Random numbers and algorithms:

I was once browsing a math newsgroup, at a certain point I found a tricky question which made me think a bit.

The question sounded like this:

• consider all the numbers between 0 and 1 (Real or Rational, there is no big difference).
• Now please extract a random number among them.
• (within 3 seconds I thought: 0.4324), done?

At this point the person highlighted the following strange thing: If a set contains infinite numbers, each of them has ZERO probability. How could you extract a number?

That made me think.

This is one of the typical paradoxes based on the fact that you are told by somebody to do something, you think that you have done it, but in reality it is not true. It is generally called false choice when used by magicians or mentalists. For example, they make you choose freely a card, but in reality your choice is guided… In this case, I think it was a genuine question.

How do you pick a random number? Let’s suppose that a human brain contains a natural mechanism which allows you to select digits randomly (like those random number generators based on quantum mechanics). Let’s also suppose that a brain can think of 100 digits per second. It took me 3 seconds to answer, so at most I had the time for thinking at a number with 300 digits: a lot! (well my brain is surely much slower…).

Unfortunately, as long as the number of digits is finite, it means that the mechanism for picking random number works on a finite set of numbers (and not on an infinite set, as it appeared from the question). So we can say that at least one of them has a probability different from zero.

Another way to see the same thing is to think that, even if I had the “possibility” to choose among an infinite set of numbers, for most of them the probability was zero. However, I don’t like very much this last interpretation. I can use it for the sake of math modelling but…. there are no infinities in real life.

Infinities

Well, this should be explained with a bit more of detail since in every course of undergrad physics, one deals with all sorts of infinities.

The potential of a point charge diverges at the center. Every time an integral is carried out we are dividing an interval in an infinite set of “points” and so on and so forth. The models requiring infinities are in fact very pervasive, but curiously enough people tend to forget that no infinite quantity has ever been measured.

The reason for this is essentially the same one that prevents a person to pick a number from an infinite set. In order to measure an infinite quantity, which needs to be described with an infinite set of digits, one person needs too much time: infinite.

Epistemology

Since I am walking in the kingdom of philosophy, please let me walk a bit further.

The practical “feasibility” of an action should always be taken into account. Sometimes people have the tendency to think too theoretically. If something is theoretically feasible, then it is as if we have already done it.

Clearly, this is not true. If an action requires too much time to be completed it might become practically impossible. The same happens if the action requires too many resources.

One of the most popular definitions of science (due to Karl Popper), suggests that a statement is scientific if it is possible to disprove it. It is a very compact definition which grasps a lot of what is science, and that is probably the reason of its popularity.

I notice however one of it limits with the following example. There is at least a mushroom in Norway, underneath which lives a gnome. It is enough to look under a mushroom to check, and the number of mushrooms in Norway is clearly finite. In this respect, there is a theoretical method to disprove the sentence, and somebody might think that for this reason it becomes “scientific”. Nonetheless, the practical feasibility of checking all the mushrooms is essentially zero. According to my opinion, the scientific value of the sentence is essentially zero too.

In general, I believe that an adjustment of this definition of science should use weights for the sentences. The more “feasible” it is to disprove the sentence, the more scientific it becomes.

An even better way to see science should include not only partial disproofs but also partial proofs and hints. Think of Cosmology for example, it is a science which deals with the whole universe (and even more…) we can easily say that most of the cosmological theories, including the more accepted ones, can hardly be disproved. On the other hand, we can collect many hints which can guide our idea of the universe and build theories which are more and more solid as a function of the hints we have. I remember once at a school of general relativity with some of the big shots around, one of the finest lecturers was George Ellis, who provided a very nice survey of many possible variations regarding the structure of the universe. His approach being rather different from those of some science popularizers who give for granted that the universe has 13.6 billion years and so on… At the same school, there was Andrei Linde, one of the fathers of inflationary cosmology. He explained his idea according to which there are many inflationary “big bangs” generating infinite universes, each with its own constants of physics. When he was asked about the strength of his theory, he simply replied that he could not prove it. However, this theory was like a nice symphony and he would stick with it unless he heard a better one. The point here is that for some complex topics, the very idea of proving or disproving becomes blurry.

I know that this modified definition of science has limits (for example, is very difficult to decide the weights defining to which extent an action is feasible). However, one might accept the fact that science is not just a block of granite, but its boundaries are more like the seashore.

P.S.: People who have seen an analog tester might claim that when you measure the resistance there is a scale with an \(\infty\) sign on it. This is a bit misleading, it simply means that the resistance under observation exceeds the max range measurable by the tester.

P.P.S: I think that, among the approaches dealing with science, the paradigms shifts by Thomas Kuhn are very appealing (but less compact than Popper ideas, and thus less easy to use).