Greatest mathematician of our times is dead.

Miscellaneous General
1

Benoit Mandelbrot RIP

What the Austrian School is to economics, the French School of Fractal Geometry of Nature was to statistics.

Mandelbrot started an entire French school of thought in statistics that quietly advanced fifty years beyond Anglo-American statistical and quantitative financial techniques, with absolutely no notice from the outside world until a Greco-Lebanese derivatives trader who studied in France popularized it.

The Austrian School of Economics and the French School of Statistics are like two brothers born in the same house, but one left from the front door and the other left from the back door. Both are founded on the following premises:

  1. Human action and behaviour is uncertain, and you can’t use mathematical models to predict humans beings on a large scale as if they are atoms and planets.

  2. You can not reject uncertainty, but accept it as a necessary reality, and work with it.

  3. We must accept the limitations of our own given field, and move beyond to a broader understanding of human life, because understanding and hedging from risks is a liberal art that encompasses all knowledge.

Both these schools were driven back by the rise of Paul Samuelson, who mathematized economics and social science completely, and then were pushed underground after Samuelsonian thinking led to the birth of the Black-Scholes-Merton model.

As a self-taught derivatives trader, I have learnt the Black-Scholes-Merton model from variois books, and was eager to implement its seemingly simple perfection of prediction in my trading. Thanks to both the Austrian School and Mandelbrot’s criticism and rejection of the normal distribution and standard deviation model, I was saved from thinking I could predict prices and make money, and before I lost more money from it, I started to work only on diversifying between extremes and taking small profits. These days, I make 40% returns per trade, and I used to break even initially, thinking these techniques worked. I think I do owe something to the fight against scientism and faith in mathematization done by these two fronts.

2

I think Benoit Mandelbrot’s greatest legacy will be making a rational, mathematical analysis of natural organic patterns possible, where before mathematicians were not interested in any of it because they did not have the tools to understand it.

He made nature scientific, so to speak.

3

Yes, RIP. Great post Prateek.

4

His fellow colleague, Nassim Nicholas Taleb, did much to advance the Austrian School through his best selling books, The Black Swan and Fooled By Randomness.

Because of Taleb, many people now know that the Austrian School is more than just a “more radically free market than Chicago” set of thinking - it’s about something different and more specific, like limitations of knowledge, unintended consequences, problems of historical analysis, and so on.

People normally criticise Austrians based on what they assume they are about - “neoliberalism” or advancing some corporate-controlled NWO - and thus they can criticise them by framing their arguments for them and refuting them, but once they are diverted towards talking about methodology, they hit a road block on a priori, limitations of knowledge, and so on. So Taleb’s book helped play its own small part in getting people to talk about those things.

What Joseph Schumpeter did for business studies, what George Polya did for mathematical method, what Benoit Mandelbrot did for statistics, what Friedrich Hayek did for economics, (and what Gary Taubes recent work did for nutrition!) all are a small part of destroying Anglo-American orthodoxy, and bringing back the superior continental European thinking. The German, Central European, and French underground thinkers, if popularized, could devastate the entire pyramid of nonsense taught in American and British universities and in other nations that imitate them.

5

Could you maybe recommend some books to read on the subject? Especially those that would relate it to economics. This is new to me, and I have no idea where to start.

6

So continental European thinking is superior to Anglo American thinking in all fields.

What did George Polya do for mathematical method btw? Not very much. And Hayek was indeed a comptetent economist who wrote a few books in a very specialized field.

What part of Anglo American orthodox mathematics did Polya and Mandelbrot destroy? Can you tell me the equation they proved false?

What is your source for any of this? Or is it original research?

7

IT IS AN EMPTY UNQUALIFIED OPINION! THAT’S ALL I HAVE!

There, I said it.

George Polya revived the heuristic method, the art of solving problems, not the technique of solving problems. While heuristic reasoning has been unpopular, have you ever asked yourself in your high school math classes, “How would I figure out the solution to this problem having no idea of the already established technique?” Basically, where would be the zero point or the starting point of problem solving?

