JSO sends us to Will You Remember Me? by the Pine Box Boys. The dessicated soul of the secularist is incapable of understanding religion. He thinks he will eradicate it. But religion, like philosophy, always buries its undertakers.
Over the last 24 hours I have been obsessing over Kant's spherical triangles. He claims that they are incongruent counterparts. Now I understand how a hand and its mirror image are incongruent counterparts. (A right hand's mirror image is a left hand.) But it is not clear to me how Kant's spherical triangles are incongruent counterparts. Supplement the above diagram with a second lower triangle that shares its base (an arc of the equator) with that of the upper triangle and whose sides are two arcs whose vertex is the south pole.
David Brightly's comment is the best I received in the earlier thread. (He works in Info Tech and I believe he has an advanced degree in mathematics.) He writes,
Not clear to me either, Bill. Why does Kant resort to spherical triangles? [To show the existence of incongruent counterparts.] Consider first two right triangles in the plane with vertices (0,0), (3,0), (0,4) in triangle A and (0,0), (3,0), (0,-4) in B. In plane geometry A and B are considered congruent, not by translation or rotation in the plane but rotation out of the plane ('flipping') with their shared edge as axis. Now think of these triangles on the sphere with edges of length 3 along the equator and those of length 4 on a meridian. The lower triangle cannot be flipped into congruence with the upper—it curves 'the wrong way'. Congruence on the sphere is more restrictive than congruence in the plane. But they are mirror images of one another in the equatorial plane. Likewise, Kant's isosceles triangles cannot be flipped into registration. Has he just overlooked that they can be slid on the sphere into alignment?
As Brightly quite rightly points out, "The lower triangle cannot be flipped into congruence with the upper — it curves 'the wrong way'." That was clear to me all along. My thought was that if you rotate the lower triangle through 180 degrees so that its southern vertex points north, it would fit right over the upper triangle. I think that is what David means when he writes, "they can be slid on the sphere into alignment."
In other words, the lower triangle needn't be rotated off the surface of the sphere with the axis of rotation being the common base, it suffices to slide the triangles into alignment and thus into congruence along the surface of the sphere.
Therefore: Kant's spherical triangles are not incongruent counterparts or enantiomorphs.
Now David, have I understood you? I am not a mathematician and I might be making a mistake.
Bill Vallicella
11 responses to “Spherical Triangles as Incongruent Counterparts?”
Consensus is no guarantee of truth. If all or most of the experts in some subject area agree that p, it does not follow that p is true. But that is not to say, or imply, that consensus has no bearing on truth. A consensus of unbiased and uncoerced experts in a field is a reliable guide to truth in that field, assuming that the consensus is real and not the fabrication of, say, climate hoaxers.
Kitsch is art's comfort food: familiar, reliable in its satisfactions, readily available, not particularly nourishing, but also not challenging to its consumers, remunerative for its producers.
'Institutional capture' by anti-civilizational wokesters is fait accompli and we are collapsing on all fronts. Imagine being a English professor facing the daunting task, not of Higher Education, but of Higher Remediation.
Buckner demands an argument from incongruent counterparts to the ideality of space. But before we get to that, I am having trouble understanding how the 'spherical triangles' Kant mentions in the Prolegomena to Any Future Metaphysics, sec. 13, are incongruent counterparts. Perhaps my powers of visualization are weak. Maybe someone can help me.
I understand how a hand and its mirror image are incongruent counterparts. If I hold up my right hand before a mirror what I see is a left hand. As Kant says, "I cannot put such a hand as is seen in the glass in the place of its original; for if this is a right hand, that in the glass is a left one . . . ." (p. 13) That is clear to me.
Now visualize a sphere and two non-plane 'spherical triangles' the common base of which is an arc of the sphere's equator. The remaining two sides of the one triangle meet at the north pole; the remaining two sides of the other at the south pole. The two triangles are exact counterparts, equal in all such internal respects as lengths of sides, angles, etc. They are supposed to be incongruent in that "the one cannot be put in place of the other (that is, upon the opposite hemisphere)." (ibid.) That is not clear to me.
