/sci/ - Science & Math

>>12714549
This is an awful point. Just because some system is practically unpredictable, doesn't mean it's indistinguishable from free will. We could still potentially have a theoretical description which is deterministic and even though we can't use it to predict large-scale stuff, we know it's right by observing it work in small scale particularly designed systems. Such a description would by a legitimate argument for determinism and hence against free will. And it has been a great argument for determinists at the time of classical physics, when it was reasonable to believe that deterministic physical laws as known in the day govern the whole universe.
Unfortunately, due to the advent of quantum mechanics, this argument no longer works. There is no inference to determinism from physics.
The point stands that you are simply wrong.

>>12673538
Even accepting the infinitist framework of mathematics your definition is way too vague to be useful and to reflect the actual modern understanding of the real numbers.
This is not least because there is no a priori good reason why the number line should have the least upper bound property.
You could just as well take the number line to be the set [math]\bar{\mathbb{Q}}[/math] of algebraic numbers, or even the smaller set of constructible numbers and do Euclidean geometry on them perfectly fine.
But suppose you really want completeness, i.e. you really want your number line to have the least upper bound property, even though you can't justify it a priori.
Then the elephant that comes into the room is the Suslin problem: is the set R the only set fitting our "number line" intuitions? I.e. is it the only nonempty totally ordered set X such that
- X is unbounded in either direction
- X is dense as an order.
- X has the least upper bound property.
and a less intuitive technical requirement that
- Any collection of disjoint open intervals is countable.
...