Concrete and Abstract

It is in principle impossible to set up a system of formulas which would be equivalent to intuitionistic mathematics, for the possibilities of thought cannot be reduced to a finite number of rules set up in advance.

Arend Heyting, 1930, quoted by Paolo Mancosu, From Brouwer to Hilbert [Oxford, 1998, p.10]

While most of the following essay will focus on concepts of concrete and abstract in Western philosophy, the Chinese characters in the title are borrowed from a similar duality in Buddhist metaphysics. In ordinary Chinese usage, the character means "an affair; a matter; an undertaking, business" [Mathews' Chinese-English Dictionary, Harvard University Press, 1972, #5787; the requivalent of , pragma in Greek philosophy]. Similarly, the character means "reason, principle; the fitness of things; right, as an abstract principle" [#3864]. Combined, the binome means "principles of action; facts and principles involved; way of doing business" [#5787:54].

The terms of have a particular technical meaning in Mayâyâna Buddhism. We have the following statement from a Tendai oral transmission text of the Eshin school, the Sanjû shika no kotogaki, "Notes on thirty-four articles." Japanese Tendai Buddhism derives from the Chinese T'ien-t'ai [Tiantai] School of Chih-I [Zhiyi, 538597 AD]:

You should understand carefully what is meant by principle (ri []) and by concrete actuality (ji []). "Principle" means that, although the dharmas have distinctions, because they are all suchness, they are resolved in the one.... With respect to principle, there are no distinctions whatsoever; the myriad dharmas are dissolved. But with respect to actuality, the myriad dharmas are not dissolved; they remain constant in themselves. [Original Enlightenment and the Transformation of Medieval Japanese Buddhism, Jacqueline Stone, Kuroda, University of Hawai'i Press, 1999, p.200]

"Actuality" is the concrete, phenomenal world, whose entities are the "dharmas" in Buddhist metaphysics. "Principle" here is the underlying reality, in which the dharmas all dissolve in "suchness," or Emptiness. In T'ien-t'ai, these are equivalent, most simply expressed in the famous words of the Heart Sutra (), , , "Form is emptiness, Emptiness is form." Although Buddhist metaphysics has its own construction of ultimate reality, the terms of the contrast between abstract "principle" and concrete "actuality" are familiar and relevant in the history of Western debate on the matter.

"Abstract" and "Concrete" are words with particular meanings in the history of philosophy and of metaphysics, but they also have quite ordinary and common meanings outside of philosophy. The most familiar and vivid is of "concrete" as a construction material. This was developed by the Romans by mixing cement, to bind, with aggregate, which could be sand, gravel, or even, in Roman practice, volcanic ash. The cement has always been at base "quick lime," i.e. calcium oxide (CaO), which, with silicon dioxide (SiO2, i.e. quartz), becomes a mineralogical family of calcium silicates. Quick lime historically is derived from limestone, itself largely consisting of calcium carbonate (CaCO3). Other additives create cements of different properties, such as standard modern Portland Cement. Depending on the aggregate used, the result can be, not just concrete, but mortar, stucco, or grout. Roman concretes could even set under water, enabling the Romans to pour habour breakwaters right down through the water, without the coffer dams that would be necessary to inspect and prepare the foundations for stone constructions.

The term "concrete," from Latin concretus, is from a verb, concrescere, meaning to "grow together," i.e. in the sense that the elements of concrete fuse together to make a stone-like solid. The opposite of this in etymology would then be "abstract," abstractus, from the verb abstrahere, to "draw away." A common meaning of "abstraction" can thus simply mean to "remove," not unlike when your covert operatives need "extraction" from the dangerous site of their activities. An "abstract" can be a summary of a much larger document, which gives a sense of its overall meaning, drift, or argument. In general, what is "abstract" will be a limited feature of what as "concrete" was a preexisting whole that is dense with such features.

The philosophical concern will thus be with the nature of this relationship between feature and whole. In the most traditional and standard sense, the tangible objects of experience and perception, and their properties, will be "concrete." "Abstraction" is what we do mentally to focus or conceive on particular features or attributes of those objects. So the barn is concrete, but its redness, for all its vividness, is an abstraction. Other features of concrete objects are their edges, sides, solidity, etc. High on the scale of abstractions are mathematical features, such as the measured size, weight, density, etc. of the concrete objects. The numbers involved in these measurements then assume a life of their own, as we deal with them as such in mathematics, and they begin to assume the status of abstract objects, whose relationship to the concrete objects becomes a question.

A great deal of the history of metaphysics and epistemology is taken up with the question of abstract objects. In the first place, since abstract features can belong to many different objects, and not just one, they have a general or universal application, which leads to the Problem of Universals. To Plato, abstract objects were the "Forms" (ἰδέαι, idéai, "ideas," or εἴδη, eídê), which existed in a separate reality, eternal and unchanging. Aristotle "immanentized" the Forms and thought that abstract objects existed within the concrete particulars. We know about abstract features (the εἴδη, eídê, "species," in Aristotle's usage, inherited by modern biology) because of a particular act of the mind, "abstraction," which perceives and conceives them.

The notion that abstract features inhere in the objects of perception and are recognized by us becomes the doctrine of Aristotelian "Realism." This was disputed in the Middle Ages by "Nominalists," such as William of Ockham, who held that universals were just "names," nomina, and that there are no real abstract entities present in objects. The argument, then, was that only the concrete individuals are what we see and know. This doctrine would be revived in popularity among the British Empiricists. While John Locke thought that "abstract ideas" were a common-sense part of epistemology, and he could think of them in terms of isolated visual features, George Berkeley more consistently followed the logic of Locke's own argument, that "ideas" were images, which are concrete, and so denied that there were "abstract ideas" at all -- despite the paradox that he must indeed have conceived of abstract ideas in order to talk about them. Indeed, Locke had some difficulty explaining how the image of a particular horse could be "used" to signify all horses, for what is it about the one horse that gives it an affinity with another? Berkeley tossed out the whole question and fell back on the Nominalist proposition, that only the name is common to all the horses. However, this tells us nothing about why the same name is used for disparate objects or, even worse, how we are to know whether the name is to be applied to some unfamiliar object. Is Fido a horse just because I call him that, or does there need to be something about Fido that will make him a horse, regardless of what I think? Thus, Socrates asked Euthyphro for that ἰδέα, idéa, that would enable him to say whether one thing is an example of piety, or that it is not. What makes something a horse will be some specific and distinct features (e.g. an odd number of toes) but not others.

Hume agreed with Berkeley, and he continued in the vein of making things worse rather than better. Neither of them liked what mathematicians talked about. Geometrical points, infinitesimals, or the infinite divisibility of the continuum were all illegitimate concepts to these Empiricists, because obviously the images we can form of such things do not match their mathematicial definitions. While one can still find people who think that this discredits the mathematics rather than the Empiricism, it should be obvious that the mathematics has productive applications, to say the least, while the Empiricism cannot offer anything remotely comparable. Certainly not in mathematics.

In the end, it is now clear that the brain reacts in specific ways to abstract features such as edges or colors. This does no more than affirm a common-sense understanding that we recognize such things and always have. They are not just names. On the other hand, it does not tell us how they inhere in objects, or why common features should inhere in multiple objects. In other words, neuroscience, while perhaps refuting Nominalism, Empiricism, and even Conceptualism (since the brain is demonstrably reacting to external phenomena), does not supply the relevant answers to the metaphysical questions. We still don't know the relationship between abstract and concrete objects, or the apparently independent status of a world of abstract objects as in mathematics.

