amimetobios

Infinity 24: Last class: review and final explication of Zeno

Last class: a review of the irrationality of the square root of 2; and of Cantor's diagonalization proof.  How the difference between infinity by addition and by division corresponds to the difference between the infinity of natural numbers (which are all under the rubric of infinity by addition) and the reals between 0 and 1, all of which may be ranged under the rubric of infinity by division.  What makes one set "larger" than another.  The idea of a list as involving the concept of "next."  The non-denumerability of the reals means that the concept of next doesn't apply to them.  Zeno's paradoxes rely on the idea of the next: the next point on space, the next time slice.  The larger order of infinity that comprises the reals means that Achilles passes the tortoise at a point between two rationals, which are the only points Zeno considered, in considering the next rational point the tortoise gets to, while Achilles is still at a previous rational point.  A similar intuition applies to trying to come up with a commensurate scale for measuring hypotenuse and leg of an isosceles right triangle.  Whatever units you divide one line segment up into, there'll never be a point which is the exact passing point, so to speak, as you go from fewer of those units than you need to measure the other line segment to more of those units than you need that you'll have exactly the right number -- no point, you could say, where one line segment is passing another at a rationally measurable distance from its other endpoint.  And so farewell to this class. (I'm actually not sure why there are 24 and not 25 classes.  I may have miscounted somewhere)

Diagon Alley

Some discussion of the nature of proof; listing rationals between 0 and 1; function vs. algorithm; question whether any list of irrationals is possible; Cantor's diagonalization proof that it isn't; discussion about 1-many correspondence between rationals and reals; approach to the idea that the power set of an infinite set is a higher order of infinity because you could do the diagonalization proof on binary expansions between 0 and 1, leading to the construction 2^n numbers not in the original set.  I am interested in what computer scientists make of the discussion we (Kenneth Foner and I in particular) had (and which I am not pretty but not fully confident about) concerning the difference between a function that picks out all primes (which would allow you to use the Sieve of Eratosthenes efficiently, in, um polynomial time [right?], and which we can't [right?]) and an algorithm which ultimately has to do it through a somewhat stream-lined brute force procedure.

Infinity 22: Newcomb's problem; Shelley

Paper assignment,* which requires a lengthy exposition of the set-up for Newcomb's problem; segue via Descartes and an exposition of the difference between romanticism and Cartesian skepticism, with Kant as a pivot, to Shelley's  Mont Blanc.  A word more about that difference: Descartes was trying to prove that it wasn't all in the mind; the Romantics were trying to prove that it more ore less was.  But they are similar (via Kant) in believing that the external world was empirical trash, that this gave them access to, or at least desire for, a supersensible externality: magnitude itself (say) and not the pseudo-magnitude of the empirical world.  End of Shelley's Mont Blanc. *Here is the paper assignment as posted to the class site:

If you weren't in class yesterday, you'll probably want to listen to the podcast, where we discussed the second paper topic at some length.

This is the short version:

An extremely acute reader of human character gives you a box whose contents are either $1,000,000 or nothing. She also offers you $10,000, which you are free to take or leave.  If she thinks you'll take the $10,000, she won't have put anything in the box she's given you; when you open it it will be empty.  If she thinks you won't take it, but will be satisfied with the mystery-box, she'll have put $1,000,000 in it, which you will find when you open it.  But you can't open the box until you either take or reject the $10,000.

She's done this hundreds of thousands of times before, and has never been wrong in her predictions as to what people would do - take the $10,000 (everyone who did got nothing in the mystery box), or leave it (everyone who did got $1,000,000 in the mystery box).  She can't see the future, though, and she has no magical powers to decide what will be in the box after you make your choice.  She's put something in the box, or hasn't, depending only on her ability to dope out your character or personality, to figure out what you will do in the situation in question.  What will you do and why?

Make your answer vivid; make the argument one about what you would do and why, not necessarily what you think a perfectly rational agent would do.  Write it, if you like, as a short story, or in whatever way you can make your own thinking most compelling, most about how you would think this out if it were really happening.  (After all, she's predicted what you would do when it really happens.)

You can, and should, think about using any of the ideas we've covered this semester: I could see a way in which practically any of them could be relevant.

Don't do any outside reading on this problem. Don't talk to your friends about it, don't look it up on Wikipedia (as some of you did for the Pascal paper).  Think this through in your own way.

Infinity 21: Klee, Kant, Shelley

Paul Klee on space and time -- a line taken out for a walk.  Back to Kant and a tedious brief exposition of the third critique.  The beautiful as the harmonious, showing the mind as tuned or tuned up, with the willing that organizes perception or the structure of appearance coming from elsewhere, viz. the beautiful object.  (This is what Kant calls the reflective rather than regulative use of reason, and is the reason the analytic of the beautiful in the  Third Critique is so important: it shows the harmony of the mind apart from its own willfulness.)  The mathematical sublime as the will engaged under the rubric of the understanding: apprehension and its relation to comprehension.  The dynamic sublime as the will engaged under the rubric of the will. Illustration via Shelley's Mont Blanc, which we begin reading.