I am a Platonist... [the equations of fundamental physics are all that is real and] we see only shadows on the wall.Allan Sandage (1926-2010), Scientific American, August 1998, p. 22.
Physicists, by and large, are Platonists who seek reality in the archetypes behind the scenes.
Non-scientists, by and large, are Kierkegaardians for whom the subjectivity of life and thought is more real than scientific models.
Allan Sandage, "Science and religion -- separate closets in the same house," Science and the Spiritual Quest, New Essays by Leading Scientists, edited by W. Mark Richardson, Robert John Russell, Philip Clayton and Kirk Wegter-McNelly, Routledge, 2002, p.61 [note].
How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?
Albert Einstein (1879-1955), 1934.
"But tell me, what would happen if one of you had the fortune to look upon beauty itself [αὐτὸ τὸ καλόν] entire, pure and unalloyed, not infected with the flesh [σαρκές] and skin [χρωμάτες] of humanity, and ever so much more of mortal nonsense [φλυαρία]? But what if he could behold the divine beauty itself [αὐτὸ τὸ θεῖον καλόν], in its unique form? Do you call it a pitiful life [φαῦλος βίος] for a man to lead -- looking that way, observing that vision by the proper means, and having it ever with him? Do but consider," [Diotima] said, "that there only will it befall him, as he sees the beautiful through that which makes it visible, to breed not illusions [εἴδωλα] but true examples of virtue [ἀρετή], since his contact is not with illusion but with truth. So when he has begotten a true virtue and has reared it up he is destined to win divine friendship [θεοφιλής]; he, above all men, is immortal [ἀθάνατος]."
Plato, "Symposium," 211e-212a, Plato III, Lysis, Symposium, Gorgias, Loeb Classical Library, translated by W.R.M. Lamb, Harvard, 1925, 1991, p.207, translation modified.
Τοῦτο τοίνυν τὸ τὴν ἀλήθειαν παρέχον τοῖς γιγνωσκομένοις
καὶ τῷ γιγνώσκοντι τὴν δύναμιν ἀποδιδὸν τὴν τοῦ ἀγαθοῦ ἰδέαν
φάθι εἶναι, αἰτίαν δ᾽ ἐπιστήμης οὖσαν καὶ ἀληθείας ὡς
γιγνωσκομένης μὲν διανοοῦ...This reality, then, that gives their truth to the objects of knowledge and the power of knowing to the knower, you must say is the idea of the good, and you must conceive it as being the cause of knowledge [ἐπιστήμη] and of truth [ἀλήθεια] in so far as known.
Plato, Republic, 508e, Republic II, translated by Paul Shorey, Loeb Classical Library, Harvard University Press, 1930, 1969, pp.102-105, color added.
In the Alex Proyas movie Knowing [2009], Nicolas Cage is asked by one of his students a question that I used to get occasionally, "What do you believe?" While this sometimes was asked after I might have thought that I had made my preferences fairly clear, I hope that it was a tribute to the general objectivity and even-handness of my presentations, rather than an exposure of the cluelessness of the student, that they might be left with such a question.
Advocating at this website the doctrines of the Friesian School, much of what I write is already about what I believe. But sometimes more specific or focused statements are in order, such as when I explained, "Why I am not a Christian" -- now to be read with "Infantile Atheism." The general explanations and expositions here may drift away from the personal existential connection and affect that are the most suitable when personal beliefs are the issue. Thus, it seems appropriate to try and present in a succinct form what this all means to me. In a word, what that's going to be is "Platonism."
But if I would present myself as Platonist, with Platonic beliefs, it isn't quite what Plato meant by it. There are modifications that reflect the progress of the history of philosophy. Yet there is something in it that is going to be essentially Platonic, and it is best to make clear just what that is, why it is important, and what the basic modifications of it are.
Platonism at its foundation includes a metaphysical doctrine that there are two levels to reality. So the doctrine is of an ontological dualism. In Plato himself, this meant the difference between, on the one hand, the World of Becoming, where we live, where things come to be and then decay and pass away, and where perfection is something that rarely, if ever, occurs, and, on the other hand, the World of Being.
That is where the perfect, eternal, and unchanging exemplars and archetypes of everything exist, embodied in the "Forms" or "Ideas" -- in Greek the Εἴδη, Eídē (neuter singular Εἶδος, Eîdos), or Ἰδέαι, Idéai (feminine singular Ἰδέα, Idéa) -- such as the Form of Beauty, or "the beautiful itself," τὸ καλὸν αὐτόν. Eîdos is translated Species in Latin (which to occasional confusion is both the singular and plural form). This then tends to represent Aristotle's use of the term, in which the forms subsist in the objects, not Plato's. "Idea" was used by Aristotle to refer to Plato's theory, creating a tradition that was followed in subsequent philosophy, until quite recently. In Modern Philosophy, "Idea" is used to mean very different things by Kant, Schopenhauer, Hegel, and others, without direct reference to Plato.
The most important thing accomplished by such a dualistic Platonic doctrine is to answer 
Hume's dilemma of "fact and value," specifically about propositions whose subjects are connected with their predicates through a copula "ought" as opposed to "is," namely that the former cannot logically be deduced from the latter. For example, just because crime exists doesn't mean that it ought to exist. And what "ought" to be, like justice in the courts, may in fact never be in fact -- hence, Diogenes looking for an "honest man." Thus, matters of fact are grounded in the World of Becoming, matters of value in Being.
What the question of "is and ought" meant to Hume, however, is commonly misunderstood or ignored -- or rejected by some for their own obscure reasons. As widely interpreted today, Hume's argument means that moral imperatives, with "ought," are based on nothing -- since empirical knowledge is only of perceptual facts -- and therefore are without justification and so without either logical or moral force. Yet Hume's point was only that they were without rational or objective foundation. As he says of a wrongful act, a "vice," like "wilful murder" (without any sense that this is anything but wrongful):
The vice entirely escapes you, as long as you consider the object. You never can find it, till you turn your reflexion into your own breast, and find a sentiment of disapprobation, which arises in you, towards this action. Here is a matter of fact; but 'tis the object of feeling, not of reason. [A Treatise of Human Nature, Shelby-Bigge edition, Oxford, 1888, 1968, pp.468-469, original spelling, boldface added]
Like other philosophers of the Scottish Enlightenment, including his friend Adam Smith, Hume believed that moral judgment is based on subjective sentiment. Thus, as we see in the quote, there is a "matter of fact" after all, but it is internal rather than external and subjective rather than objective. While the modern reaction to the "subjective" is that it is arbitrary and idiosyncratic, that is not the way Hume or the other Scots saw the matter. There is no less certainty in the morals of sentiment than there is with anyone who can cite some kind of external moral authority.
If Hume is portrayed as a relativist or a trendy pseudo-Nietzschean nihilist, or as giving comfort to these ideas, by those engaged in either polemics or apologetics, they have not understood him. Thus, Kant said of Hume's critics that they "were ever taking for granted that which he doubted, and demonstrating with zeal and often with impudence that which he never thought of doubting..." [Prolegomena to Any Future Metaphysics, p. 259, Lewis White Beck translation, Library of the Liberal Arts, 1950, p.6]. Kant understood Hume better than most purported partisans and followers of Hume in the 20th century.
Yet the drawback of Hume's doctrine is that there is, indeed, no rational justification for moral sentiments. How do we deal with disagreements? Hume himself was not content with this lacuna. Since he had explained our belief in causality in terms of the habits engendered in our mind by the experience of the regularity of natural laws, he could make a similar appeal to the source of morality. The force of Custom through the course of history insensibly produces in us the corresponding feelings.
It should be remembered that a recognized source of much British law at the time was custom. This made Hume, although a Skeptic and an atheist, a political conservative, to the confusion and consternation of everyone since then who cannot imagine anyone containing in one mind the conjunction of such a selection of convictions. Yet it is perfectly consistent with Hume's system, as was recognized by no less than Thomas Jefferson. The common misreadings and misunderstandings of Hume may serve to psychologically protect the sensitive and delicate minds who cannot put "atheist" and "conservative" together in the same thought.
But there remains a gap. Obviously we cannot logically derive the "ought" of morality from the "is" of history. This would violate Hume's own foundational insight. Hume faced a similar problem with causality. Although we apply the principle of causality as a matter of "necessary connection," such necessity cannot be logically deduced from the contingency we find in the mere occurrence of "constant conjunction." As he says, "There is a step taken by the mind which is not supported by any argument or process of the understanding" [Enquiry Concerning Human Understanding, Shelby-Bigge edition, Oxford, 1902, 1972, p.41].