With Kant came the next step. Kant thought that mathematics was based on intuitions (Anschauungen), acts of immediate knowledge or cognitions, of space and time. In what might be thought to be a version of Conceptualism, Kant believed that these intuitions derived from the mind itself and were the forms, the conditions, imposed by the mind on perceptions. The need for such a theory followed from Kant's view that the logical basis of both arithmetic and geometry was synthetic, which meant that it was not a necessity of logic ("analytic") but, as it was formulated in propositions, these could be denied without generating contradictions. In other words, something like the Fifth, Parallel Postulate in Euclid could be contradicted, as had previously been done by Saccheri and would subsequently be done by Gauss, Bolay, and Lobachevski, and a consistent system of geometry nevertheless could be constructed. This is rarely understood in either popular or academic discussions of geometry. Instead, it is typically asserted that the successful construction of non-Euclidean geometries refuted Kant's theory, as though Kant had denied that such geometries were logically possible. Since Kant asserted quite the opposite, this means that both academic philosophers and popular science writers do not understand the meaning of "synthetic," even though the definition, clearly stated, actually dates back to Hume. In fact, the construction of non-Euclidean geometries does refute Hume's, not Kant's, view of geometry -- since Hume thought that the axioms were "relations of ideas," that is, the equivalent of what Kant called "analytic."

Since Kant did believe that Euclidean geometry was true, these worthies might reasonably ask how then that could be the case. While revealing that they don't know the rest of Kant's theory any better than they know about analytic and synthetic propositions, we do owe them an answer -- which is that the synthetic first principles of mathematics rest on the abstract intuitions just mentioned.

We must also admit the peculiarities of Kant's theory. Kant also uses "intuition," Anschauung, to mean perception, or at least the sensible, empirical content of perception. But our intuitions of space and time are "pure" and not empirical, which means we do not need to observe any actual objects of perception in order to imagine and reason about space and time. Since geometers have always reasoned about space and time without such observations, and still do so (perplexing the uninitiated by talking about infinite dimensions and curved spaces), Kant is describing something with some historical and psychological validity. Some modern philosophers like the idea that geometry is no more than a kind of empirical science, but the practice of mathematicians continues to ignore the applications that geometry, including non-Euclidean geometry, may have as parts of physical theories. But there are paradoxes. How is a mathematical "intuitionism" subject to verification, or falsification? If Kant's pure intuition is imposed by the mind, what does it have to do with reality or the world? And if, somehow, we are "intuiting" mathematical truths, how is this different from the intuitionism that we find in Empiricism?

"Intuitionism" as a meta-mathematical doctrine is now associated with L.E.J. Brouwer (1881-1966) and his controversy with David Hilbert, Georg Cantor, etc. Brouwer's Intuitionism shares some features with the epistemology of the Empiricists, including some strange scruples about the use of logic (i.e. the rejection of indirect proof). Most importantly, it includes a Nominalistic denial of mathematical realism. This makes for a poor fit with Kant's mathematical intuitionism, since, despite positing a subjective origin of mathematical intuitions, Kant's theory nevertheless is one of Empirical Realism, in which, although being subjective in some sense in relation to things in themselves, it is realistic in relation to the phenomenal world. Because Brouwer's sympathies are largely constructivist and conventionalist, this is not what Kant would have had in mind.

Some of the motivation for Brouwer's Intuitionism, however, is understandable and important. Hilbert's formalism, in which the meaning of mathematical terms drops out of the equation (literally), leaves mathematics floating as a kind of empty logical structure. We should find that disturbing even beyond the peculiarities of Brouwer's system, and it is a point relevant to Kant's theory. Brouwer was also unhappy with Cantor's trans-finite numbers, whose basis, meaning, and application remain perplexing, and where not a great deal of progress seems to have been made since Cantor, even with the proof that the Continuum Hypothesis is axiomatically independent of the rest of Set Theory. Nevertheless, I don't think Kant would have any objection to mathematical reasoning about infinities, if this can be done with consistency. Cantor's achievement was to find a way to do that. Kant's intuitionism addresses a different issue. Similarly, the scruples of Brouwer about indirect proofs or the Excluded Middle (p v ~p) do not seem helpful. Since the existence of irrational numbers and the non-existence of a largest prime were originally established by indirect proofs, which rely on the principle of the Excluded Middle (as does ordinary logic), one wonders how much of traditional mathematics Brouwer contemplated tossing overboard. Indeed, there are people who don't like irrationals, for much the same reasons that the Empiricists objected to basic definitions in geometry (such as a point and a line); but the fundamental drawback of all of this sort of thing as philosophy of mathematics is that it fails to save the phenomena, i.e. it seems to strive to discredit rather than explain what are otherwise well established principles and results in mathematics, and even logic. The purpose of this seems a little confused. (On the other hand, the status of imaginary numbers appears to enjoy a paradoxical sanctity that moves people, who may otherwise take Brouwer and Intuitionism seriously, to dismiss those who have questions about it as ignorant of mathematics! Imaginaries, which have no common-sense correlate, are particularly challenging to an intuitionistic mathematics.)

A Kantian philosophy of mathematics will have no objection to Hilbert's formalism except that it needs a pied à terre to ground it in truth and reality. That is where mathematical intuition comes in. It grounds the axioms of geometry or, in arithmetic, what Popper might call "basic" propositions. Thus, the confidence of Ancient, Mediaeval, and early Modern mathematicians in Euclidean geometry did not rest on their respect for the authority of tradition, or the blindness to any obvious falsehoods in Euclid, but simply on what seemed to be the case in space as they were able to imagine it. Worries about the Parallel Postulate were because it looked too complicated in comparison to the other first principles of geometry, which suggested that it was a theorem of the system and so could be demonstrable on the basis of the other axioms and postulates. Hence attempts over the centuries to prove it. The attempts were not because anyone thought it was false or actually needed to be proven.

The most revealing attempt at proof would be, indeed, the indirect proofs of Gerolamo Saccheri (1667-1733). By denying the Parallel Postulate, Saccheri in effect began the construction of non-Euclidean geometries -- although he does not receive mention in this regard nearly enough. And one of his results was revealing indeed:  If there are no parallel lines, which is one way to deny the Parallel Postulate, this implies that all lines are finite in length. This is the geometry of the surface of a sphere, or of a gravitationally "closed" universe. For a long time, cosmologists and philosophers were ecstatic that this was established by Einstein's Theory of General Relativity, and that it answered key problems in cosmology, such as Kant's Antinomy of Space. Now, however, it is clear that Einstein's theory established nothing of the sort, while the observational evidence, and the theory that has been catching up to it, posits nothing less than "flat" Euclidean space in the observable universe (although people still try to smuggle in a closed universe as something that exists outside of the observable universe, something that calls into question their commitment to empiricism).

Nevertheless, we do not need to worry about the outcome of current physical debates. The question here is about geometry on its own terms. And in deducing that all lines are finite in length, Saccheri believed that he had accomplished his indirect, i.e. a reductio ad absurdum, proof. After all, what could be more absurd than to say that all lines are finite? Any line of a finite length can in turn be extended indefinitely, in fact infinitely, in either direction. Under what circumstances could a line not be extended and remain finite in length? Well, when we look at the surface of a sphere, any attempt to draw a straight line circles the globe and rejoins itself (making a "great circle"). There is a limit to how long any lines can be. In terms of Einstein's universe, the idea was that if we head off in one direction in space, we end up coming back around to where we started. Such a universe is "finite but unbounded." We never find an edge, and there is nothing to stop us from going forward indefinitely, but we will end up retracing our steps.

