Is the word "Empirical" meaningless?

[inocente]

However, is it possible to observe what begins at position 17,387,594,880 in the decimal expansion of pi without relying upon theory?

I’m thinking perhaps Hume’s Fork might substitute ‘relations of ideas’ for your ‘theoretical’, and ‘matters of fact’ for your ‘empirical’. In which case:

Relations of ideas are abstract and necessary truths, known a priori by deduction, e.g. arithmetic. Matters of fact are concrete, contingent truths, known a posteriori by induction, e.g. physics.

In the language of possible worlds, a relation of ideas doesn’t depend on history and so is true in all possible worlds, whereas a matter of fact may only be true in one or in some possible worlds, for instance in a world with a particular physics.

The question arises: suppose a piece of math M is found by deduction. Suppose also that the Theory of Everything is found, which explains all physics and therefore all nature in our world, yet it has no use for M. After 100,000 years of diligent searching, no application is ever found for M. So there is no empirical evidence for M ever occurring in nature.

Some would argue that M is true only in the sense that it follows from given axioms. We can feed data into M and observe the results, but that’s just playing with symbols. M is merely a relation of ideas, an empty truth since it occurs nowhere as a matter of fact in nature and so tells us nothing about reality.

Now pi occurs in physics and so we could guess its value to a few d.p. and confirm by empirical evidence from nature. But certainly nowhere near 17,387,594,880 d.p. accuracy, in which case the 0123456789 sequence can only be observed as a result of a formula constructed by humans, as a relation of ideas, never confirmed by observation of nature as a matter of fact. Imho.