On why Newton’s theory did not derive from observation.

I was back home this afternoon. I let this churn in my brain for a while, then passed on the churning to my sister. After a lot of pacing we figured out how difficult this is when you don’t recognize what’s being said and how obvious it seems after a lot of pacing.

(Chapter 8)

// My third point—the contention that it is logically impossible to derive Newton’s theory from observations—follows immediately from Hume’s critique of the validity of inductive inferences, was pointed out by Kant. Hume’s decisive point may be put as follows:

Take a class consisting of any number of true observation-statements and designate it by the letter K. The statements in the class K will describe actual observations, i.e. past observations: thus we designate by the letter K any class whatsoever of true statements about observations actually made in the past. Since we have assumed that K consists only of true statements, all statements in the class K must also be consistent statements, and, furthermore, all statements belonging to the class K must be compatible with one another. Now take a further observation-statement which we shall designate by the letter B. We assume that B describes some future, logically possible, observation; for example, that B tells us that there will be an eclipse of the sun tomorrow. Since eclipses of the sun have already been observed, we can be certain that a statement B, asserting that there will be an eclipse of the sun tomorrow, is a statement which, on purely logical grounds, is possible; that is to say, our B is self-consistent. Now Hume shows the following: if B is a self-consistent observation-statement about a possible future event, and K any class of true observation-statements about past events, then B can always be conjoined with K without contradiction; or, in other words, if we add a statement B about a possible future event to statements in K we can never arrive at a logical contradiction. Hume’s finding can also be formulated as follows: no logically possible future observation can ever contradict the class of past observations.

Let us now add to Hume’s simple finding a theorem of pure logic, namely: whenever a statement B can be conjoined without contradiction to a class of statements K, then it can also be conjoined without contradiction to any class of statements that consists of statements of K together with any statement that can be derived from K.

And so we have proved our point: if Newton’s theory could be derived from a class K of true observation-statements, then no future observation B could possibly contradict Newton’s theory and the observations K.

Yet it is known, on the other hand, that from Newton’s theory and past observations we can logically derive a statement that tells us whether or not there will be an eclipse of the sun tomorrow. Now if this derived statement tells us that tomorrow there will be no eclipse of the sun, then our B is clearly incompatible with Newton’s theory and the class K. From this and our previous results it follows logically that it is impossible to assume that Newton’s theory can be derived from observations.