Frege’s first great idea, slightly refined through subsequent mathematical experience, amounts to this: Concepts are functions, specifically functions from a domain of niladic objects (possibly including truth-values, in which case the concept is logical) to a codomain of truth-values. The choice of emphasis and slight restatement may suggest a couple of related ways of looking aslant at a now-familiar (too-familiar) idea.
First, regarded from its underside (the side of objects) Frege’s thesis means that a particular object, the truth-value object, is promoted to the level of a transcendental by which the other objects of the world can be indexed according to their concepts and vice versa. I prefer saying that it is promoted to saying that it is posited, postulated, or constructed for this purpose, but my point is not to emphasize the pre-existence of this object whatever that would mean in the mythical time of logical constitution, but rather its independence, as existent (that is, as a genuine referent) from the orders in which it finds its function (or sense).
Indeed, we are always in danger of misleading ourselves when we imply that an object has been constructed for a purpose, rather than speaking of trying out a fit of purposes to objects, whether the latter be assembled or discovered or both. One might well follow L. Fraser in referring here to ‘mathematical existentialism’, for from this angle it becomes easy to see how diagonalization adds a nuance* to Sartre’s dialectic of the creation/creature relation precisely in recognizing an instance of it in the relation of a formal system to its theorems: either the system does not really produce anything other than itself, or else what it produces is not bound in the imagined way by the purposes, intents, and knowledge of the system or its scribes. In neither case is there a creature-essence such that the existent can be essentially characterized by being the creature of its creator. What pertains to number, truth, and being thus exceeds the relations of creation, provability, and theoremhoood. And, as I read Plato to be reminding us in the Charmides, we can just as well read “referring” for “producing”. The crucial difference is not between knowledge of the pre-existent production of the new, but between the trivial and the nontrivial, in relations of knowledge as in those of production.
*The added nuance results from that typical behavior of diagonalization in illuminating philosophical problems, which I have called reversal. One begins by recognizing in diagonalization an application of certain philosophical concepts, here, the Sartrean dialectic of creation. This first step is valuable on both sides as far as it goes: the concept is enriched by its embrace of a far-flung example, while our ability to put a variety of intuitive spins on the formalism is proportionally enriched. What I find repeatedly in philosophical experience, however, is that diagonalization never remains at the level of the example in any discourse, but moves from an example to the crucial example, to the form itself, so that it finally comes, in the end, to reverse the flow of interpretation: adding or subtracting something from the conceptual system in which intuition first interprets it in each case of its recognition, making a modification both radically new and recognizably essential, thus solving Meno’s paradox anew each time, for each system, in perhaps the only way possible, and establishing an irreducibly dialectical mode of inference. But to keep to the current case, here the outcome is to recognize syntactic and semantic aspects of the interface between two existents, which play no explicit role in Sartre’s analysis, as recognizably demanded by that analysis.
Now, the submerged role of this object in constituting the two-dimensional lattice of concepts and objects remains unacknowledged in Frege, but it is precisely the debt incurred in the institution of the transcendental which is paid back, inevitably, by the diagonalization out of this lattice. (For Frege, denying the debt, its consequences are nothing short of disastrous.) The repayment entails recognizing that the elevated object is isomorphic to various objects that are not constructed, or at least not bound by the intent of any particular construction. Once again, it’s by the diagonal theorems that we know that i) The number Two in its recognition as classical truth-value-object is inseparably entangled with everything else about the number Two that is not intended in this usage, and ii) that the latter is strictly inexhaustible. Inseparability from an inexhaustible excess is one form taken by the repayment of Reason’s debt.
Note that the language of promotion, though prompting apt questions about the contingency and lifespan of a transcendental is not incompatible with recognizing that the promotion is also entirely justified, since it is the One and at least some of its immediate neighbors which must function as elements of a truth-value object, if cognition, which is to say compression, is to be possible. The dialectic here is one of necessity and contingency, and especially (Sartre) of the necessity of contingency, but not of a simple inversion of logicist necessity into empiricist contingency. It’s natural to want to appeal to the infinite in order to overcome this necessity of contingency, to say, for instance, that the One etc. are only the initial elements in the construction of a transcendental that can be extended into the transfinite. The diagonal theorems, however, anticipate and block this restoration, all at once, and from the outset.