So what is the problem with this picture? Well, if we imagine the path taken on the sphere or in Einstein's space, we see a curved line closing a circle. But, ex hypothese, the Parallel Postulate is about straight lines. The intuitionistic geometer, in the broadest sense, will object that curves and circles are not straight lines. The rejoinder of the non-Euclidean advocate will be that the lines are geodesics, which is the equivalent of straight lines in the given geometry. However, even this qualification does not alter the fact that lines that are manifestly straight to our imagination, or in illustration, will not return and close on themselves. Even if a non-Euclidean geometry were physically true of the world, its terms nevertheless defeat our imagination. And it is not just a subjective matter. Non-Euclidean geometries cannot be illustrated with models or projections without the use of curved lines, which is why such spaces are traditionally and quite candidly and honestly referred to as "curved."

Since what Saccheri did troubles the conventional narrative about non-Euclidean geometry and its history, he is typically ignored in accounts of them. But this gives us the meaning of Kant's theory. It is in "pure intuition" that we imagine what a line can do, and this was always the basis of the historical confidence in all the axioms of Euclid, including the Parallel Postulate. If the axioms of geometry, in turn, are synthetic, as Kant says, they can be contradicted and alternative "geometries" can logically be constructed. But this will be, in root and branch, "counter-intuitive" in the fullest sense of the word. As we know from many other matters in mathematics, people can become accustomed to the most bizarre things; but it never helps any meta-mathematics or philosophy of mathematics to decide that something bizarre or counter-intuitive really isn't, just because in some way it "works." And there is nothing merely subjective about this. We can cook up definitions in which a curved line is "really" straight, but this does not alter, not just how we imagine them, but how they can be shown. And if it must be admitted that non-Euclidean geodesics cannot be shown in the proper terms in the illustrations, models, and projections that we see, this would seem to involve an admission that the space before us, mentally or physically, is Euclidean, regardless of what is hypothesized about it in astrophysics or cosmology.

And this is where Kant may cross paths with Brouwer's Intuitionism. In the epigraph to this page we see Arend Heyting saying, in 1930:

It is in principle impossible to set up a system of formulas which would be equivalent to intuitionistic mathematics, for the possibilities of thought cannot be reduced to a finite number of rules set up in advance.

Here now we get the notion that, as a system of formulas is necessarily abstract, the object of an intuitionistic mathematics will be concrete, returning us precisely to the theme of this essay. If mathematics, including both geometry and arithmetic, is founded on the "pure intuitions" of space and time, as Kant thought, what would we expect that this would mean for the future of mathematics? It would depend, indeed, on the density of features in the concrete intuition.

As it happens, we must postulate that the density of such features is practically infinite; and this is a fundamental feature of the distinction between abstract and concrete. This turns up in one respect in controversies about naming. It has been common to say that, given Frege's distinction between sense and reference, that "sense determines reference," i.e. that the abstract features of meaning in the "sense" of a name will uniquely determine and specify the individual referred to by the name. However, counterexamples have been adduced in which individuals with identical attributes must be regarded as different. Indeed, this is what we should expect. The principle of Leibniz, of the "identity of indiscernibles," which means that things with the same attributes must be the same thing, only holds because Leibniz denies the existence of space, which is the very thing that, since Aristotle, has allowed different individuals of the same attributes to exist. While, after Einstein, it has been popular to say that Leibniz's metaphysics of space has been vindicated, because it allows for the Relativity of Einstein's physics, the people holding this position mostly do not seem to realize that space simply does not exist according to Leibniz. On the other hand, the infinite density of attributes that actually belong to each monad in Leibniz can only be thought by the infinite mind of God. This is probably more metaphysics (and theology) than supporters of Einstein might otherwise want to bite off -- they say silly enough things when they try to invoke Spinoza.

If the density of features in mathematical intuition is practically, or actually, infinite, then we would expect nothing less than the truth of Gödel's Proof of the Incompleteness of Mathematics. There are always going to be additional features upon which an expansion of mathematics can be based -- Group Theory now seems to be the hot property. Since Gödel's actual proof is a reductio ad absurdum, it does not tell us what Incompleteness really means, i.e. we don't know what enables mathematics to be expanded indefinitely. No one has worried much about the metaphysics that would be required by this. The infinite features of mathematical intuition, and nothing else, do that job -- unless mathematics is about nothing, and Incompleteness just means that you can do anything, forever. The latter is probably what most (conventionalist) philosophers of mathematics would prefer, making it up as you go along.

For the purpose here, the actual metaphysics of space is of secondary importance. What is noteworthy in Leibniz is that the feature of the concreteness of individuals in space is transfered to the monads. While God conceives all those features in their abstract clarity, we are left with them as the concrete and obscure muddle that we perceive in the sensible images of spatial objects. This is something that we can only clarify, and can never clarify completely, by conceiving the features one after another. Since we only have finite minds, and will never know things like God, Leibniz must posit a kind of intuition for our knowledge. At the same time, where Aristotle had seen matter as contributing both extension and concreteness to the abstract form of an object, Leibniz eliminated both space and extension, with concreteness following from the infinite features of a monad. This was only paralleled in Aristotle by the beings of pure form, including God and the celestial intelligences, whose form made them both unique of their kind and unextended. As we have seen, something cannot be unique of its kind, in principle, without the potentially infinite features of something truly concrete. God, as an infinite being, might easily be thought to qualify in that respect, but it is more difficult to construe such existence in terms of the intelligences, the angels into which Christianity transformed those beings, or the human souls that Christian Aristotelians, like St. Thomas, added to this category of beings.

In each of these cases, there is a move to turn the abstract into the concrete, something we also see in Hegel, where the individual as such dies and is overcome by the species, which absorbs the concreteness of all the individuals it subsumes, living, dead, or potential, culminating in the Divine Presence on Earth, namely the Prussian State (the wet dream of every modern statist). In Plato, we imagine that the infinite features of the concrete each and all "participate" in a particular Form, so that the nature of concreteness is due to the number of Forms that make each feature of that nature possible. Thus, in the history of metaphysics, we see philosophers trying to reconcile the concreteness of individuals with the abstract features that make them what they are and by which they can be identified.

The most curious case of this may be in Aristotle; for the matter that renders an abstract form into something concrete does so without, ultimately, any apparent content of features. The ultimate form of matter, "prime matter," is a pure potential that has no evident structure in Aristotle's system. We can fix this up by conceiving of intermediate grades of matter, so that the "accidental" attributes that are inessential to, say, an individual human are contributed by the particular matter that differentiates the person, which has already given the proximate matter some actual features, beginning with the elements of earth, air, fire, and water. There is, however, a residual paradox here, that all form consists of abstract features that are, in their own right, abstract, while what it is that renders form concrete, the matter, ultimately has no features itself. Prime matter, by definition, is something that doesn't actually exist, since it is pure potential. So what is this potential? In the end it receives little enough respect, since God is without it, generating the further paradox, and an awkward circumstance for Thomists, that God, far from being the Almighty of familiar religions, is himself, free of matter, without the power or potential to do anything that he is not already doing. He is pure actuality, ἐνέργεια, enérgeia.

