277. How a priori knowledge is really possible. I believe that mathematics, logic, and philosophy all include and presuppose some basic — that is, primitive, starting-point-providing — and authoritative rational intuitions that constitute authentic a priori knowledge of objectively necessary truths, such that those rational intuitions are

(i) intrinsically compelling or self-evident,

(ii) cognitively virtuous, and also

(iii) essentially reliable, or absolutely skepticism-resistant.

More precisely, however, the beliefs included in those rational intuitions are factive or world-involving and modally grounded.

That is, they are beliefs that are inherently connected to necessary-truth-makers for those beliefs.

Furthermore, the cognitive capacities or mechanisms yielding self-evidence for those beliefs track truth in the actual world and also counterfactually across all relevant nomologically possible and metaphysically possible worlds.

And any explicit or implicit denial or rejection of those beliefs would be self-stultifying in the strongly normative sense that human rationality itself would then be impossible, including also skeptical human rationality.

Hence we categorically ought not to reject them, insofar as we are rational human animals.

In short, these basic authoritative a priori rational intuitions — constituting self-evident, cognitively virtuous, and also essentially reliable, or absolutely skepticism-resistant, a priori knowledge of objectively necessary truths — are robustly normative conditions of the possibility of human rationality, and implicit even in every attempt to reject these rational intuitions for any intelligible or defensible reason whatsoever.

278. And that’s not all.

I also believe that, starting with these basic authoritative a priori rational intuitions of objectively necessary truths, then mathematicians, logicians, and philosophers can also rationally construct non-basic, and non-authoritative a priori rational intuitions.

These intuitions are not completely convincing, not intrinsically compelling, and therefore not self-evident; and they are neither essentially reliable, nor absolutely skepticism-resistant.

But at the same time, they remain fairly convincing, fairly compelling, and therefore fairly evident; and they are also fairly reliable, and fairly skepticism-resistant.[i]

They thereby effectively extend their foundational corpus of basic authoritative a priori knowledge to a fairly secure non-foundational constructed corpus of a priori knowledge, thus making rational progress in mathematics, logic, and philosophy.

279. Of course, a postmodern anti-rational nihilist skeptic could still choose to reject all of these intuitions, whether basic authoritative rational intuitions, or non-basic constructed rational intuitions, for no defensible or intelligible reason whatsoever — as it were, just for the hell of it.

So at least as a form of emotional self-expression, postmodern anti-rational nihilist skepticism is really possible.

And, to be sure, someone’s striking an attitude, or acting-out some passion, is always psychologically or sociologically fascinating.

Nevertheless, for all its psychological or sociological interest, that sort of skepticism is philosophically perverse and pointless.

An attitude struck, or a passion acted-out, is not an argument made.

280. The original Benacerraf Dilemma, as formulated by Paul Benacerraf in 1973, is about the apparent impossibility of reconciling a “standard, uniform” semantics of truth in natural language with a “reasonable” epistemology of cognizing true statements, when the relevant kind of true statement to be semantically explained is mathematical truth and the relevant kind of cognition to be epistemologically explained is mathematical knowledge:

As an account of our knowledge about medium-sized objects, in the present, this is along the right lines. [A reasonable epistemology] will involve, causally, some direct reference to the facts known, and, through that, reference to those objects themselves…. [C]ombining this view of knowledge with the “standard” view of mathematical truth makes it difficult to see how mathematical knowledge is possible. If, for example, numbers are the kinds of entities they are normally taken to be [namely, platonically abstract objects], then the connection between the truth conditions for the statements of number theory and any relevant events connected with the people who are supposed to have knowledge cannot be made out.[ii]

Clearly, the original Benacerraf Dilemma puts the real possibility of authentic mathematical a priori knowledge in jeopardy.

281. But there is an even more fundamental problem about the real possibility of authentic a priori knowledge that I call The Generalized Benacerraf Dilemma:

(1) All knowledge is factive, that is, all knowledge contains an objective truth-making component, so all a priori knowledge whatsoever is factive, especially including a priori knowledge in mathematics, logic, and philosophy.

