Occasional reflections on Life, the World, and Mathematics


or, What Dr. Tortoise said to Prof. Achilles

I remember reading a biologist — I’ve forgotten who it was — remarking that, in comparison to some other fields, arguments in biology rarely turn venomous, because no matter how certain you may be, based on current knowledge, you can expect that within a few years the question will be definitively resolved one way or the other. If you are indeed correct, science will not suffer for your having indulgent your colleague’s error, and neither will your reputation. And you might be wrong. It’s very difficult to hold to absolute confidence in your own beliefs when they concern a matter of fact, and the fact will be revealed. It is a bit like the logic of keeping the peace through nuclear deterrence.

This was most trenchantly put by the great German mathematician David Hilbert, who famously commented on Galileo’s “cowardly” recantation in the face of the Inquisition:

He was not an idiot. Only an idiot could believe that scientific truth needs martyrdom — that may be necessary in religion, but scientific results prove themselves in time.

But this requires that there is a firm external standard of evidence that can conclusively decide a question, independent of current opinions or authority. People will naturally try to interpret the evidence in a way favourable to their prejudices and pet hypotheses, but there has to be the possibility of a conclusive disproof. This is hard but possible in biology, certainly possible in physics. In mathematics such conflicts can barely get started, the standards of proof are so high, and so clearly defined.

What about economics? Certainly from the perspective of an outsider, and based on my reading of my favourite popular economics authors and bloggers, it seems that at least the most politically applicable high-stakes areas like macroeconomics can easily turn into a replay of Lewis Carroll’s “What the Tortoise said to Achilles“.* No matter how conclusively an approach has failed, either logically or empirically, its proponents can simply return to the Wall Street Journal editorial page, or the lecture circuit, and repeat the errors that have just been disproved. As Carroll demonstrated, it’s not obvious how a logical argument can actually compel a change in belief, much less in action.

The dialogue might go something like this:

“That beautiful First Proposition of Keynes!” the Tortoise murmured dreamily. “You admire Keynes?”
“Passionately! So far, at least, as one can admire a treatise that won’t he published for some millennia to come!”
“Well, now, let’s take a little bit of the argument — just two steps, and the conclusion drawn from them. Kindly enter them in your notebook. And in order to refer to them conveniently, let’s call them A, B, and Z: —
(A) Tripling the monetary base, in the absence of a liquidity trap, must lead to high inflation.
(B) The monetary base has tripled, and there has been low inflation.
(Z) Our economy must be in a liquidity trap.
Readers of Keynes will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, must accept Z as true?”
“Undoubtedly! The youngest graduate student in an economics department — as soon as economics departments are invented, which will not be till some two thousand years later — will grant that.”
“And if some reader had not yet accepted A and B as true, he might still accept the sequence as a valid one, I suppose?”
“No doubt such a reader might exist. He might say ‘I accept as true the Hypothetical Proposition that, if A and B be true, Z must be true; but, I don’t accept A and B as true.’ Such a reader would do wisely in abandoning economics, and taking to football.”
“And might there not also he some reader who would say ‘I accept A and B as true, but I don’t accept the Hypothetical ‘?”
“Certainly there might. He, also, had better take to football.”
“And neither of these readers,” the Tortoise continued, “is as yet under any logical necessity to accept Z as true? They could think that the Federal government is falsifying the inflation statistics.”
“Quite so,” Achilles assented.
“Well, now, I want you to consider me as a reader of the second kind, and to force me, logically, to accept Z as true.”
“I’m to force you to accept Z, am I?” Achilles said musingly. “And your present position is that you accept A and B, but you don’t accept the Hypothetical –”
“Let’s call it C,” said the Tortoise.
“– but you don’t accept
(C) If A and B are true, Z must be true. ”
“That is my present position,” said the Tortoise.
“Then I must ask you to accept C.”
“I’ll do so,” said the Tortoise, “as soon as you’ve entered it in that note-book of yours. Of course no one is falsifying inflation statistics! What else have you got in it? Plenty of blank leaves, I see!” the Tortoise cheerily remarked. “We shall need them all!” (Achilles shuddered.) “Now write as I dictate: —

(A) Tripling the monetary base, in the absence of a liquidity trap, must lead to high inflation.
(B) The monetary base has tripled, and there has been low inflation.
(C) If A and B are true, Z must be true.
(Z) Our economy must be in a liquidity trap.

“You should call it D, not Z,” said Achilles. “It comes next to the other three. If you accept A and B and C, you must accept Z.”
“And why must I?”
“Because it follows logically from them. If A and B and C are true, Z must be true. You don’t dispute that, I imagine?”
“If A and B and C are true, Z must he true,” the Tortoise thoughtfully repeated. “That’s another Hypothetical, isn’t it?
And, if I failed to see its truth, I might accept A and B and C’, and still not accept Z. mightn’t I?”
“You might,” the candid hero admitted; “though such obtuseness would certainly be phenomenal. Still, the event is possible. So I must ask you to grant one more Hypothetical.”

And so on.

Paul Krugman has been incredibly good at explaining (in real time!), with posts such as this one, what ought to count as definitive tests in economics. But it’s not clear how  the field could enforce a standard of evidence, if the advocates of austerity with impeccable credentials want to keep saying “And why must I?” What do you do when there are understandings crucial to a field of study, and the salaries of a significant portion of the practitioners in the field depend on them not understanding them?

The sequel Economic Notes from Underground almost writes itself!

* For those who don’t know this, it is a treat in itself, and served as one of the foundational texts of Douglas Hofstadter’s Gödel, Escher, Bach, the book that most inspired many of my mathematical contemporaries in our teenage years.

Comments on: "Mutually Assured Logical Destruction" (1)

  1. Reblogged this on misentopop.

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