# Who thinks probability is just a number? A plea.

January 16, 2014 10 Comments

Many people think – perhaps they were taught it – that it is meaningful to talk about the unconditional probability of ‘Heads’ (I.e. P(Heads)) for a real coin, and even that there are logical or mathematical arguments to this effect. I have been collecting and commenting on works which have been – too widely – interpreted in this way, and quoting their authors in contradiction. De Finetti seemed to be the only example of a respected person who seemed to think that he had provided such an argument. But a friendly economist has just forwarded a link to a recent work that debunks this notion, based on wider reading of his work.

So, am I done? Does anyone have any seeming mathematical sources for the view that ‘probability is just a number’ for me to consider?

I have already covered:

There are some more modern authors who make strong claims about probability, but – unless you know different – they rely on the above, and hence do not need to be addressed separately. I do also opine on a few less well known sources: you can search my blog to check.

What about George Boole?

I comment on his laws of thought https://djmarsay.wordpress.com/bibliography/rationality-and-uncertainty/logic/booles-laws-of-thought/ .

He might seem to assume that there always exist precise objective (aleatory) probabilities, but I haven’t seen him make any argument, mathematical or otherwise, in support of such a proposition. Can you suggest anything?

What he does do is argue against precise logical or subjective probabilities, as in my summary above.

So, what about him 😉 ?

One more thing comes to my mind which could be relevant:

Physicist Scott Aaronson argues at http://www.scottaaronson.com/democritus/lec9.html that the core of Quantum Mechanics is just a generalization of probability theory.

He further argues: “if you want a universe with certain very generic properties, you seem forced to one of three choices: (1) determinism, (2) classical probabilities, or (3) quantum mechanics”

So one might argue:

1.) Probability Theory is a generalization of logic (as argued by Jaynes, see also http://bayes.wustl.edu/ )

2.) Quantum Mechanics is a generalization of probability theory

Thanks. Scott sheds some light on the mystery that is QM. I get that the classical universe (that we see) is just a projection of some more complicated universe, much in the same way as classical probability might be a projection of some more complex uncertainty. But it seems to me that most QM (as in the Copenhagen interpretation) applies classical probability and tries to gloss over any problems. David Deutsch seems to me to be an exception. One problem is that the Bayesian updating rule is non-relativistic, so one has a clash between relativity and QM. DD writes around this, and it seems to me that some of his work can be read as supporting the view that weight of evidence is a more fundamental (and reliable) concept than classical probability / information (which only make sense within epochs), and so QM seems to me to demand a reformed concept of probability. But, as far as I know, physicists still seem to calculate classically and then have various rules of thumb to work around any problems that this might cause, so they are pushing for any reforms. Of course, their advantage is that they can do experiments, and so do not need to rely on this aspect of mathematics is not. But I do nudge any that I meet, and enquire about any change.

I trained at Stanford under Ron Howard. He was very, very careful to note that there is no such thing as an unconditional probability; his preferred syntax for the probabilit of x was {x | E}, in which E represented all of the experience of the decision maker (or the decision maker’s agent) at the time of the probability assessment. This was back when formulas were written on paper with mechanical pencils, and the act of conditioning all of the assessments created muscle memory of the importance of this concept.

Mr.Marsay I have never read the classics outside of having Jayne’s on my bookshelf is there any book you would recommend on probability before all else? It’s going to take me a long time to work through all that I have. What reading order would you recommend?

SE, I had a quick look at your reading list and saw little that I had read myself. You seemed to have summary works rather than original work. In case your reading seems to give any authorities for the view that probability is just a number could I ask you to consult my blog for any contrary quotes, and let me know if you come across an argument that I haven’t covered?

As for suggesting a book, it seems very much to depend on ‘where you are coming from’. If you found something you liked I may be able to suggest something that you might also like to read. But there is no general book.

“SE, I had a quick look at your reading list and saw little that I had read myself. You seemed to have summary works rather than original work. In case your reading seems to give any authorities for the view that probability is just a number could I ask you to consult my blog for any contrary quotes, and let me know if you come across an argument that I haven’t covered?”

Yeah sure will do.

Have any of you ever heard the names Andrey Kolmogorov or Norbert Wiener? They are the founders of modern mathematical probability theory. Kolmogorov in particular laid the mathematical foundations of pure probability with something called measure theory; if you are a mathematician then you will have necessarily studied it. Wiener in his rigorous work on Brownian motion as a measure in function space (developed further by Paul Levy, PhD advisor of Benoit Mandelbrot) along with Kolmogorov also laid the mathematical foundations for stochastic process theory. You should be addressing these people if you are serious about probability theory.

Thanks for your comment. If you search my blog for Kolmogorov you will see my comments on his ‘foundations’. Basically, l get that by analogy we should expect upper and lower measures, but I have not found an argument by Kolmogorov that these measures must always be the same. Similarly, although I have read some of Weiner, Levy and Mandelbrot I have not found any argument by them that I must be a stochastic process. Please could you let me have any specific references? There is a strong probability that I may reply by blog 😉

Thanks again.