Author Topic: Neo-Pythagoreans are at least half wrong ...  (Read 262 times)

Offline Baruch

Neo-Pythagoreans are at least half wrong ...
« on: September 29, 2018, 11:56:08 AM »
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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.
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Re: Neo-Pythagoreans are at least half wrong ...
« Reply #1 on: October 01, 2018, 02:31:42 PM »
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Pi is a non-trivial algebraic number.

I was under the impression that pi is a non-algebraic number.
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Offline Cavebear

Re: Neo-Pythagoreans are at least half wrong ...
« Reply #2 on: October 01, 2018, 02:57:34 PM »
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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). 
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Re: Neo-Pythagoreans are at least half wrong ...
« Reply #3 on: October 01, 2018, 03:09:58 PM »
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.
« Last Edit: October 01, 2018, 04:00:00 PM by Unbeliever »
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Offline Cavebear

Re: Neo-Pythagoreans are at least half wrong ...
« Reply #4 on: October 01, 2018, 03:58:02 PM »
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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?
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Re: Neo-Pythagoreans are at least half wrong ...
« Reply #5 on: October 01, 2018, 03:59:23 PM »
Sure, the square root of two, among many others, is both irrational and algebraic.
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Offline Baruch

Re: Neo-Pythagoreans are at least half wrong ...
« Reply #6 on: October 01, 2018, 07:51:03 PM »
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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.
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Offline Baruch

Re: Neo-Pythagoreans are at least half wrong ...
« Reply #7 on: October 01, 2018, 07:51:47 PM »
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OK. that makes sense.    Can an irrational number ever be algebraic?

Square root of two is irrational, and algebraic (solution of a polynomial).
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Re: Neo-Pythagoreans are at least half wrong ...
« Reply #8 on: October 01, 2018, 07:52:31 PM »
Hell, not many transcendental numbers can even be identified! LOL


Though they make up the bulk off the number line.
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Re: Neo-Pythagoreans are at least half wrong ...
« Reply #9 on: October 01, 2018, 07:54:26 PM »
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Offline Baruch

Re: Neo-Pythagoreans are at least half wrong ...
« Reply #10 on: October 01, 2018, 07:54:57 PM »
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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.
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Re: Neo-Pythagoreans are at least half wrong ...
« Reply #11 on: October 01, 2018, 08:02:10 PM »
I was under the impression that there's an uncountable infinity of transcendental numbers.
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Offline Baruch

Re: Neo-Pythagoreans are at least half wrong ...
« Reply #12 on: October 01, 2018, 09:34:42 PM »
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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".
« Last Edit: October 01, 2018, 10:56:47 PM by Baruch »
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Re: Neo-Pythagoreans are at least half wrong ...
« Reply #13 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?
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Offline Baruch

Re: Neo-Pythagoreans are at least half wrong ...
« Reply #14 on: October 03, 2018, 03:56:26 PM »
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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 ;-)
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