It is essential to an understanding of Hume that we realize he does not deny the validity of that "step." This means that we understand and apply causality as a matter of necessary connections -- which results, for Hume (not me), in the rejection of chance, free will, and miracles as violations of natural causation -- and that we regard and apply morality as a matter of imperative obligation. Modern Hume apologists enthuse over the atheism and the arguments against miracles but then ignore or misconstrue everything else.
Kant wanted to supply an objective correlate to explain Hume's subjective certainties. But Plato had already done so. The necessity of natural law, as well as the imperative obligation of morality, was vested in the World of Being. But where is this so-called "World of Being"? Aye, there's the rub.
Consequently, I would adopt the Kantian modification of Plato in metaphysics. The ontological dualism we face is not between the World of Being and the World of Becoming, but between Phaenomena and Things-in-themselves. This means that the foundation of necessity and value is not removed from the world we inhabit, but it is just that world itself, in itself, outside, not this order of Being, but outside of consciousness and the conditions and limitations that its construction imposes on our experience. [note]
Kant, like Hume, is also commonly misunderstood. Thus, if we read him as saying that things-in-themselves are unknowable, this introduces the paradox that, nevertheless, Kant seems to know that they are there. This was sufficient for the self-described school of "Neo-Kantians" of a century ago to dismiss the reality of things-in-themselves altogether. This overlooked a key part of Kant's argument, that the existence of external objects does not depend on the activities of the mind.
When the Neo-Kantians eliminated things-in-themselves, they may not have noticed that the result would be a Hegelian Phenomenalism, a "Consciousness Only" doctrine (like the Buddhist Yogācāra School) in which each mind is part of the meta-consciousness of a World Mind -- the only way to avoid solipsism. This is not a reasonable way to defuse the paradox of Kant's theory.
On the other hand, it is equally damaging to try and retrieve Kant's position by construing the system as no more than a version of the "transcendental realism" (Kant's term) of Descartes, in which things-in-themselves are the truly existing things, while the phenomenal world is simply a subjective phantasm of the world in our minds. This substitutes for the unknowability of things-in-themselves the venerable Cartesian Problem of Knowledge, which leaves us unable to say how the external world has communicated itself to the interior of our minds.
Schopenhauer did not help by comparing Kant to Berkeley, whose "subjective idealism" (Kant's term again) denied the existence of external objects and relied on the causality of God to maintain consistency in the fictitious existence of an external world. Whatever is supposed to have happened to the trees, or the bears, in the woods, when we are not there, God keeps track of it.
No, Kant's own theory is self-described as "empirical realism" together with "transcendental idealism." This double barreled doctrine is confusing and paradoxical enough in itself, as I have considered elsewhere; and I suspect that it is not the best construction of what Kant wanted, or should have wanted, to get at. The clearest part of it is "empirical realism": The phenomenal world that we perceive is real and objective, and we are directly acquainted with external objects in most of the commonsense ways that are ontologically and scientifically significant. Descartes is put to rest.
What "transcendental idealism" is supposed to mean is the problem. "Transcendental" applies to things-in-themselves, as in the "transcendental realism" of Descartes. "Idealism" means we possess it as an internal representation -- with the original empirical "ideas" of John Locke. Consequently, these two terms together look oxymoronic, with "transcendental" referring to the external and "idealism" to the internal. How can we have both?
We get both because of some obscure postulates of Kant's thought. He always believed that the forms of Reason applied directly to things-in-themselves or, as Kant puts it, to "things in general." And this was because Reason is our pipeline to things-in-themselves, perhaps because it is the core of our own being as things-in-ourselves. Kant never does spell this out very explicitly, and not surprisingly, since we would like to know how he knows this. Our experience of the phenomenal world is then based on a certain construction applying the forms of Reason in perception.
This gives us an objective world, but as a fragment, leaving out facets of things which nevertheless continue to gnaw and bubble, after a fashion, at the roots of Reason and of our own being. We see a world that is real both objectively and in its own terms but is actually only a fragment of a larger reality, just as in the common modern understanding of the world we see material things all around us but do not see the particles, atoms, molecules, and force fields, like gravity, that constitute and control all that matter.
This is an obscure postulate of Kant's system because, not only does he never spell it out very well, but he never gives us anything like a proper argument for the theory. The famous Transcendental Deduction of the Critique of Pure Reason is an argument for the objective application of concepts like causality to phenomena, but it is not an argument for any residual application of causality to things-in-themselves. Yet such an application is essential for Kant's larger understanding.
The concepts that Kant believes Reason continually throws up about the transcendent, although he calls them part of "Dialectical Illusion," are nevertheless necessary, essential, and unavoidable products of Reason. Even if they are "illusion," this does not mean they are illegitimate things -- "a philosophical term [that] is employed without any meaning or idea," in Hume's words -- that can be dismissed entirely from consideration. They are clues; they tell us something; and by and by they will be cashed out.
One of the clearest features of phenomenal reality as a fragment of a possible larger reality is that no objects exist as unconditioned entities. Everything affects and so is conditioned by everything else. People talk about spacecraft leaving the Earth's gravity; but the range of the force of gravity is infinite. Nothing can ever leave it, even at a range of billions of light years. And since the mass of the Earth was already present in the singularity of the Big Bang, its mass has already affected every other quantity of mass in the Universe.
Such a view of the world has a striking parallel in Buddhist metaphysics, in the doctrines of "relative existence" and "dependent origination": the character of everything depends on its relations, causal relations, to everything else (also a feature of Whitehead's metaphysics). Nothing exists independently; and this is actually the source of the vulnerability and suffering that characterizes the life of sentient beings. To be free of suffering requires an unconditioned reality, which in Buddhist terms is Nirvāṇa, 
.
Snap. In Kant's metaphysics, concepts of unconditioned realities are what Reason keeps throwing up out of "Dialectical Illusion." And these are not always about transcendent objects. The Universe itself, as a whole, is an unconditioned object (barring multiverses, which as a whole would then be unconditioned anyway); and it generates is own paradoxes, about whether it is finite or infinite in space and time. Yet the illusion remains illusion, an undetermined speculation, even for the Universe, until some datum can ground it, the way that sensation grounds our knowledge of the phenomenal world. And Kant thinks he knows what the unconditioned datum is: Morality. And it isn't just that morality issues an unconditioned command, the Categorical Imperative, but that this command presupposes and implies other unconditioned realities, namely the "Ideas" of God, Freedom, and Immortality.
And so with Kant we actually come full circle from Plato. The dualism of phenomena and things-in-themselves, which also embodies a dualism between conditioned and unconditioned realities, is the foundation of the difference between Hume's "is" and "ought," fact and value, indicative and imperative. As Kant wanted to answer Hume about the origin of necessity in causation, he also wanted to answer him about the origin of obligation in morality. Indeed, as the moral goal of Hume's philosophy is often ignored in its popular nihilistic interpretations, the moral goal of Kant's, in parallel to Hume, is also neglected. Indeed, if we follow the common impression that Kant's philosophy stands or falls on the success of the Transcendent Deduction of the Categories, then its larger goal has already fallen, since even the success of the Transcendental Deduction is irrelevant to the elements that constitute morality and its meta-ethical support in epistemology and metaphysics.
Kantian modifications of Plato's metaphysics, while preserving the benefits and avoiding some of the pitfalls of that system, now suffer from their own pitfalls. Kant's conception of "Reason" turns out to be a thin reed upon which to rest the full weight of the epistemology, metaphysics, and value theory that we require. For Kant, as it happens, wishes to construe reason as no more than the formal rules that become evident in logic, with a couple of characteristics that we can comfortably attribute to those rules. Nowhere is this lack of substance so evident as in ethics, where Kant heaves and strains to provide some positive content for morality using little more than the concepts of rule-making, universality, and consistency. I have examined the woeful inadequacy of this elsewhere.
