“Ask and it will be given to you; seek and you will find; knock and the door will be opened to you.” — Matthew 7:7

Michael Emmett Brady’s paper *Keynes, Mathematics and Probability: A Reappraisal* is a bizarre piece of work. In it he reads things into Keynes work on economics in a manner that is not dissimilar to someone reading another person’s fortune into tea leaves.

In the paper Brady claims that in Keynes’ *Treatise on Probability* the author stopped trying to make ‘point estimates’ of probabilities — that is, estimates that crunch out a single number — and instead moved toward trying to make ‘interval estimates’ — that is, estimates that place the probability of an event in a position among a series of other events. In this part of the essay Brady’s scholarship is up to scratch and his conclusions convincing.

But then Brady starts to read this into Keynes’ later work as if the original author were leaving secret clues for Brady to pick up. At this point Brady goes in off the deep end. Here is an example of one of the passages of the *General Theory* that Brady quotes,

The object of our analyses is… to provide ourselves with an organized… method of thinking our particular problems; and, after we have reached a provisional conclusion by isolating the complicating factors one by one, we then… allow for the probable interactions… This is the nature of economic thinking. Any other way of applying our formal principles of thought… will lead us to error. It is great fault of symbolic pseudo-mathematical methods of formalizing a system of economic analysis… that they expressly assume strict independence between the factors involved and lose all their cogency and authority if this hypothesis is disallowed.

Now most people would take this quote as proof that Keynes — as those who worked with him on economic issues like Joan Robinson insisted — was largely against the overuse of mathematics in economics. Brady, however, underlines the word ‘pseudo’ and claims that Keynes was actually hinting that a whole new system of ‘real’ mathematics should be used for economic thinking. Brady writes,

These two quotations, along with a similar quotation from p. 275 of the GT, have been widely quoted and horribly misinterpreted. Keynes’ objection, on both p. 275 and pp.297-98 of the GT, is to pseudo-mathematical methods, which ignore feedback and interactive effects and assume strict independence among the variables. Obviously, such assumptions would lead to simple, unrealistic linear models. Keynes, on the other hand, was simply more advanced in his understanding of the need to model such economic phenomenon non-linearly. (p18)

Ask yourself a question: is a linear model that ignores feedback and interactive effects really a pseudo-mathematical model? Is the mathematics involved ‘fake’ in some way? Of course it is not. It may be a poor representation of the system it is trying to model but that doesn’t make the mathematics ‘fake’. Keynes knew this, of course. He was, however, engaged in rhetoric. Calling such applications ‘pseudo-mathematical’ was not an accusation that the mathematics was ‘false’ and that all that needed be done is to apply ‘real’ mathematics, rather it was a rhetorical flourish to point out the pretentiousness of the technique.

So, why doesn’t Keynes keep his ‘real’ mathematics a secret from his reader? Well, Brady has an answer and it is as follows,

Unfortunately, Keynes does not point out that he is not opposed to the use of numbers, only their misuse as point estimates, in this essay. However, since he had discussed his inexact approach or applied it in ten chapters in the TP (chapters 5, 10, 14, 15-17, 20, 25, 29-30), his omission is understandable. (p19)

Well, I don’t think its at all understandable. What Brady is telling us is that Keynes was keeping his true feelings about mathematics a secret. Why would he have done this? Why not just say “If the reader would like to know how to interpret what I have just said mathematically please see the following chapters of my *Treatise on Probability*…”? Brady never tells us. He assumes that its normal for authors to bury clues in various texts for him to find and interpret.

Now, here’s where Brady’s paper turns from tragedy to farce… In the very next sentence he writes,

In conclusion,

I find absolutely no textual evidence to support the claim, made in the following two sections, that Keynes was either opposed to the use of mathematical methods or that he himself was ignorant and error prone in the use of such methods. (ibid — My Emphasis)

Brady says that he finds no textual evidence to support Keynes’ opposition to the use of mathematics in economics. But he also finds no textual evidence to support Keynes’ championing of his own interval approach. And then he concludes that the latter exists anyway! Bizarre stuff.

What Brady is really doing is picking at words in the text like a Talmudic scholar, ignoring the fact that there is no evidence whatsoever that Keynes ever applied his interval approach to probability to any economic problems and then finding a trail of breadcrumbs that any non-biased reader would see he is himself dropping as he makes his way through Keynes’ oeuvre.

Did Keynes’ early work on probability influence his thoughts on economics? Yes, of course. Anyone who reads his famous essay on econometrics — *Professor Tinbergen’s Method* — can see that clearly. But his work on probability did no more than allow him to appreciate the limits of using mathematics to understand complex systems where humans interact. Otherwise why didn’t he write a response to Tinbergen showing how you could use his own interval approach to do the econometrics properly? Well, Brady has a fairly wild imagination so I’m sure he can come up with some reason or other.

I was familiar with the work of Keynes the mathematician and scientific eminence grise long before realising that he was also the well-known economist. Reading his General Theory it seemed to me entirely consistent with his Treatise on Probability, but written for economists, not mathematicians.

Looking at it again, the first two sections of chapter 12 (on expectations) seem to me to support the view that Keynes’ approach to economics was founded on his mathematics: it was no secret.

Are there any quotes where Keynes really does denigrate the use of (appropriate) mathematics? I have seen claims like yours, but never clear primary quotes.

Prof. Brady puts Keynes’ interval-valued theory of probability centre stage. This is clearly important, but I am not myself clear that it is always entirely adequate: Keynes considered others.

Perhaps the most important point that you raise is the distinction between ‘real’ and ‘pseudo’ mathematics. I struggle to make the distinction clear, but there is one, and it matters. ‘Real’ mathematics is a genuine, informed, attempt to do something appropriate. Pseudo mathematics is an attempt to look good, without so much concern for the consequences. Keynes, I think, was in favour of the former. We can all be against the latter.

It might also be a good idea for economists to be more familiar with his Treatise. (Happy New Year!)