A little rant about AC...

So. Before we get started. Math is based on logic. The most important result of this fact is that there must be some axioms-- a starting point for the logical "gears". So, there must be a few things which are accepted as true, without argument, in order to "prove" anything. The really nice thing about this is you can use different axiom sets for different purposes; for example, the Peano Axioms are the axioms for number theory; any set theory capable of producing arithmetic will have axioms that imply the Peano Axioms.

The problem with this forced reliance on axioms is that mathematical truth is based to some small extent on human intuition. And human intuition of mathematical concepts is notoriously fickle.

One point that is still a hot topic amongst foundational researchers, and amongst those who spend their time discussing the philosophy of math, is called the Axiom of Choice. The "normal" statement (which gives the axiom its name) is a bit technical, but there's a completely equivalent statement: A non-empty Cartesian product of non-empty sets is non-empty. In other words, if we have a bunch of non-empty sets, and we take the set of all tuples (ordered lists) of these sets, we have a non-empty set. An example: X={1,2}, Y={1,3} Z={3}, X×Y×Z={(1,1,3),(1,3,3),(2,1,3),(2,3,3)}. (For the more interested reader, the first statement of AC in the wikipedia article mentions a choice function. Any point in the cartesian product encodes such a choice function. If we have a non-empty cartesian product, we have a choice function.)

The debate is mostly about whether or not this statement is intuitively true-- can we say it can be placed with the "obvious" axioms? It seems to make sense to do so, but it leads to a lot of counterintuitive results. The most famous such result is the Banach-Tarski "paradox", which says it is possible to take apart a sphere to create 2 spheres whose sizes are each equal to that of the first. Let's repeat that: Start with one sphere of a certain volume. Split it into 2 in a very clever way. Now you have two spheres, each with volume equal to the first. The "clever way" of splitting the sphere requires the axiom of choice (in a way I'm not sure I have the background to understand.)

The problem, of course, is that counter-intuitive starts at infinity, not at AC. We can split the set of all even numbers into two copies without choice (pull out 2,6,10,... then divide those by two and add 1, and simply divide the rest by 2), so why is a sphere less intuitive? How is it intuitive that there are as many rationals as there are integers? How is it intuitive that you have 0 probability of selecting an algebraic number from the reals, despite the fact that they are dense?

I don't mind people rejecting choice for certain work: constructive logic is incredibly useful for CS, but it explicitly contradicts choice. What I mind is people bringing up "counter-intuitive" results that are no more counter-intuitive than results that have long been taken for granted, because we're so used to seeing them.

My last sentence reminds me of another problem with the whole discussion: Human intuition is so fickle! The results I mentioned are not considered counter-intuitive to most working mathematicians, because the results are so fundamental. In addition, in the same breath that they say "AC leads to counter-intuitive results", they talk about how certain people haven't built up an intuition for these sorts of foundational results. Perhaps none of us have built up an intuition for certain results?

Copycenter

So, I guess I should point out that unless noted otherwise, or the work is not by me, everything on this blag is licensed under the CC-by license. In other words, do anything you want with it as long as you attribute me (Cory Knapp) in any distribution or modification in a way that does not imply that I endorse your use of the work, without explicit permission to do so, more here:

I'd like to get into a discussion about copyrights here, but I just don't care enough... Let me just say, I prefer copycenter to copyleft, and I prefer copyleft to copyright... I think open source (and the non-software equivalents) is the right thing to do in a "give to charity" sort of way, not in a "don't kill" sort of way...

Remind me...

To write a story about Feynman diagrams. And a poem about candles. I'll know what you mean.

War! Huh! What is it good for?

Actually... Absolutely nothing. So, I hear people bring up the economic benefits of war every so often, mostly regarding how WWII "got us out of the depression." before I talk about how little sense this makes, I'd like to point out that the US had mostly recovered from the depression before 1940. Last I checked, the US didn't get involved until December 1941, so time disagrees with this theory.

Now, let's talk about wars. Specifically, let's talk about WWII. People talk about how American industry was mobilized for the war. This is true, but there's a subtle fallacy at play here. Jobs, in and of themselves, do not add to the economic vitality of a nation. If they did, we could have everyone working rolling rocks up hills, and letting them fall down again, and we'd have a booming economy. What does add to economic vitality is the creation of capital. Capital is a good which can be used to make more goods. In other words, a booming economy is an economy which is one which is increasing its capacity to produce.