Kant's system thus stands in need of further modification, and we get this, first of all, from 
Jakob Fries. Fries actually retained Kant's ideas about Reason as an inner faculty of the mind mysteriously connected to the transcendent; but otherwise we lose the formalistic and logicist conception of Reason and return to something more like Plato's theory of concrete and positive content for rational knowledge. This might open the door for an intuitionistic take on Plato's theory of Recollection, as that had been developed by the Neoplatonists using Aristotelian epistemological principles. But Fries avoided that. Instead, we get a version of the virtue of Plato's own theory, that if Recollection means memory, then this is neither self-evident nor intuitively certain in its truth. How do we distinguish veridical memories from confabulations? The answer comes in two parts.
Over the years people have expressed puzzlement at the idea of knowledge that is "non-intuitive" but "immediate." Doesn't "immediate" mean intuitive? It can. But the alternative, of immediate knowledge that is not intuitive, should be familiar to anyone with a basic education in Greek philosophy. This is Plato's epistemology, that we have knowledge, forget it, use it without thinking, and then rememeber it. If the idea of our having knowledge of which we are not aware is perplexing, then people need to go back and study the Meno. The terminology is new, but the idea is not. All that is different is the elimination of the first two steps in Plato's epistemology. Our knowledge is there, non-intuitively, not because we learned it in the "interim state," between reincarnations (which is Plato's full theory), but because it is built into consciousness at its construction, in what Kant called "synthesis." Thus, anyone puzzled by Friesian epistemology needs to start fresh in the history of philosophy. See here.
The telling modern analogue of all this may be found with the grammatical rules of natural languages, which are used with ease by the native speakers of any language, yet may be complex and obscure beyond the ability of anyone but grammatical or linguistic specialists to say what they are. Indeed, native speakers may insensibly use forms that they then deny ever using, or positively affirm that they use forms that they never do. The actual rules for the formation of the regular plural in English, in terms of both phonology and morphology, are something likely never encountered outside a formal venue in Linguistics, as I indeed first encountered them in a Linguistics class as a graduate student at the University of Texas in 1975 [note].
In arguing that behind the rules of natural languages is a "universal grammar" that is common to all languages and innate to the mind, Noam Chomsky concluded (before he became a vicious lunatic fringe leftist) that the Rationalists had been correct, from Descartes to Leibniz (all the way back to Plato), about innate knowledge.
Leonard Nelson, did a better job with this than Fries did. First of all, Nelson realized that what 
Socrates did was the clue in this direction.
Socrates asked questions, and this forced people to reflect on what they were saying and doing. They actually may not have been aware of what they were saying and doing. Indeed, it is by such reflection that we become aware of the presence of our non-intuitive knowledge, since it has worked its way, like the grammar of a language, into our otherwise pre-reflective opinion, statements, and actions. We need to begin examining those. Plato had believed this himself, and consequently he adapted the method of questioning from Socrates into the "Socratic Method" introduced in the Meno, whereby the products of Recollection were drawn out from the interlocutor. Yet Nelson himself introduced an intuitionistic element at this point.
Nelson thought that the elements of non-intuitive knowledge, elicited by Socratic Method, are then recognized by "abstraction"; and we have our result. However, although there must be a function of the mind -- this is even studied in neurology now (we can watch the brain do it) -- whereby abstract features are consciously recognized, this does not mean that propositions involving them are thereby shown to be true. Karl Popper quite properly criticized Fries and Nelson for apparently believing that there was a subjective element of certainty involved here, which unfortunately then undercuts the whole tendency of Friesian theory away from intuitionism and provides no better way than any other intuitionism to resolve disputes.
Popper's own insights show the way to finish Nelson's own theory. Socrates, again, provides the paradigm. As I have discussed elsewhere, Socrates did not carry out a "regress of reasons," after the manner of Aristotle, looking for premises -- Nelson was still thinking in those terms. Instead, Socrates examined the logical consequences of the beliefs that he elicited from interlocutors, using them as premises and looking for contradictions. This accomplishes what Popper thought happens in science: Falsification, the original Socratic Ἔλεγχος, Elenchus, "cross-examnation, disproof, refutation."
A contradiction means that there is falsehood among the premises that logically imply the contradictory propositions. This does not tell us which premises are false. In fact, they may all be false. But, with further examination of additional beliefs, this enables us to begin a winnowing process. As Plato, Nelson, and Popper all would have thought, only the truth ultimately allows a system without contradictions.
Of course, "ultimately" is the key word there. A system may be logically consistent but still be entirely false. This does not involve a "coherence" theory of truth. Apparent consistency does not mean we are finished. There is no room for complacency in Socratic Method. Even something that looks complete and satisfactory must be the target of restless inquiry, trying to see if some new information or perspective will introduce contradictions and problematize the whole, just as Popper thought that we must always be thinking of new ways to seriously test scientific theories. The restlessness of Socrates, who kept questioning everyone, until he was killed for it, is again the precedent.
In terms of the original theory of Fries, it must be kept in mind that the uncertain, fallible, and corrigible nature of the result that we get from Socratic Method is not itself anomalous or falsifying -- it ought to be an approximation, a kind of prototype. Immediate knowledge provides the referents for concepts and propositions, but it is not itself conceptual. Concepts constitute mediate knowledge, and that is always fallible and corrigible. Nelson's notion about knowledge being drawn by "abstraction" from the Socratic datum, while perhaps containing an element of truth, is no longer even necessary for the theory.
As with Popper's view of science, it does not matter what the provenance is of the propositions that we examine. We may reflect on our own beliefs or actions. We may examine the propositions elicited by Socratic inquiry from others. Or we may simply try to think up something new, although Socrates did like to stick with what people actually believed. What we will do with this data, then, is compare it with other propositions that will come from all such sources. After all, we must seek out all comers if we are to expect a severe test, in Popper's terms, of any views or conjectures we entertain. One of the best ways to protect bogus theories is the hermeneutics of ignoring falsifying evidence. This is often done insensibly but is thus the betrayal effected by a complacent or stultified mind.
With Friesian epistemology, as completed by Nelson, with the help of Popper, the metaphysics begins to look a bit different. We no longer need the internal faculty of Reason as posited by Kant or even Fries. First of all, it is not necessary, since, as noted, the provenance of the propositions we examine is irrelevant. Second, a simpler construction becomes possible. Both Plato and Kant accepted an internal source for rational knowledge in order to separate that from the intuitive stream of perception. For Plato, this went along with the separation of the World of Becoming from the World of Being. For Kant, there is still a Cartesian remnant, that our relation to the physical world is mediated by causally induced sensation, and a kind of Platonic ghost, that our relationship to transcendent reality, even if that is the ontological presence of things-in-themselves, must be from within, like Platonic memory. None of this is now necessary.
Instead, the conjunction of Kant's empirical realism with Fries's non-intuitive knowledge allows something new: We are already looking at things-in-themselves. We are just not yet aware of it. The "transcendental" (i.e. a priori, to use an alternative in Kant's ambiguous terminology) non-intuitive knowledge that we have is like a hidden message or signal concealed in the overt signal of perception and conscious mental activity.
There is already a hint of this in Plato. Beauty is something that we certainly see right now. It is common in modern thought, of course, to relegate beauty to our own subjective response and to dismiss its real presence in external objects. Plato, on the other hand, despite his theory of the internal source of our knowledge, also believes that value has an external presence, as the entities within the World of Becoming derive their reality from "participation" in the eternal Forms. Thus, the beauty in objects is really there, as the objects themselves participate in Beauty Itself. This is what we see, as Plato waxes eloquent in a passage I have quoted more than once at this website:
Now beauty [κάλλος], as we said, shone bright among those visions, and in this world below we apprehend it through the clearest of our senses, clear and resplendent. For sight [ὄψις] is the keenest of the physical senses [αἰσθήσεις, singular αἴσθησις], though wisdom [φρόνησις] is not seen by it -- how terrible [δεινός] would be our love [ἔρως] for it, if such a clear image [εἴδωλον] of wisdom were granted as would come through sight -- and the same is true of the other beloved [ἐραστά, singlular ἐραστόν] objects; but beauty alone has this privilege [μοῖρα], to be most clearly shown [ἐκφανέστατον] and most lovely [ἐρασμιώτατον] of them all. [Phaedrus, 250D, after R. Hackford, Plato's Phaedrus, Library of the Liberal Arts, 1952, p. 93, and the Loeb Classical Library, Euthryphro Apology Crito Phaedo Phaedrus, Harvard University Press, 1914-1966, p. 485]
Plato, of course, takes such a vision of Beauty as a clue to the Form of Beauty, which can only be accessed in its purity through Recollection. In Friesian terms, however, the beautiful object does not just participate in the Form in some relationship outside our reckoning, it is already the instantiation and the non-intuitive portal to a mode of value, in this case beauty, of the transcendent thing-in-itself. Since it is non-intuitive, degrees of awareness of it may vary. Indeed, not everyone sees the beauty of nature, art, fashion, or music (not even Kant). Taste may be cultivated just as knowledge is; and various unfortunate considerations or influences may warp opinion into bizarre, disturbing, or ridiculous forms, for which we must then again call on the aid of Socrates to winnow out the chaff.
