Motives
In The Foundations of Arithmetic, Frege argues for logicism, the view that the truths of arithmetic can be derived from the laws of logic. Frege distinguishes two motivations for his foundational work in arithmetic:
- mathematical motives (§§1 & 2)
- philosophical motives (§§3 & 4)
Mathematical Motives
The mathematical project is to restore the “old Euclidean standards of rigor” by providing proofs of arithmetical claims that have hitherto been regarded as self-evident—proofs which satisfy two conditions:
- every assumption is explicitly stated; and
- every inferential transition is in accord with an acknowledged rule (in the logic of the Begriffsschrift, detachment or the substitution rule).
If the target mathematical claims are indeed self-evident, what’s the point? Frege cites four related reasons for his project of proving the obvious:
- The renewed impulse of rigor in geometry and analysis has already shown itself mathematically fruitful by revealing the “limits of validity” of certain important theorems.
- By making explicit the inferential principles that implicitly guide our judgments we may arrive at “general methods” of forming concepts and proving theorems that may help us resolve open mathematical questions.
- By reducing the number of judgments that are accepted without proof we achieve an economy that is valuable in its own right.
- Even if the truth of a thought is beyond doubt and the limits of its validity is perfectly clear, it is still a mathematical advance to prove it-in this way we reveal the pattern of logical dependencies among thoughts thereby clarifying the content of our mathematical theories.
If one is going to dedicate one’s lifework to proving self-evident arithmetical claims reasons one and two may seem a little thin. There is no guarantee at the outset that the limits of validity of important theorems will be established or that any general methods of concept formation or mathematical proof will be discovered. The economy put forth by reason three is merely an aesthetic virtue of which Frege makes little theoretical use. Frege’s real motive lies with the thought that discovering proofs where proof is available is a genuine mathematical advance.
Philosophical Motives
What’s the epistemological status of arithmetical truths? Are arithmetical truths analytic a priori, synthetic a priori, or synthetic a posteriori? A closely related question that Frege considers is how are the numbers given to us? Kant, in the Transcendental Aesthetic of The Critique of Pure Reason, claims that: “It is [only] through the medium of sensibility that objects are given to us and it alone provides us with intuitions.” Numbers however, if they exist at all, are invisible and intangible. How then could they be given to us through the medium of sensibility? One of the goals of The Foundations is to militate against this Kantian claim. Frege will argue that the numbers are given to us not through the medium of sensibility whether through pure or empirical intuition, but rather through the medium of reason.
Kant distinguishes on the one hand:
- a priori judgments
- a posteriori judgments
and on the other:
- analytic judgments
- synthetic judgments.
The distinction between a priori and a posteriori judgments concerns the grounds or type of justification for such judgments. A posteriori judgments are judgments that are grounded in or justified by the deliveries of experience. Thus, for example, we can only be justified in believing that some particular piece of jade is green on the basis of empirical intuition. The judgment that this piece of jade is green is thus a posteriori based as it is on the delivery of experience. However, we may be justified in accepting certain judgments independently of the contingent course of our experience. Such judgments are a priori. Mathematical judgments are prime candidates for being a priori, though some, such as Mill, have doubted this.
Analytic judgments are grounded in the content of the judgment. We only need to apprehend the constituent concepts of the judgment that all bachelors are unmarried males to recognize its truth. Synthetic judgments, on the other hand, cannot be justified on the basis of their content alone. Empirical truths are the prime examples of synthetic truths. Many philosophers, such as Kant, have held that coming to accept a synthetic judgment constitutes a genuine advance in our knowledge whereas accepting an analytic judgment does not. Synthetic judgments are in this sense substantive.
These distinctions intersect:
| Analytic | Synthetic | |
| A priori | ||
| A posteriori | NA |
There is no analytic a posteriori—if the content of a judgment suffices for its justification there is no need for supplementation by empirical intuition.
At the center of the Critique is the question: How are synthetic a priori judgments possible? Kant is convinced that we have a priori knowledge that isn’t grounded in sensible experience, for example, our knowledge of space and time, causality. We can only know a priori what we contribute to experience, but then how can such knowledge be substantive? Our synthetic a priori knowledge is grounded in pure intuition. Intuition, for Kant, is a singular cognitive representation—it is a direct cognitive relation to an object as opposed to propositional awareness of a thing. Intuitions divide into empirical and pure intuitions. Empirical intuition depends on contingent features perception. Pure intuition on the other hand does not so depend. Rather pure intuition is necessarily presupposed in every act of empirical intuition. What is intuited in pure intuition is the structure that is presupposed and makes possible ordinary empirical intuition. A priori knowledge is only synthetic if supplemented by pure intuition.
