get on without them.”23 But the attempt to lay down basic principles governing these entities has revealed that the most natural way of understanding them, that is, Frege’s way, in terms of a purportedly-analytic principle of extensionality, is not workable. It is in this sense that Frege’s clarity about the importance of value-ranges, and the precision of his attempt to lay down their fundamental principles, has been salutary for the development of modern foundational theories and for an understanding of their philosophical significance. Current foundational work, following the lead of Cantor and Zermelo, focuses on the axiomatic presentation of theories of sets, dropping Frege’s claim to the analyticity of those axioms. The philosophical question of what such foundational strategies can tell us about the nature of mathematical truth, of mathematical objects, and of mathematical knowledge is one that continues to be a subject of fruitful debate, and one that turns in part on the question of which of Frege’s aims, in attempting to found arithmetic on value-ranges, must be given up in the modern setting.
Finally: an especially important thread in the influence of Frege’s logicism concerns the impact of that thesis on modern empiricism. Here the influence is twofold. First of all, Frege’s logicist notion of the reduction of truths about a given subject-matter to truths about something arguably more simple serves as a model for later attempts to reduce the empirical sciences to something closely linked to the immediate contents of experience. In this regard, Frege’s influence on both Bertrand Russell and Rudolf Carnap is direct, and through them the Fregean conception of theoretical reduction, or descendants of it, continues to influence empiricists and their critics today. The second central influence of Frege’s logicism on this movement was that the logicist thesis offered hope to early twentieth-century empiricists that mathematical knowledge might be subsumed under the umbrella of logic, and hence no longer stand as a prima facie obstacle to the empiricists’ denial of synthetic a priori knowledge. In this, the tradition that follows Frege goes further than Frege would have gone himself: Frege is no empiricist, and holds that geometry (unlike arithmetic) offers a clear instance of the synthetic a priori. Though Frege’s own attempt to demonstrate the truth of his logicist thesis was, as discussed, a clear failure, the question of whether that logicist thesis, or some near relative of it, is in fact defensible remains open.24
11.3.2 Language and Mind
Frege’s theory of sense and reference has had an enormous impact on twentieth- and early twenty-first-century philosophy of language and mind. Most significant here have been his views that each significant piece of language
has a sense that’s grasped by competent speakers, and that it is in virtue of expressing this sense that a piece of language refers to the item that it in fact refers to. Put together, these two views about sense give rise to a thesis about the connection between mental states and language that Frege himself did not dwell on: the thesis that what our words refer to is determined by what we, the speakers of those words, understand, or represent, when we use them. So stated, the thesis is not entirely clear, depending as it does on exactly what one means by “understand” or “represent,” and also on what it is for such a represented entity to determine the reference of a word. While different accounts of exactly what Frege has in mind when he speaks of “grasping” the sense of a sentence will yield different verdicts on the degree to which Frege can be said to explicitly agree with different variations of the theme here, the fundamental idea is relatively straightforward: the Fregean view, not unreasonably so-called, is the view that our terms refer to entities in virtue of those entities’ satisfaction of properties that we speakers explicitly associate with those words.
The Oxford Handbook of German Philosophy in the Nineteenth Century Page 43
