Still one of my favorite novels of all time, especially because of the rich prose, the subtle account of burgeoning female sexuality, and astute observations of life in middle America.

]]>Weirdly enough, my best friend in college and I both read Narcissus and Goldmund at around the same time, so that was interesting. And, between the two of us, I thought of myself as more of a Narcissus and he a Goldmund. He was a bit more free-spirited than me, and closer to nature. He was also more of a loose cannon. One night, he was drunk and depressed, and tried to hurl himself out of the 2nd story window of our house. I was in the top bunk so I couldn’t do anything. Ironically, the guy who grabbed him and potentially saved his life was a gay guy; my roommate had homophobic tendencies and had just been in the process of screaming slurs at him. The next morning, my roommate and I walked with a girl friend to go get breakfast. We passed a mural painted in the style of the Mexican Day of the Dead, with the text, “It’s not death that’s tragic, but rather, the times we die when we’re alive.”

Anyway, interesting that someone could be Goldmund in one relationship and Narcissus in the other.

]]>So Josh, again, the Inquisitor defends *his own policies* while, yes, tacitly also condoning the tendency of his subjects to be enthralled by them. Famously: the Inquisitor insists that Jesus should *not* have declined to grant the Devil’s three temptations in the Gospels—as in declining He offered in exchange only “some promise of freedom which men in their simplicity and their natural unruliness cannot even understand, which they fear and dreadâ€”for nothing has ever been more insupportable for a man and a human society than freedom.”

The Grand Inquisitor defended the actions of the masses (millions) who chose to be subservient under his tyrannical rule. Meanwhile, Jesus argued on behalf of those rare citizens (ten thousand) that rejected authoritarian rule and chose to be free.

I think the point was that Jesus wanted people to follow him, and to be virtuous, on their on initiative, and not because they were told to.

Paging @benediamond

]]>All games seem chaotic to an untrained observer. But some games are easily solved (checkers, 9×9 go, connect 4), while other games certainly are not solved (chess, 19×19 go, connect6). I might compare a simple “market” to a solved game, and a chaotic market to an unsolved game.

The point of this analogy, then, is to show that your question isn’t as silly as you make it out to be. Many games seem hard to beat, but still, there is in fact a threshold, which is obscure to a game-player, but fairly obvious to a computer scientist.

To your point, again, even if game cannot be solved, there still exist strategies for success, for both a computer scientist and a game-player.

]]>Allais’ paradox addresses the utility of decisions. What in particular is it that we consider to be *utile*?

If it’s helping people, then Allais’ paradox applies well. We have a 1% chance of helping tons of people, vs. a 100% chance of helping a moderate amount of people.

Unfortunately for me, the probability distribution is probably roughly similar for heroism, to Ben’s point. So, it would be hard for me to deflect allegations that I chose medicine for this reason.

On that note, is the distinction between these as clear as we’d like it to be? It feels good to run into a burning building and save people, and to be recognized as a hero afterwards, since this recognition is evidence that we have been altruistic, a recognition that we have likely evolved to pursue.

]]>I think we can remain agnostic regarding Searle’s arguments and questions of intelligence, for now. It still merits asking whether or not we could build a computer that would pass a Turing test. This would certainly be an important achievement, and we could address the Chinese room issue once we come to it.

Now, towards building this computer. You argue that it’s “fairly safe to say that the computers we use do not share the same underlying set of physical properties”. I’d consider this alone to be a big claim, and I’d be curious to hear more about your reasoning. But I think, in retrospect, that this was really the question I was getting at. Do brains and computers use the same tools (perhaps the same hardware, if not software)? If so, great. Ostensibly, building a “brain” presents a practical problem, albeit a huge one, but not a theoretical question. (Aside: for a problem this vast, the distinction between practical and theoretical might become meaningless). If not, though, the interesting question becomes: what *is* the brain made out of? Quantum…stuff? Higher dimensions? You mention that musings like these lack credibility given our limited understanding of the universe. But then what *can* we say about the brain’s makeup? If we’re able to say that it’s made of “more than computers”, we should be able to say what that “more” is.

Again…4 years later, still confused. Appreciate your input though.

