Just Finish the Quote! John von Neumann is Sad.
The Quote
I’m doing research into pseudo-random number generation as part of a larger research project, and I ran across a quotation I haven’t seen in a few months:
This is incomplete:
“Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin.” ~ John von Neumann
Good luck finding a citation or context for the quote (I was nice and did all of the hard work for you). By itself, it sounds like a condemnation against all generation of random numbers using deterministic methods. Sure enough, it’s easy to find dismissive uses of the quote. For example, here is a search of Reddit for the text “in a state of sin” [Yes, one of the results is in response to an old post of mine, and no, that was hardly the straw that broke the Jake's back].
Expanding on the Quote
Let’s add the second half of the quote and see how it changes.
This is better:
“Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number– there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.”
~ John von Neuman, “Various Techniques Used in Connection With Random Digits.”
Now we see that the famous sentence isn’t necessarily a condemnation. It is a setup for a logical statement. The second sentence carries the meat and potatoes, and it’s never included!
The quote is merely pointing out the obvious: there are no actual deterministic random number generators. If you were to create a method to find random systems, the system isn’t random because it was generated by your method!
What first seemed like a condemnation now looks a little like a concession.
Next, let’s look at the context: “Various Techniques Used in Connection With Random Digits.” Unfortunately, I can’t find the text of the article, but I WAS able to find a book that references von Neumann’s statements extensively: Image and Logic, a Material Culture of Microphysics. Some scanned pages are missing around the particular quotation, but the context is loud and clear: von Neumann is developing and describing ideas on the “Monte Carlo method”, a method that depends heavily on the use of pseudo-random number generation.
John von Neumann and the Monte Carlo Method
The Monte Carlo method, in general, is the use of random trials to determine a value. If you run enough random trials, the theory is that you should eventually develop an approximation. Not only that, but by using statistical tools, you will be able to develop a confidence interval.
Did you ever do the 
approximation test with toothpicks when you were a kid? Surprise, you were tricked into using a Monte Carlo method! Your teacher wasn’t just wasting time with another class activity, you were learning the foundations of an important method for calculation.
von Neumann made significant contributions to the field. The most germane to our discussion is the “Middle-Squared method“. The method is so easy that any reasonably educated 5th grader with a calculator should be up to speed on its use in 5 minutes. Let’s say we want an n digit random number. We take any number of n digits and square it. We then take the n middle digits. That is your new random number. Rinse and repeat.
You’ll notice that this method has significant problems. For example, if your sequence has well-placed zeros, your “random numbers” will suddenly look a lot like, well, a bunch of zeroes. John von Neumann, according to Wikipedia, was well aware of these shortcomings as well as the crudeness of his method. He decided to use it anyways, particularly because it was fast on “modern” hardware (modern circa 1950).
In Conclusion
The quote is not a condemnation of those who attempt to deterministically generate random numbers. It is a concession as to the limitations of pseudo-random methods. John von Neumann himself is prolific in early literature as well as early practical use of Monte Carlo methods, so it is hard to imagine that he would disavow his own work so strongly, especially considered the fact that in correspondence with Householder, another mathematician working on Monte Carlo methods, von Neumann is quoted as saying that random numbers only need to be “random for practical purposes.”
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Reader Comments
Very enjoyable post on von Neumanns’s logical thought. He is most definitely right about not being able to have a truly random number when an algorithmic formula was used to create that number. It is not ‘true’ randomness, per von Neumann.
This makes me ponder whether it is possible that random happenings such as worlds developing or the make-up of the constellations are ‘true’ randomness? If I go with von Neumann, then the universe is not randomly set up at all, but rather through logic and physics it is set up through “strict arithmetic procedure”.
“The method is so easy that any reasonably educated 5th grader with a calculator should be up to speed on its use in 5 minutes.”
Sadly, we know this isn’t the case for 5th grade Lisette.
Unless there can be physical manifestations of transfinite sets. The assumption is that all form can be mapped onto arithmetic which can be reduced further to the simple elements n and ~n. Anything that cannot be so reduced cannot be represented arithmetically. Colours have this property. There are three primary colours. This set cannot be reduced further and so cannot be represented in terms of n and ~n. So the human mind can directly apprehend transfinite sets.
If there is an infinite number of possibilities at a juncture then determinism breaks down.
After reading the initial “condemnation” the programmer in me said “but… but… but…” until I kept reading your post. Yet another reason to not take quotes out of context!!