If you're having math problems, I feel bad for you, son...

[napoleon_in_rags]

26. Now there's some interesting stuff.

I didn't know about the darts... Yeah, I agree with all you're saying. My thing is that a number has to have a finite definition somewhere, and those systems of representation where a number has a precise and finite definition is where it belongs. So 1/3 belongs in the rationals, but not what I call the decimals, which exclude 0.333...

Here's what you say that really interests me though:

C and d are empirically measured quantities, so that's a separate issue of precision.

and

since they do not rely on empirical measurement, but were derived via other maths

Its striking me how odd it is that these two things are conceptually separate. We both share the sense the pi is a well defined number, definable through an infinite series. But we both know that empirical measurement is limited, and pi can never be measured beyond a certain number. So pi has this divine transcendent existence, but where is that? Where does it exist? Far more bothersome to me is the idea that the exclusively uncountable numbers (can be thought of as decimals with infinite random decimals, so they can't be specified in finite time in any system.) are also thought of as real, even though none of them can ever be represented completely in this world. So belief in them is faith based... No let me rephrase that, we can prove their existence plural. But rather the belief in any one of them can never be proven by construction, in fact I can't even think of a logical way to prove that a SINGLE given such number exists that feels right to me. They always exist in collections.

Crazy as it sounds, I'm starting to feel like the real numbers don't make much sense.

I'm thinking back. Why do we have them? We live in a universe where if we have 4 apples are in a basket and you add 3, then you have 7 in the basket, so 3+4 = 7. If we lived in a universe where the same operation yielded 8 apples, it would be a different universe, where 3+4 = 8. So our math comes from our universe. The argument for the reals, as it would be made in Newton's time, was that if a particle of light were moving across space, it must be moving in a smooth continuous line not skipping to the next discrete point, but crossing all points, including those that would be specified by uncountable reals. But we now know that doesn't happen, the light particle doesn't even have a location until a measurement is made and collapses it down to a point. And because the precision of that measurement is limited, its necessarily discrete and finite. But before that, it was just probability wave. (Sayeth quantum mechanics which I make no claim to fully understand at all.) But the point is, the universe doesn't even seem to give a damn about the uncountable numbers at its fundamental level, why should we?

I do know quantum mechanics is intimately related with information theory, and I have a childishly simple idea of what information is that has served me well: if you have a probability space of what might be, information is a function that collapses it down to a smaller space of what is, or more generally makes a new space with a smaller information entropy than the first space. So if that's how QM says the foundations of the universe are, probability spaces undifferentiated until measured, I wonder if you couldn't construct math from foundations based on the same principles. You could just say the uncountable numbers only exist as undifferentiated probability space until somebody actually provides a precise, finite definition of the number they are talking about. It would be so nice to get rid of the two infinities and just have one. Such a math would include the mathematician as observer, acting to collapse the spaces down through pieces of information. Pi and 0.999... might exist more as theories, which predict the finite outcomes of measurements rather than being thought of as "existing" with infinite definitions...

Anyway, I'm just sort of drinking my coffee and dreaming out loud. Sorry to write an essay to you, but you got me thinking...