A second way of looking aslant at Frege’s thesis likewise applies the principle that identity is not a one-way relation. When we say that concepts are functions, at least if we want to know what we are saying, and not merely to promulgate a riddle, we import (productively) into logic, the substantive mathematical problems concerning the nature and behavior of functions.
Now, the stock objection is that this is a fallacious conversion. The assertion is partititve, says the textboook Fregean: All concepts are functions, but of course not all functions are concepts. Addition*, for instance, is not a concept (e.g., Heck and May, 2011, pp. 126-66). The correct reply, as it now seems to me, is to grant that we can take the thesis as partitive, but only by being forced to recognize that this partitive logic, taking the concept as its proper sphere, now selects from a pre-existing ideal matter of functions, becoming a derivative activity rather than a presuppositionless one, while any identification of it with first philosophy is blocked. So whether we choose a broader or a narrower conception of the concept, the systematic importance is the same: in recognizing that concepts are functions, the first philosopher or ontologist, whether or not she decides to call herself a logician, is forced to take on the problems of functions as such. Work on the problems of functions as such will either be classified as belonging to an ontology prior to logic, or else as part of a radical expansion of logic, a “greater logic”.
*Ironically, the function x+1 seems like the most natural choice for a fixed-point-free map taking over the role of classical negation in more general contexts. (That is to say, since generality is itself a contextually-relative notion, in contexts which are more general in a particular context.) Diagonalization, taken as a name for the whole structure essential to the diagonal theorems, is, after all, about the distribution of fixed points: if not over here (a fixed-point-free map), then over there (in the diagonal map in the narrow sense).
So our choice is then a curious one: We can remain logicists and accept the startling converse, “All functions are concepts” – even if we necessarily don’t know how to use nearly any of them as concepts, even if we therefore don’t intuitively recognize or want to recognize most of them as concepts. (You would expect this to be Deleuze’s line, but as we know, it isn’t, and instead we find the rather inert distinction in What is Philosophy? between the scientist’s ‘creation of functions’ and the philosopher’s ‘creation of concepts’.) Or else we can save the letter of the proposition, “All concepts are functions but not all functions are concepts” by losing the systematic significance of “concept” and “logic”, and demoting logic to a branch of mathematics, the latter henceforth the ontologist’s primary concern.
Frege’s thesis, “Concepts are functions”, then, far from representing the inaugural contribution to logicism he intends, and as which it is generally received, provokes a dilemma that brings the ideology of logicism to its close at the instant of its historical inception, or an instant before. Whether by expanding logic to include all of its foundation in mathematics, or by showing itself to be a founded specialization of the latter, logic testifies to the primacy of the mathematical for ontology. Logicism, like the Pythagoreanism whose trajectory it wonderfully recapitulates, is, from the beginning, “between two deaths”. Though its recognition as such can only be retroactive, the first clear articulation of logicism is equivalent to its self-refutation, again like Pythagoreanism, and ultimately for exactly the same reason. (There are too many functions!).
And in this lies the uniqueness and supreme usefulness of Pythagoreanism and logicism both, for, while they are doubtless ideological to the core, no mere ideology is capable of self-destruction. It doesn’t matter how many absurdities, say, Catholicism piles one atop the other: it will never be capable, in recognizing a contradiction as its own being, of reduplicating the feat of its God in learning to die. But Pythagoreanism and Fregeanism are guided by a consistency which makes it possible for them to provoke and discover their own impasse in the face of a mathematical Real which they can neither assimilate nor reject. From one angle, they are immediately-surpassed ideological formations; from another they remain as permanent subsystems of any thinking relation to that Real. They belong, negatively but forever, to a dialectic which can be nothing other than their negation.
