Probability And Induction: The Very Foundations Of Science

I have been downloading a cartload of books on my new Kindle lately, since I really enjoy the idea of walking into the subway carrying a rather inconspicuous, very light, yet incredibly large library with me. One of these books is Samir Okasha’s Philosophy of Science: A Very Short Introduction, which I’m reading because I intend to review and promote it. Samir writes very clearly, and this short introduction is very useful for the general curious reader (and, frankly, some scientists of my acquaintance could use it too).

Anyway, Samir devotes quite a bit of space in chapter 2 of his book to Hume’s problem of induction, which is fundamental to our understanding of how science -- indeed, reasoning in general -- works. Seems like the kinds of things that readers of this blog enjoy sinking their teeth into, so here we go.

The problem itself is well known: induction is the most common type of reasoning we all use (the other fundamental kind, deduction, is used largely within formal logic and mathematics), and it consists of generalizations from a series of observations. So when we say, for instance, that we are confident that the sun will rise tomorrow, this is not because we have a logical proof that it cannot be otherwise, but because we have seen it rising every day and we have no reason to think that tomorrow it will be otherwise.

As Okasha points out, we literally stake our lives on this sort of inductive reasoning, for instance every time we bet that a car will turn to the left if we rotate the steering wheel counterclockwise. (By the way, it won’t do to claim that you expect the sun to rise or the wheels to turn because you understand the mechanism: your understanding of the mechanism is itself built on a series of inductions, it is not that there is a logical necessity for solar systems or cars to work in the particular way they do work.)

The problem is that, according to Hume, there is no rational justification for induction! You see, if I’d asked you why you use inductive reasoning, pretty much the best you can do is to reply that it has worked in the past. Which is an argument based on induction. Which means you are begging the question, in philosophical terms, engaging in circular reasoning.

This may seem yet another example of philosophers engaging in intellectual masturbation, but the more you think about it the more Hume’s problem grows on you, and becomes disturbing. To quote Okasha: “If Hume is right, the foundations on which science is built do not look quite as solid as we might have hoped.” Oops.

Several ways have been proposed out of Hume’s dilemma, none of them particularly successful. I’d like to briefly discuss here the idea -- presented by Okasha in some detail -- that the concept of probability might rescue science and reason from the problem of induction. It goes something like this: granted that induction (unlike deduction) does not guarantee truth. Perhaps, however, we can rephrase what induction allows us to do in terms of probable statements. That is, we don’t really mean that we know that the sun will rise tomorrow, or that the car will turn to the left. We mean to say that, based on past experience, we think there is a high probability that those events will happen again in the future. (Incidentally, since deduction does guarantee truth, why not use it instead? Because deductive reasoning has to start with two or more premises, and at least one of those premises is arrived at via experience, not from first principles. Which means that even deduction itself has to rely on induction, at some point or another. The mystery deepens...)

Now, the problem is that philosophers have pointed out that there are at least three concepts of probability, so we have to see which, if any, of them is going to be helpful to dispel Hume’s ghost. The first way to think about probability is as a measure of the frequency of an event: if I say that the probability of a coin to land heads up is 50% I may mean that, if I flip the coin say 100 times, on average I will get heads 50 times. This is not going to get us out of Hume’s problem, because probabilities interpreted as frequencies of events are, again, a form of induction -- we generalize from a few observations to a broader range of events instead of all possible events, but the type of reasoning is the same.

Secondly, we can think of probabilities as reflecting subjective judgment. If I say that it is probable that the coin will land heads up, I might simply be trying to express my feeling that this will be the case. You might have a different feeling, and respond that you don’t think it's probable that the coin will lend heads up. This is certainly not a viable solution to the problem of induction, because subjective probabilities are, well, subjective, and hence reflect opinions, not degrees of truth.

Lastly, one can adopt what Okasha calls the logical interpretation of probabilities, according to which there is a probability X that an event will occur means that we have objective reasons to believe (or not) that X will occur (for instance, because we understand the physics of the solar system, the mechanics of cars, or the physics of coin flipping). This doesn’t mean that we will always be correct, but it does offer a promising way out of Hume’s dilemma, since it seems to ground our judgments on a more solid foundation. Indeed, this is the option adopted by many philosophers, and would be the one probably preferred by scientists, if they ever gave this sort of thing a moment’s thought. (The statistically savvy among you may have noticed that this concept of probability is not the standard frequentist one common in classical statistical analysis, but more akin to either likelihood or Bayesian methods.)