(2) If all a priori knowledge is factive in that it contains an objective truth-making component, then what rules out the possibility that its factive component is nothing but the result of a cosmic accident or massive coincidence, in that its truth-maker is merely accidentally connected to rational human belief and justification in the actual world (which is the classical Gettier problem,[iii] now extended to a priori knowledge), and also introspectively cognitively indistinguishable from connection with falsity-makers in relevantly similar possible worlds (which is the neo-classical new evil demon global skepticism,[iv] now extended to a priori knowedge)? Let us call this “the possibility of cognitive-semantic luck.”

(3) If nothing rules out the possibility of cognitive-semantic luck, then a priori knowledge of any kind whatsoever is impossible.

(4) There are only two possible candidates for ruling out the possibility of cognitive-semantic luck:

either (i) non-naturalism about the objective truth-makers and their connection with rational human beliefs,

or else (ii) naturalism about the objective truth-makers and their connection with rational human beliefs.

(5) Consider non-naturalism about the objective truth-makers and their connection with rational human beliefs — for example, as per classical Rationalist platonism, Cartesian innate clear and distinct ideas of real essences, grounded in God’s existence and non-deceitfulness, Leibnizian pre-established harmony, etc. This puts the truth-makers outside of space and time, and renders their connection with rational human beliefs a metaphysical mystery. Hence non-naturalism about the objective truth-makers and their connection with rational human beliefs does not explain how rational human a priori knowers can stand in a non-accidental, global-skepticism-resistant connection with the known truth-making objects of a priori knowledge.

(6) And now consider naturalism about the objective truth-makers and their connection with rational human beliefs. At least prima facie, naturalism can account for how rational human knowers can stand in a non-accidental, global-skepticism-resistant connection with the known truth-making objects — for example, via some or another causally reliable connection. But naturalism cannot explain how rational human beliefs can be either necessary or a priori. Indeed, on the contrary, what naturalism shows is that those rational human beliefs are contingent and a posteriori, as per either Lockean-Humean classical Empiricism or Quinean radical Empiricism. Hence naturalism about the objective truth-makers and their connection with rational human beliefs does not explain how rational human a priori knowers can stand in a non-accidental, global-skepticism-proof connection with the known truth-making objects of specifically a priori knowledge.

(7) So, since the possibility of cognitive-semantic luck cannot be ruled out, then a priori knowledge of any kind whatsoever is impossible, including a priori knowledge in mathematics, logic, philosophy, morality, axiology, linguistics, semantics, etc.

282. The original Benacerraf Dilemma seems to entail that objective mathematical necessary truth on the one hand, and rational human a priori knowledge of objective mathematical necessary truth on the other hand, are mutually incompatible.

In order to solve this problem adequately, I think that we must adopt two contemporary Kantian doctrines.

283. First, we must reject the classical platonic conception of abstractness, which says that something is abstract if and only if it has a mind-independent, substantial existence in a separate, non-spatiotemporal, non-natural, non-sensory, causally irrelevant, and causally inert realm.

And in its place, we should put a non-platonic, Kantian conception of abstractness, which says:

X is abstract if and only if X is not uniquely located and realized in actual spacetime, and X is concrete otherwise.

By “X is uniquely located and realized in actual spacetime,” I mean that X is exclusively embodied or incarnated at and exclusively embodied or incarnated in, and thereby fully occupies, one and only one actual spacetime volume.

Then this conception of abstractness is saying that something is concrete if and only if it is uniquely located and realized in actual spacetime, and abstract otherwise.

More specifically, according to this conception, whatever is either multiply located, multiply realized, non-actual, or non-spatiotemporal will count as abstract.

What makes this conception of abstractness non-platonic, above all, is its comparatively liberal approach to what will count as abstract.

It in fact includes the platonic conception of abstractness — under the special constraint of radical agnosticism about platonically abstract objects in particular and noumenal objects more generally, whereby we know a priori that we cannot know whether they exist or do not exist.

But this conception of abstractness is also significantly less restrictive than the platonic conception, robustly non-dualistic, and fully compatible with causal relevance.