Note that “all functions are concepts” doesn’t mean, contrary to our experience, that any given function can be grasped as a concept. It both implies and explains the exact opposite. Most functions cannot be used as concepts in a given situation in exactly the same way in which “most” numbers are uninteresting. (But in that case there is a least, but being the least uninteresting number is an interesting property, etc. cf. Berry’s paradox and the examination paradox.) Uninterestingness, like indefinability, is evanescent in exactly the manner of Sartre’s contingency, though Sartre, of course, does not recognize the computational basis of this phenomenon, or that it enables, and even demands, a computational account of the for-itself. Diagonalization fulfills the promise of Sartre’s dyadic phenomenology in providing the rigorous solution, which Sartre anticipates only through his sketch of pure reflection, to the problem of how such evanescences can be known to the phenomenologist. As I show elsewhere, the problem and its solution are isomorphic to those of Meno’s paradox.
The right mapping between Frege and Sartre, then: between the ideal of the in-itself-for-itself and a system which would bring together the intensional and extensional sides of functions as such (not only functions restricted to a domain gerrymandered to get this correspondence to work out and baptized “logical”). We thus wind up shifting our ground from both analytic logicism and phenomenology, whose sterile oppositions only make sense from within their joint refusal of mathematical ontology. This two sidedness is metalogical difference. The negation which holds together the two sides in the greatest intimacy yet without possibility of synthesis is metalogical negation, of which diagonalization is the witness.
How should this account modify our reception of Frege’s second great idea, that concepts about number are second-order concepts? (“…The content of a statement about number is an assertion about a concept.”) Again, largely by opening this bridge for two-way traffic, so that rather than the intended logicist reduction of number, we have a permanent contamination of logic, and through it of ontology, by number. The result is a Platonism far more radical, and I think, far more interesting, than Frege’s so-called Platonism, in which first-order concepts are flash-frozen without dialectical transformation.
Few things give me more joy, by the way, than reflecting on Frege’s assertion, its grounds and consequences, and on the limits of the identity it proposes (and which Frege would like to keep domesticated) between number and (some) higher concepts. Though it claims to be able to work from given concepts, indifferently mathematical or nonmathematical, without transformation, it suffices to provide a royal road to the investigations of the mathematical dialectician to call this into question. Meanwhile, immediately, it gives to mathematical dialectic the key to the relation between Socratic elenchos, about aboutness and the difficulty of speaking about the concept, and the Platonic ontology, i.e., the essential relation of concepts to number, which relation is constitutive of any useful theory of forms-ideas.
How does Frege block the return traffic? By observing that, in respect of their unsaturatedness, “functions differ fundamentally from numbers.” But this is quite mistaken. Not only because, as I maintain, reversibility between the roles of concept and object is permanently characteristic of number in each of its local domains, but because an excess of unsaturatedness has been mistaken for a lack. We recognize the unsaturatedness of ordinary first-order concepts(IsBlue, and the like), precisely because we see how to compete them. That there is one way to fill in the blank makes the blank both conspicuous and manageable. But what gives the illusion of common sense to Frege’s observation about numbers it is that the ways in which they can function as, well, functions, are as numerous as the functions themselves. This excessive unsaturatedness disrupts the logicist’s implicit or explicit typing scheme. By being able to be connected in too many ways to other mathematical objects, numbers place second-order demands on the indexing of concepts, types, etc. This excess of relatedness in number is misread by the logicist as a lack of relatedness, except insofar as the number is taken up by a proper concept.
But the problem is not restricted to number. Note that rather, there must be a certain unsaturatedness of the object generally, expressed in Frege’s own context principle, though it is not directly exchangeable with the unsaturatedness of the concept, in absence of a higher function for coordinating the two. Nevertheless, we should not speak of the object as saturated, but as unsaturated in a different place, in a different order, than is what counts as the concept. (Perhaps we should speak of the object as unsaturated to the left, and the concept as unsaturated to the right, and consequentially of left- and right-adicity, or similar.)