I have considered elsewhere the bias in the metaphysics of philosophers, beginning with Plato and Aristotle, for the perfect aspect. This is the unchanging fixity by which abstract objects are conceived, whether the transcendent Forms of Plato or the immanent forms of Aristotle -- with versions, as we have seen, in Leibniz, Hegel, etc. This invests all genuine reality in these things, and it eliminates the imperfect aspect of change and power, just as Aristotelian matter drops out of existence when we get to God. Yet if Plato's World of Being means anything sensible, it must be as the source of all that is possible, just as above I referenced Plato's theory in terms of all the features of concrete individuals "participating" in the Form that gives them their reality, i.e. that makes them possible. This should be a very significant clue for us in resolving these problems. For, what if the structure of possibility, which in Plato is in the separate World of Forms, and in Aristotle seems to come out of nowhere, is itself in the power that Aristotle was unable to provide to God and that the Neoplatonists in turn simply identified with non-existence? This is what Whitehead was looking for when he condemned Western metaphysics as based on nouns rather than verbs and made the existence of his "actual entities," the equivalent of Leibniz's monads, transient rather than permanent -- setting off enthusiastic comparisons of this to the transient dharmas of Buddhist metaphysics.

An analogy that we now have is with the wave function in quantum mechanics. The square of the wave function gives us the sum of the possible states that may occur once we have observed the quantum system. Thus, for example, a wave may be a particular electron. Since a wave is extended in space, we can observe the paradoxes that the electron may seem to have been in two places at once, or that it can interfere with itself and produce a diffraction pattern. On the other hand, if we look for the electron, as a particle, the square of the wave function tells us the probability of finding the particle at a particular location -- were probability gives a numerical value to possibility. Thus, all the possible locations of the electron, as a particle, are summed in the wave function. In macroscopic terms, as Einstein realized, this can mean that the wave could extend to cosmological distances, and that its instantaneous collapse, once we observe the electron, can violate the postulate of Special Relativity that nothing can travel faster than the velocity of light. Bell's Theorem has born out that violation.

What does the physics mean for metaphysics? Just that we now see a conception of structured possibility. Not everything is possible. What comes-to-be is constrained. The only things allowed to happen are those that conform to laws. In physics, the structure is, in general, all the laws of physics, and, specifically for quantum mechanics, Schrödinger's Equation. In metaphysics, we get the constraints imposed by all the modes of necessity, including those of logic, the perfect aspect, and ontological principles such as that of Causality. Thus, if a macroscopic wave function represents all real possibilities, it will possess a structure not unlike the Forms of Plato's World of Being, yet it belongs at a level of reality that is not unlike the matter of Aristotle's metaphysics. It is, of course, a peculiar matter, very far from the solidity and simplicity of Democritean Atoms, much to the distress of the modern materialist, who would rather that the question is as simple as it was for Dr. Johnson.

If we want to know how the structure of possibility exists in its ontological place, this is not so easily answered. The wave function itself is something that, once we look at it, evaporates into the specifics of particle location. In the same way, behind the phenomenal presence of particles, the structure of possibility belongs to the Kantian level of things in themselves. A Kantian interpretation of quantum mechanics has aleady been considered elsewhere; but all it amounts to is a limitation on our knowledge such as is already evident in the uncertainties and indeterminacies of quantum mechanics.

At one time the uncertainty of quantum mechanics could be ascribed, as a limitation on our knowledge, to the subjective conditions of observation. By looking at a system, we disturb it. However, this could not explain how individual electrons often seemed to be in two places at once. Subjective conditions of observation thus turned into an indeterminacy of the states themselves. Exactly how to think about this has been a matter of perplexity and controversy ever since. The physical reality of the electron as a wave is the simplest and most direct construction. As an extended object, the wave can indeed be in two places at once, or go through more than one slot in a diffraction grid. For some reason, however, the reality of the wave seems to make people nervous, perhaps because it does not seem right that physical reality should operate one way at one level -- outside observation -- and another way at another level -- once an observation has occurred. Yet this is exactly the "Complementarity" principle of Niels Bohr, whose "Copenhagen" interpretation of quantum mechanics has repeatedly defeated the more realistic interpretations of Einstein and others.

The sense that the physical reality must be either really one or the other, wave or particle, has driven most of the suggestions about the matter. But there was already a system with two levels of reality that worked pretty much the same way, namely Kantian metaphysics. And now the move that we can make is to collapse into each other, not just the wave function as the sum of possibilities, but both Plato's World of Forms and Aristotle's Prime Matter. This paradoxical union overcomes the Perfect Fallacy in the conception of "form" in both Plato and Aristotle, resolves the ultimate emptiness, interpreted as Nothingness by the Neoplatonists, of Arisotle's conception of matter, and provides an ontological position for abstract objects, without the multiple paradoxes that would otherwise accrue to them. Thus, the necessity that these Greeks attributed to the objects of knowledge, and so indirectly to knowledge itself, becomes no more than an artifact, as in Kant's Principle of the Possibility of Experience, that the possibility of actual things must be necessary. Indeed, considering that Plato's World of Becoming is actual and his World of Being necessary, the factor that seems to drop out of the equation is the possible. Aristotle supplied the deficiency, through matter, but this remained an ontologically poor relation, destined to drop out of the mix when it came to God, the intelligences, and -- as reformulated by Aquinas -- souls. Since this deprived God of the element of power, otherwise essential to the religious conception, such a metaphysics actually made for a poor fit with what we might expect.

Instead, we surrender the Platonic linkage of necessity to the pseudo-actuality of the Forms and couple it instead with the power and possibility concealed behind the veil of Aristotelian matter. The benefits of this are multiple. Now, the necessities of logic, metaphysical principles like Causality, and natural law are all artifacts of the necessities that possibility imposes on the actual. Not just anything can happen. A concrete, phenomenal reality cannot violate the Principle of Non-Contradiction. Concrete, phenomenal events cannot happen without a Cause. And concrete, phenomenal events cannot happen without conforming to the laws of nature that we come to know through physics, chemistry, etc. Mathematics bridges this span, dependent as it is on logic but supplying much of the language used by fundamental physics. Indeed, the sanguine physicist may expect that eventually all the irrationalities of physics, especially the apparently arbitrary constants of nature, will ultimately be derived from pure mathematics in the ultimate mathematical "Theory of Everything." As it is, even the Gravitational Constant, postulated by Newton himself, is not known with very much precision. At the same time, the abstract objects of mathematics have proven a wild card through which actual mathematicians, typically Platonists of some sort, find the Skeptical metaphysics of contemporary philosophers the most dissatisfying, while those philosophers themselves may be no more than condescending towards the mathematicians. This is an ironic situation in an Analytic tradition where mathematicians, particularly Kurt Gödel, have often been revered as gods.

The union of Kant, Plato, and Aristotle (which also would have been the sort of thing the Neoplatonists were looking for) resolves one of the most curious things about Platonism, something that bothered Plato himself. For, after all, if there is to be a Form for everything, then there are Forms of beds, television sets, duck calls, toilets, and, in Plato's own examples, hair and dirt. Somehow, the Eternal and Unchanging Exemplar and Archetype of the television, sitting in the World of Being, does not ring very true, or even sensible. But there is no doubt that televisions are possible, and they are possible precisely because of the laws of nature that they exploit to display the transmissions and images that they do. Indeed, televisions as artifacts, which have been assembled through human creativity, do embody a certain level of necessity, namely that of the perfect aspect; and this is the particular manifestation of the general necessity found in the conditions that make them possible in the first place. And this is something that must be kept in mind. Necessity is more than just a matter of logic through mathematics and physics, so that the conventional nature of human artifacts, art, customs, or language, while not going back to a particular Platonic Form, nevertheless hit multiple notes of necessity in terms of their logic and physical possibility and in the actuality of their presence or usage. The passage of something from the possible of the imperfect aspect to the actual of the perfect is itself part of the structure of possibility.