But I cannot take credit for this particular view of beauty as a result of developing concepts in Kant and Fries. 
Schopenhauer beat me to it. Schopenhauer's own theory of "Ideas" expresses much the same thing. The "Ideas" are the world free of space and time, and Schopenhauer says that they directly represent, as aesthetic entities, the Will, which is the thing-in-itself.
Now, Schopenhauer doesn't have much of an epistemology to go along with this, which Fries supplies; and we also have the paradox, characteristic of Schopenhauer's thought, that the Will, whose effects seem so horrible, causing all the suffering of the world, nevertheless appears in the beauty of art. To be sure, tragedy portrays the evils of life in a beautiful way; but other forms of beauty, including beauty in nature, conceal rather than reveal the horrors beneath -- both Schopenhauer and Nietzsche, in his own pessimism, see us alternatively as protected or deceived by fair appearances. Be that as it may, Schopenhauer promotes an aesthetic realism that is deliberately reminiscent of Plato but which, in any form, had been abandoned by Kant and not fully restored by Fries or Nelson. Ironically, for all Kant's immanentalization of Plato's World of Being into things-in-themselves, the divide between the transcendent and the phenomenal is more absolute in his theory and detaches us from direct acquaintance with beauty.
| Cognac | .95398 |
| Blue Curaçao | 1.0963 |
| White Creme de Cacao | 1.1229 |
| Parfait Amour | 1.1269 |
| Crème de Menthe | 1.1320 |
| Grenadine | 1.3427 |
The column of water thus might be thought of as a pousse-café, in which liqueurs of different densities float in layers, as we see at right. The column, in different colors,
| Good Will | z |
|---|---|
| +5ℏ | |
| Justice | +4ℏ |
| Right | +3ℏ |
| Good | +2ℏ |
| Beauty | +ℏ |
| Holy | 0 |
The Platonism I have in mind here thus has abandoned Plato's Forms as a separate order of objects or the World of Being as a separate reality. After all, you often can see into the depths, and the colors below may come through to the surface. Beauty, which Plato saw as a clue to the transcendent, is now more like a window, albeit pehaps more translucent than transparent, ἐν αἰνίγματι ("in enigma," "darkness").
But what remains preserves a kind of Kantian ontological duality because the Forms (the "depths") represent (1) matters of value whose modality escapes the factual character of phenomena, and (2) any possible character of things, especially sacred things, as unconditioned realities. It retains the impersonal and inactive character of the Forms, which only serve in some Aristotelian fashion as the end, the telos, τέλος, of all things. How do we dive into the depths? Can we swim within pure justice, pure goodness, pure beauty (even erotic beauty?!), or pure holiness? This is undoubtedly what we are drawn to, like the Neoplatonists ascending the Declension of Being -- as below -- although swimming with Dejah Thoris definitely works for me.
βλέπομεν γὰρ ἄρτι δι᾽ ἐσόπτρου ἐν αἰνίγματι,
τότε δὲ πρόσωπον πρὸς πρόσωπον·Videmus nunc per speculum in enigmate,
tunc autem facie ad faciem.For now we see through a glass, darkly;
but then face to face.1 Corinthians 13:12
One might, however, ask why something like the Form of the Good would then not simply be God. Indeed, Aristotle's own God is impersonal and inactive, qualities passed over or wished away by someone like St. Thomas, who was adapting Aristotle to Christian theism. Indeed, I have myself critiqued Plato's Forms as an example, a prime one, of the "perfect fallacy," whereby ultimate reality is conceived in only fixed terms, leaving out any aspect of power or possibility. The Form of the Good plus power and activity -- in short, Will -- provides the elements we might need for a theistic conception.
It is better to retain the impersonal character of the Forms for at least three reasons:
Indeed, out in the desert, the Israelites missed the "flesh-pots" of Egypt and made a Golden Calf to appease the Egyptian gods. But no one would miss anything about the German death camps, certainly not the food -- although I sometimes wonder if the use of the Nazi Martin Heidegger in subsequent philosophy has a bit of the Golden Calf about it.
Instead of the ten terrible Plagues visited on Egypt, the only sense of Divine Wrath that ordinary Germans might feel would have been the rain of terror and destruction from the sky in the Allied bombing campaign, whose own morality and effectiveness were disputed then and now.
As a matter of faith, the power of prayer will endure, but any empirical demonstration of its effectiveness will continue to elude study. Indeed, miracles as violations of the regularity of nature would thereby, by definition, not seem suitable for an empirical or scientific investigation or confirmation, when the regularity of nature and of natural law are themselves the object of science.
With Platonism, on the other hand, we expect to find that the world is imperfect. Ugliness, wrong, ignorance, hatred, death, folly, and evil are all around us in such profusion that it takes little effort to find them, if they have not already been brought directly to us in the most unasked, unwelcome, and unpleasant ways. Theologically, we would be left, of course, with the question why an omnipotent God allows all this to go on -- the Problem of Evil. In Platonism, it is no less than the nature of things: the world cannot measure up to the ideals embodied in the Forms.
Platonism is, to be sure, left with a question parallel to the Problem of Evil, which is why phenomenal reality should exist to be as intrinsically imperfect as it is. This is a less acute question for Platonism than for theism; for inactive or impassive Forms are not responsible for whatever actions or lack of actions have effected the visible arrangement of things. Buddhism is in a similar situation, since it addresses the nature of things as these are presented to us and makes no effort to explain why the world was originally structured as it was -- unless we see the suffering of the world as due to our own karma, which really only kicks the question back a step, since we then want to know why the law of karma even exists.
In the context of the Kantian version of Platonism here, I might say that the imperfect nature of the phenomenal world may be due to our own nature as conscious beings. In the form that we possess it, consciousness comes into being and passes out of being, daily in terms of sleep and waking, but then with the greatest portent in terms of birth and death. In death, we face the possibility of our own non-existence; and otherwise what we see in the world is a great round of coming-into-being and passing-out-of-being, generation and corruption, growth and decay.
As the Buddha taught, we see the transient nature of all things, which pass into nothing, even as we often observe with horror, alarm, and grief that those closest to us can pass away, often in unexpected and terrible ways. If our conscious existence thus introduces the possibility of non-existence into us, it may be the element that infects all of the phenomenal world with the scourge of death and annihilation. After all, many who grow old welcome death, like Benjamin Franklin or Mark Twain,
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| Sir Thomas Brock (1847-1922), Eve, 1900, Tate Britain, London |
We might like to know why we are in this situation. Like the story of Adam and Eve, it would seem to involve some kind of "Fall" from a condition where we would not be threatened by non-existence. Platonism and Buddhism might toy with the idea that such a Fall is a consequence of ignorance, but there is no reasonable explanation why ignorance, or any other defect, would be present in a pre-lapsarian reality.
The Adam and Eve story itself contains the paradox that the couple, innocent of the knowledge of good and evil, could not know that it would be evil to disobey the commands of God -- while this story and that of the Revolt of the Angels seem to contradict the omniscience and benevolence of God, who seems surprised, puzzled, and angry by the events in the Garden, and who should have realized that the misadventures of his creations would visit untold suffering on future generations of entirely innocent beings.
The Platonist or Buddhist is reasonable to confess his ignorance in the face of this tangle, while the theist can only awkwardly claim, like Leibniz, that God has deliberately done it all for the best in terms of a Plan whose details and strategy defeat our understanding. This leaves the theist with no more positive an explanation than the Platonist or Buddhist, but then his faith is repeatedly subjected to the insults and outrages that misfortune and the march of folly and violence constantly visit upon the world.
Also, it is then awkward for the theist to turn around and claim that "intelligent design" explains the world and proves the existence of God, when he is thus otherwise at a loss to explain either the intelligence or the design evident in the moral valence of events. Genuine edification for Platonist, Buddhist, and theist alike, as with the Confucian, is then simply to do good, without expectation of reward or of usefulness in a divine Plan.
