metalogic/metaethics 2

i trace the grounds and consequences of a wager that metalogic is ethics.

Platonic dialectic & knowledge-trivialization

In Platonic dialectic as I currently see it, arguments typically regarded as directly asserting the existence of X are often actually taking (roughly) the form, “If you adopt a theory (an ontology, a semantics, an epistemology, etc.) much simpler than one that includes X or equivalent, you wind up trivializing knowledge.” That these would be arguments against trivialization distinguishes them importantly, I think, from transcendental arguments, which are typically concerned to show the possibility of knowledge in a more positive sense. (It would be interesting to know whether my contrast is based in an insufficiently-uncharitable reading of the transcendental tradition.)

“Codes have no surplus value.” (Sellars)

There’s more than one way of taking this line, certainly, and it seems part of Sellars’ style to pack an invitation to a dense problematic into that tossed-off phrase, as well as teasing his audience with a clear allusion to forbidden material, but I’m inclined to say that if S more clearly marked that codes as such, across a crucial threshold (which I like to call ‘nontriviality’ but has many apt names) do come charged with a surplus value – with which subjects are perhaps co-constituted, but which is neither produced by them, individually or socially, nor is entirely under their control – he would be less tempted to fall back on positing a semantic power/faculty for us, rather than construing the acquisition of a shared first language (“Does he speak Greek?”*) as the platform from which we are able to realize that there are semantic problems.


*Meno 82b.  Ἕλλην μέν ἐστι καὶ ἑλληνίζει;  The καὶ, I would argue, is epexegetical: to be able to speak with him (to “Hellenize”) is to be Greek in the only sense relevant here for Plato’s Socrates.)

Could all functions be concepts? A note on the dialectic of logicism (Frege/Sartre)

1.

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 versaI 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. 

2.

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.

3.

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.)

from Lautman 1939

A very rough translation (really just Google plus a first pass at corrections) of the closing paragraphs of Lautman’s half of this 1939 joint presentation with Cavaillès to the SPdP, entitled “Mathematical Thought” (La Pensee Mathematique), which, for whatever reason, are edited out of the Goldhammer translation in the Balibar/Rajchman/Boyman ed. anthology, French Philosophy Since 1945, which is the only place I know of the text’s being available in English. (Whew!) Anyway, to me these lines seem important, and to differ in minor but interesting ways from the statements of the same theses by Lautman elsewhere. (For these, see, of course, S. Duffy’s translation of Mathematics, Ideas, and the Physical Real.) So having poked at them a bit, I offer them to the stranger-gods of the web. Corrections are entirely welcome; probably half the people who’d be inclined to read this are more proficient (and definitely more efficient) Francophones than yours truly!