In addition to its highly problematic assumption that there are humanly knowable objects that are not only causally inert but also completely causally irrelevant, what the platonic conception mistakenly assumes is that multiple location, non-actuality, and non-spatiotemporality are all necessarily equivalent with one another, so that platonic abstractness includes them all as necessarily conjoined features.

But in fact they are logically independent features of things: hence the correct, non-platonic conception of abstractness includes them disjunctively, not conjunctively.

What makes the conception of abstractness I favor not only non-platonic, but also specifically Kantian?

In Kantian terms, X is concrete if and only if X is

either (i) what Kant calls an “appearance,” which is the “undetermined object of an empirical intuition” (CPR A20/B34) — “undetermined” in that it is not fully specified as to its contingent or essential properties —

or else it is (ii) what he calls “a real object of experience” (CPR B289–291), the fully-determined and thus fully specified object of an objectively valid and true empirical judgment (the judgment of experience), and

(iii) X is abstract otherwise.

So X is abstract if and only if X is neither what I call a veridical appearance[v] nor a real object of experience in Kant’s sense.

284. Second, I think that we must also adopt contemporary Kantian versions of Mathematical Structuralism and mathematical authoritative rational intuition.

Mathematical Structuralism says that mathematical entities are not independent substances of some sort, but instead are nothing more and nothing less than relational positions or roles in a larger mathematical theory-structure.

Kantian mathematical structuralism, in turn, says that mathematical theory-structures are mind-dependent, or ideal, yet also manifestly real, entities that are abstract in the non-platonic, Kantian sense, and are veridically represented in pure or a priori intuition (Anschauung) via the productive imagination (produktive Einbildungskraft).

Correspondingly, mathematical authoritative rational intuitions, as I am understanding them, are self-evident, cognitively virtuous, and essentially reliable a priori conscious pattern-matching graspings of some proper parts of a larger mathematical theory-structure, via our direct conscious experience, in spatiotemporally-framed schematic sense perception, memory, or sensory imagination, of — in effect — David Hilbert’s basic objects of finitistic mathematical reasoning:

[A]s a condition for the use of logical inferences and the performance of logical operations, something must already be given to our faculty of representation, certain extralogical concrete objects that are intuitively present as immediate experience prior to all thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that can neither be reduced to anything else nor requires reduction. This is the basic philosophical position that I consider requisite for mathematics and, in general, for all scientific thinking, understanding, and communication.[vi]

This kind of direct conscious experience is equivalent to what Kant calls the cognitive construction of a sensible form (Gestalt) in pure or a priori intuition (Anschauung) via the productive imagination (produktive Einbildungskraft):

Although these principles [of mathematics], and the representation of the object with which this science occupies itself are generated in the mind completely a priori, they would still not signify anything at all if we could not always exhibit their significance in appearances (empirical objects). Hence it is also requisite for one to make an abstract concept sensible, i.e., display the object that corresponds to it in intuition (Anschauung), since without this the concept would remain … without sense, i.e., without significance. Mathematics fulfills this requirement by means of the construction of the sensible form (Gestalt), which is an appearance present to the senses (even though brought about a priori). In the same science, the concept of magnitude seeks its standing and sense in number, but seeks this in turn in the shapes, in the beads of an abacus, or in the strokes and points that are placed before the eyes. The concept is always generated a priori, together with the synthetic principles of formulas from such concepts; but their use and reference to supposed objects can in the end be sought nowhere but in experience, the possibility of which (as far as its form is concerned) is contained in them a priori. (CPR A239–240/B299)

It is also equivalent to the cognitive construction of what the cognitive psychologist Philip Johnson-Laird calls mental models.[vii]

We could also call it the cognitive construction of mental diagrams, mental pictures, structural imagery, or schemata.

Whatever we call it, the main claims I am making here are these:

(1) Mathematical necessary truths directly express proper parts of larger mathematical theory-structures.

(2) Mathematical rational intuitions are self-evident, cognitively virtuous, and essentially reliable a priori conscious pattern-matching graspings of some of those proper parts of those very structures, by means of the cognitive construction and manipulation of sensible forms given in Kantian pure or a priori intuition and constructed by the productive imagination.

