Using math puzzles to illustrate the nature of science


(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 nth term in the series would be 2n-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 2n-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 2n-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 2n-1 but the alternative formula 2n-1+(n-1)(n-2)(n-3)(n-4)m is only satisfied if m=0 and thus also gives us 2n-1. So have we arrived at a unique formula? The answer is no because now we can generate a different alternative formula 2n-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 pq, 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.

Comments

  1. Bruce says

    All this sounds great to me, and I’m looking forward to your book.
    But you need to finish editing this line:

    aesthetic criteria that may not be command universal agreement.

  2. --bill says

    The mathematical puzzles you present involve social constructions–the implicit restrictions on solutions are due to the social context of the puzzle. The mathematical restrictions (that whatever solution is proposed fits the initial data) are slight. To what extent does this part of the analogy carry over to one of the sciences (like physics)? What are the relative weights of social constructions and data in a physical theory?

    Also: you cite Lakatos–which book/article is this? You might want to take a look at the writings of Niccolò Guicciardini on Newton.

  3. Johnny Vector says

    This is really good. I do think it’s important that you eventually get around to the “perverse to withhold provisional assent” point. Otherwise woo-wooers will misinterpret this as “see, science can’t tell us anything!“. But I expect you will get to that later.

  4. Mano Singham says

    -bill,

    Lakatos’s is quoted from his article Science and Pseudoscience that appears in his collected works The Methodology of Scientific Research Programmes edited by Worral and Currie

  5. Mano Singham says

    Johnny,

    Yes, I will definitely be making the case against a permanent state of unjustified agnosticism on some issues.

  6. Lassi Hippeläinen says

    I’ve always disliked those next-in-sequence tricks. Any answer is correct. Just take the given numbers, add a random point, and fit a polynomial that passes through all of them.

    These puzzles do have one thing in common with scientific thinking: an implicit reference to Occam’s Razor. The question should be “what is the simplest algorithm that produces this sequence?”

    But you still need to define your metric for simplicity, e.g. the shortest computer program. There’s lots of mathematical fun in that direction, even without going into philosophy of science…

  7. Rob Grigjanis says

    Mano, can you think of an example in which there were two competing theories, with the same predictive power and domain of validity, and where the discarding of one of them was based on subjective reasons?

  8. Mano Singham says

    Rob,

    It is impossible to quantify predictive power, which is part of the problem when it comes to choosing one theory over another. Instead another subjective criterion, the importance of the prediction, is often used.

  9. John Morales says

    I actually had thought about the puzzle before reading on, and I didn’t have a clue.

    The answer is clever and makes sense after the fact — but I can’t help feeling like it’s somehow cheating: the numerical sequence is 15, 15, 15… what is changing is the notation, not the number.

    (Had the same feeling once with a similar “outside the box” problem, where the sequence was the first few digits in (English) alphabetical order)

    PS Bases don’t need to be integral…

  10. Holms says

    The irritating thing with this pattern, and something that happens to dovetail neatly into your point, is that there is a coincidence that has nothing to do with the actual pattern as explained in the solution. The numbers starting with a 1 are all separated by gaps of 1; the numbers starting with 2 are separated by gaps of 2; the numbers starting with 3 are separated by gaps of 3. That coincidence has the appearance of a pattern, and had me completely off track.

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