So, the construction of the factories and machinery to create the weapons of war was economically healthy because it created capital, but it ends here. All of those factories went to work building supplies for the war. What we then have is capital-- raw goods (mostly ore and oil) and processed goods (metal alloys, gasoline, rubber) -- being turned into finished products. All well and good, but these products are leaving the economy: To go be used (and destroyed) in a war.

What, then do we have? Capital that is not being used to create more capital. Capital which is being used exclusively to push products out of the economy. We were wasting capital.

And this capital waste rears its ugly head in the shortages and rationing. Capital was leaving the economy at an enormous rate. The market response is hyperinflation: If goods are rare (as they will be if all capital is leaving the economy), goods are expensive. This also happened in Europe after WWI, because all of their monetary capital was going to reparations (Germany, Austria), or repaying loans (Allied powers). (To be fair, Weimar monetary policy didn't help the hyperinflation.)
This hyperinflation can be curbed by fixing prices, but as we saw in the 70s (and every other time prices have been fixed), this leads to shortages. The rationing in the US was a response to the shortages. These shortages were caused because the capital was leaving the country.

In case you hadn't caught this, rationing is not a sign of a healthy economy. It is quite the opposite.

Next. The Marshall Plan. Total US aid in Europe was over 1.2 Billion. A healthy economy does not need billions of dollars pumped into it. Money from the US even came after the European economy had started turn around: After a few years of mass shortages. So a few years after the war, the economy, while recovering, was still in shambles. Should war have brought about a boom?

Finally, take a look at all the poorest nations in the world. They all have one thing in common: They have been war-torn for at least 10 years. In almost all of these conflicts, the war started for socia-political reasons, and after a short time, much of the fighting became centered around mines. Why? Because the combatants ran out of money, and need a way to finance the war. But if a war is good for the economy, they shouldn't need half-working mines to fund their wars.

Poem

It's been a while since, I've given you a poem, eh?

***

Multitudes, multidudes in the valley of derision!
But the glory of the Lord is near in the valley of decisions,
and a wicked wind blows through the sea of visions
and revisions before the taking of toast and tea.

In His house we come and go,
talking, oh, of Michaelangelo.

Remember when David danced in his ephod?
Well, I stood by and scoffed:
The king of Israel naked (with the slave girls!)
for all the world to see!

And centuries later:
"Your mind is not far from the kingdom of Heaven."
My mind is close! O my soul, rejoice!
But, O soul, alack!
It seems you've fled the winter of despair,
and won't be coming back.

These pedals on a wet black bough,
they'll fall off soon, any day now--
drifting to the ground
like nameless faces, wandering through the crowd;
While my mind, lofty, in the clouds,
crashes into a mountain, and comes tumbling down.


***
There will be another one shortly.

Summer goals

Here's a tentative list of things I want to get done this summer... As with every break, I hardly expect to get all of it done:

• Understand Grothendieck's Axiom, and why it is (or isn't) important.
  
• Write a decent Binomial Heap in Haskell.
  
• Read Type Theory and Functional Programming or Proofs and Types.
  
• Read these lecture notes on Categorical Logic.
  
• Read all the minor prophetic books (and Daniel).
  Read all of the Epistles.
  
• Get comfortable with functors, monads and arrows in Haskell.
  
• Study for GRE. Especially, read Apostal's calculus.
  
• Make headway into the graph problem Dr. Pelsmajer gave me.
  
• Work with Beckman on something interesting...
  
• Pray more!
  
• Read The Tragical History of Doctor Faustus and Das Faustbuch.
  

There's more, I'm sure... There's always more.
Edit: Oh, dear! I got none of those done... sigh.

From qntm.org

You will not be able to stay home, blogger.

You will not be able to dial up, log in and cop out.

You will not be able to watch the revolution unfold on your RSS feed because the revolution will not be tweeted.

The revolution will not be tweeted; the revolution will not cost ninety-nine cents from iTunes; the revolution will not appear on Fark, Digg, Reddit or Metafilter, nor be brought to you by Randall Munroe, Ben Croshaw, Jack Thompson, Ron Paul or Stephen Colbert. The revolution will not be tagged "nsfw" or locked for editing by newly-registered users due to persistent vandalism.

The revolution will not have rounded corners because the revolution will not be tweeted.

*****
Read the whole thing here. Also read all of Fine Structure.