Imagine the southern triangle detached from the sphere and rotated through 180 degrees so that the south vertex is pointing north and the base is directly south. Now imagine the southern triangle place on top of the northern triangle. To my geometrical intuition they are congruent!
So, as I see it, hands and gloves are chiral but Kant's spherical triangles are not.
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral.
A chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χείρ (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the Greek ἐναντίος (enantios) 'opposite' + μορφή (morphe) 'form'.
Bill Vallicella
6 responses to “Kant, Spherical Triangles, and Incongruent Counterparts”
In an e-mail, a correspondent poses a problem that I will put in my own way.
BV is alone in a room facing a standard, functioning mirror and he is looking at a man, the man in the mirror. Call that man MM. So in this situation, BV is looking at MM. The question is this. Is BV numerically the same as MM? Or is BV numerically different from MM?
Surely it would be absurd to claim that there are two men in the room, the one facing the mirror and the one in (or behind) the mirror. The sensible thing to say is that MM is a mere image of a man, not a man. And of course it is the image of BV, not of any other man. Accordingly, when BV looks into the mirror, he sees himself via a mirror image. Now most people will stop right here and go on to something else. But philosophers are a strange breed of cat. They sense something below the mundane surface and want to bring it into the light.
Suppose BV points in the direction of the mirror and exclaims, "That's me! Look how beat-to-hell I've become!" But if MM is a mere image of a man, and not a man, then BV is not pointing at himself, the man BV, but at a mere image. This suggests, contrary to the point made in the immediately preceding paragraph, that there is a man in the mirror and that he is identical to BV! In the situation described, we seem to have good reason to affirm both of the following propositions despite their collective inconsistency:
1) BV is pointing at an image, not a man. (Because there is only one man in the room.)
2) BV is pointing at a man. (Because BV is pointing at himself, and BV is a man.)
This has got to be a pseudo-problem, right? Well then, dissolve it!
A Variant Puzzle
Perhaps the following variant of the puzzle is clearer. BV holds up his right hand and looks at it in the mirror. With the index finger of his left hand BV points to the hand in the mirror and says, "That is a beautiful hand!" With that same index finger he then points to the hand he is holding up and says the same thing. Pointing as he is in two different directions, BV is pointing at two different things, each of which is a hand. But then BV has two left hands and one right hand, for a total of three hands — which is absurd. Why two left hands? Because the hand in the mirror is a left hand being the incongruent counterpart of the right hand BV is holding up.
Incongruent counterparts are discussed by Kant in no less than four places, twice in his pre-Critical writings and twice after 1781. More on this later.
Polarization in a physical body has to have a limit lest the polarized body break apart. (Imagine the distance between Earth's North and South poles — 8595.35 miles — increasing indefinitely.) It is no different with the body politic. We will eventually break apart or be broken apart by an external force (how about the ChiComs in cahoots with the Russkis?) if our political polarization continues. United we stand, divided we fall; come on now people, let's get on the ball. We won't of course.
Time was, when I read Mona Charen and George F. Will with quite a bit of approval. But then Trump came along and both lost their minds. Here is Will over at The Washington [Com]Post. Take a gander at the comments to gauge the level of present political polarization.
Dear old Mona's latest outburst anent Trump is such that I cannot bring myself to sully my site by linking to it.
Do you want to hear some sane and characteristically brilliant commentary by a lion of the law? Here is Alan Dershowitz on the Trump indictment. (HT: Vito Caiati)
Elliot C. asked me about tropes. What follows is a re-post from 30 March 2016, slightly emended, which stands up well under current scrutiny. Perhaps Elliot will find the time to tell me whether he finds it clear and convincing and whether it answers his questions.
…………………………..
A reader has been much exercised of late by trope theory and other questions in ontology. He has been sharing his enthusiasm with me. He espies
. . . an apparent antinomy at the heart of trope theory. On the one hand, tropes are logically prior to objects. But on the other hand, objects (or, more precisely, the trope-bundles constituting objects) are logically prior to tropes, because without objects tropes have nowhere to be – without objects (or the trope-bundles constituting objects) tropes cannot be. Moreover, as has I hope been shown, a trope cannot be in (or constitute) any object or trope-bundle other than that in which it already is.