So the imagination-based cognitive phenomenology of mathematical authoritative rational intuition is a perfect mirror of the structuralist ontology of the truth-makers of a priori mathematical beliefs.

285. The simplest example of what I am talking about here is the one I used in Thinking For A Living 13, namely reading and adequately understanding the symbol sequence, “3+4=7,” while looking carefully and thoughtfully at this stroke diagram,

| | | + | | | | = | | | | | | |

and also assertorically saying “Three plus four equals seven” to yourself.

Here, the phenomenological structure of your conscious experience internally mirrors the content of the proposition you are thinking and asserting, and in turn there is also a non-accidental and indeed necessary conformity between the content of the proposition and the underlying mathematical structure of the manifest natural world, of which the stroke diagram is one highly salient instance.

The productive imagination is in play precisely to the extent that you are able consciously to scan the stroke diagram, then consciously reproduce it in short-term memory, and then consciously manipulate it in certain definite ways with the same epistemic force.

For example, the operations of the productive imagination would be phenomenologically manifest and salient if you now were now, self-consciously, to generate in your mind a corresponding diagram for “2+3=5,” and then also come to know this truth objectively a priori via rational intuition.

The actual existence of the Kantian productive imagination in precisely this sense of a phenomenologically-robust image-generating, image-scanning, image-reproducing, and image-manipulating function of the conscious rational human mind, has been empirically well-confirmed in classic cognitive-psychological work by Roger Shepard and others.[viii]

286. In any case, the metaphysical ground of the necessary conformity between mathematical authoritative rational intuitions in the human mind on the one hand, and mathematical structures in the manifest natural world outside the human mind on the other hand — a necessary conformity which suffices to close the gap between justification and truth, and thereby guarantees essentially reliable a priori knowledge of objective necessity — is none other than the realistic idealism I spelled out in Thinking For A Living 4:

63. Thesis 1: A world that cannot veridically appear, a world “in itself,” a noumenal world, is logically possible but not really possible. (The Real Impossibility of a Noumenal World)

64. Thesis 2: Necessarily, if the manifestly real world exists, then the specific characters of its basic structures systematically correspond to the specific characters of the innate structures of the rational human cognitive and practical capacities. (World-to-Mind Conformity)

65. Thesis 3: Necessarily, if the manifestly real world exists, then if rational human cognizers/agents had been/were differently constituted as to their innate cognitive or practical capacities, the manifestly real world would have been/be correspondingly differently constituted as to its basic structures. (World-to-Mind Covariance)

66. Thesis 4: Necessarily, if the manifestly real world exists, then if some rational human cognizers/agents were to exist, they would be able to know or change that manifestly real world to some salient extent, by means of the normal operations of their innate cognitive or practical capacities. (Mind-to-World Access)

67. Thesis 5: Even if any or all rational human cognizers/agents were to go out of existence, nevertheless it is really possible for the manifestly real world not only to remain in existence but also to retain all the specific characters of its basic structures. (The Mind-Independence of the Manifestly Real World)

68. Thesis 6: Necessarily, if the manifestly real world exists, then if some rational human cognizers/agents were to exist, they would all be able to know or change that world in essentially the same ways. (The Objectivity of the Manifestly Real World)

69. Thesis 7: Necessarily, if the manifestly real world exists, then for some but not all spacetime locations L in the manifestly real world, if any given rational human cognizer/agent — call it Bob — were to have been/be actually present, cognizant, and active at L, then the manifestly real world would have been/be differently constituted at L than it would have been/be had Bob not been present, cognizant, and active at L. (The Observer-Dependence of Some Proper Parts of the Manifestly Real World — for example, Heisenberg-style quantum mechanical effects.)