That is a forgotten art, for technique is about duplicating technique, which was formed by duplicating technique and so on.

Heuristic reasoning is, for the lack of a better word, an a priori method of problem solving. If and when you knew nothing in advance of how the problem is solved, how would you solve it? At that point, it’s no longer just mathematics. It’s a liberal art. At this point, you need to know how and why your mind works and how it stumbles upon a solution.

Having read two books on heuristic reasoning, I suddenly found myself answering tough Olympiad level problems with ease, like, “How will you inscribe an isosceles triangle in any circle?”

About Mandelbrot, he gave a heterodox counter to the Gaussian models of prediction. What he showed was that while the Gaussian model presumes that extreme events are less likely, he showed that this assumption is greatly false - just because extreme events haven’t happened in the past doesn’t mean they can happen in the future. The truth is that the Gaussian curve should have been inverted - ordinary events are rare, and extreme events are more common. We don’t know it given the context of a brief moment in time, but in the long run, there are nothing but extreme events and they happen more often than we predict. That’s an internet summary - to explain all of it would require long path requiring many small steps, which you can get from the books of Taleb and Mandelbrot.

Either way, the Gaussian model destroyed its own self when Long Term Capital Management went bust using Gaussian techniques. Just the way Mandelbrot put his money where his mouth was, and did quite well in the stock market for decades, Hayek also put his money where his mouth was, and quickly withdrew his money before the Depression, while his more optimistic friend Keynes lost much of his stock market wealth in the big crash. That’s one of my simpler criteria for seeing who is right.

8

There is a pretty big rift in the foundations of probability and statistics between the frequentists and the Bayesians. This rift is metaphysical, not formal. That’s what’s nice about formalism, there’s never anything to argue about once you agree on a formalism. But that just pushes the problem back to choosing which formalism to use in the first place. Is probability only definable as a ratio? Mises was influenced by his brother’s view of probability (Richard von Mises), frequentism. Frequentism is the idea that probability is only ever a ratio and nothing else. It doesn’t reflect anything about the future or expected beliefs or any such nonsense.

However, this idea assumes that the problem of induction is, in fact, insurmountable when exactly the opposite is the case. I think it’s not an exaggeration to say that the problem of induction has been solved, by Ray Solomonoff. This little article is too brief to really understand the concept if you haven’t had prior exposure to it but it’s a great summary of the idea and will give you a launch point from which to perform further investigation on your own.

Given that there is no problem of induction, the prohibition on speaking about future probabilities - the essence of frequentism - is purely artificial. It may be a good formalism for paleontology or any other science whose domain is defined to lie strictly in the past. There can, of course, be no contradiction between a frequentist theory of the past and a Bayesian theory of induction… both must be consistent with one another. Bayesianism does not imply a denial of frequentism (as a description of past events) but frequentism does imply a denial of Bayesianism. It’s an interesting lapse in Mises’s usually rigorous thinking that he characterized probability as concerned solely with past events when his subject of study was specifically about the future… future human action. Taken to its limit, the frequentist view of probability actually contradicts the Austrian conception of cause and effect.

Clayton -

9

My field is computer engineering so I find it easiest to understand the limits of probability theory by trying to apply it to computing devices. What is the probability of a page-swap at any given time, for example? Page swaps are completely deterministic events but, even given the code executing on a system, trying to determine the probability of a page-swap is impossible. Now, marbles drawn from an urn are not best modeled as a computer. Classical probability theory suffices. But biological organisms are computers, so classical probability theory breaks down when applied to biological organisms (including humans and the human economy) for the same reasons that it breaks down when applied to computers.

Clayton -

10

Interesting stuff. Thanks, Clayton.