Frege and Kant on Analyticity
Kant provided two distinct characterizations of analyticity:
- A judgment is analytic just in case the subject concept contains the predicate concept.
- A judgment is analytic just in case its denial is self-contradictory.
Four points:
- On the first characterization, our understanding of analyticity is only as clear as our understanding of conceptual containment. But talk of containment is metaphorical at best. Unless the literal content of containment is made explicit, we will lack a clear understanding of analyticity.
- Kant thinks of concepts as comprising a checklist of features.
Kant takes it for granted that falling under a concept is having a
number of definable features. Empirical concepts will be a concept
of an A where A picks out a kind of thing directly encounterable in
experience; moreover to be an A is to have empirically observable
features, F1, F2,…,Fn, that are logically independent. A given
thing falls under a concept just in case that thing instantiates
each of the features on the list. Given this we can think of
conceptual containment in the following terms. A judgment is
analytic just in case the list of features associated with the
predicate concept is a subset of the list of features that an
object must posses in order to fall under the subject concept.
While this clarifies talk of containment a number of problems
remain:
- Presupposes the subject/predicate analysis and is thus subject to Frege’s criticisms;
- Kant writes as if concepts are always concepts of encounterable particulars. He makes no allowances for relational concepts or for concepts of unobservable things-theoretical concepts.
- The empirically observable features on the list are supposed to be logically independent, but not all empirical concepts fit this pattern, e.g.,: to be an uncle is to be the brother of the foreparent of somebody. To be a brother to someone just is to have a foreparent in common so the features brother and foreparent are not logically independent in the way that Kant requires.
- Not all concepts can be understood as a list of features. Frege contends that not a single arithmetical concept has this form. Later Putnam will argue that natural kind concepts cannot be understood as lists of features.
- The two characterizations are nonequivalent. Kant’s characterization of analyticity in terms of conceptual containment presupposes an oversimplified view of judgment-this account only applies to universal affirmative judgments, i.e., judgments of the form: All $A$s are $B$s. But many judgments, e.g., $23 > 17$, do not take this form. Not only do singular judgments not take this form but neither do existential judgments: there is a square root of 16. The notion of analytic judgments as judgments whose denials are self-contradictory provides a broader notion of analyticity. The notion of logical consistency is not restricted to universal affirmative judgments, and so the second characterization has a broader range of applicability. So not only are different concepts deployed in Kant’s two characterizations, but they are not even extensionally equivalent.
- The second characterization, (an analytic judgment as a judgment whose denial is self-contradictory) is only as good as your underlying logic. Kant however still accepts the term logic inherited from Aristotle.
Frege’s reconstrual of Kant’s notion of analyticity at once resolves these difficulties and reconciles the distinct characterizations. In this regard, Frege’s hermeneutic achievement is, as Jamie Tappendon puts it, of Talmudic proportions:
A truth is analytic just in case it can be transformed into a logical truth by the substitution of synonym for synonym.
Where for Frege, a logical truth is a truth that can be proved from the logical laws alone (the Laws of Truth).
So consider:
All bachelors are unmarried males.
If the expressions “bachelor” and “unmarried male” are indeed synonyms (if they share the same conceptual content), then by compositionality we may substitute the latter in for the former without altering the thought expressed and thus arrive at a truth of logic:
All unmarried males are unmarried males.
The denial of a logical truth is self-contradictory, and so Frege’s characterization is faithful to the spirit of Kant’s logical characterization of analyticity. Moreover, that logical truths are arrived at by the substitution of synonym for synonym explicates Kant’s talk of conceptual containment.
