Loeb Theorem Cartoon

I can’t say I could describe this theorem after reading this highly entertaining cartoon, but it certainly offers a great deal of food for thought.


Singularities and Lunatic Fringes

I copied the following passage from Imre Lakatos’ unfinished dialog, “Proofs and Refutations,” which recasts the debate over Euler’s conjecture that: for all regular polyhedra, V – E + F = 2, where V stands for the number of vertices, E the number of edges, and F the number of faces on any polyhedra.

I think that if we want to learn about anything really deep, we have to study it not in its “normal,” regular, usual form, but in its critical state, in fever, in passion. If you want to know the normal healthy body, study it when it’s abnormal, when it’s ill. If you want to know functions, study singularities. If you want to know ordinary polyhedra, study their lunatic fringe. This is how one can carry mathematical analysis into the very heart of the subject … If you want to draw a borderline between counterexamples and monsters, you cannot do it in fits and starts.

Lakatos’ full dialog richly explores the central problem of mathematical (which is to say “logical”) proof /and/ does so by considering a particular geometric “monster” brought to life by Euler. There are so many wonderful ideas delineated by this imaginary dialog that it proves the wealth mathematics has to offer us all.

Mathematical Induction

I’ve been reading the lucid textbook Introduction to Discrete Mathematics by Matousek and Nesetril. From the point of view of an English teacher, it really is a wonderful read. In the early chapters M&N cover basic set notation, functions, relations, equivalences, and mathematical induction. Now, mathematical induction has always given me a hard time; it’s the concept that basically kept me from pursuing a formal math education beyond calculus. But something about M&N makes it all suddenly clear to me.

Here’s my new understanding of mathematical induction:

A proof by induction consists of five basic parts: (1) hypothesis, (2) an initial test, (3) a lemma, (4) a secondary test discharging the lemma, and (5) an inductive step. Prove: the sum of all natural numbers 1 through n equals n(n + 1)/2 (Matousek and Nesetril, pg 23, exercise 1(a))

(1) 1 + 2 + … + n = n(n + 1)/2    hypothesis

(2) Let n’ = 1. Thus, 1 = 1(1 + 1)/2 = 1 is true.    initial test

(3) Let n = n’ + 1.    lemma

(4) Thus, Σn’ + n(n + 1)/2 = 1 + 1(1 + 1)/2 = 1 + 2 = 3 is true.    secondary test & discharge

(5) Σn = n(n + 1)/2 holds for all integers n, n > 0.    inductive step

Simple, eh? You could calculate the sum of all positive integers 1 through n and match each summation against n(n + 1)/2, but if you did that you’d essentially repeat steps (1) – (4) endlessly … which is why we can safely proceed to (5).

Pretty cool. Now I feel I can return to mathematics in my spare time.

Definition Machine

Procrastinating again, I have been reading “Mind as Software of the Brain” by Ned Block (available here). Here’s a fun passage:

Defining a word is something we can do in our armchair, by consulting our linguistic intuitions about hypothetical cases, or, bypassing this process, by simply stipulating a meaning for a word. Defining (or explicating) the thing is an activity that involves empirical investigation into the nature of something in the world.

One can’t really argue with that. Block uses the example to point out that the Turing test examines intelligence in the first sense (whether an observer would call a machine intelligent) and not in the second sense (whether an investigator would find a machine to be intelligent); and he sets up the rest of his argument nicely with this simple distinction.

I bring up the example – and not the argument – because I want to take it on a tangent. Let’s say we build a machine that actually is intelligent. Would it define an item “j” intuitively or through an investigation?

The definition machine which defines “j” intuitively might match “j” against a list of known items to find the corresponding definition; where there is no match, the machine adds “j” to the list and provides a new definition (perhaps in terms of the other definitions). The definition machine which defines “j” following an investigation of the item may describe its concrete properties, its uses, and its material composition; the machine decides which features rank most prominent and pegs the definition of “j” to those features.

Interestingly, it’s hard to think of these as two separate machines – in part because our own minds utilize both processes – but examples of each exist. A search engine such as Google might represent an intuitive definition machine (Google simply finds things and puts them in a list, ready for near-instant reordering), and the Mars rover might represent an investigative definition machine (the rovers plot their own course across the Martian landscape while analyzing soil, sending results back to Earth). Now, imagine you put Google search algorithms on the Mars rovers: you might end up with an artificially intelligent definition machine. It won’t be able to do much besides make known certain facts about its environment, but it will do this extremely well.

This is a fun, slightly sci-fi topic that I think I’d like to return to.