I really need to get brushed up on the materials talked about in this thread. :stuck_out_tongue:

11

Giant_Joe,

If you haven’t read Taleb’s book, The Black Swan, I strongly encourage you to do so. While I thought that the book was highly informative (and a great read at that), I do think that Taleb got carried away glorifying himself, which impeded the flow of the book at times. He also questionably praised Keynes, essentially calling him a genius, which I still don’t understand to this day, considering Taleb classifies himself as a “libertarian” (but maybe I’m biased ).

With that being said, if you can get past a few minor points, it’s a great book IMHO.

12

^

In the Black Swan, Taleb explained what he meant when he uses certain words, and he tends to give them his own definition.

Taleb is an “academic libertarian”, a believer in breaking academic orthodoxy. Not a libertarian per se. However, he does give credit to the Austrian School for some ideas with which he is in agreement.

Clayton, I remember Murray Rothbard mentioning that Richard von Mises’ theories on probability debunked many of Keynes’ theories on probability. Of course, while Ludwig and Richard rarely agreed, both were supposedly quite brilliant and original thinkers, and you can just see that continental Europeans have so much natural superiority to Anglo-Americans on so many fronts. It’s not that Keynes’ Treatise on Probability was a bad work at all; it may have been his main strength, since it was also helped by the fact that he was a person of philosophical background.

13

@Prateek: I’m not familiar with Keynes’s work on probability but I would be surprised if Keynes was a friend to Bayesianism. Frequentism is usually the view held by those with a positivist bent.

Clayton -

14

@PrateekSanjay

I’m not a philosopher, and I’m not an expert on method, so I don’t know what you’re referring to regarding the “underground thinkers”. Is it rationalism?

With Hayek, there appears to be an important distinction regarding Hayek vis-a’-vis his representing Austrian economics—in particular its methodology.

Here’s a few words on Austrian method from Hoppe (from “Murray Rothbard: Economics, Science, and Liberty”):

15

@patreek Sanjay: I’ve been looking for some good books on Heuristics! Can you tell me which ones you read?

16

bump

Some good info in this thread. I came back to it to write down some names. Given the benefits I’ve gained from my anti-mainstream views of politics, economics, and nutrition, I think I’ll soon delve into alternative problem solving techniques and alternative finance.

If I get caught up in doing AI work, (it’s a possibility) I’ll be looking into that stuff you mentioned, Clayton.

Thanks again, people.

17

Can you concisely state what the problem of induction is?

An Inconsistent Argument In Human Action.
18

I remember when I was studying the calculation argument, I coudn’t help but feel there would be problems similar to those expounded in chaos theory for any social planner who attempted to plan a complex economy without prices. My memory is fuzzy though, so I don’t remember the details, I should look over my old notes.

I think chaos theory would be a great educational tool(or at least something for most of the econ grad students who are ex physicists to consider) in ilustrating to economists the limitatons of calculational methods. We have long known in physics only a small subset of problems can be solved analytically, with most others requiring computational methods. Yet chaos theory has made us even more humble in recognising the limitations we cannot avoid in principle for predicting the evolution of complex systems.

Again, as fancy pants as economists like to be with their maths, their ignorance of chaos theory is telling.

19

I’m surprised the AI community hasn’t invested more interest in this.

Clayton -

20

Yes. Why should the future be like the past?

That the universe has behaved in an orderly, lawlike manner in the past does not constitute evidence that it will in the future. Hume shows the way. If you say that we can believe that the future will be like the past because past futures have always been like past pasts, Hume’s reply is that you must still assume that the future will be like the past to get from that observation to the belief that future futures will be like future pasts.

Restricting oneself to strings*, Solomonoff induction resolves this problem by asking what are the future states of the most likely Turing machines, given the output string which has been seen so far. By weighting these future states, we can assign meaningful probabilities to the future behavior of whatever is generating the output we are observing. Nick Szabo has written a pretty nice summary of the idea, here.

Clayton -

*In the computer theoretic sense, i.e. a string of characters drawn from a finite alphabet