Despite this achievement of rational reconstruction, Frege does emphasize one major difference Kant’s conception of analyticity and his reconstruction of it in light of the new logic. Kant held that analytic judgment could not extend knowledge. This is most clearly seen if we consider Kant’s characterization of analyticity in terms of conceptual containment. Consider the judgment:
All bachelors are unmarried males
The judgment is analytic because the features, unmarried and male, are on the list associated with the concept bachelor. If we possesses the concept of bachelor, and this involves the recognitional capacity to identify things as bachelors with reference to the associated list of logically independent empirical features, then it is no advance in our knowledge to be told that all bachelors are unmarried males. One knows that in advance merely by possessing the concepts in question. Judgments that constitute a genuine advance in knowledge must, for Kant, be synthetic. An advance in knowledge always requires the supplementation of intuition whether empirical or pure. Frege, on the other hand, held that the acceptance of certain analytic truths could constitute a valuable extension of our knowledge. Kant could not see how analytic judgments could extend knowledge because of his impoverished view of concept formation given his allegiance to the old logic.
Frege gives the following geometrical analogy. Consider a plane. Kantian concepts characterized by lists of features can be understood as defining regions of that plane. Thus the concept male defines a region of the plane. Within that region are all those things that are male. The concept unmarried defines another region within which are all those things that are unmarried. On the Kantian conception, once you have a basic stock of concepts new concept are arrived at by the operations of intersection and inclusion. The concept bachelor, that is the concept of all things unmarried and male, is thus represented by their intersection. Forming new concepts is always a matter of exploiting the boundaries of the regions defined by concepts antecedently given. Frege claims, however, that with his new logic, the possibility exists of drawing new boundaries.
The key is Frege’s treatment of generality and the associated method of concept formation. Recall, Frege’s fundamental insight is that one begins with the conceptual content of a possible judgment and then analyzes that content into a function and argument. Statements of generality are understood as involving second level concepts, that is, concepts that take first level concepts as arguments. These first level concepts needn’t be antecedently given. One may arrive at them by first substituting variables in for (multiple) occurrences of singular terms in a sentence and then binding these by a quantifier. The concept denoted by the complex predicate isn’t necessarily definable in terms of the operations of inclusion and intersection. When Frege speaks of drawing new boundaries, he has in mind definitions that essentially involve generality such as his definition of the ancestral in the Begriffsschrift.
How do such definitions constitute valuable extensions of our knowledge? Defining novel concepts in this way, licenses us to draw inferences that we weren’t explicitly licensed to draw before accepting the definition. Accepting, what Frege describes as a fruitful definition, thus constitutes an advance in knowledge since it provides warrant for beliefs that we weren’t previously justified in holding. Analytic judgments, such as definitions essentially involving generality, the so-called fruitful definitions, can thus constitute a genuine advance in knowledge.
Frege and Kant on Apriority
Not only has Frege reconstrued the Kantian distinction between analytic and synthetic, but what has been less well appreciated is that Frege has also reconstrued the distinction between a priori and a posteriori judgment. For Kant, a priority was an epistemological notion. What counts as a priori was in the first instance a cognitive act, a bit of knowledge. And to call a cognition a priori is to say that it somehow prior to or independent of experience (for Kant, empirical intuition). That is, at least ever since Kant, philosophers have recognized a fundamental epistemological distinction between that knowledge which is based on experience or empirical intuition or observation and that which is somehow not.
Given this, Frege’s characterization of the a priori ought to strike us as peculiar. For it contains no reference whatsoever to experience, empirical intuition, observation or any other experience-theoretic category. For Frege:
A truth S is a priori just in case there exists a proof of S that does not depend on any basic facts about particular objects, that is, just in case there exists at least one proof of S that involves only general truths as premises.
Frege seems to have provided a logical characterization of what has been previously construed as an epistemological notion. Whereas tradition speaks of experience and observation—modes or sources of knowledge and belief, Frege speaks only of general laws and facts about particulars. How then can Frege’s answer to the question of the a priority of mathematical truths as Frege understands the notion, even count as an answer to the traditional questions about the epistemological status of mathematical truths, the questions that Plato, Leibniz, Mill and above all Kant were at pains to answer in their remarks about mathematical practice? Hasn’t Frege just changed the subject here?
(Paul Benacerraf, in “Frege, The Last Logicist”, has suggested that Frege has indeed changed the subject. He argues that the only sense in which Frege has answered any of the traditional philosophical questions about mathematics is by radically refashioning them so that they admit of mathematical answers. Frege only recasts his mathematical work in wooly philosophical verbiage to give the appearance of continuity with the tradition.)
Has Frege really just changed the subject? Can we interpret Frege’s remarks about a priority so that his work can be brought in line with traditional philosophical questions about the epistemological status of mathematical belief? Gideon Rosen has recently suggested a resolution to this difficulty.
