https://www.popsci.com/quantum-hall-computer-simulation
There is math and there is computable math. What is ruled out is that the universe is computable math (aka algorithm generated). This is even true the real number line. There are numbers which have a repeating infinite expansion (rational numbers), and there are numbers which have a non-repeating infinite expansion. These are the transcendental numbers. But there are two kinds of transcendental numbers ... the algebraic and the non-algebraic. Some infinite expansions can be computed (aka Pi). But not all of them can. The set of rational numbers are trivial algebraic numbers. Pi is a non-trivial algebraic number. But there are gaps in the real number line in spite of the infinite number of algebraic numbers. These are the non-algebraic numbers, and there is an infinite number of them. The combination of the two sets, is continuous. In so far as reality is analog, then it is not computable, it can't be a computer simulation. There is a difference between emulation and simulation. A digital system (computer) can simulate an analog signal, in some respects. But not in all respects. That would be an emulation. Why? Because the derivative of a digital signal has spikes, but the derivative of the analog signal it is simulating, is smooth. An analog system is used to simulate a digital system (aka that is what a computer is doing there are no true digital systems).
So the universe may be mathematical, but it is larger than what a computable mathematics will allow.
Quote from: Baruch on September 29, 2018, 11:56:08 AM
Pi is a non-trivial algebraic number.
I was under the impression that pi is a non-algebraic number.
Quote from: Unbeliever on October 01, 2018, 02:31:42 PM
I was under the impression that pi is a non-algebraic number.
If I understand correctly, PI is an irrational number that cannot be expressed as a fraction (a ration). I'm not quite sure how that relates to algebra (I understood geometry better than algebra).
Well, irrational numbers, like the square root of two, are algebraic, because they can be the solution to an algebraic equation. But a number like e (the base of natural logarithms) or pi cannot ever be the solution to an algebraic equation, and so, though they are irrational they are transcendental, or non-algebraic.
Quote from: Unbeliever on October 01, 2018, 03:09:58 PM
Well, irrational numbers, like the square root of two, are algebraic, because they can be to solution to an algebraic equation. But a number like e (the base of natural logarithms) or pi cannot ever be the solution to an algebraic equation, and so, though they are irrational they are transcendental, or non-algebraic.
OK. that makes sense. Can an irrational number ever be algebraic?
Sure, the square root of two, among many others, is both irrational and algebraic.
Quote from: Unbeliever on October 01, 2018, 02:31:42 PM
I was under the impression that pi is a non-algebraic number.
Sorry. My bad. Substituted algebraic for computational. But the argument still holds, even though an algebraic number (saw square root of two) is easier to compute that Pi, but they both can be computed. Not all transcendental numbers (which Pi is) can be computed.
Quote from: Cavebear on October 01, 2018, 03:58:02 PM
OK. that makes sense. Can an irrational number ever be algebraic?
Square root of two is irrational, and algebraic (solution of a polynomial).
Hell, not many transcendental numbers can even be identified! LOL
Though they make up the bulk off the number line.
https://www.youtube.com/watch?v=Swm8tTLWirU
Quote from: Unbeliever on October 01, 2018, 07:52:31 PM
Hell, not many transcendental numbers can even be identified! LOL
There are a countable infinity of them (same as rational numbers). There is a system for naming any rational number (finite decimal expansion). All other numbers, irrational or transcendental have only a finite system of naming. Otherwise you can name the last digit of Pi.
I was under the impression that there's an uncountable infinity of transcendental numbers.
Quote from: Unbeliever on October 01, 2018, 08:02:10 PM
I was under the impression that there's an uncountable infinity of transcendental numbers.
Rationals and algebraic numbers are countable infinities.
Countables: because of mapping, all of these are cardinal equivalent (which is the first weird thing about infinities)
N (the natural numbers) (by definition)
Z (the integers)
Q (the rational numbers)
The algebraic numbers
The set of finite words over a finite alphabet. - this cardinal comparison shows that all countable infinities can be properly labeled/named.
Which transcendental numbers? By definition of the above line, these numbers which aren't algebraic ... can't be named by any finite system. That pretty much makes them "uncountable".
Apparently there are an infinite number of infinities, from Aleph-null on up to aleph-infinity. But is the number of infinities a countable infinity or an uncountable infinity?
Quote from: Unbeliever on October 03, 2018, 01:49:22 PM
Apparently there are an infinite number of infinities, from Aleph-null on up to aleph-infinity. But is the number of infinities a countable infinity or an uncountable infinity?
This is why mathematicians gave up on human language ;-)
Quote from: Baruch on October 01, 2018, 07:51:03 PM
Sorry. My bad. Substituted algebraic for computational. But the argument still holds, even though an algebraic number (saw square root of two) is easier to compute that Pi, but they both can be computed. Not all transcendental numbers (which Pi is) can be computed.
"easier to compute that Pi, but they both can be computed". You can compute Pi?
No, pi can't be computed, only approximated.
Quote from: Unbeliever on October 04, 2018, 02:01:48 PM
No, pi can't be computed, only approximated.
Thank you. I was pretty sure an irrational number couldn't be computed (by definition), but math discoveries surprise me frequently. If I read that someone found a way to prove calculating primes, I wouldn't be too surprised.
Quote from: Cavebear on October 04, 2018, 04:02:21 AM
"easier to compute that Pi, but they both can be computed". You can compute Pi?
One digit at a time. Not the whole of it. For a non-computable transcendental number, you can't even compute one digit at time (convergence is too slow).