]]>Dostoevsky’s above quote continues: “I acknowledge humbly that I have no faculty for settling such questions, I have a Euclidean earthly mind, and how could I solve problems that are not of this world? And I advise you never to think about it either, my dear Alyosha, especially about God, whether He exists or not. All such questions are utterly inappropriate for a mind created with an idea of only three dimensions.” The apparent conflation persists in modern Dostoevsky scholarship, for instance in (even the title of) Liza Knapp’s otherwise highly compelling The Fourth Dimension of the Non-Euclidean Mind; Time in Brothers Karamazov or Why Ivan Karamazov’s Devil Does not Carry a Watch. Knapp writes: “Ivan was unable to accept the harmony of God’s universe because he was unable to understand the mystery of time – time being the ‘fourth dimension’ from which his three-dimensional, Euclidean mind barred him.”

In short, there is nothing non-Euclidean per se about higher-dimensional space, and neither need non-Euclidean geometries be higher-dimensional.

The appropriate modern framework within which to explore these ideas is perhaps Riemannian geometry, a field of math which studies Riemannian manifolds, curved and higher-dimensional generalizations of the standard three-dimensional Euclidean space we know. (I’ll speak about projective space later.) Einstein’s general relativity — in which gravity is nothing but the curvature of four-dimensional spacetime — is perhaps Riemannian geometry’s greatest modern vessel.

Curvature and dimensionality endure a double dissociation in Riemannian geometry, as they naturally should. We may have arbitrary-dimensional spaces which are perfectly flat (in which two parallel lines never meet). We can also have two-dimensional spaces which are curved (and in which they do).

The dissociation between these two notions is tempered and complicated by a few auxiliary facts. Curvature *does* involve extra dimensions, in the naive sense that a curved manifold of *k* dimensions can only be placed into a flat space if that flat space has has more than *k* dimensions (this is easy to visualize when *k* = 2). On the other hand, the higher dimension, say *n*, of this “ambient” space has nothing to do with the “intrinsic” dimension of the manifold, and is less important mathematically. (A smooth curved surface in 3-dimensional space is 2-dimensional, not 3-dimensional.) Thus this higher ambient dimension is not as relevant mathematically.

Projective space (one sort of space for which it could be said that two parallel lines meet, and perhaps that to which Dostoevsky referred) presents another sense in which these notions become complicated. I have avoided discussing projective space because it belongs not to the field of Riemannian geometry, but rather to that of algebraic geometry. In particular, projective space supports no notion of curvature; two parallel lines meet here instead because of the intrinsic construction of the space.

While projective spaces of any dimension — even 1, 2, and 3 — exist, projective spaces too require extra dimensions in a sense relevant more psychologically than mathematically. Indeed, projective space (typically) operates over the complex numbers, which require twice as many dimensions to depict as do the reals. Furthermore, the construction of projective space again requires an extra dimension, which too becomes irrelevant mathematically. In short: projective space is a strange world, and even if its mathematical dimension is small, its “human” dimension is always larger.

It remains to decide how best to interpret Ivan. The most mathematically consistent thing for him to say — let’s imagine Ivan as a modern mathematician — would perhaps have been that this world is a four-dimensional manifold (which we can’t visualize) with nonzero curvature (which is even harder to visualize), in which who’s to say what will come of any two parallel lines. Furthemore, time and space form indistinguishable continuum and gravity is the curvature of this continuum.

All of this goes to say, perhaps, that I am splitting hairs here — and that Dostoevsky (and his modern scholars) could have expressed their thoughts just as well by replacing references to the “Euclidean mind” and to “higher dimensions” by a generic appeal to the mathematical physics we simply cannot understand. I might as well, further still, perform this substitution myself and accept the point as given. This I do. In any case, it’s interesting to flesh out the matter mathematically.

“All such questions are utterly inappropriate for” my mind as well. And even if I came to understand them, there is always plenty more of the incomprehensible in math and physics to be had.

See the post above?

]]>This flimsy prospect is easily put to rest. It is enough to find an algebraic cycle and a map for which the image of in is nonzero. (This would imply that is not torsion.) To this effect, we introduce the map induced by pairing any class with the square of the hyperplane class and then taking the resulting 0-cycle’s degree.

The image of under this map is none other than the degree of , which is of course nonzero.

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