(As I mentioned some time ago, I am working on my next book that is tentatively titled The Paradox of Science. From time to time, I will try out ideas from it on the blog, suitably modified to make the blog posts self contained. Readers get the benefit of a sneak preview and I hope to get feedback from readers as to clarity, correctness, style, etc. Note that the book is aimed at the interested layperson and not the many experts who read this blog so put yourself in their shoes when reading. This post is the first of such offerings. Enjoy!)

The website Fivethirtyeight has a weekly feature called The Riddler where puzzles of a mathematical sort are presented and the solution given the following week. Here is one such problem stated in its entirety:

Complete this series:

10, 11, 12, 13, 14, 15, 16, 17, 21, 23, 30, 33, …

I want to use this simple purely mathematical puzzle to illustrate an important insight about science. I will give the solution below the jump and then discuss the relation to science.

Here is the solution given the next week by the creator of the site:

The missing numbers are 120 and 1,111. The sequence is the number 15 written in different number bases, or radices. Specifically, it’s the number “15” written in base 15, 14, 13, and so on down until base four. The missing numbers were 15 written in base three and base two.

These kinds of puzzles are common. I agree that the solution given is one possible solution. But is it the only solution? After all, we were simply given a set of numbers with no explanation of their origins. For all we know, they could have been produced by a random number generator and so the next numbers in the series could also be random. So there could be an infinite number of solutions.

But of course, there are implicit rules that are understood by puzzle-makers and puzzle solvers alike that govern such puzzles and going outside those rules is frowned upon. The main one is that the given set of numbers was produced by a closed form formula or a rule or an algorithm and the subsequent numbers should also be products of the same method. A secondary expectation is that the predicted numbers be of the same type as the ones given, so in this case one would expect them to be integers and likely positive ones.

So the puzzle is really asking us to find the formula (or rule or algorithm). But is there a unique formula? Again the answer is no. In fact, there are an infinite number of formulas that can be generated that would produce the original series of numbers and then differ in the subsequent numbers.

To understand this, let me replace the puzzle by a simpler one because the argument is the same in both cases but easier to write out.

Complete this series:

1, 2, 4, 8, …

One could again argue of course that these numbers were produced by some random process and hence there is no reason to expect any particular number to follow. But if one is told that there does exist an underlying pattern and that this set was produced using a closed form formula or some kind of algorithm, then one can reasonably infer that the numbers given are the first four obtained by starting with 1 and doubling each term to get the next one. The fifth term would then be predicted to be 16. The general formula for producing the n^th term in the series would be 2^n-1 and hence the predicted next number in the sequence, the fifth, would be 2⁴=16. Such an answer would usually be deemed correct in standard logic tests.

But is that the only possible prediction? The answer is no. Take the formula 2^n-1+(n-1)(n-2)(n-3)(n-4)m, where m can be any number at all. This formula also predicts correctly the first four terms but for the fifth it predicts 16+24m and since m can take any value, the fifth term is undetermined. If we set m=0 we get the earlier answer of 16 but there is nothing that forces us to make that choice. If m=1, then the fifth term becomes 40. If m=2, then the fifth term is 64. Since there are infinite possible values of m, we have an infinite number of formulas that work as well as 2^n-1.

But suppose we enlarge the starting data set to be explained by the formula and add a fifth term so that it is now 1, 2, 4, 8, and 16. This also satisfies the formula 2^n-1 but the alternative formula 2^n-1+(n-1)(n-2)(n-3)(n-4)m is only satisfied if m=0 and thus also gives us 2^n-1. So have we arrived at a unique formula? The answer is no because now we can generate a different alternative formula 2^n-1+(n-1)(n-2)(n-3)(n-4)(n-5)m (where again m can be any number) that satisfies the new starting data set of five terms but predicts a different sixth term. In general, however large you make the given initial sequence, there will always be an infinite number of formulas that can be constructed to fit it that will differ in their subsequent terms and there is no compelling reason to pick one. In fact, it is not just an infinite number of values of m that is the problem. I could if I wished construct an infinite number of formulas as well. Of course, one could invoke Occam’s Razor or some other principle of economy or elegance or parsimony in favor of one, but those are aesthetic criteria that may not be command universal agreement.

The analogy with science is close to exact. People tend to believe that data can uniquely constrain scientific theories. When he proposed his laws of motion and gravitation, Isaac Newton made the claim that his laws were forced on him by the data and thus free of any theoretical speculation. As Imre Lakaots says, “Newton himself thought that he proved his laws from facts. He was proud of not uttering mere hypotheses: he only published theories proven from facts. In particular, he claimed that he deduced his laws from the ‘phenomena’ provided by Kepler.” (Imre Lakatos, p. 2)

But Newton’s claim was false. Theories are always underdetermined by data, however large the data set may be. However much data we collect, there is never just a single theory that explains them. We can always find an infinite number of alternative theories that explain the same set of data. If one takes each of these theories and makes predictions for some future experiment and then carry out that experiment and generate more data (similar to the way we increased the number of elements in the starting series of numbers), the result may result in the elimination of some or even many of the competing theories. But there will emerge a different set of an infinite number of alternative theories that explain the enlarged data set. While we can invoke other criteria to reject all but one theory, these will be based on aesthetics or other subjective criteria and there are no compelling reasons that demand that we use them.

(It is also possible that Newton’s statement may not have represented his genuine feelings and been merely a clever argumentative move on his part because he knew that his invoking of gravity as an innate force that was just there and did not have a mechanical explanation was a reversion to an older way of thinking that had been rejected by his contemporaries, and indeed his critics said that adopting it would mean a return of science to the Dark Ages. By saying that he had no choice in the matter, he could deflect the charge that he was indulging in metaphysical speculations and claim that his knowledge was certain, thus meeting the standard of infallibility that was prevalent in the field of theology that was the other major source of knowledge at that time.)

Newton’s theories were not forced on him by the data, whether he genuinely thought so or not. And neither are any of our present day theories forced on us. They are the result of severely restricting the range of possible theories. But this is not done arbitrarily or even consciously. As we shall see later in this book, it is instead the result of a very long sequence of judgments that have been made through history that results in giving us, at any given point in time, what seem like a limited set of alternative theories to choose from and that data can help us decide. There are good reasons for what may seem on the surface like deliberately placing blinkers that restrict one’s vision. It is, paradoxically, one of the sources of science’s strength and success.

Reverting to our mathematical analogy, it is possible to add on sufficiently strict restrictions on the range of possible closed form formula or algorithmic solutions such that a unique solution is forced on you. For example, by requiring that the solution must be of the form p^q, where p and q are integers, then the starting sequence of 1, 2, 4, 8 does indeed predict that p=2 and q=n-1. (Proving that a solution is unique is not easy and I leave open the possibility that some ingenious mathematician may come up with an alternative formula that works as well but I cannot think of one.) Similarly in science, one could restrict the range of competing theories to just two or a handful and then use the data to eliminate all but one. This is usually the model presented in science textbooks about how one theory emerges as the dominant one. But such situations are contrived. Given a set of data, however large, and complete freedom to construct theories to explain them, scientists can come up with an infinite number.