First of all, the Perfect Fallacy mentioned above, while perhaps allowing a theistic interpretation for something like the Form of the Good, can also go off in a different direction. What is actual among the Forms is what is possible in the world. The whole Platonic World of
Being thus represents no more than the sum of all possibilities. In Aristotle's metaphysics, the form of things is their actuality, while matter represents the power and the possibility of things changing and becoming different. This produced the paradox, awkward for St. Thomas, that the most actual thing, which would be God, is simultaneously immaterial, which is good, but also without power, which is bad.
Yet Aristotle's theory has remained the only treatment of possibility as such in Western philosophy. Where modern quantum mechanics has tangled with the same issue, it has provided the interesting theory that possibility consists of the "wave function" that is collapsed into discrete actualities by acts of observation. The whole Platonic World of Being is like a vast quantum wave function, summing all the possibilities of reality. But then we get the "many worlds" interpretation, where every possibility corresponds to a different actual universe -- a theory of possibility separately advanced in philosophy by David Lewis (1941-2001). Such theories grotesquely violate Ockham's Razor, without a shadow of motivation or evidence apart from the desire to provide a metaphysic of possibility; and they become reductionistic, replacing possibility with quantified actuality -- using Saul Kripke's "possible worlds" modal logic, which, however, he did not mean to have imply the existence of the possible worlds.
A Kantian metaphysics can only deal with this by way of an Antinomy. The Forms are both absolutely perfect and actual, in their own way, but also perfect possibility, with a power to generate the world, not as a matter of divine deliberation, choice, and creation, but more like the 
Neoplatonic Sun that spontaneously and intrinsically radiates Being out into the darkness of Not-Being, which consequently is intrinsically imperfect thereby, with an ontological declension producing ever more imperfect hypostases.
We should expect such an Antinomy among things-in-themselves, and it is thus not surprising that we have difficulty understanding how the Forms could be, in Aristotle's terms, both God (in Aristotle's impersonal mode) and Prime Matter simultaneously. Yet that is what the case requires, even as I think it is possible to propose a Kantian interpretation of quantum mechanics.
With a momentary detour back to epistemology, we can use the Neoplatonic radiation of Being for an analogy with non-intuitive immediate knowledge. In those terms, visible light would be the factual side of phenomenal reality. It is what we immediately see. The value side of phenomena would be like infrared or ultraviolet light. These are electromagnetic wavelengths that are invisible to the naked eye. This is like the non-intuitive character of the right, the good, the beautiful, and the holy. These are prima facie invisible; but they have an effect. Ultraviolet light produces sunburns, and infrared produces heat. Value, of course, has different kinds of effects, just as grammatical rules are insensibly applied in the use of natural languages. The effects of value, insensibly applied, emerge on reflection, just as was solicited by Socrates.
This brings us to a further Antinomy. Beginning with Parmenides, most Greek philosophers argued that Nothing or Not-Being was not a legitimate concept and not a real ontological possibility. Thus, if our "Fall" is into a realm tainted by Not-Being and subject to evil because of it, the whole business is actually an illusion. This is no less than what Buddhism or Advaita Vedanta would say; but then, once again, we are without an explanation why such an illusion should have occurred or why, for the time being, the unpleasant nature of our existence is an all-too-real reality. Shãnkara's Advaita Vedanta applies the Four-Fold Negation to the phenomenal world (Māyā), as Buddhist metaphysics applies it to all conceivable reality, that it neither exists, nor doesn't exist, nor both, nor neither. This is an Antinomy with a vengeance; but if it applies even to the very concepts of Being and Not-Being, the metaphysical sweep may be more devastating than Shãnkara or Nagārjuna imagined.
A fundamental principle of Kantian philosophy is that the attempt to formulate a systematic theory of transcendent objects generates contradictions -- Antinomies. Kant thought that some of the Antinomies could be resolved by moral considerations, producing the "Ideas" of God, Freedom, and Immortality as "Postulates of Practical Reason." While I do think that moral considerations require free will, I am also of the opinion that the principle of Ontological Undecidability and the metaphysics of negative transcendence are sufficient to produce independently a sound theory of free will. A theory of God, however, or immortality, inevitably involves conceptions well beyond the limits of the phenomenal world. Each of them becomes hopelessly tangled in the paradoxes of God's goodness and his freedom, of possibility and actuality, and of Being and Not-Being. If these simply cannot be resolved, where does that leave us?
In the first place, morally it leaves us at no disadvantage whatsoever. If an ontological dualism equivalent to Plato's provides separate foundations, metaphysically and epistemologically, for fact and value, then no nihilist, antinomian, or libertine can take any solace from the theory. We are precisely in the situation of Confucian ethics, which offers neither threats nor promises for the fulfillment of duty. Therefore, the task at hand is evident; and we may consider that in terms of "need to know," that we know as much as it necessary -- as long as we can resolve disputes about the actual content of moral obligation.
This leaves us with Kant's question, What can we hope? In his treatment, immortality was the reward of virtue. This is not the right idea, for it violates something that should be basic to Kantian ethics, the difference between categorical and hypothetical imperatives, i.e. the difference between morality and prudence. If I am good simply to earn immortality, then I would be a fool to act otherwise and my determination is a matter of prudential self-interest. This is a reasonable but not a morally laudable motive.
Promises of heaven and threats of hell mean that we act out of personal calculation and/or coercion. Thus, truly moral reasoning, which does what is good for its own sake, does not take into consideration reward or punishment. In the absence of knowledge of the hereafter, and whether divine punishment or reward even exist, one must face the chance that moral behavior will actually earn no recompense. We must be content with the satisfaction of having done the right thing, as in the following statement of karmayoga from the Bhagavad Gita, which expresses the same kind of deontological principle as in Kant or Confucius.
Karmaṇy evādhikāras te mā phaleṣu kadācana /
mā karmaphalahetur bhūr mā te sango 'stv akarmaṇi,Set thy heart upon thy work, but never on its reward.
Work not for a reward; but never cease to do thy work.The Bhagavad Gita, 2:47, Juan Mascaró translation [Penguin Books, 1962, p.52]
Therefore, substantive or verifiable knowledge of the hereafter or of divine providence cannot exist if there is to be a difference between morality and prudence. The centuries of preachers and theologians who posit divine reward and punishment as necessary for morality have actually got it all backwards. They appeal to the wrong motives and, given the uncertainties of the metaphysics, they create a moral hazard that loss of confidence or belief in the transcendent entities, God and the soul, will apparently remove the grounds for moral conscience. A crisis of faith can produce a Nietzsche rather than a Confucius.
If we know what we are to do but do not know why, I have elsewhere considered that this is a situation of "need to know," such as we find in military and intelligence operations. Characteristically, we do not even know why we don't need to know what we don't. Indeed, if the difference between morality and prudence depends on our ignorance, then we do have a reason -- although we would then want to ask why there needs to be a difference between morality and prudence, between principle and calculation. That there is a difference, with principle more admirable, is widely appreciated, despite the confusion of the two in much traditional morality and religion; but there is an irreducible mystery to it.
At the same time, a world of nothing but Kantian moral duty is a bleak place. We can certainly hope for more, and not just in the quantitative terms that Kant offers. A world of beauty is described by Plato, Fries, and Schopenhauer. This is something more conducive to positive joy and engagement, and each of them means more by it than the mere beauty of appearances. Plato thinks that beauty carries us up to the Forms; Fries thinks that religion has a positive content of beauty that relates to things-in-themselves; and Schopenhauer sees beauty as quieting the Will in a way analogous or suggestive of the quieting that produces holiness, renunciation, and salvation.
In that, Schopenhauer is the only one with a hint of something like salvation or of something like the practices of actual religions, where he admired Christianity, Hinduism, and Buddhism precisely for their ascetic bent. With Plato, Fries, or even Kant, not to mention Nelson, it is a little hard to see what salvation would be, if it is not just moral redemption, something that doesn't really compute in terms of beauty, if that is to be our extra-moral content of religion. No, there is an extra dimension to it all in Schopenhauer, and we see it in what is famously represented as his "pessimism." The "pessimism," however, is no more than a recognition, as in Buddhism, of the inescapable suffering and tragedy of life. We laugh for some days in the sunlight, but in time we are surely overtaken by the night. And we wonder, What was that all for?