Il me semble que le problème des rapports de la théorie des Idées et de la Physique pourrait être étudié de la même façon. Considérons par exemple le problème de la coexistence de deux ou de plusieurs corps ; c’est là un problème purement philosophique dont nous dirons que Kant l’a plutôt posé que résolu dans la troisième catégorie de la relation. Il se trouve néanmoins que, dès que l’esprit essaie de penser ce que peut être la coexistence de plusieurs corps dans l’espace, il s’engage nécessairement dans les difficultés encore insurmontées du problème des n corps. Considérons encore le problème des rapports du mouvement et du repos. On peut poser abstraitement le problème de savoir si la notion de mouvement n’a de sens que par rapport à un repos absolu ou si, au contraire, il n’y a de repos que par rapport à certains changements ; mais tout effort pour résoudre de pareilles difficultés donne naissance aux subtilités de la théorie de la Relativité restreinte. On peut également se demander à laquelle des deux notions de mouvement et de repos il faut attacher un sens physique, et c’est un point où s’opposent la Mécanique classique et la Mécanique ondulatoire. Celle-là envisage l’onde comme un mouvement physique réel ; pour celle-ci au contraire, l’équation d’onde n’apparaît plus que comme un artifice destiné à mettre en lumière l’invariance physique de certaines expressions par rapport à certaines transformations. Il apparaît ainsi que les théories de Hamilton, de Einstein, de Louis de Broglie, prennent tout leur sens par référence aux notions de mouvement et de repos dont elles constitueraient la véritable dialectique. Il se peut même que ce que les physiciens appellent une crise de la Physique contemporaine, aux prises avec les difficultés des rapports du continu et du discontinu, ne soit crise que par rapport à une certaine conception assez stérile de la vie de l’esprit, où le rationnel s’identifie avec l’unité. Il semble au contraire plus fécond de se demander si la raison dans les sciences n’a pas plutôt pour objet de voir dans la complexité du réel en Mathématiques comme en Physique, un mixte, dont la nature ne pourrait s’expliquer qu’en remontant aux Idées auxquelles ce réel participe. It seems to me that the problem of the relations of the theory of Ideas and Physics could be studied in the same way. Consider for example the problem of the coexistence of two or more bodies; this is a purely philosophical problem that we say that Kant posed rather than solved in the third category of relation. It is nevertheless found that as soon as the mind tries to think what may be the coexistence of several bodies in space, it necessarily engages in the yet-insurmountable difficulties of the n-body problem. Or again consider the problem of the relations of movement and rest. One can pose abstractly the problem whether the notion of movement is meaningful only in relation to absolute rest, or whether, on the contrary, there is rest only in relation to certain changes; but any effort to resolve such difficulties give rise to the subtleties of the special theory of relativity. One can also wonder to which of the two concepts of movement and rest one must attach a physical sense, and this is a point where classical mechanics and wave mechanics are opposed. The former considers the wave as a real physical motion; for the latter, on the contrary, the wave equation appears only as an artifice to highlight the physical invariance of certain expressions with respect to certain transformations.  It thus appears that the theories of Hamilton, Einstein, Louis de Broglie, become meaningful by reference to the concepts of movement and rest of which they would constitute the genuine dialectic. It may even be that what physicists call a crisis of contemporary physics, struggling with the difficulties of the relations of the continuous and discontinuous, is only a crisis relative to a pretty barren conception of life of the mind where the rational is identified with the unit. [ne soit crise que par rapport à une certaine conception assez stérile de la vie de l’esprit, où le rationnel s’identifie avec l’unité.] To the contrary, it seems more fruitful to wonder whether the goal of reason in the sciences is not rather to see, in the complexity of the real, in Mathematics as in Physics, a mixed [un mixte], whose nature can be explained only by going back to the ideas in which this real participates.
On voit ainsi quelle doit être la tâche de la Philosophie mathématique et même de la Philosophie des sciences en général. Il y a à édifier la théorie des Idées, et ceci exige trois sortes de recherches : celles qui ressortissant à ce que Husserl appelle l’eidétique descriptive, c’est-à-dire ici la description de ces structures idéales, incarnées dans les Mathématiques et dont la richesse est inépuisable. Le spectacle de chacune de ces structures est à chaque fois plus qu’un exemple nouveau apporté à l’appui d’une même thèse, car il n’est pas exclu qu’il soit possible, et c’est la seconde des tâches assignables à la Philosophie mathématique, d’établir une hiérarchie des Idées et une théorie de la genèse des Idées les unes à partir des autres, comme l’avait envisagé Platon. Il reste enfin, et c’est la troisième des tâches annoncées, à refaire le Timée, c’est-à-dire à montrer, au sein des Idées elles-mêmes, les raisons de leurs applications à l’Univers sensible. One sees here what the task of mathematical philosophy, and even of the philosophy of science in general, must be. One has to build the theory of Ideas, and this requires three kinds of investigations: [first] the ones that fall within what Husserl calls descriptive eidetics, which is to say, the description of those ideal structures which are incarnated in Mathematics and whose wealth is inexhaustible. The spectacle of each of these structures is each time more than just another example provided in support of the same thesis, since it is not excluded that it is possible (and this is the second of the tasks assignable to mathematical philosophy), to establish a hierarchy of ideas and a theory of the genesis of ideas from one another [les unes à partir des autres], as Plato envisioned. It remains finally (and this is the third of the announced tasks) to rework the Timaeus, that is to say, to show, in the Ideas themselves [au sein des Idées elles-mêmes], the reasons for their applications to the sensible universe.
Tels me paraissent être les buts principaux de la Philosophie mathématique. These seem to be the main goals of mathematical Philosophy [or, “philosophy of mathematics”: Philosophie mathématique].

groups-as-eide hypothesis: a summary restatement

I have long suspected that:

  1. Identity: Groups are the forms in which the maximum of identity between terms and relations is realized.
  2. Ideality: This particular kind of identity is criterial for ideality, so that groups are eide.
  3. Plurality: Significantly, this maximal identity is only realized in each case of a concrete plurality of individual groups, not in a synthesis of all groups (DNE), and not by the concept of a group in general. (Groups behave more like Platonic ideas-forms than the post-Kantian Idea/Concept.) 
  4. Participation: A concept of participation seems to fall naturally out of this conception of groups, when we add relations to nongroups (as when we speak of the group of a space, etc.)

the argument for negative foundations

…stripped-down, given in the course of a remark on Sartre. The context is relatively arbitrary; the argument isn’t.