The limitation on our knowledge of abstract objects in a Kantian metaphysics itself goes down to a kind of indeterminacy in the objects. Thus, Kant's view was that any attempt to construct a system of transcendent objects, perhaps like the Platonic World of Being, generates Antinomies. This already rules out Platonic Forms as independent objects in their own right. Kant did believe that morality motivated the resolution of some of the Antinomies into "Postulates of Practical Reason," but the nature of his argument for something like God does not inspire confidence in the soundness of his approach. Nevertheless, Kant overlooked something that may resolve at least one Antinomy, that of freedom and determinism.

Thus, as I have examined in Ontological Undecidability, the world is very different depending on whether we examine it from without or from within, that is, from an external or an internal perspective. In fact, these cannot be separated. From an internal perspective we realize that our existence as conscious beings is embodied in a transient, fragile, private inner space. From that position, we look out at the world, and reflect on ourselves, but we also lose that position every day in sleep, or in other forms of unconsiousness. From an external perspective, we regard the world as independent objects entirely separate from ourselves, among which our own body figures, with exactly the same status as other things. We expect that even sleep, unconsciousness, or death does not remove the body from its place among others. How things happen is very different from these two perspectives. Externally, there are physical laws that sharply constrain how we can act and that establish necessities, like food, and dangers, like falls, of which we must be aware and govern our acts accordingly. Internally, we find ourselves deliberating about the actions we should take to deal with these external realities.

We then notice that our actions are also constrained by necessities of an entirely different sort. What we do may be good or bad. Our actions may be morally right or wrong. This is a very different business in comparison to how things happen entirely on their own in the external world. What happens there simply happens; and nature does not pause to deliberate whether to enforce the law of gravity at a particular moment or not. Internally, therefore, we discover our freedom; and this is not simply a feature of things in themselves, as Kant would have construed it. And it does embody a kind of Antinomy, since the nature of our free deliberations, on our intentions, purposes, and actions, is available to us in our own internal space, in paradoxical contrast to the remorseless, relentless, and indifferent mechanism of natural law in the external world. However, such an Antinomy, however curious and paradoxical, does not result in a practical contradiction in life. The laws of nature are conditiones sine qua non that cannot be violated but also present us with plenty of alternatives for action nevertheless. Thus, we have little sense that we will be able to raise the dead, but at the same time we can choose to be killing people, or allowing them to die, when there are alternative courses of action. We have not done wrong by failing to raise the dead, but we may indeed have done wrong if we have previously killed them.

This difference is a feature of the nature of necessity itself. With physical necessity, there are no alternatives. You drop the weight, and, barring some clever, deceptive arrangement (as a magician might do), it will fall. In the wrong circumstances, you will fall, perhaps to painful, ruinous, or fatal consequences. With moral necessities, on the other hand, you may fail in your duty and choose not to do the right thing. This also may have painful, ruinous, or fatal consequences, although even morally right action may be vulnerable to the same disadvantages (producing dilemmas). Moral necessity is thus imposed in knowledge but must be freely chosen and assumed in action. But, just as we may see the necessities of logic attenuated in force in physics, there are degrees of necessity in moral freedom also. Moral duty, strictly speaking, is an imperative, an unconditioned command; but most actions are not done with a narrowly moral purpose. You order the pizza that you like and avoid anchovies (or in Japan, squid), not because they are morally wrong, but because, to you, they taste bad (or with squid, like rubber) -- or, at least, their taste doesn't fit in with the way you want a pizza to taste. Taste, therefore, is subject to aesthetic variety; and this is not a matter of imperatives. In this way, Friesian philosophy allows for optatives, matters that we wish for, which here have been expanded to multiple modes of value. Thus, you urge a friend not to smoke because, even though this may not bother you (and you may know that most "second-hand smoke" studies are fraudulent), it is something that is clearly bad for his health. This is an exhortation, not a command, and thus not best called an "imperative" or even an "optative," but a hortative. He does not have a duty to do what is best for himself (although Kant would say he does), but he would be well advised to do what is best for himself. By the same token, urging your friend to order anchovies for his pizza, while also a kind of exhortation, expresses an aesthetic preference that cannot be construed as contributing to the substantive well being of the friend -- unless you think you know something about anchovies (or squid) that the rest of us don't.

The internal and external duality with which we are confronted in the world is specifically addressed elsewhere. Where Aristotle believed that all natural objects were subject to both efficient causes and final causes, i.e. purposes, Western philosophy did not improve on Aristotle's theory of purpose (i.e. teleology) and, in the development of the methods of natural science, the whole dimension of purpose has dropped out of the world, often even in the analysis of our own minds (e.g. in Behaviorism), where subjective purposes, at least, may reasonably be said to exist self-evidently. In the theory here, neither internal nor external can be preferred over the other, which leaves something like the Kantian Antinomy but otherwise puts it in the form of a duality with which we are actually already familiar and comfortable (more or less) in the considerations of ordinary life. Thus, can we explain behavior both deterministically, which we see in psychiatric and pathological explanations, or in terms of freedom, which is the basis of moral judgment, legal sanction, and our expectations about the actions of most people around us. The boundary between the two is actually specified in legal terms in the definition of insanity. The insane do not know the difference between right and wrong and are not morally responsible for their actions. They are not in control of themselves -- which is certainly the impression we get of many wandering the streets of Santa Monica or Manhattan (where at the corner of 57th Street and Eighth Avenue I recently saw a fellow shouting, "Shut the fuck up!" to one passerby after another).

With the metaphysics of abstract objects blocked out, we must turn to the epistemology. Kant's mathematical intuitionism, while vividly applicable to the axioms of geometry, does not otherwise always seem to work even in mathematics and becomes very awkward indeed in metaphysics or ethics, despite the Aristotelian tradition of considering metaphysical truths to be self-evident and the popuarlity even today of intuitionism in ethics. The approach here is through the distinctively Friesian doctrine of non-intuitive immediate knowledge. Although this theory was dropped by the later students of Leonard Nelson, as part of an ill-advised strategy of presenting Nelson as "an Analytic philosopher before his time" (thereby rendering his thought redundant and superfluous, even as he remains unknown to most Anglo-American academics), and I have often found colleagues professing not to understanding what it would mean, the conception actually goes directly back to Plato, who believed that there are things we know that we do not know that we know, because, in Platonism, they were known but have been forgotten. This is familiar and clear enough in the history of philosophy that there is something disingenuous about protestations that the doctrine cannot be understood. It is also familiar now in Linguistics, where it is obvious that people use language without always, or sometimes ever, being explicitly aware of the grammatical rules that they apply. We expect that the rules of natural languages are learned usually in covert ways, but most people in philosophy should always be aware of the Chomskian theory that a "universal grammar" is innate and that it is the basis, not just for all the grammars of natural languages, but for our infantile ability to quickly learn those languages in the first place.

