Random Conversation

(Laughter has been removed)

MC: hmm. I'm demoting you
CK: ?
MC: you are no longer Master Commander of Hypothetical Operations.
CK: But if you don't demote me, think about how great everything would be!
MC: Your title is now Chief Executive of Jello Affairs.
CK: But!!! I'm not jiggly enough!
MC: hmm... actually, I don't know if I like that title... Jello Executive, Fruity Faction.
CK :I do, however, know that every conditional with a false antecedent is true... and I am responsible enough to only imagine badass scenarios.
MC: mostly so it abbreviates to JEFF
CK: If I had not been demoted, you would be richer than google.
MC: a googol dollars!!!
CK: If I were currently MC of HO, then yes.
MC: hmm
CK: (Hurray vacuous truths!!!) Allowing mathematicians to make vacuous promises since 1000BC
MC: right, could have doesn't actually imply causal effect, does it?
CK: Well, it's that If x Then y is always logically true when x is false. Because the implication is only broken when x is true and y is false. I had a prof who was in the habit of using vacuously true cases for the base case of an induction. Like, a statement about edges in a graph; his base case would have no edges...
CK: Why did I get demoted, by the way?
MC: glitch in the payroll system.
CK: Ah. Well; can't be helped.
MC: actually, we introduced the glitch after the fact
CK: I can't blame anyone, can I.
MC: there was a glitch in the name placards and they came out wrong.
CK: Well, if the name placard says so, it must be so.
MC: so we demoted/promoted people accordingly. It only made sense.
CK: Of course.
MC: we didn't want to waste the money we spent printing them.
CK: A company needs principles if it's to run smoothly. Principals? I don't know which.
MC: well, it needs both; who else is going to turn the hamster-wheel-power-generator?
CK: Right. This is why you can promote, and I'm only the JEFF. Best conversation ever, by the way.
MC: no, they only let me demote. I don't have authority to promote. They only give that authority to the janitor's secretary.
CK: I see. That seems sensible.
MC: I'm not sure this conversation would make any sense were I to read through it after forgetting the fact itself.
CK: You forget that it makes no sense now.

The mathematician on the street...

That last post was a bit of frustration about an ongoing discussion of AC/CH on the FOM mailing list. Not everything about the discussion has been quite as frustrating as the whole discussion-- namely, some fantastic quotes have come from it. Here are some of my favorite in posting order (I think they slowly get less and less technical...):

"All that we have here in this quasi-paradox is confirmation that reals are not a perfect model of dart throwing and vice versa." -Thomas Lord

"It would be useful to provide some rationale why the continuum having cardinality aleph_1 leads to more unusual results than, say, the Banach-Tarski paradox. Furthermore, I would like to know why you think these results should lead us to reject the continuum hypothesis but not the axiom of choice. Finally, I would be interested to know what has led you to conclude that most 'mainstream mathematicians' find your arguments convincing." -Lasse Rempe

"They are not claiming to have an argument formalizable in ZFC; they are merely claiming that mathematicians have overreacted to the results of Banach-Tarski, Godel, and Cohen by throwing out too much of their intuition about assigning measures to subsets of R^n." -Joe Shipman

"If you think it's an interesting question to investigate plausible extensions of ZFC that settle CH, then you're already a dyed-in-the-wool f.o.m.er." -Tim Chow

"But in my experience, if you pick a random mathematician who is not already interested in f.o.m., there's at least a 50% chance that you'll have to remind them of the definition of a well-ordering of the reals and of its relationship to the axiom of choice." -Tim Chow

"So you do not accept AC in the same way you accept the other ZF axioms? That's fine, but it's not the position of the mathematician in the street." -Joe Shipman

" I don't think the mathematician in the street will respond, 'Gee, since I accept AC as gospel, I am forced to blame these pathologies entirely on CH!'" -Tim Chow

"For starters, [the mathematician on the street] is unlikely to be able even to list the axioms of ZF, but he or she will know AC explicitly, precisely because it is known to have some strange consequences." -Tim Chow

(Included because of how wrong it is): "I think we are in danger of forgetting that not only do most mathematicians-in-the-street not believe AC, most of them have no intuitions about it and cannot state it even roughly, let alone have any idea how to use it." -T Foster

"Someone who does not know such basic material cannot be called 'a mathematician' (neither in the street nor anywhere else)." -Arnon Avron

"I am reminded of a time in graduate school [...] when I delivered my self of the opinion that cardinal trichotomy was, intuitively, OBVIOUSLY true and that the Well-Ordering Theorem was, intuitively, OBVIOUSY very fishy." -Allen Hazen

"This certainly circumvents the use of AC, but I submit that it is somewhat contrary to the mathematical practice of *not* equipping structures with non-canonical stuff that is extraneous to their essence. You could define a vector space as something that comes equipped with a basis, or a manifold as something that comes equipped with an embedding in R^n, or a group as something that comes equipped with a homomorphism to an automorphism group of something, etc." -Tim Chow

"mathematicians tend to replace the use of existential statements by the introduction of skolem functions. This is such a common procedure that they do not even notice that they are using AC when they do so." -Arnon Avron

"In particular we agree that these street mathematicians (one pictures them performing Hilbert's Nullstellensatz while passers-by drop coins in their hat) are enumerating witnesses to countability rather than countable sets." -Vaughn Pratt

"Depriving the street mathematician of her witnesses is like depriving a boxer of his fists. [...] Why should she care that foundationalists make things harder by killing off her witnesses?" -Vaughn Pratt

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.