How might a trope theorist plausibly respond to this? Can she? [My use of the feminine third-person singular pronoun does not signal my nonexistent political correctness, but is an anticipatory reference to Anna-Sofia Maurin whom I will discuss below. 'Anna-Sofia'! What a beautiful name, so aptronymic. Nomen est omen.)
What are tropes?
It is a 'Moorean fact,' a pre-analytic datum, that things have properties. This is a pre-philosophical observation. In making it we are not yet doing philosophy. If things have properties, then there are properties. This is a related pre-philosophical observation. We begin to do philosophy when we ask: given that there are properties, what exactly are they? What is their nature? How are we to understand them? This is not the question, what properties are there, but the question, what are properties? The philosophical question, then, is not whether there are properties, nor is it the question what properties there are, but the question what properties are.
On trope theory, properties are assayed not as universals but as particulars: the redness of a tomato is as particular, as unrepeatable, as the tomato. Thus a tomato is red, not in virtue of exemplifying a universal, but by having a redness trope as one of its constituents (on the standard bundle version of trope theory) or by being a substratum in which a redness trope inheres (on a nonstandard theory championed by C. B. Martin which I will not further discuss). A trope is a simple entity in that there is no distinction between it and the property it ‘has.’ 'Has' and cognates are words of ordinary English: they do not commit us to ontological theories of what the having consists in. So don't confuse 'a has F-ness' with 'a instantiates F-ness.' Instantiation is a term of art, a terminus technicus in ontology. Or at least that is what it is in my book. More on instantiation in a moment.
Thus a redness trope is red, but it is not red by instantiating redness, or by having redness as a constituent, but by being (a bit of) redness. So a trope is what it has. It has redness by being identical to (a bit of) redness.
It is therefore inaccurate to speak of tropes as property instances. A trope is not a property instance on one clear understanding of the latter. First-order instantiation is a dyadic asymmetrical relation: if a instantiates F-ness, then it is not the case that F-ness instantiates a. (Higher order instantiation is not asymmetrical but nonsymmetrical. Exercise for the reader: prove it!) Suppose the instantiation relation connects the individual Socrates here below to the universal wisdom in the realm of platonica. Then a further item comes into consideration, namely, the wisdom of Socrates. This is a property instance. It is a particular, an unrepeatable, since it is the wisdom of Socrates and of no one else. This distinguishes it from the universal, wisdom, which is repeated in each wise individual. On the other side, the wisdom of Socrates is distinct from Socrates since there is more to Socrates that his being wise. There is his being snub-nosed, etc. Now why do I maintain that a trope is not a property instance? Two arguments.
Tropes are simple, not complex. (See Maurin, here.) They are not further analyzable. Property instances, however, are complex, not simple. 'The F-ness of a' – 'the wisdom of Socrates,' e.g. — picks out a complex item that is analyzable into F-ness, a, and the referent of 'of.' Therefore, tropes are not property instances.
A second, related, argument. Tropes are in no way proposition-like. Property instances are proposition-like as can be gathered from the phrases we use to refer to them. Ergo, tropes are not property instances.
One can see from this that tropes on standard trope theory, as ably presented by Maurin in her Stanford Encyclopedia of Philosophy entry, are very strange items, so strange indeed that one can wonder whether they are coherently conceivable at all by minds of our discursive constitution. Here is one problem.
How could anything be both predicable and impredicable?
Properties are predicable items. So if tropes are properties, then tropes are predicable items. If the redness of my tomato, call it 'Tom,' is a trope, then this trope is predicable of Tom. Suppose I assertively utter a token of 'Tom is red.' On one way of parsing this we have a subject term 'Tom' and a predicate term '___ is red.' Thus the parsing: Tom/is red. But then the trope would appear to have a proposition-like structure, the structure of what Russell calls a propositional function. Clearly, '___ is red' does not pick out a proposition, but it does pick out something proposition-like and thus something complex. But now we have trouble since tropes are supposed to be simple. Expressed as an aporetic triad or antilogism:
a. Tropes are simple. b. Tropes are predicable. c. Predicable items are complex.