ὁ ἄναξ οὗ τὸ μαντεῖόν ἐστι τὸ ἐν Δελφοῖς
οὔτε λέγει οὔτε κρύπτει ἀλλὰ σημαίνει.The lord whose oracle is in Delphi
neither speaks nor conceals but gives a sign.Heraclitus of Ephesus, quoted by Plutarch, De Pythiae oraculis 21, 404 E, The Presocratic Philosophers, G.S. Kirk & J.E. Raven, Cambridge, 1964, p.211
  Scio enim quod redemptor meus vivat, et in novissimo de terra surrecturus sim, et rursum circumdabor pelle mea et in carne mea videbo Deum.
I know that my Redeemer liveth, and that He shall stand at the latter day upon the earth. And though worms destroy this body, yet in my flesh shall I see God.
οἶδα γὰρ ὅτι ἀένναός ἐστιν ὁ ἐκλύειν με μέλλων ἐπὶ γῆς,
ἀναστήσει δὲ τὸ δέρμα μου τὸ ἀναντλοῦν ταῦτα· παρὰ γὰρ Κυρίου ταῦτά μοι συνετελέσθη.
For I know that he is eternal who is about to deliver me, to raise up upon the earth my skin that endures these: for these things have been accomplished to me of the Lord.
Job 19:25-26 |
there is something at once uncanny, terrifying, and overwhelming: the Holy.
We may get the feeling that the beautiful and the sublime may fill the world, but the holy cannot be entirely accommodated by nature. Even the Greek gods, who in general seem rather a part of Nature, sunk in it and expressing it, nevertheless are able to act in very unnatural ways. Their powers, however those often seem circumscribed and particularly focused, are well beyond what mortals can accomplish. They are supernatural. How much more the God who breaks out of Nature entirely, whose every power is supernatural and unlimited. This will be an awesome Deity indeed.
It is thus not the Entity, the object that is the metaphysical construction of an unconditioned reality, that is the clue to extraordinary meaning. It is the Quality (Qualitas, Quale, pl. Qualia), what Otto called the "holy," the "sacred," or the "numinous." The Quality, in Friesian terms (Ahndung), carries us past what reason and theory can formulate. Fries simply erred in stopping at aesthetic value, which is insufficient to the phenomenon.
Is the focus of this Quality the God of Abraham and Isaac? The Dharma Body of the Buddha? The Form of the Good? The Tao? 
Brahman and Ātman? These are all what we might say Kant's Dialectic of Reason can throw up out of the feverish churning of the unconscious -- if, indeed, Kant's imagination and perspective had not been bound by his time and place. Then, if we are, as it happens, unable to resolve such disparate possibilities, we are left with the Quality alone, like a faint music in the background of phenomena, or, indeed, sometimes something rather close and terrifying.
In Pure Land Buddhism, the Buddha Amitābha and his Court may come for those who by their devotion have gained rebirth in his Western Paradise. While only the dying person may see the full splendor of the Buddha's descent, others in attendance, or even strangers and passersby, may hear unearthly music and see strange lights or purple clouds or fogs in the area.
This is rather like the situation I see us in. Religions offer a system of reality, which may be incoherent to merely unprovable, and a system of ethics and values, which may be edifying to horrifying, but the heart of any religion is found more in mystery and ritual, often involving things like pollution, characteristics that the enlightened dismiss as nonsense.
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| "L'Estasi di Santa Teresa," Gian Lorenzo Bernini (1598-1680), 1647-1652, Cornaro Chapel, Church of Santa Maria della Vittoria, Rome, 2019 |
Schopenhauer says, in great art we can see the bliss, the knowledge, and the triumph of the blessed -- as in Herman Hesse's Siddhartha, Ānanda, meeting Shākyamuni after his Enlightenment, sees the mysterious smile that we still can see on all his statues.
With St. Teresa of Ávila, in Bernini's unearthly L'Estasi di Santa Teresa, we see rather more than a smile. We see the ecstasy and rapture indeed, what St. Teresa herself called a swooning bliss, which many compare to an orgasm, even as the arrow held by the angel may be phallic. But who is to say that a sexual orgasm is not itself an imitation of something else, something more profound indeed?
As Blair Brown imagines in Altered States, what if her husband (William Hurt) really did "get it on with God"? Once the crude jokes are set aside, might not the union of mates in sex be just a pale shadow of union with the Transcendent, whatever its character? Bernini's Santa Teresa looks in itself to be, as Schopenhauer says of such art, "a complete and certain gospel," ein ganzes und sicheres Evangelium. Based on St. Teresa's own articulate descriptions, it is not hard to see what her religion meant to her, and the rewards she drew from it. Centuries of philosophers, often trapped in Stoical or Nihilistic solipsism, have little to offer in comparison.
While I withhold commitment to a particular religion, or to God, or even to a non-theistic Buddha, others may find that dissatisfying. After all, St. Theresa was Catholic.
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| Der Wanderer über dem Nebelmeer, "ὁ Τηλεπατητικὸς above the Sea of Fog," 1818, by Caspar David Friedrich (1774–1840), Hamburger Kunsthalle |
At the same time, it is not difficult to criticize most religions for moral failures in their teachings or practice. Or to criticize the contradictions or incoherence in their doctrine. Indeed, Antinomies in the transcedent are inevitable. Some people carefully evaluate religions on their moral teaching or even their aesthetics, looking for rational recommendation or just a good feeling. But this cannot persuade all, or escape refutations.
So, in the end, we need to take Heraclitus seriously. What lies at the boundary between reason and faith is a Sign, σημεῖον, Arabic آيَة, ʾāyah. This can take many forms, from the bullets missing Samuel L. Jackson, to St. Paul being struck blind on the road to Damascus. In each of those, of course, we see something else. Jackson felt "the touch of God," and Paul actually heard the voice of Christ. That is always something very personal and very specific. It cannot be experimentally reproduced by science. And it is not going to be the same from one person to another. John Travolta was immune to "the touch of God" -- and paid for it with his life. Even the Sign given by the Oracle at Delphi does not convey the proper meaning in itself. Themistocles, but not King Croesus, registered the correct interpretation of the Oracles they received [note].
Whether or not all religions are equally "good" or "right" in some sense, this may be irrelevant to their ultimate appeal. A powerful religion, I would say, is one that embodies a particular intuition into the transcendent, and onto one or more of its logically isolated features. Built around that may be more or less coherent elaborations and doctrinal, moral, or ritual additions. This may be attractive to different persons in different ways, and may even be misused for immoral purposes by some of them; but the core of durable religions is likely to be something genuine, something that defeats reason and yet holds the promise of ultimate meaning and purpose in life, and for the whole world. Otherwise, like Nietzsche, we stare into the Void, and its Madness [note].
Allāhu nūru s-samāwāti wa-l-ʾarḍi maθalu nūrihi ka-miškātin God is the light of the heavens and the earth. The parable of his light is a niche wherein is a lamp, the lamp is in a glass, the glass as it were a glittering star, lit from a blessed olive tree, neither eastern nor western, whose oil would almost glow even if untouched by fire, light upon light. God guides to his light whom he wishes.
ʾal-Qurʾān, Sūrah 24, Verse 35; the "Verse of Light"
A Deuteronomy of Kant-Friesian Metaphysics
Allan Sandage (1926-2010) was one of the more important observational astronomers of the 20th century. Following (often literally) in the footsteps of Edwin Hubble (1889-1953), he not only continued work on the cosmological problems initiated and explored by Hubble, but he completed work on The Hubble Atlas of Galaxies [1961] -- which illustrates the variety and types of galaxies observed by Hubble, and which Hubble had left unfinished at his death.
Although his confession of Platonism above does not necessarily sound like orthodox Christianity, Sandage announced that he had become a Christian in 1983. And while this meant he became concerned about the relation of religion to science, the quotes I use here do not specifically address any religious issues. Indeed, distinguishing the approach of Kierkegaard from the science addressed by Sandage would seem to preclude a path towards religion. But this may mean that Sandage did not actually reject the "subjectivity of life," only that he wanted to distinguish it from physics and science, as in the title of his essay, "...separate closets in the same house." Maintaining a sharp distinction can preclude the follies of the Creationists.