In those terms, we may observe that children quickly grasp the Principle of Causality. You touch the stove once, get burned, and the sensible child will never make the same mistake. Yet this process of learning is the equivalent of applying the metaphysical principles that (1) everything has a cause, (2) like causes have like effects, and (3) the laws of nature are uniform and invariant. Sometimes the child, or even the adult, will check out a phenomenon more than once, usually carefully and tentatively; but that will typically be just to confirm the regularity. As Karl Popper notes, this is not a classical case of induction, which supposedly, as understood from Aristotle to Bacon to Hume, requires a large body of uniform data, but more a process of "conjectures and refutations," as Popper has described in the actual practice of science. This child appears to already know the metaphysical principles, which supply the framework for assessing individual cases, like the physics of heat in the kitchen.

While this would be no surprise to Plato or Kant, or to any other Rationalist, it is something to befuddle more recent philosophers, who keep paradoxically, even incoherently, invoking Hume, who demonstrated, ostensively to the satisfaction of everyone quoting him, that induction as traditionally understood cannot prove anything like what had been expected of it. One begins to wonder about generations, now, of philosophers who fail to see that the authoritative figure they confidently cite denies the very thing they are trying to assert and support (i.e. that scientific truth is based on induction). Analytic philosophers characteristically miss the point and misunderstand Hume's doctrine. The ironic fact of this was appreciated by F.A. Hayek but not by any academic philosopher I have ever noticed. They suffer, indeed, from the false dilemma described by Nelson, that knowledge is either immediate and intuitive or consciously learned. This false dilemma matches neither the child's application of causality nor Chomsky's theory of universal grammar. It is of a piece with the preposterous and question-begging argument of the Empiricists that there are no abstract ideas because "ideas" are images and images are always concrete.

Thus, while the child does not know that the stove is hot, or that heat causes pain, he needs no more than the most tentative confirmation of his experience in order for these truths to be firmly fixed in his mind and indeed in his spontaneous behavior. In truth, apart from establishing caution about stoves, the child may exhibit positive terror of the threatening object because of such experiences. In short, the average child naturally understands the principles of causality just cited and needs very little input from experience in order to apply them firmly [note].

As I have discussed extensively elsewhere, the phenomenon of understanding is rather different from the way Kant and other philosophers have presented it. Understanding is not discourse or language use. It is instantaneous. When one understands something, one understands it all at once. Discourse is something else. Explaining one's understanding is another process, and this can involve considerable labor, difficulty, error, and even failure. Nor is understanding to be identified with any specific discursive expression, for the procedure to verify that someone actually has understood something is to require that they explain it in different ways, sometimes several different ways, and certainly never as the repetition of some stereotypical expression or conventional definition. Similarly, to explain something to someone who does not understand it, multiple explanations are required again, limited only by the goal of conveying the understanding. We hope from the target of these explanations a moment of epiphany, when they say, "Now I get it!" But of course, some people, like Keynesians, never get it. There is also the interesting dynamic described by Thomas Jefferson:

..the more a subject is understood, the more briefly it may be explained. [letter to Joseph Milligan, April 6, 1816]

As it happens, Jefferson was referring to the work of Jean Baptiste Say, which he thought was an improvement, for brevity and clarity, over Adam Smith. But his point is correct and revealing. As understanding improves, one understands better how to explain it. This is a valuable gift for teachers, which means that it is a bad sign when some simply read the same lecture notes to students semester after semester (which I have seen done, to soporific effect).

Thus, the abstract knowledge that is implicit in Plato's theory of Recollection, or in the Friesian theory of non-intuitive immediate knowledge, is something that enters into experience by way of its being spontaneously understood in its concrete application. The child does not need to be told that the oven is dangerous, once he has been burned by it; and the implicit truths of causality immediately govern his actions, and reactions. As Hume says, all reasonings about matters of fact rely on these truths.

A vivid illustration of the immanence of understanding in perception can be found in the tricks or illusions of Gestalt psychology. The image at right is alternatively seen as two faces or as a central vase or chalice. Which of these we see at a given moment is not a matter of conscious decision or choice. We recognize the faces and then we recognize the chalice. This is said to be an alternation of figure and background, but the alternation depends on one's ability to recognize faces and chalices. We see these abstract forms in the image, but we would not see anything of the sort if we did not know about faces or chalices. Individuals from cultures without vases or chalices could not possibly see something that they never learned about. The image will not shift from one background to another for them. On the other hand, there is considerable neurological evidence now that facial recognition is hard-wired in the brain. Infants begin to focus on and recognize faces quite early. At the same time, the neurological system for recognizing faces can be damaged and fail, which leaves individuals, such as the eponymous Man Who Mistook His Wife for a Hat of Oliver Sacks unable to distinguish (visually) between persons and inanimate objects.

These phenomena would be music to the ears of Kant -- who otherwise doesn't seem to have had much respect for music. The abstract forms of understanding are literally seen in the image, which consequently has been generated by the mental activity of synthesis, by which the concrete qualia are imbued with abstract rules and forms. We subsequently look at the image and think, "now faces, now chalice," as concrete pattern recognition shifts from one to the other. And, just as infants recognize faces and brain damage can prevent such recognition, it is not difficult to understand how something like Kant's "categories" -- causality, substance, etc. -- are sewn into perception spontaneously by synthesis, with us subsequently to rediscover them there. The learned nature of chalices, as opposed to the innate knowledge of faces or causality, doesn't make much difference in the perceptual phenomena. There is simply a different epistemological and even ontological basis for the forms in question.

Something like causality, of course, is not seen in quite the same way as pattern recognition. That the stove has caused the burn, and not something else, involves an element that must be learned. The certainty of the child is not that the stove has burned but that something has caused the burn. That the stove is adjacent to the realization of the burn in time and space puts the matter beyond much doubt; but the matter is still not one of absolute certainty -- hence tentative experiments in confirmation. The reason for that is the difference between the principle of causality itself, that everything is caused, and the individual laws of nature (in this case, thermodynamics) that mediate things like the transfer of heat as burns are effected. Not only must such laws be learned, but it may take centuries for even the most basic, e.g. of mechanics and gravity, to be puzzled out.

In mathematics, what is intuitive and what is not make a complex pattern. The ability to intuitively visualize space is essential to ordinary life, since it enables us to imagine routes of travel, the geometry of machines, the tying of knots, or even the patterns of weaving, sewing, and stitching. While in both traditional belief and recent research, men have some advantage with spatial orientation, it is worth noting that the traditional feminine activities of sewing involve spatial complexities that are often incomprehensible to men. One hopes that unbiased research will eventually illuminate how this works. At the same time, we should realize from these examples how the axioms of geometry were first regarded as self-evident. We can reason about what we can visualize, and, as Kant appreciated, our basic understanding of space is based on this. The Greeks, discovered, however, that many unobvious things can be teased out of what seem to be intuitively true axioms. There is nothing self-evident about the Pythagorean Theorem, whose explanation is deceptively simple. At the same time, basic arithemetic, like 1+1=2, also is so intuitively obvious that no one until the 20th century thought that deriving it from axioms was even necessary. When that was finally done, in Set Theory, the axioms themselves were not easily understood, let alone the sorts of things that could be regarded as self-evident. Trying to teach arithmetic as Set Theory, starting in the 1960's, something called the "New Math," all but destroyed mathematics education in the United States for many years. Since those in education still have difficulty admitting, or perhaps understanding, what went wrong, mathematics education still has not recovered.

