In experiments, induction, justificationism on 20/08/2011 at 7:29 am

A philosophical problem has the form: I don’t know my way about. (Ludwig Wittgenstein)

Up until the late 19th century every observation was compatible with Newton’s theory of gravity. All these observations are also compatible with Einstein’s General Theory of Relativity. Two quite different theories were compatible with the same set of observations; therefore, one cannot know they have derived true theories from observations.

Assume we have a long series of numbers. They go on: 2, 4, 8 … What is the next number in the series?

16 works (double the last number), but so does 14 (add two, then four, then six) and 10 (alternate between adding two and adding four) works as well. And what about 21.333… (square, then divide by one; take that number then square, then divide by two…)? Or what of (X2), (X4), (X8), (X9), (X10), … Each answer assumes a specific rule, which can be as complicated as you want. The moral of the story?

Any finite series allows for an infinite number of rules.

Perhaps some heuristics might eliminate some rules and leave a much smaller number of viable rules. I don’t think this can work. For instance, parsimony, while very helpful, doesn’t imply truth, since the rule the series is following could be complex with many assumptions or it could be simple with very few assumptions. Parsimony cannot be known a priori to track the truth; therefore, it can only be known a posteriori — and only retroactively, for it may track truth only in some cases. In short, the nature of any particular rule doesn’t provide a way to determine a preference.

Our beliefs about what number comes next in the series cannot work, since subjective degrees of certainty say nothing about truth. Confirming observations affect the way we feel towards the rule, but they do not affect the rule simpliciter. Confirmations do not make theories more true. If the rule were true, we would expect to see these observations … but this is true for an infinite number of theories. Therefore, the nature of our beliefs when directed at our rules cannot determine a preference.

Confidence would be useful only when confident in the rules that correctly predict the next number in the series, and yet confidence may be held with all rules that have merely predicted the finite series of numbers. It is not a priori true, therefore it must be a posteriori true — but once again it can be know to be true in any case only retroactively: we can tell if our confidence matches the true rule only by exhausting the numbers in the series. And if the series is infinite …

Now, assume that we’re dealing with the results of experiments designed to test some scientific theory. No more mathematical rules; now we’re dealing with theories that predict mass, degree, distance, etc. Furthermore, because of the nature of experiments, there’s always a margin of error. We don’t have 2, 4, 8, but 2.43, 4.71, 8.54 …

Any finite set of observations allows for an infinite number of theories that (approximately) predict the finite set of observations.

One might think that by increasing the number of tests we will get closer to the truth. This cannot be successful, since an infinite number of theories are equally (approximately) corroborated by the current set of tests at any time t. Therefore, even if we knew where we are and how we got here, we cannot know where we are going.

Conclusion: confirming observations tell scientists nothing new about the theories they entertain.


  1. I wouldn’t necessarily jump to that conclusion. At the very least it might say that it is a set of rules that still works rather than what does not work. So maybe one can claim at least a little knowledge that it isn’t one that does not work. Just as 2, 4, 6, 8… can falsify the idea that it’s multiples of 2 by the next number being 9.

    It also appears that Wittgenstein is toying with the idea of uniqueness. If you’re looking for a little bit on this topic, there’s a proof for first order logic by Michael Titelbaum titled Not Enough There There that argues that there cannot be a 3 place evidentiary function that uniquely determines which hypothesis is the correct one. Hence, there is not a single unique hypothesis supported by any one given data set. It’s an extension of the grue problem if you’re interested. (yet another article to pile on to your list :)

    • James,

      I agree with you on the negative theology aspect of ‘knowing’ — we know that we are wrong. Of course, this ‘knowing’ is in no way justified, since in science, that ‘9’ may in due course be revised. And yet, even with that assumption of ‘knowing’, tentative though it may be, it does not entail that confirming observations tell us anything new about our explanatory theories.

      I’ll have to put the article on my list, then. It sounds a bit complicated for me to understand (I’m hopelessly inadequate to comment on mathematics), but I’ll give it a try.

      • He breaks it down into fairly straight forward argumentation. The first order logical proof is in the appendix, but the main paper does not utilize much technical prowess to navigate. Nevertheless, it is a devastating paper for understanding evidentiary relationships.

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