The limbs of the antilogism are each of them rationally supportable, but they cannot all be true. Individually plausible, collectively inconsistent. The conjunction of any two limbs entails the negation of the remaining one. Thus the conjunction of (b) and (c) entails ~(a).
We might try to get around this difficulty by parsing 'Tom is red' differently, as: Tom/is/red. On this scheme, 'Tom' and 'red' are both names. 'Tom' names a concrete particular whereas 'red' names an abstract particular. ('Abstract' is here being used in the classical, not the Quinean, sense.) As Maurin relates, D. C. Williams, who introduced the term 'trope' in its present usage back in the '50s, thinks of the designators of tropes as akin to names and demonstratives, not as definite descriptions. But then it becomes difficult to see how tropes could be predicable entities.
A tomato is not a predicable entity. One cannot predicate a tomato of anything. The same goes for the parts of a tomato; the seeds, e.g., are not predicable of anything. Now if a tomato is a bundle of tropes, then it is a whole of ontological parts, these latter being tropes. If we think of the tomato as a (full-fledged) substance, then the tropes constituting it are "junior substances." (See D. M. Armstrong, 1989, 115) But now the problem is: how can one and the same item — a trope – be both a substance and a property, both an object and a concept (in Fregean jargon), both impredicable and predicable? Expressed as an aporetic dyad or antinomy:
d. Tropes are predicable items. e. Tropes are not predicable items.
Maurin seems to think that the limbs of the dyad can both be true: ". . . tropes are by their nature such that they can be adequately categorized both as a kind of property and as a kind of substance." If the limbs can both be true, then they are not contradictory despite appearances.
How can we defuse the apparent contradiction in the d-e dyad? Consider again Tom and the redness trope R. To say that R is predicable of Tom is to say that Tom is a trope bundle having R as an ontological (proper) part. To say that R is impredicable or a substance is to say that R is capable of independent existence. Recall that Armstrong plausibly defines a substance as anything logically capable of independent existence.
It looks as if we have just rid ourselves of the contradiction. The sense in which tropes are predicable is not the sense in which they are impredicable. They are predicable as constituents of trope bundles; they are impredicable in themselves. Equivalently, tropes are properties when they are compresent with sufficiently many other tropes to form trope bundles (concrete particulars); but they are substances in themselves apart from trope bundles as the 'building blocks' out of which such bundles are (logically or rather ontologically) constructed.
Which came first: the whole or the parts?
But wait! This solution appears to have all the advantages of jumping from the frying pan into the fire. For now we bang up against the above Antinomy, or something like it, to wit:
f. Tropes as substances, as ontological building blocks, are logically prior to concrete particulars. g. Tropes as properties, as predicable items, are not logically prior to concrete particulars.
This looks like an aporia in the strict and narrow sense: an insoluble problem. The limbs cannot both be true. And yet each is an entailment of standard (bundle) trope theory. If tropes are the "alphabet of being" in a phrase from Williams, then they are logically prior to what they spell out. But if tropes are unrepeatable properties, properties as particulars, then a trope cannot exist except as a proper ontological part of a trope bundle, the very one of which it is a part. For if a trope were not tied to the very bundle of which it is a part, it would be a universal, perhaps only an immanent universal, but a universal all the same.
Furthermore, what makes a trope abstract in the classical (as opposed to Quinean) sense of the term is that it is abstracted from a concretum. But then the concretum comes first, ontologically speaking, and (g) is true.
Interim conclusion: Trope theory, pace Anna-Sofia Maurin, is incoherent. But of course we have only scratched the surface.
Pictured below, left-to-right: Anna-Sofia Maurin, your humble correspondent, Arianna Betti, Jan Willem Wieland. Geneva, Switzerland, December 2008. It was a cold night.
Bill Vallicella
9 responses to “Trope Troubles: An Exercise in Aporetics”
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