First notice that “experience” in philosophical discourse has a number of distinct senses. For our purposes it is enough to distinguish the following two. The notion of experience that is at stake in discussions of the epistemological status of mathematical truths is a normative notion. Experience in the normative sense is that which a posteriori knowledge depends for its justification. We may distinguish another sense of experience, call it phenomenal experience. Phenomenal experience is characterized by the subjective character of experience that is immediately available to consciousness. The two notions are distinct. Suppose that there could be zombies, intelligent creatures whose cognitive access to the world lacks an associated phenomenology. If such creatures are indeed possible and they are indeed capable of forming beliefs about the world given whatever cognitive faculties they possess, then at least some of their beliefs would be governed by the normative sense of experience even though they lack phenomenal experience.
There is a potential problem in adequately characterizing normative experience. So far, we have been content to characterize it as that which has a distinct normative function, that is, that on which a kind of knowledge, a posteriori knowledge, depends for its justification. But surely we would be right to be disappointed if we are then told that a posteriori knowledge is that which depends on experience in the normative sense. We have gone around in a circle and have provided neither an informative characterization of normative experience nor a posteriori knowledge. Is there any way out of the circle?
The classical empiricism of Berkeley and Hume can be seen as providing a way out of the circle. The classical empiricists avoid this difficulty by providing a substantive identification of the normative sense of experience. They contend that normative experience just is phenomenal experience (and so are committed to the impossibility of zombies). Frege would reject any such identification. Phenomenal experience concerns ideas, in Frege’s sense of the term, mental images, and Frege is at great pains to argue that ideas lack any normative significance and so phenomenal experience is not even a candidate for identification with normative experience. Nonetheless, Gideon Rosen has suggested that Frege can be understood as implicitly following the model provided by classical empiricism for breaking out of the circle. That is, Frege is himself providing a substantive identification of normative experience, though not the one proposed by Berkeley and Hume:
Frege’s Theory of Experience Experience in the normative sense just is the source of knowledge of basic facts about particular things. To be a “delivery of experience”’ or a judgment directly “based upon” experience just is to be an epistemically basic judgment about a particular thing.
One further remark is called for. By particular Frege does not mean object in his mature vocabulary, that is, that which is denoted by a non-defective singular term. Otherwise geometry would come out as a posteriori in Frege’s sense. But Frege consistently throughout his career held with Kant that geometry is synthetic a priori. In particular, he held that geometrical knowledge is based upon pure intuition of space or of geometrical objects such as points and lines. Aren’t such pure intuitions of geometricalia epistemologically basic judgments about particulars? In the Foundations, Frege explicitly denies this, that is, he denies that geometrical objects are particulars. For Frege, objects, that which are denoted by non-defective singular terms, divide into particulars and, as we shall call them, pseudo-particulars. Whereas concrete objects encountered in empirical intuition and the objects of arithmetic that, according to Frege, are given to us through the medium of reason, are particulars, geometrical objects such as points and lines are pseudo-particulars. The distinction is roughly this: particulars have intrinsic features that individuate them one from another. Thus seventeen has properties unique to it that will allow us to distinguish it from all the other numbers. Pseudo-particulars, on the other hand, constitute classes of intrinsic duplicates. If we prove something about a point given in pure intuition we may validly infer that the condition holds good of all relevantly similar points, that is all points in the relevant class of duplicates. Geometrical knowledge does not come out as a posteriori for Frege since geometrical objects are not particulars in the sense that figures in Frege’s characterization of normative experience.
There remain a number of important questions about this interpretation. Let me mention only the following: Is Frege’s distinction between particulars and pseudo-particulars a logical distinction or is it purely metaphysical? And if it is purely metaphysical what grounds are there for drawing this distinction? Larger interpretive issues hang on this matter. Thus for instance, Crispin Wright, in providing an interpretation of Frege’s Context Principle attributes to him what Wright describes as the syntactic priority thesis—roughly, that ontological distinctions are made on the basis of prior grammatical distinctions. If the distinction between particulars and pseudo-particulars is purely metaphysical then Wright’s attribution of the syntactic priority thesis is a misattribution. This matter is not at all clear. Frege does indeed characterize pseudo-particulars in inferential terms suggesting that the distinction is logical in character—proving something about a pseudo-particular amounts to a proof about the whole class of relevantly similar pseudo-particulars. Unfortunately making this precise turns out to be surprisingly difficult to do. And so it is an open question whether the distinction is logical or metaphysical, and whether the syntactic priority thesis is properly attributable to Frege.