70. Thesis 8: Necessarily, if the manifestly real world exists, then rational human cognizers/agents are not only logically (analytically, weakly metaphysically) possible but also really (synthetically, strongly metaphysically) possible. (Anthropocentricity 1)

71. Thesis 9: Necessarily, if rational human cognizers/agents had not been really possible, then the manifestly real world would not have existed. (Anthropocentricity 2)

72. Thesis 10: It cannot be the case that both (i) the manifestly real world exists, and also (ii) rational human cognizers/agents are really impossible.(Anthropocentricity 3)

287. I now also have in hand a general template for solving The Generalized Benacerraf Dilemma.

The Generalized Bencarraf Dilemma, you will recall, generalizes The Original Benacerraf Dilemma to any kind of a priori knowledge whatsoever.

It does so by pointing up, on the one hand, the logical, semantic, metaphysical, and epistemological clash between two basic authoritative philosophical rational intuitions about the need to rule out the possibility of cognitive-semantic luck, and, on the other hand, the fact that the truth-makers of knowledge are either non-natural or natural.

Having just sketched a basic solution to The Original Dilemma, I can now solve The General Benacerraf Dilemma by simply generalizing the basic solution in the following way:

For a priori knowledge of any kind K whatsoever —

Postulate the ten basic theses of realistic idealism as a metaphysical backdrop.
Adopt a Kantian version of Structuralism for K.
Adopt a Kantian version of Intuitionism for K.
Explain the sufficient justification (including, especially, the essential reliability) of K-type authoritative rational intuition in terms of Kantian Structuralism and Kantian Intuitionism, against the metaphysical backdrop of realistic idealism.
Work out the cognitive phenomenology of self-evidence for K-type authoritative rational intuition.

288. To be sure, the specific details of carrying out this five-part theory for, say, logical a priori knowledge, moral a priori knowledge, axiological a priori knowledge, linguistic a priori knowledge, semantic a priori knowledge, etc., and finally philosophical a priori knowledge, are going to be somewhat complex.

But in each case, working out all those specific details really is just a high-powered philosophical engineering problem, for which the general template remains the same.

So I think we can reasonably conclude that The Generalized Benacerraf Dilemma has, in its essentials, been solved, and also that we now know how authentic a priori knowledge is really possible.[ix]

NOTES

[i] I also believe that at least some non-basic rational intuitions are authoritative. But that refinement isn’t necessary for the point I am making right here.

[ii] P. Benacerraf, “Mathematical Truth,” Journal of Philosophy 70 (1973): 661–680, at pp. 672–673.

[iii] See E. Gettier, “Is Justified True Belief Knowledge?” Analysis 23 (1963): 121–123.

[iv] See S. Cohen, “Justification and Truth.” Philosophical Studies 46 (1984): 279–295.

[v] There is an important distinction between (i) a veridical appearance, aka an objective appearance, or an Erscheinung, and (ii) a mere appearance, aka a subjective appearance, or a Schein. See R. Hanna, “Kant, Radical Agnosticism, and Methodological Eliminativism about Things-in-Themselves,” Contemporary Studies in Kantian Philosophy 2 (2017), available online at URL = <https://www.cckp.space/single-post/2017/05/10/Kant-Radical-Agnosticism-and-Methodological-Eliminativism-about-Things-in-Themselves>.

[vi] D. Hilbert, “On the Infinite,” in J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1897–1931 (Cambridge, MA: Harvard Univ. Press, 1967), pp. 367–392.

[vii] See P. Johnson-Laird, Mental Models (Cambridge: Harvard Univ. Press, 1983).

[viii] See, e.g., R. Shepard, “The Mental Image,”American Psychologist 33 (1978): 125–137; R. Shepard and S. Chipman, “Second Order Isomorphisms of Internal Representations: Shapes of States,” Cognitive Psychology 1 (1970): 1–17; R. Shepard and L. Cooper, Mental Images and their Transformations (Cambridge: MIT Press, 1982: and R. Shepard and J. Metzler, “Mental Rotation of Three-Dimensional Objects,” Science 171 (1971): 701–703.

[ix] For a fully elaborated version of this argument, see R. Hanna, Cognition, Content, and the A Priori: A Study in the Philosophy of Mind and Knowledge (aka THE RATIONAL HUMAN CONDITION, Vol. 5) (Oxford: Oxford Univ. Press, 2015), PREVIEW, chs. 6–8.