While mathematics poses its own interesting problems, our concern must focus on our knowledge in morality and matters of value. This is what originally concerned Socrates and thus motivated the whole of the later development of the theories in Plato. It is Socrates who discerned that people may be applying moral principles, not only that they are not consciously or explicitly aware of, but that they may forthrightly deny in their explicit assertions. This phenomenon is by no means confined to ancient discourse. Instead, there are systematic examples of it in modern thought, for instance in the common case of moralistic relativism; and we have no difficulty identifying it in the views of otherwise quite sophisticated thinkers. For example, we find Jacob Bronowksi asserting that certain and absolute value judgments lead to things like the Nazi death camps. However, Bronowski's own moral condemnation of the Nazis and their death camps is clearly no less certain and absolute than whatever modality the judgments of the Nazis may be thought to have possessed. Socrates would have loved it; for his own approach was to question people until he had exposed the contradictions that were implicit in their beliefs and assertions. Bronowski owed his readers and viewers an account of his own moral judgments and their foundation. Yet, as a philosopher of science, his approach was entirely innocent of the moral branch of philosophy; and his thought was morally naive and even, not to put too fine a point on it, infantile.

What Socrates discovered, then, is that people can apply moral principles, in judgment and action, that are contrary to their own conscious understanding and inconsistent with their professed account of that understanding. One approach to this would be to use the unconscious, in a Freudian or Jungian sense, to explain this as the manifestation of a relatively autonomous, but hidden, side of the mind, expressing itself regardless of our conscious self. Or, we could follow Plato, whose explanation was that it was the application of principles that are innate to the mind, and which remain active in understanding and behavior, but that have not been brought to conscious and reflective knowledge. Meanwhile, people repeat things that they have heard, or accept some half-baked theory, because there is otherwise some reason for them to believe that these things are true. Bronowski clearly had come to think that tolerance, which was a political and moral virtue he valued and endorsed, required some sort of uncertainty or relativism to be true. Since uncertainty could be linked to quantum mechanics, he could think that this fit in nicely with the lessons of his philosophy of science. That his own certain and absolute moral commitment contradicted this approach escaped his reflection. It would have required a Socratic cross-examination to bring out the problem. The bitter truth is that the racism and social Darwinism of the Nazis, and of the still celebrated philosopher Nietzsche, had a lot more to do with the science of their day (however misguided it may now seem) than Bronowski may have wanted to admit, while they certainly agreed with Nietzsche that concern for the weak or respect for the rights of others, i.e. the content of traditional ethics, was of a piece with the "slave morality" of Judaism and Christianity. The impression that Bronowski leaves with us, that the absolutism of traditional ethics was the source of the problem, is thus entirely unjustified and misdirected. Indeed, the traditional values of the "slave morality" are precisely those upon which Bronowski implicitly relies in his insensible and uncritical invocation of them.

The method developed by Socrates was to expose the contradictions in the sort of tangle that we find in the contrasting thought and action of someone like Bronowski. What Plato added to this was the sense that a Socratic examination could winnow out the principles that are indispensibly used by everyone because they are the common heritage of transcendent truth. Both Plato and Leonard Nelson thus believed that this "Socratic Method" would quickly progress to full knowledge. However, it is not that simple. Even when we find someone insensibly applying principles of which they may not be aware and may consciously dislike, this does not mean that these are the cognitively or morally correct principles. Falsehoods can also have become things that are unconsciously applied. Thus, a Platonic confidence that the truth exists as non-intuitive immediate knowledge must be tempered by a Popperian form of uncertainty. The result is always fallible and corrigible, which means that Socratic testing, looking for contradictions, must continue indefinitely. The other dimension to it is that we may have no way of ever knowing how well we understand the principles. Truth is one thing, and in some cases it may even be uncovered with some ease, but meaning is something else. Jefferson's phrase, "the more a subject is understood," means that mere truth is not enough. Better understanding is a further process; and we may in time be surprised as further dimensions of meaning emerge. This is the proper phenomenon of the hermeneutic cycle.

Between intuition and learning there is thus the intermediate ground of knowledge that is not learned but that is not intuitively present. Since such a conception goes back to Plato and is, after a fashion, familiar to anyone who has read or been responsibly informed about Platonic doctrine, it is surprising that an equivalent theory in modern philosophy is really not to be found -- outside the Friesian School -- and that people express perplexity or incredulity when informed about the theory of non-intuitive immediate knowledge. But this provides the background and foundation for any modern Socratic inquiry. And it is a sufficient explanation for how knowledge of abstract truths is apparently to be found in the beliefs or actions of people who actually are not directly aware of them, yet insensibly apply them in manifest confidence.

In Leibniz, each monad contains a representation of the whole universe in all of its history. The bizarre nature of this extraordinary doctrine, generally ignored by people who think that Einstein proved Leibniz correct for the metaphysics of space, is mitigated by the provision that only God is able to consciously think such a representation in all clarity and detail. For the rest of us, the representation, to a greater or lesser extent, is collapsed into the confusion and murk of concrete perception. Leibnizian perception is thus merely confused thought.

Here, there is perhaps a single point of similarity with Leibniz. In an obscure fashion we implicitly understand a set of metaphysical and value principles, held as non-intuitive immediate knowledge, just as each monad represents the universe, but this includes no knowledge of actual phenomena. There are also other significant differences. Like Leibniz, Kant and the Friesians viewed this representation as a subjective matter, present either in the substantially independent monad, on the one hand, or in the faculty of reason, on the other. Each of these tends to reproduce a Cartesian paradox of solipsism, where its relation to the external world or objective reality is open and unexplained. Instead, I would say that whether the abstract objects of metaphysics, mathematics, morality, or aesthetics are internal or external is Undecidable, because each is an equal and opposite reflex of the other within immediate representation. Where Leibniz viewed the monads as substances, i.e. separable and independent, a proper Kantian does not know what counts as substance among things in themselves, which means that we are given, not the material substance of external reality or the spiritual substance or soul of internal reality, but the inherent dual nature of phenomenal representation itself, which is at once the world and the internal representation of it.

Our concrete perception is thus, all at once, the phenomenal world and, on reflection, the content of our own conscious existence. This turns inside out the traditional metaphysical dilemma of "materialism," favoring the external, against "idealism," favoring the internal. Both vanish into the inherent relation contained in phenomenal reality. We are thus not compelled, like Kant, to spin both metaphysics and morality out of the gossamer threads of the forms of logic, an exercise that I expect few have found credible since its proposal. Instead, we can imagine something superficially more intuitive, in a Platonic or Neoplatonic sense, by which the objects themselves are available, with the qualification that they are present non-intuitively. This means that we are not directly aware of them, but that their presence becomes evident on reflection, when we see that they have been insensibly used or applied, just as the rules of natural grammar are applied by people who have no instruction, and indeed no clue, in what those rules may be. We develop our understanding, in which a body of knowledge emerges, is corrected, and expands until we have a conscious grasp of metaphysics, ethics, etc. Thus, we begin with the "concrete phenomena," , and gradually become aware, by reflection and examination, of "abstract principle," . If we derive from this some knowledge of metaphysics and ethics, or even mathemtics and aesthetics, and other things, this will then be something that is rather different, and more, than the Buddhist appreciation of the Emptiness of the dharmas. Yet the ultimate Kantian limitation on the knowledge of things in themselves nevertheless may still be conformable to Emptiness; and the final term of the system here is the category of numinosity, which is without discursive content.