Okasha warns his readers, however, that even the logical interpretation of probabilities runs into both philosophical and mathematical problems, but we shall leave that for another time. Let me conclude with another quote from Samir’s book, which to me encapsulates the whole point of doing philosophical analysis: “Like most philosophical questions, these questions probably do not admit of final answers, but in grappling with them we learn much about the nature and limits of scientific knowledge.” Indeed.

Anyway, Samir devotes quite a bit of space in chapter 2 of his book to Hume’s problem of induction, which is fundamental to our understanding of how science -- indeed, reasoning in general -- works. Seems like the kinds of things that readers of this blog enjoy sinking their teeth into, so here we go.

The problem itself is well known: induction is the most common type of reasoning we all use (the other fundamental kind, deduction, is used largely within formal logic and mathematics), and it consists of generalizations from a series of observations. So when we say, for instance, that we are confident that the sun will rise tomorrow, this is not because we have a logical proof that it cannot be otherwise, but because we have seen it rising every day and we have no reason to think that tomorrow it will be otherwise.

As Okasha points out, we literally stake our lives on this sort of inductive reasoning, for instance every time we bet that a car will turn to the left if we rotate the steering wheel counterclockwise. (By the way, it won’t do to claim that you expect the sun to rise or the wheels to turn because you understand the mechanism: your understanding of the mechanism is itself built on a series of inductions, it is not that there is a logical necessity for solar systems or cars to work in the particular way they do work.)

The problem is that, according to Hume, there is no rational justification for induction! You see, if I’d asked you why you use inductive reasoning, pretty much the best you can do is to reply that it has worked in the past. Which is an argument based on induction. Which means you are begging the question, in philosophical terms, engaging in circular reasoning.

This may seem yet another example of philosophers engaging in intellectual masturbation, but the more you think about it the more Hume’s problem grows on you, and becomes disturbing. To quote Okasha: “If Hume is right, the foundations on which science is built do not look quite as solid as we might have hoped.” Oops.

Several ways have been proposed out of Hume’s dilemma, none of them particularly successful. I’d like to briefly discuss here the idea -- presented by Okasha in some detail -- that the concept of probability might rescue science and reason from the problem of induction. It goes something like this: granted that induction (unlike deduction) does not guarantee truth. Perhaps, however, we can rephrase what induction allows us to do in terms of probable statements. That is, we don’t really mean that we know that the sun will rise tomorrow, or that the car will turn to the left. We mean to say that, based on past experience, we think there is a high probability that those events will happen again in the future. (Incidentally, since deduction does guarantee truth, why not use it instead? Because deductive reasoning has to start with two or more premises, and at least one of those premises is arrived at via experience, not from first principles. Which means that even deduction itself has to rely on induction, at some point or another. The mystery deepens...)

Now, the problem is that philosophers have pointed out that there are at least three concepts of probability, so we have to see which, if any, of them is going to be helpful to dispel Hume’s ghost. The first way to think about probability is as a measure of the frequency of an event: if I say that the probability of a coin to land heads up is 50% I may mean that, if I flip the coin say 100 times, on average I will get heads 50 times. This is not going to get us out of Hume’s problem, because probabilities interpreted as frequencies of events are, again, a form of induction -- we generalize from a few observations to a broader range of events instead of all possible events, but the type of reasoning is the same.

Secondly, we can think of probabilities as reflecting subjective judgment. If I say that it is probable that the coin will land heads up, I might simply be trying to express my feeling that this will be the case. You might have a different feeling, and respond that you don’t think it's probable that the coin will lend heads up. This is certainly not a viable solution to the problem of induction, because subjective probabilities are, well, subjective, and hence reflect opinions, not degrees of truth.

Lastly, one can adopt what Okasha calls the logical interpretation of probabilities, according to which there is a probability X that an event will occur means that we have objective reasons to believe (or not) that X will occur (for instance, because we understand the physics of the solar system, the mechanics of cars, or the physics of coin flipping). This doesn’t mean that we will always be correct, but it does offer a promising way out of Hume’s dilemma, since it seems to ground our judgments on a more solid foundation. Indeed, this is the option adopted by many philosophers, and would be the one probably preferred by scientists, if they ever gave this sort of thing a moment’s thought. (The statistically savvy among you may have noticed that this concept of probability is not the standard frequentist one common in classical statistical analysis, but more akin to either likelihood or Bayesian methods.)

Okasha warns his readers, however, that even the logical interpretation of probabilities runs into both philosophical and mathematical problems, but we shall leave that for another time. Let me conclude with another quote from Samir’s book, which to me encapsulates the whole point of doing philosophical analysis: “Like most philosophical questions, these questions probably do not admit of final answers, but in grappling with them we learn much about the nature and limits of scientific knowledge.” Indeed.

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