If an all knowing being offered you a very long list of digits, claiming those are the N first digits of pi, you could never verify it if N is large enough.
The mere notion of an irrational number is irrational, because you can effectively do nothing with irrational numbers that you couldn't do with rational numbers.
Their only purpose is to act as placeholders for objects that have some relationship with actual rational numbers while you take a little detour, until you circle back to a rational number that you can actually work with.

>>16726690 (OP)
To describe the laws of the universe we need differential geometry/calculus among other things and those simply don't work without complete metric spaces.
>The mere notion of an irrational number is irrational
Consider a right-angled triangle with unit side lengths, then the length of the hypotenuse will be irrational. Do you consider such an object to be "irrational"?
>you can effectively do nothing with irrational numbers that you couldn't do with rational numbers.
Ok, that's just blatantly false. The fundamental theorem of calculus doesn't hold with rationals for one.

If you are making the point that constructive methods necessarily use rational numbers due to the finiteness of our resources, then yeah no shit. You are not making as deep of a point as you think you are.

>>16726707
>To describe the laws of the universe we need differential geometry/calculus among other things and those simply don't work without complete metric spaces.
we can have complete discrete spaces. The whole "quantum" thing should be enough of a clue at this point.
>Consider a right-angled triangle with unit side lengths, then the length of the hypotenuse will be irrational. Do you consider such an object to be "irrational"?
the length of the hypothenuse is an object defined by the lengths of the sides (1 and 1). We don't need to involve a non rational number except to label the object. Do you disagree?

>If you are making the point that constructive methods necessarily use rational numbers due to the finiteness of our resources, then yeah no shit.
My point is that the finiteness of our resources is proof limits our ability to know irrational numbers. As I said, I can't even check if a big list of numbers is really a sequence of digits of pi or not, and that should be a hint that I'm headed the wrong way.

>>16726715
>we can have complete discrete spaces.
Can you describe the universe with complete discrete spaces? Can you perform e.g. fluid simulations to test aerodynamic properties of a car with discrete spaces?
>The whole "quantum" thing should be enough of a clue at this point.
Why? Whether the universe is fundamentally continuous or discrete is not at all a settled debate.
>the length of the hypothenuse is an object defined by the lengths of the sides (1 and 1). We don't need to involve a non rational number except to label the object. Do you disagree?
I do disagree. What numerical value would you assign to the length of the hypotenuse because your suggestion of definition of length is abstract nonsense. We can make (an approximation of) such a triangle in the real world and measure an actual numerical value of the length of the hypotenuse, this is not a labeling of the object.
>My point is that the finiteness of our resources is proof limits our ability to know irrational numbers. As I said, I can't even check if a big list of numbers is really a sequence of digits of pi or not
A big list of numbers is necessarily finite which means we can check exactly if it matches the digits of pi. Your misconception is that the big list is allowed to be arbitrarily large but our computational resources to check it aren't.
>and that should be a hint that I'm headed the wrong way.
Why? Shouldn't the fact that the way we use irrational numbers to explain and make very reliable predictions on almost everything happening in the world and the universe be a hint that it is the right way?

>>16726715
>The whole "quantum" thing should be enough of a clue at this point.
No, quantum is the concatenation of quantity and continuum since each quantity is based on the statistical inference from nearby quantities across an entire spacetime continuum. Continuum means a continuous field which is the opposite of discrete field which doesn't use adjacent points to calculate the value of each point in the field that each stands on its own instead of being directly integrated into an entire continuum of points.

>We don't need to involve a non rational number except to label the object.
Yes, that is the whole point, you can't come up with a clear valid numerical label for the hypotenuse without using irrationals, but when you use irrationals you have an actual number instead of a series of unfinished formulas to label the line with.

>I can't even check if a big list of numbers is really a sequence of digits of pi or not
We have mapped out hundreds of trillions of digits of pi, though, you can can definitely check if the sequence occurs within the first hundred trillion digits of pi.

>>16726707

Does the fundamental theorem of calculus work for groups? And why don't they show up in umbral calculus?

>>16726739
>Can you perform e.g. fluid simulations to test aerodynamic properties of a car with discrete spaces?
Aren't numerical simulations always occuring in discrete spaces unless you're actually solving the calc, be it manually or with a CAS?
Not a finitist retard, just nitpicking.

>>16727473
Yes, I should have phrased it differently I guess. What I mean is that the theory of fluid flow is based upon physics PDEs is based upon analysis is based upon completeness of real numbers. The theory does not work without complete metric spaces.
Then, after you have the theoretic model sorted out, you start to perform calculations/simulation and space and time are discretized because a digital computer can only work in discrete steps.

>>16726690 (OP)
but professor nosenstein said infinite sets exist because god told him they do.

>>16726690 (OP)
I know how much π is, it's π. If anything is flawed, it's base 10.

>>16726690 (OP)
Exactly how large is large enough?
Like, if N was seven, I think I could do it.