Quote from: Unbeliever on October 04, 2018, 02:01:48 PM
No, pi can't be computed, only approximated.
Bad use of synonyms. An approximation isn't the same as a computation. You can with finite time, compute the millionth digit of Pi. An approximation is ... Baruch is approximately 6 ft tall. Rational numbers are simply easier to compute than algebraic numbers, and algebraic numbers are easier to compute than computable transcendental numbers like Pi. Non-computable transcendental numbers are harder still, so hard you can't get an approximation of any of the digits (infinitely slow convergence).
Quote from: Baruch on October 04, 2018, 07:10:24 PM
Bad use of synonyms. An approximation isn't the same as a computation. You can with finite time, compute the millionth digit of Pi. An approximation is ... Baruch is approximately 6 ft tall. Rational numbers are simply easier to compute than algebraic numbers, and algebraic numbers are easier to compute than computable transcendental numbers like Pi. Non-computable transcendental numbers are harder still, so hard you can't get an approximation of any of the digits (infinitely slow convergence).
I think it was obvious he was speaking both geometrically and fractionally. Fractionally, Pi is close to 22/7 but closer to the fraction 355/113. Geometrically, such things can often be described by enclosing larger and smaller polygrams but at upper limits, the maths get hideous.
Quote from: Cavebear on October 07, 2018, 06:40:52 AM
I think it was obvious he was speaking both geometrically and fractionally. Fractionally, Pi is close to 22/7 but closer to the fraction 355/113. Geometrically, such things can often be described by enclosing larger and smaller polygrams but at upper limits, the maths get hideous.
Correct on what you say. But I think he implied something else.
Quote from: Baruch on October 07, 2018, 08:51:24 AM
Correct on what you say. But I think he implied something else.
OK, what was it?
Quote from: Cavebear on October 07, 2018, 10:14:04 AM
OK, what was it?
He hasn't bothered to respond, so we must wait ...
Quote from: Baruch on October 07, 2018, 11:41:11 AM
He hasn't bothered to respond, so we must wait ...
OK, what do you THINK he implied?
Quote from: Cavebear on October 07, 2018, 12:22:23 PM
OK, what do you THINK he implied?
Lack of education in number theory. Something that applied to me, only a few years ago. Though earlier in this string, I had a faux pas.
There is the old joke about approximation ... a mathematician and an engineer are trapped in a room with an attractive other person. The rule is, you can approach the attractive person, by dividing the remaining distance in half. The mathematician gives up immediately. The engineer knows that an approximation is good enough. Mathematicians are bound by rigor in their work, like that example.
Or we can say that Unbeliever was speaking like an engineer. I one-upped with the rigorous version.
Quote from: Baruch on October 07, 2018, 01:41:56 PM
Lack of education in number theory. Something that applied to me, only a few years ago. Though earlier in this string, I had a faux pas.
There is the old joke about approximation ... a mathematician and an engineer are trapped in a room with an attractive other person. The rule is, you can approach the attractive person, by dividing the remaining distance in half. The mathematician gives up immediately. The engineer knows that an approximation is good enough. Mathematicians are bound by rigor in their work, like that example.
Or we can say that Unbeliever was speaking like an engineer. I one-upped with the rigorous version.
The old fractional approach argument was nonsense to begin with. When it got to a fraction of a second, all you had to do was measure the next full second. Problem solved.
Quote from: Cavebear on October 07, 2018, 02:02:33 PM
The old fractional approach argument was nonsense to begin with. When it got to a fraction of a second, all you had to do was measure the next full second. Problem solved.
Spoken like an engineer. Not a logician/mathematician. The other guy is playing for the other team.
Quote from: Baruch on October 07, 2018, 02:44:47 PM
Spoken like an engineer. Not a logician/mathematician. The other guy is playing for the other team.
Engineers build bridges; math guys calculate Pi. And that's a softball comment for you...
Quote from: Cavebear on October 07, 2018, 02:51:05 PM
Engineers build bridges; math guys calculate Pi. And that's a softball comment for you...
It is said that just 7 digits of Pi would be enough to make a sphere as smooth as a ball bearing, the size of the Earth. Math is more nerdy.
Quote from: Baruch on October 07, 2018, 02:55:19 PM
It is said that just 7 digits of Pi would be enough to make a sphere as smooth as a ball bearing, the size of the Earth. Math is more nerdy.
Math is ALWAYS nerdy. If you CAN calculate it, you should. And even if you can't.
If neo-Pythagoreans are half wrong, does that mean they are half right?
Quote from: Unbeliever on October 07, 2018, 05:25:10 PM
If neo-Pythagoreans are half wrong, does that mean they are half right?
Bingo ... if you have H3 etc.
If I measure my thumb, as 1 inch long (when it is 2 inches long) I am half right, half wrong. Or just don't give a shit to be accurate.
Quote from: Unbeliever on October 07, 2018, 05:25:10 PM
If neo-Pythagoreans are half wrong, does that mean they are half right?
Well, you COULD half wrong AND half wrong again. But the implication is half of both.
Neo-Pythagoreans may be half-wits, but half a wit is better than none! LOL
Quote from: Unbeliever on October 10, 2018, 01:31:20 PM
Neo-Pythagoreans may be half-wits, but half a wit is better than none! LOL
No chance. Go divide by zero!
Quote from: Unbeliever on October 10, 2018, 01:31:20 PM
Neo-Pythagoreans may be half-wits, but half a wit is better than none! LOL
Um, maybe. But half-wit is sort of pre-Gallileic thinking. Maybe Brunistic.