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Newton knew what he had done. He was no accidental writer. A parabola, of course, is a curve that keeps on going – and that meant that at the end of a very long and very dense book, he lifted off again from the hard ground of daily reality and said, in effect, look: All this math and all these physical ideas govern everything we can see, out to and past the point where we can’t see anymore. Most important, he did so with implacable rigor, a demonstration that, he argued, should leave no room for dissent. He wrote “The theory that corresponds exactly to so nonuniform a motion through the greatest part of the heavens, and that observes the same laws as the theory of the planets and that agrees exactly with exact astronomical observations cannot fail to be true.” (Italics added).

To make his ambitions absolutely clear Newton used the same phrase for the title of book three. There his readers would discover “The System of the World.” This is where the literary structure of the work really comes into play, in my view. Through book three, Newton takes his audience through a carefully constructed tour of all the places within the grasp of his new physics. It begins with an analysis of the moons of Jupiter, demonstrating that inverse square relationships govern those motions. He went on, to show how the interaction between Jupiter and Saturn would pull each out of a perfect elliptical orbit; the real world, he says here, is messier than a geometer’s dream.

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

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.