The Blinding Insight

Why I Am a Platonist

Meaning and the Problem of Universals, A Kant-Friesian Approach

Meaning and Naming in Michael Devitt and Kim Sterelny's Language and Reality, MIT Press, 1999


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Concrete and Abstract, Note

Does a child similarly understand moral principles in the same spontaneous way? This is not at all obvious, and it begins to tangle us in the ancient debate whether people are innately good (as Confucians would say), innately bad (as the Legalists would say), or something else. Many people observing children on the playground, or after reading The Lord of the Flies [William Golding, 1954; movie, 1963], might come away with the impression that children are naturally savages and would grow up in the same mode without discipline and instruction. The English jurist Blackstone explained the incompetence of minors in terms of their "imbecillity of judgment" [Commentaries on the Laws of England, Volume 1, p.424]. And it is hard to disagree that children have a poor sense of prudence and can be infuriatingly self-centered -- before we even get to the actual bullies on the playground. The lack of prudence, of course, is a function of not knowing what things, like stoves, are dangerous. That bullies have no respect for others is a moral disability that often persists into adulthood, where it can land them in prison or, in milder cases, merely make the lives of those around them a living hell.

On the other hand, we sometimes find surprising levels of moral awareness in children. Recent testimony about this curiously comes from P.J. O'Rourke, about his own childhood:

We'd started with a conscience. When I was just old enough to be allowed to go around the block on my tricycle, I pedaled up the driveway of Mrs. Furstein, who was arty. Behind her garden shed she had a pile of the kind of rocks, brought from the seashore, that were considered artistic when arranged in a garden. I climbed the chain-link fence around the shed, boosted a rock over the smooth folded ends of the fence top, put the rock on the back step of my tricycle, and pedaled up the driveway of Mr. Biedermeyer. Then, fulfilling some mission of the imagination that I can't recall, I hid the rock in Mr. Biedermeyer's two-car garage, behind the boat. I did this twice and was wracked with guilt.

Bitter self-reproach kept me awake whole halves of hours past my bedtime. One night I finally got up, went down to the kitchen in my pajamas, faced my mother, and told her everything.

Purloined Furstein rocks smuggled to the Biedermeyer garage makes no sense, and it must have made even less sense when recounted by a four-year-old. But Mom said the right thing, "I'm sure you didn't mean to." And I didn't mean to do all the things I did in the 1960s and '70's either. [The Baby Boom -- How It Got That Way And It Wasn't My Fault And I'll Never Do It Again, Atlantic Monthly Press, 2014, pp.84-85]

Whether children are savages or have inchoate consciences, what this begins to sound like is that morality is in something of the same position as a natural language. Feral children, growing up without language, are never able to learn a language properly. Whether feral children are similarly disabled morally is a matter for empirical study; but let us assume that they are. They must receive some kind of discipline and instruction in order to begin morally maturing -- even that as mild as from P.J. O'Rourke's mother -- just as they must be exposed to some kind of language in order to develop full linguistic capacity.

What children will then be taught raises the next question. Just because some moral principle is taught and learned does not mean that it is right. Otherwise we have commited both the genetic fallacy and Hume's fallacy of confusing "is" and "ought". Children can be made to behave by instruction and punishments, but this does not mean they possess moral understanding. They have simply learned a moral system as a form of prudence, in the same vein as they learn that the stove burns and the knife cuts. Young children can quickly learn to make claims about what's "fair," but then we may notice that this is often to complain and whine in the self-centered way that we might otherwise expect. At some point, they should acquire some respect for others and some conscientiousness about themselves.

That tends to dawn, in traditional expectation, if it ever does, with adulthood, which historically will mean the "age of reason" at 13 or 14, when we get the Bar Mitzvah or the First Communion. These introduce the adolescent into the adult community, socially and religiously, and where sexual maturation usually meant that arrangements for marriage began to be considered. It also has traditionally meant entering the workforce. Apprenticeships might have begun at 6 or 7 and will be completed in early adolescence. The apprentice, in Mediaeval Europe, then becomes a "journeyman" and may move around, from one master craftsman to another to learn his trade more thoroughly. Peasants do not move around but do marry. Beginning in Athens, no one would have been considered mature enough to vote until their 21st year. The extraordinary modern delays of education and marriage are only possible, of course, when the average lifespan now approaches 80, rather than 40. That the modern voting age has dropped to 18 is contrary to this trend and would perplex the Ancients exceedingly.

The age of reason in moral terms becomes the age of moral autonomy, in that mature moral judgment does not take the truth of what is received or instructed as a given. There is a corrosive form of this, in which an agent decides that moral duty is empty, but also the positive form in which we know that getting something morally right is our responsibility. That is where non-intuitive knowledge comes in; for having awakened one's moral judgment, one must then observe its own dynamic. Respect for the dignity and will of others is something that may be asserted to us, but it will lack moral force until we, after a fashion, feel it. Following the Enlightenment Scottish moralists, including Hume, we might call this a moral "sentiment," but its non-intuitive nature is evident in its obscurity. When Kant says that we must treat ourselves, as well as others, as ends also and not as means only, the obscurity here is not only what it means to treat anyone as an end but how, if this refers to the dignity and autonomy of others, this can possibily be applied to our treatment of ourselves. Deciding things for myself may be respecting my autonomy, but what duties does my dignity impose on me? This has been a source of perplexity to interpreters of Kant ever since, and Schopenhauer, at least, washed his hands of the whole business as too confused.

But that is just the point. The obscurity and confusions of moral knowledge bespeak its non-intuitive nature. If there is nothing to be made out in that confusion, then a nihilism that rejects immediate moral knowledge may be warranted. But if we can pursue a Socratic examination and begin to make out consistent principles, then there is some sense and some hope that the immediate knowledge, and moral truth, is there. Kant's own principle is an excellent start, with corrections, and has already advanced beyond the implications of the Golden Rule, which, after all, would justify Sadism in the judgment of the Masochist.

This returns us to the linguistic analogy. Non-intuitive immediate knowledge is analogous to Chomsky's "universal grammar." Now, linguists like Derek Bickerton believe that in certain circumstances the "universal grammar" can become the default grammar of an actual kind of langauge. The kind of language is a "Creole," which is a language that has grown up among the children of parents who speak Pidgin languages. Pidgins are languages that develop among adult speakers of different languages, who never learn each other's native languages (and may not be able to, as adults usually have difficulty learning another language to a native level of competence), but who develop a common vocabulary that enables them to communicate, even without the advantages of a formal grammar. Children who are raised with a Pidgin spoken around them nevertheless appear to spontaneously supply a grammar, which has features common to Creole languages around the world, however isolated they have been in origin from each other.

The moral analogy is that most people who receive moral instruction retain that system into adulthood, like the grammar of a learned natural language. Moral autonomy, however, means that, by questioning, one begins to feel out the default grammar of morality. The analogy is more perfect when the moral system is not one of rules and principles but merely of a vocabulary. A Pidgin language of morality is thus something like Aristotle's virtue ethics, in which virtues and vices appear in relative isolation from other virtues and vices. With a moral vocabulary, the inquisitive agent, trying to systematize the vocabulary, will approach a Creole language of morality. This is something like what we see in Plato's Republic, where an investigation into the term "justice" results in a rule, that justice is each having what is his. This is not at all satisfactory as a rule for justice, with the glaring open question what it would mean for something to be "mine," but the dynamic here is clear. The Pidgin vocabulary leads to the Creole rules, and the great moralist, of whom there have been far too few, makes some progress in improving our understanding of morality.

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