Galileo Galilei (1564-1642), "Discourse on Floating Bodies," 1612
The problem with this Platonist view of mathematics -- one that [mathematician Edward] Frenkel, going on in a misterioso vein, never quite recognizes as such -- is that it makes mathematical knowledge a miracle. If the objects of mathematics exist apart from us, living in a Platonic heaven that transcends the physical world of space and time, then how does the human mind "get in touch" with them and learn about their properties and relations? Do mathematicians have ESP? The trouble with Platonism, as the philosopher Hilary Putnam has observed, "is that it seems flatly incompatible with the simple fact that we think with our brains, and not with immaterial souls."
Perhaps Frenkel should be allowed his Platonic fantasy. Jim Holt, "A Mathematical Romance," The New York Review of Books, December 5, 2013, review of Love and Math: The Heart of Hidden Reality, by Edward Frenkel, p.29 (Holt "writes about science and philosophy," p.3; Putnam clearly relies on the certainty of ontological materialism)
...I will state my own position dogmatically in order to avoid minor misapprehension. I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.
G.H. (Godfrey Harold) Hardy (1877-1947), A Mathematicians's Apology [Cambridge U. Press, 1940, 2019, pp.123-124]
I hold it to be true that pure thought can grasp reality, as the Ancients dreamed.
Albert Einstein, "On the Method of Theoretical Physics," 1933, Graham Farmelo, The Universe Speaks in Numbers, How Modern Math Reveals Nature's Deepest Secrets [Basic Books, 2019, p.1] The shift from a Platonic to a Kantian construction of ontological transcendence also makes a difference for philosophy of mathematics. The consequences of such a move are something that I examine in relation to the question of Imaginary Numbers. While for matters of value, Kantian metaphysics increases the reality of value in appearances, in mathematics what is significant is that it decreases the ultimate reality of mathematical abstractions. This is because, as also in the case of a Kantian theory of universals, and in a Kantian Quantum Mechanics, we postulate an intermediate state between concrete phenomena and things-in-themselves.
The precedent for this is at the cognitive level of what Kant called "pure intuition," and it is where we examine in our imaginations the empty space of pure geometry, the logical relationships of concepts (including numbers), or the quantum wave function. This difference mitigates the paradoxes of Plato's metaphysics that arise in relation to my discussion of Questions about Socrates; and it could even provide an ontological space, if need be, for the "non-existent objects" posited by Alexius Meinong (1853-1920).
At the same time, we could get by, perhaps, without Kant's own theory of mathematics. That is because arithmetic begins with number, and number is an artifact of universals. Unless there is more than one tree, trees do not need to be counted. And when trees need to be counted, you need a system of numbers to count them. Eventually, you need to consider what number you use when the question is "How many trees?" and the answer is "None." Every language can answer that question, but only in India and with the Maya did "none" become a number, i.e. "zero," from Sanskrit A Kantian modification of Platonism exposes some of the problems with an epigraph on this page, namely the statement of Jim Holt in The New York Review of Books that Platonism would make "mathematical knowledge a miracle." He wonders "how does the human mind 'get in touch' with" mathematical objects if they are in the transcendent Platonic world. Perhaps he knows, but does not deign to mention, Plato's own view that we are carried to a vision of that world between lives, before being reincarnated, as set out in the Meno. Or perhaps he does not know, or regards the theory as too absurd to consider. I suspect, however, that he is entirely ignorant of the Neoplatonic modification of this, in which the intuition of mathematical objects is available to us right now, because our reality radiates from the transcendent and is not separated from it, as Plato would have thought. Instead, Holt chooses to ridicule the subject of his review, the mathematician Edward Frenkel, with references to "ESP" and "his Platonic fantasy." Come to think of it, the Neoplatonic theory could indeed be presented as Extra-Sensory Perception.
At the same time, Holt obviously overlooks other philosophical theories of mathematical knowledge, such as that of Kant. No transcendent Platonic world in Kant's theory; but Holt acknowledges its existence no more than the provision in Plato that answers his question, or the doctrines of Neoplatonism. Although The New York Review says that Holt "writes about science and philosophy," I see no evidence from his article that he has much interest, or perhaps even awareness, in how mathematical knowledge is treated in the history of philosophy.
fīhā miṣbāḥun-i l-miṣbāḥu fī zujājatin-i z-zujājatu ka-ʾannahā kawkabun
durriyyun yūqadu min šajaratin mubārakatin zaytūnatin
lā šarqiyyatin wa-lā γarbiyyatin,
yakādu zaytu-hā yuḍīʾu wa-law lam tamsas-hu nārun,
nūrun ʿalā nūrin, yahdī llāhu li-nūri-hi man yashāʾu.
Copyright (c) 2012, 2013, 2014, 2015, 2016, 2018, 2019, 2020, 2021, 2023, 2024 Kelley L. Ross, Ph.D. All Rights Reserved
Why I am a Platonist, Note 1
Why I am a Platonist, Note 2;
MathematicsI expect a terrible rebuke from one of my adversaries, and I can almost hear him shouting in my ears that it is one thing to deal with matters physically and quite another to do so mathematically, and that geometers should stick to their fantasies, and not get involved in philosophical matters where the conclusions are different from those in mathematics. As if truth could ever be more than one.
, śūnya, by way of 
, ṣifr, in Arabic. But if number is an artifact of universals, then all we need for a philosophy of mathematics is a theory of universals. I have already linked to the Kantian theory there.
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| Kurt Gödel and Albert Einstein walking to work at the Institute for Advanced Study |
But the "problem," not with a "Platonist view of mathematics," but with its denial, is already evident in a question that Holt quotes from Einstein, "How can it be... that mathematics, being after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?" [ibid.]. Indeed. Not an original mathematician himself (he had to be taught tensor calculus by a friend), Einstein does not take the step that apparently is so obvious to "great" mathematicians (with Plato and Kant) and infer that mathematical structures, however they are accessible to us, must underlie the natural phenomena themselves. Otherwise, it is beyond coincidence that the world just turns out to be amenable to mathematical description. What luck. Of course, this was originally the Pythagorean conviction, long before Plato and his Forms -- and is taken up by physicist Max Tegmark, in the epigraph above -- although Tegmark is a bit off the deep end. Plato would not have said that the world "is mathematics" -- behind it all is the Form of the Good.
The icing on the cake of Holt's dismissal of Platonism, however, is his reference to Hilary Putnam, whose refutation of the doctrine "is that it seems flatly incompatible with the simple fact that we think with our brains, and not with immaterial souls." It isn't just that Putnam will give not a moment's credence to the existence of an immaterial soul, about which many questions will certainly arise, it is his obvious comfort and certainty with the foundational ontological axiom that there exists the substratum of a material brain to our thought. He is a reductionistic materialist and obviously has not experienced a moment's doubt on the issue, despite the collapse of good, solid 19th Century Democritean Atomism into the bizarre real/virtual, smeared and uncertain world of quantum mechanics -- where, to Einstein's horror, Bohr asserted that "nothing exists" until it is observed (by an immaterial soul?). How anyone, let alone a presumably informed philosopher (an Emeritus Professor of Philosophy at Harvard University), can appeal to the Alice in Wonderland state of physics as a "simple fact" validating materialism is remarkable. But this is often what we are faced with in academic philosophy, despite the supposed concern of analytic philosophers to be informed about science, and the responsibility of any philosopher to be aware of the development of metaphysics in the history of philosophy. Putnam does not seem to measure up in either respect, and it is a "simple fact" that his opinion provides too thin a reed to warrant Holt's confidence.
Dr. Tegmark himself gives us what Kant would call a full "transcendental realism" of mathematics, with no more ontological sophistication than could already be associated with the Pythagoreans themselves. One reference to Kant, that "we have no access to what Immanuel Kant called 'das Ding an sich'" [op.cit., p.9], reveals the superficial nature of his knowledge of Kant, who believes that we do have direct access to things-in-themselves through the forms of Reason and, in particular, the Moral Law, which then determines the specific resolution of the Ideas of Reason -- God, Freedom, and Immortality. But it is not unusual in popular treatments of Kant to say that we can know nothing about things-in-themselves. While even Plato places matters of value, i.e. the Form of the Good, beyond the real objects of mathematics, only Kant provides an account for the difference and the distinction, with mathematics not grounded in things-in-themselves at all.
If we ask most people how to form the regular plural in English -- as opposed to irregular plurals, like "child, children" or "woman, women" -- the answer would probably be, "Add an 's'."