Philosophy or Mathematics
Is the Foundations a work of philosophy or mathematics? This is a matter of some exegetical controversy. Paul Benacerraf (“Frege, the Last Logicist”) has argued that Frege only answers the philosophical questions by reconfiguring them to have a mathematical answer. Thus according to Benacerraf, the Foundations is primarily a work of mathematics, despite its informal character. But Joan Weiner (Frege in Perspective) has argued that Frege’s mathematical motivations aren’t genuinely mathematical at least by the standard of late nineteenth century mathematics. Frege’s work is thus, in the first instance, philosophical. Perhaps the truth lies somewhere between-perhaps, for Frege, the question, “Is the Foundations a work of philosophy or mathematics?”, lacks clear sense:
What’s the epistemological status of arithmetical truths? The question is tantamount to deciding whether arithmetical truths are analytic, synthetic a priori, or a posteriori. Given Frege’s refashioning of the Kantian categories, an adequate answer to this philosophical question will involve an answer to Frege’s mathematical question—to what extent are arithmetical truths provable on the basis of logic alone? Thus the philosophical questions of the Foundations depend on the answers to mathematical questions. Indeed immediately after describing his mathematical motives Frege writes in §3: “Philosophical motives too have prompted me to enquiries of this kind”:
The answers to the questions raised about the nature of arithmetical truths—are they a priori or a posteriori? synthetic or analytic?—must lie in this same direction. For even though the concepts concerned may themselves belong to philosophy, yet, as I believe, no decision on these questions can be reached without assistance from mathematics—though this depends of course on the sense in which we understand them.
In the last line of this passage Frege explicitly recognizes that he has given new meaning to Kant’s distinctions between a priori and a posteriori, and analytic and synthetic. Frege’s point in this passage is that once the Kantian categories are understood as he understands them, then establishing the epistemological status of arithmetical truths requires a mathematical investigation into the foundations of arithmetic of the kind that he is presently undertaking.
Frege contends that discovering a proof where proof is available is always a mathematical advance even if the limits of validity of the theorem is perfectly clear and the theorem is universally regarded as self-evident. How can this be? How can proving self-evident theorems whose limits of validity are perfectly clear constitute a mathematical advance? Two answers:
- In revealing the logical dependencies among arithmetical thoughts, one explicitly articulates their content thus clarifying the subject matter of arithmetic. (Frege is here relying on the Begriffsschrift conception of content as substitutional invariance of inferential role.)
- One function of proof in mathematical practice is to provide justification for the acceptance of mathematical theorems. For Frege, justification is an objective matter. For Frege, Kant’s epistemological categories apply primarily to thoughts and only derivatively to judgments (in virtue of the thoughts they express). Moreover the justificatory relations revealed by proof are objective relations among thoughts that need not reflect the psychological process by which we arrived at judgment. Frege recognizes two sorts of justificatory relations—logical consequence and probabilistic relations. Frege is thus using the notion of proof in a sense more general than deduction. He speaks of proofs of a posteriori truths, but he is not following Leibniz in holding that all truths are disguised a priori truths derivable from certain general laws such as the principle of sufficient reason. Rather they admit of proof in the sense that they are confirmed or disconfirmed to a certain degree given the deliveries of experience. Justification is an objective matter and consists in bringing opinion in line with the objective grounds of the truths they concern. To be justified in one’s mathematical opinion is to bring one’s mathematical beliefs in line with the order of objective dependency that is revealed by mathematical proof. Discovering proof where proof is available is a mathematical advance insofar as the justification of mathematical opinion depends on it.
Notice 1 and 2 are philosophical motivations. 1 essentially involves a philosophical thesis about the content of our mathematical judgments, indeed, about the content of judgment in general; and 2 essentially involves philosophical claims about the nature of epistemological justification. In this way Frege’s mathematical motivations depend on philosophical reasons. The question, is the Foundations primarily a work of philosophy or a work of mathematics lacks a clear answer because the mathematical and philosophical motivations are inextricably intertwined.
Summary
- The project of the Foundations is to establish that arithmetic is derivable from the Laws of Logic
- Frege has interdependent mathematical and philosophical motivations for this project
- Frege refashions the Kantian categories (analytic, syntehtic, a priori, a posterori) but remains faithful to their spirit
