The Ambrosini Critique

I’ve enjoyed John Quiggin’s discussion of decision making under uncertainty in the context of preemptive war. The whole discussion was triggered by Judge Posner’s first post on his weblog.

In response to Prof. Quiggin’s first post, I commented:

I hope you’ll elaborate on the second point about war being a negative sum game. Intuitively, this seems wrong. I would think it easy to construct an example, from history or otherwise, of a war with positive gains in sum. Off the cuff, imagine a war were the attacked lay down their arms without a shot fired. The victors march in, institute democracy, grow the economy and are adored by the conquered. Wouldn’t this ‘war’ be positive sum?

Also, your arguments on this point seem to just have the effect of increasing the costs as discussed in Posner’s post. You’ve discovered some ‘hidden costs’ in the Posnerian calculation. That’s fine. Realize those costs and then redo the calculation.

Later in the comment thread, I’m chided:

Will, that’s silly. Of course we can construct an imaginary example of a ‘win-win’ war; it’s just that the preconditions very rarely obtain in reality. The neocons are rightly considered fools for kidding themselves that Iraq was an exception.

If the past teaches us anything, it is that war tends towards ‘lose-lose’ with occasional ‘win-lose’. The likelihood of ‘lose-lose’ has to enter our prior calculations. And as others point out calculations geared to ‘win-lose’ ignore the welfare losses of the losers.

Yes, the example was silly, but it was meant to illustrate that some such war exists. We’re talking about imagining future states of the world such that we can create policy (waging war being a policy). To allow my example is to see what the “neocons” where up to when they were planning this war. They, and I, believe there is such a thing as a zero-sum or positive sum war. To dismiss the possiblity out of hand, is to miss the point and to misunderstand why we’re in Iraq.

He then made a second post, where he linked to a well written paper on the precautionary principle. I commented:

Thanks for the paper. For this budding economics student, the discussion of more and more general models of uncertainty was enlightning. I think I’ll print it out and use it as a cheat sheet.

Your analysis is VERY dependent on how the decesion maker determines which state of the world is “status quo” and what would be an “innovation”. Where you see preemptive war in Iraq as an uncertain innovation, Bush sees uncertainty in the status quo, i.e. an unknown (and unknowable?) connection between Saddam’s WMD and terrorist willing to use them. This would be analoguous to the “innovation” that you observe in the doubling of green house gas emmisions if we maintain the status quo.

Similarly, there is some fudgy-ness in the statement of the incompletness hypothesis. “Incomplete estimates will generally be over-optimistic.” Your over-optimism may be my over-pessimism.

I’m reminded of Rumsfeld’s famous line about unknown unknowables and some such. What rule of nature implies only bad stuff is more unknowable?

We’ll see if anyone responds.

Prof. Derden gave a talk on math philosophy at the HSU math colloquium. He was addressing the issue of what numbers really are and his talk was a response to a commonly held belief (in philosophy) that numbers can’t be sets. Besides being sets, he wants to argue that numbers are a particular type of sets called Russell sets. Below is an excerpt of an email I sent the professor. I’ve yet to receive a reply:

As I saw it, your talk consisted of one of two things. There was a strong argument in favor of using Russell sets as a model for the natural numbers OR there was a relatively weak claim that sets, in general, can’t be in the set of “things numbers aren’t.”

I’m particularly fond of the stronger argument. That there would be a correspondence between all sets of pairs and the things we count as two seems to jive with my intuition of what numbers are. When I say “two”, I just mean all those things that I can pair together. Two is, in a sense, a pronoun for all pairs of things (or a synonym maybe?). And when I say “two plus two equals four”, I’m saying something like “two, for example, sheep, plus two, for example, rattle snakes, equals four things.”

Given the title of the talk was “What Natural Numbers Must Be,” its clear you intended to make the stronger claim. In any case, you at least argued the weaker claim. That there is a plausible argument at all for Russell sets as a model for numbers, means that sets COULD be such a model.

The word ‘model’ is tripping me up. Isn’t a model just an approximation of the real thing? Models are built to mimic the behavior of what is being modeled. As long as the model behaves as the thing being modeled (at least in the area of concern), it is said to be a good model. In this sense, the flight of fixed wing airplanes is a good model for the flight of birds. Of course, physics has shown that they are far from the same thing. In the same sense, the natural numbers can be modeled by sets, but that doesn’t mean that they are sets.

This is the same point Benacerraf makes in his essay. I couldn’t find the Benacerraf’s essay online but I found this article discussing the article. From that secondary article, “Benacerraf explains that to characterise the numbers is only to describe the structure, without any identification of the individual elements, and that this is why numbers are not objects at all.” So the model of the numbers helps us describe the structure of the numbers, but it shouldn’t be confused for the real deal.

Also, I can think of one more problem with trying to understand what the numbers really are. In your talk, you mentioned that the Platonists ‘push’ the location of ideas into the mind of god. Oh yeah, where’s he?

Similarly, if we ever find out what the number really are, won’t we just be pushing the problem? For example, let’s say, at your talk, we decided that the numbers were Russell sets. But then an intrepid undergrad would ask, in his required email follow-up, “Thanks Prof. Derden for letting us

know what numbers are. Ok, so what are Russell sets?” To which you

spend another colloquium talk discussing and then narrowing the possibilities. Suppose, at the end of that talk we all decided that Russell Sets are really something called blarbs. Then we’d have to explain what the blarbs are, then what those things are, and so on into an infinite regress. Where would this all end?

David Arnold’s discussion of technology in the teaching of Multivariable Calculus was a quick introduction to Matlab, Geometer’s sketch pad and Symbolic toolbox (in Matlab). Much of the talk was also a ‘geewiz’ exposition of some cool things you can do you in Geometer’s sketch pad. I left the discussion not having a good sense for how these technologies might be used to replace (or if they should be used to replace) current curriculum.

Overall, after reviewing Mr. Arnold’s web site, I’m impressed by his extensive use of technology in the teaching of Calculus. He’s even done away with exams and grades exclusively based on performance in labs/quizzes and homework.

The question for me is: is this appropriate. Mr. Arnold has demonstrated that technology can replace a more analytical approach, but should it? For example, you can use the sketch pad to find the arc length of the cycloid without the need of any the analytical tools you traditionally learn in second year calculus. Do you lose anything by orienting yourself to the geometric interpretation in the technology?

One might argue that the analytical tools were invented 100’s of years ago in lieu of tools like the geometer’s sketch pad. After all, the inventor’s of these tools had specific geometric problems in mind (i.e. the orbit of planets) when inventing the Calculus. Thus, its okay to dispense with the analytics to be replaced by high powered technology.

The problem with that argument is that the analytic tools have general application. Not all problems have a geometric interpretation… for example, what picture can you draw that will help you solve calculus problems involving more than 3 variables? The tools are there to help you abstract to these higher dimensions were intuition fails you. Tangentially, some people are more comfortable “pushing symbols” than working with geometry. Personally, I have a hard time imagining shapes in 3 dimensional spaces.

In the end, I applaud Mr. Arnold’s work. My question to him: how does the technology help students get beyond their intuitions of geometry to help equip them with more general analytical tools?

Exactly.