## Proper use of math

15 September, 2016 at 08:33 | Posted in Economics | 18 CommentsBalliol Croft, Cambridge

27. ii. 06

My dear Bowley,I have not been able to lay my hands on any notes as to Mathematico-economics that would be of any use to you: and I have very indistinct memories of what I used to think on the subject. I never read mathematics now: in fact I have forgotten even how to integrate a good many things.

But I know I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very unlikely to be good economics: and I went more and more on the rules — (1) Use mathematics as a short-hand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can’t succeed in 4, burn 3. This last I did often.

I believe in Newton’s Principia Methods, because they carry so much of the ordinary mind with them. Mathematics used in a Fellowship thesis by a man who is not a mathematician by nature — and I have come across a good deal of that — seems to me an unmixed evil. And I think you should do all you can to prevent people from using Mathematics in cases in which the English language is as short as the Mathematical …

Your emptyhandedly,

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Probability theory, for one, has developed hugely in the last 110 years, but I guess that we would agree that the last sentence of Marshall’s still applies: a probability theory that is as short as commonplace perceptions in dealing with lack of certainty is not doing economics any favours.

On the other hand, Keynes’ 1910 Treatise (and subsequent work in the same vein) seems to me to provide an ‘engine of enquiry’, although in my experience Marshall’s (4) leads to example that many mainstreamers will regard a highly improbable, like Taleb’s Black Swans. So there can be a lot of hard work to convince them that the examples really are important!

Comment by Dave Marsay— 15 September, 2016 #

How can probability theory be considered anything else but a waste of time? While it may be able to predict an ensemble average, it can never predict the outcome of the next event.

Comment by Henry— 15 September, 2016 #

Without the theory one is relying on ‘common sense’, which isn’t always a good idea.

Wikipedia cites Kolmogorov as a key source for contemporary mainstream probability theory. But as I note at https://djmarsay.wordpress.com/bibliography/rationality-and-uncertainty/probability/probability-classics/kolmogorovs-foundations-of-probability/

economists (among others) tend to go well beyond the mathematics. It thus seems to me that a proper appreciation of the various theories of probability are a good anti-dote to those who seek to apply them well beyond what is theoretically justifiable – even in the face of some clear counter-examples.

I’m also a little surprised that you can’t think of any situations in which even naïve probability calculations might be useful 😉

Comment by Dave Marsay— 15 September, 2016 #

Depends on what you mean by “useful”.

Probability is a figment of the human imagination, particularly the mathematician’s imagination.

Comment by Henry— 15 September, 2016 #

“Without the theory one is relying on ‘common sense’, which isn’t always a good idea.”

So what is the point of a theory of probability if it can’t tell me the outcome of the next event?

Comment by Henry— 15 September, 2016 #

Probability theory is useless because “it can’t tell me the outcome of the next event”?

I wouldn’t bet on that.

Comment by dnm— 16 September, 2016 #

I’m not sure whether you are having some fun and playing with words or you are serious.

If you are serious, please show me what you would bet on and how you rationalize your bet.

Comment by Henry— 16 September, 2016 #

@Henry

Are you saying that professional poker players, for example, don’t know the likelihoods that various hands will come up during play?

Comment by dnm— 16 September, 2016 #

@dmm

Knowing the likelihood that a particular set of cards might appear in your hand is not the same as knowing what set of cards will appear in your hand. Just because you think you know what set of cards might land in your hand does not mean that that set of cards will land in your hand. In the end you just don’t know. Presumably every player at a table thinks he knows the likelihood of a particular hand appearing yet not everyone can win, only one can win and not only because he has the best hand but because he is the best at convincing everybody else that he has the best hand.

Comment by Horace— 16 September, 2016 #

Weather forecasters can’t tell us what the weather will be tomorrow, but I find their forecasts useful. On the other hand, when financial advisers talk about ‘risk’ I find it useful to appreciate that there are important differences between their notion of probability and mine.

Incidentally, the two areas have very different notions of probability, a difference that matters.

Comment by Dave Marsay— 16 September, 2016 #

Weather forecasters do forecast the weather and their forecasts are useful. However, do they assign probabilities to their forecasts and would those probabilities be useful? I would assert not.

Imagine you’re standing next to a roulette table in a casino, chips in hand. The operator is about to toss the ball into the spinning wheel. Will it finish on red or black? Imagine also you have in your hand a computation engine programmed with every probability theory since Bernoulli. It can run a trillion computations in the time an electron spins once. Do you think such a machine could tell you what the outcome of the next spin will be?

Imagine you’re a mining entrepreneur about to push the button on a billion dollar copper mine. Do you think any probability theory will be able to tell you the average price of copper for the first year of production, let alone for each year of the life of the mine?

Comment by Henry— 16 September, 2016 #

Wikipedia has an article on probabilistic forecasting, and your favourite search engine will turn up examples from various nations.

(I agree that probability theory is irrelevant to many circumstances, and is often mis-applied in many more. But it does have some uses. I generally calculate a probability whenever the audience is expecting one, but strive to put it into the proper co ntext, drawing on the appropriate theory. Probability is rarely just a number. And I suspect that there is a lot more to the mathematics of probability than some critics think. E.g., there is theory after Bayes! )

Comment by Dave Marsay— 16 September, 2016 #

“Incidentally, the two areas have very different notions of probability, a difference that matters.”

Do you mind explaining the difference?

Comment by Horace— 16 September, 2016 #

Horace, suppose that Fred has tossed a coin lots of times, and half the time it has been heads. My experience of weather researchers is that they will ask a mathematician for advice on whether the probability of heads is necessarily a half (it isn’t). On the other hand, most financiers will resist all advice and insist that it is a half. They even have a definition of risk that seems to me to be decidedly dodgy. In the first place, probability is a tool in the search for usefulness. In the second it is dogma. I approve of the first but not the second.

Comment by Dave Marsay— 16 September, 2016 #

“Wikipedia has an article on probabilistic forecasting, and your favourite search engine will turn up examples from various nations.”

Can probabalistic forecasting tell me whether it will rain tomorrow at 3.21 pm?

Can probabalistic forecasting tell me whether the next spin will result in red or black”

Can probabalistic forecasting tell me what the average price of copper will be in 2017?

Comment by Henry— 16 September, 2016 #

“……whether the probability of heads is necessarily a half (it isn’t)”

Could you explain this some more?

“…..In the first place, probability is a tool in the search for usefulness. ”

I still can’t see how probability theory is useful. Just because something is assigned a probability doesn’t make it useful. No-one knows what a future outcome will be. Assigning a probability doesn’t make it any more certain. Weather forecasters make predictions – their predictions seem to be getting better. This has nothing to do with probability theory – it is to do with their understanding of atmospheric physics and the advances in the modelling of near surface atmospheric processes.

It seems to me that usefulness and dogma go hand in hand. In order to invoke probability theory you have to believe in its efficacy. This is an act of dogma.

P.S. I apologize for the inadvertent appearance of my alter ego, Horace.

Comment by Henry— 16 September, 2016 #

awesomely sensible

Comment by JKH— 15 September, 2016 #

“Wikipedia has an article on probabilistic forecasting, and your favourite search engine will turn up examples from various nations.”

What good was probability theory to the “masters of the universe” resident at Long Term Capital” that almost brought the world’s financial system to it’s knees in 1997/98?

What good was probability theory to the insurers that lost their shirts following the advent of Hurricane Katrina?

Comment by Henry— 17 September, 2016 #