Close. One problem is that this "s" is not always pronounced as an "s." It is in "pot, pots," but not in "dog, dogs." There, the "s" comes out as a "z," as though the word were "dogz." So how does that work? Well, the "s" is pronounced as a "z" after vowels (e.g. "pizza, pizzas") and after voiced consonsants, which means sounds that are not vowels but where the vocal cords nevertheless vibrate (e.g. "lid, lids"). For that explanation, of course, you need to know which consonants are voiced and which are not, something that is only likely to come up in Linguistics, or in languages where the sounds are organized in those terms, like Sanskrit.
But there's more. The plural of "church" is not "churchs" but "churches." This "es" is prounounced "ǝz," where the vowel, a "schwa," is "reduced" and indefinite. Why does this happen? Well, "es" is added when the word ends in a consonant that is a sibilant or an affricative. A sibilant is an "s," "z," "sh" ("š"), or the sort of "z" in "azure," which in other languages can be written "zh" or "ž." An affricative, in English at least, will be a sibilant that follows a "t" or "d." Thus, the "ch" in "church" is the sound "tsh" or "tš." We never see it written that way in English, but it may be in other languages. Finally, the English "j" is "dzh" or "dž." We see "garage," "garages," where the second "g" may be "ž" or "dž."
Other languages have other kinds of affricatives. "Ts" can occur as a sound in English ("pots"), but it is not a phoneme, i.e. it does not occur, move, or get written as a unit, as "z" and "c," both pronounced "ts," are in German. We also find "dz" elsewhere as a phoneme. German has a sound written and pronounced "pf," which foreigners find most difficult to say at the beginning of words (e.g. "Pfund," "pound"). A most interesting sound is the "kṣ" that we find in India, as in the word "Kṣatriya," This makes for multilingual fun with the Spanish name "Xavier." This is from a Basque place name, Etxeberria, meaning "new house." "Xavier" can be pronounced like "Javier" in Spanish, but the "x" in Basque is an "sh" in all its occurrences, as it is in Catalan, Galician, and Portuguese -- for the issue of the many languages of Iberia, see here. In English "Xavier" may be pronounced as in French, with a "z" for the "x" -- this is standard -- or English speakers may prefix an "e" to make the "ks" pronounceable in English, saying "Exavier." Since there is no "e" there, this is a little awkward, although prefixing a vowel to make consonant clusters pronounceable is something we see frequently in Latin, Spanish, and Arabic. And "Xerxes" is generally pronounced "Zerxes."
After that excursus: So to understand how the regular plural is formed in English, one must know the difference between vowels and consonants, between voiced and unvoiced consonants, what sibilants and affricatives are, and, at least, what a vowel like a "schwa" is. Yet people can, and always have, spoken fluent English, and produced correct pluralizations, without knowing any of these things. The rules here may be part of no conscious knowledge, and initially bewildering when explained.
As it happens, the language most closely related to English is Frisian, which is spoken in the north of the Netherlands, i.e. in Friesland. It is, of course, not closely related to Modern English, but to Old English, which makes it seems very different from English now. Plurals in Frisian can involve adding an "s" or adding an "en" or "n." The rule is simple. "En/n" goes after stressed syllables, "s" after unstressed. The "en" plural ending we still see in at least one place in English, in "children" (German Kinder). There it does come after a stressed syllable -- the only syllable in the word "child," actually.
In one of the occasionally meaningful statements about language that Wittgenstein makes, there is a difference between knowing a rule, self-consciously, and following a rule, perhaps without any awareness of it as such. In languages, people often think they know a rule, like to avoid "splitting" infinitives, but then systematicly ignore that in speaking. They may be surprised, or even indigant, if their contradictory behavior is pointed out to them. Of course, Socrates got killed for doing things like that.
Samuel L. Jackson, as Jules Winnfield, and John Travolta, as Vincent Vega, have very different fates in the movie, as a consequence of their lives being spared when bullets, fired at them from point blank range, all fail to hit them. To Jackson, this was "divine intervention," while to Travolta, it was a "freak occurrence."
As a result, Jackson abandons his life of crime, while Travolta, now alone, ends up being killed by a character played by Bruce Willis. Indeed, Travolta's fate is associated with pollution. Three times in the movie, Travolta is lingering in a lavatory while something bad is happening outside. Chronologically, the first is when a robbery begins in the coffee shop where he and Jackson are having breakfast, and Jackson is in a standoff with the robbers. Second is while Uma Thurmond's character is overdosing on Travolta's own heroin, thinking it is cocaine. And third is when Travolta emerges from the bathroom to find Willis pointing his own gun at him. In the movie, the chronology is shuffled, so that we already know Travolta will be killed even as we are seeing the robbery in the coffee shop.
There is nothing so profound in any other of Quentin Tarantino's movies, where the issue of salvation does not figure. Jackson is saved by God. Or, at least he would think so. At the same time, John Travolta's character does not see the Sign, or the "touch of God," and falls victim to the "wages of sin": Stipendia enim peccati, mors [Romans 6:23].
It is irrelevant that this is merely a Hollywood movie. It is significant that it is not intentionally made to be either religious or non-religious. The theme of salvation can easily be overlooked by any viewer; and it is not often that reviewers notice it.
Even as art may be inspired, as even perhaps good art must be inspired, the art itself can then pass on its inspiration. Movies have inspired many people to do many things, good and bad. Propaganda movies, like those of Leni Riefenstahl, rely on such powers, but then overlap with the devices of sophistry, with goals of deception. The overt goals of Pulp Fiction are trivial enough, but you never know how unintentional messages, even noble ones, may intrude. The "touch of God" might actually have been on Tarantino, without him realizing it.
"Whom the gods would destroy, they first make mad," seems to appear no earlier than 1854, repeated by Henry Wadsworth Longfellow in "The Masque of Pandora" [1875].
However, Sophocles already said:
Evil appears as good in the minds of those whom a god leads to destruction. [Antigone, 620–623] There is no doubt that Nietzsche became mad. Its etiology is unknown. Scholarly opinion likes the idea that it was from tertiary syphilis. However, Nietzsche is not known to have had any sexual contacts. The speculation, therefore -- and it is only that -- is that the disease was from a prostitute Nietzsche visited in his youth.
C.J. Jung, on the other hand, thought that Nietzsche's madness was the effect of his own thought. That would fit the principle in Sophocles. Nietzsche's philosophy certainly was something where "evil appears as good." Nietzsche's claim that Judaism "inverted" noble virtues into "evil," the "slave revolt in morals," therefore reversing aristocratic values, was what Nietzsche did himself, reversing the kindness and compassion of consideration for the weak and oppressed into what he thought the noble would consider mean and "bad." However, "nobility," in common parlance, is to protect, not take advantage of, the weak or helpless. There is nothing "noble" about a predator, unless it is no more than an animal -- and we do see Nietzsche rather admiring the "beast."
Whether to madness or to "destruction," ἄτη ("bewilderment; bane, ruin"), that was certainly Nietzsche's fate. If effected by the gods, it was poor recompense -- or justice -- for Nietzsche's devotion to them. They may have a better sense of nobility than he thinks, and we see the judgment they visited, not unlike Ezekiel 25:17.
Why I am a Platonist, Note 3;
Plurals in English
, the name of a member of the warrior caste. The "ṣ" in this case is a "retroflex" sound described on the page about Sanskrit linked above. The word "Kṣatriya" then corresponds to "xšayaθiya" in Old Persian, which means "king." This turns up in the name of the Persian King Xšayarša, which got rendered into Greek as Xerxes, Ξέρξης. The "x" in the Persian words is like "ch" in German. The affricative Greek "ksi" (ξ) can occur as a phoneme in English, "x" (e.g. "box, boxes") but not at the beginning of word.
Why I am a Platonist, Note 4
Using Pulp Fiction [1994] for an example of a divine Sign or a conversion experience may seem to trivialize such things. However, art, of whatever sort, is itself largely a matter of inspiration, and its meaning can be as hidden as that of any divine Sign.
Why I am a Platonist, Note 5;
Nietzsche's Madness
Nietzsche, who so despised Judaism and Christianity, and whose ideal was the pagan Greek gods, nevertheless ends up rather hoist on his own petard.
τὸ κακὸν δοκεῖν ποτ᾽ ἐσθλὸν
τῷδ᾽ ἔμμεν ὅτῳ φρένας
θεὸς ἄγει πρὸς ἄταν·