[…] Contra Priest and others, I’m not embracing paradox or dialetheism, so when I say that X is a witness to the proposition  “P is contradictory”, and deploy this syntax in discussing the relation of the event of pure reflection to the ontological thesis that the in-itself-for-itself is a contradiction, I mean first of all that it negates that ideal. It doesn’t paradoxically both-negate-and-affirm it or show it to be contradictory-but-assertable or anything like that.  At this point, rather, the spell is broken, even if we don’t know how (yet) to put the breaking of the spell to use, or to say “positively” what is there negatively shown.  To the extent, though, that the ideal shown to be contradictory seems to be an inevitable posit of reason, we have to look more closely at what is being affirmed and what is being negated.  For one thing, there is nothing contradictory about reason’s having to posit a hypothesis which falls to reductio. In fact, I have an argument (let’s call it the “Argument for Negative Foundations”) that any single true genuinely ontological proposition has to be negative in precisely this way.  It has to have the form, “X does not exist” (where X is some name for the Absolute). Why? Because meaning is differential. The only way to provide a characterization is by contrast, by drawing a distinction, and with what can Being be contrasted? Only with what essentially does not exist. The ancients were therefore by no means mistaken to maintain that a characterization of being must proceed by way of reference to a perfect being; they simply mistook the sign, taking for theology what is properly atheology. This is why it is valid for Sartre to maintain the structural, ontological importance of the nonexistence of God, despite the apparently reactive position that it seems, to the friends of some supposedly-pure affirmation, to leave him in, and why the corresponding post-Cantorian form of the same proposition, is at the head or heart of each of Badiou’s major works.  (As you can see, my preferred solution to the problems of reason’s dissatisfaction and delimitation is neither Kantian, nor Hegelian, nor Heideggerian.)

Minimal Sartre

The two propositions essential to reading Being and Nothingness correctly:

  1. The identity of in-itself and for-itself is just the in-itself. Id(es,ps) = es. 
  2. The relation of in-itself and for-itself is just the for-itself. Rel(es,ps) = ps. 

Notes:

  1. It takes these two propositions to say “positively” what one says “negatively” in a single proposition: that the in-itself-for-itself is a contradiction.*
  2. These propositions do not depend on adopting standard Sartrean models of the in-itself and for-itself, but apply to any sufficiently fundamental duality in which intentionality plays a role. (Thus their usefulness and potential interest beyond Sartre studies.)
  3. The for-itself’s problem, at the level of desire/value, is likewise twofold. It wants its identity to lie inside itself as for-itself and not (just) as in-itself. Again, it wants to establish a mediating relation between itself and the in-itself while it already is its own relation to the in-itself (and this is all the relation it’s going to get). This failed internalization and externalization, respectively, are not two distinct desires, but two ways of developing the contradictory desire to be in-itself-for-itself in the consistent setting of thought.
  4. Because of his adherence to the standard model (his own), Sartre is unable consistently to articulate the central concept of his own system, pure/purifying reflection. But on the nonstandard model which brings together Socratic-Platonic cognitivism with Sartrean detotalization, pure reflection is the process by which desire is transformed by metalogical duality to the extent that it learns how so to be transformed.

*This impossibility claim is the organizing thesis of the a priori atheological dialectic of Being and Nothingness, but no one, including Sartre, has stated in a satisfactory way what precisely makes the in-itself-for-itself contradictory. On my suggestion, the in-itself-for-itself turns out to be one of the characteristic totalities, in which negation has a fixed point, which stand as the hypothesis for reductio in diagonal arguments. The two logical rules given above are indispensable to the presentation of this isomorphism, and to that of the corresponding one between pure reflection and diagonalization.