Author Topic: Arithmetic for Cavebear  (Read 1736 times)

Offline Baruch

Arithmetic for Cavebear
« on: November 01, 2017, 06:48:03 AM »
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This continues the 1+1=2 discussion.
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Offline Cavebear

Re: Arithmetic for Cavebear
« Reply #1 on: November 04, 2017, 01:58:32 AM »
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This continues the 1+1=2 discussion.

My 9th grade Algebra 2 teacher offerred an automatic "A" to anyone who could prove 1+1=2.  Of course it can't; it's an axiom.

But I tried.  I offerred counting mechanisms.  1+1+2 coins.  Ordinal thinking.  Subtractive thinking (2 coins and remove one).  But he admired my efforts.  He said 10th grade plane geometry was more my style.  And it was.  I understood that from day one. 
Atheist born, atheist bred.  And when I die, atheist dead!

Offline Baruch

Re: Arithmetic for Cavebear
« Reply #2 on: November 04, 2017, 10:44:04 AM »
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My 9th grade Algebra 2 teacher offerred an automatic "A" to anyone who could prove 1+1=2.  Of course it can't; it's an axiom.

But I tried.  I offerred counting mechanisms.  1+1+2 coins.  Ordinal thinking.  Subtractive thinking (2 coins and remove one).  But he admired my efforts.  He said 10th grade plane geometry was more my style.  And it was.  I understood that from day one.

Of course I was the opposite.  Always like algebra/equations.  Nearly flunked geometry.  Still don't know why.  Too bad you didn't have the Internet back then, you could have given a better answer ... the Peano Axioms.

You are not allowed to view links. Register or Login  ... now if you could also play the piano ...
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Re: Arithmetic for Cavebear
« Reply #3 on: November 04, 2017, 10:46:05 AM »
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My 9th grade Algebra 2 teacher offerred an automatic "A" to anyone who could prove 1+1=2.  Of course it can't; it's an axiom.

But I tried.  I offerred counting mechanisms.  1+1+2 coins.  Ordinal thinking.  Subtractive thinking (2 coins and remove one).  But he admired my efforts.  He said 10th grade plane geometry was more my style.  And it was.  I understood that from day one. 

Surely you can prove that with a compass and a rule.
Winner of WitchSabrinas Best Advice Award 2012


We can easily forgive a child who is afraid of the dark; the real
tragedy of life is when men are afraid of the light. -Plato

Offline Baruch

Re: Arithmetic for Cavebear
« Reply #4 on: November 04, 2017, 01:49:23 PM »
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Surely you can prove that with a compass and a rule.

Yes, bisection of an angle.  Treat each section as being of unit length.  Division by 3 is impossible however, by compass and ruler (straight edge with no units).  This was like the scandal about the square root of two, but they could prove that in ancient times, but the division of an angle into thirds was not proven impossible until 1837 ...

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Has to do with number theory.  There are numbers that are inaccessible to any algorithm, not just the restrictive compass/rule algorithms.  BTW ... this is actually why real AI is impossible ... approximating real numbers with rational numbers is good enough for engineering, but not good enough for AI.  Real AI requires those non-algorithmic numbers (which is no problem with any analog system).  Analog signals are used to approximate digital numbers in electronics ... but you can't do it the other way around with complete accuracy.  The two sets of numbers are incommensurable (different classes of infinity).
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Offline pr126

Re: Arithmetic for Cavebear
« Reply #5 on: November 05, 2017, 12:59:56 AM »
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“Men occasionally stumble over the truth, but most of them pick themselves up and hurry off as if nothing ever happened.”  - Winston Churchill

Offline Baruch

Re: Arithmetic for Cavebear
« Reply #6 on: November 05, 2017, 01:08:03 AM »
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That is what computer science is.  You generate a string of ones and zeros, or transform such a string into a different string.  That is it, all of it.  This is why reality isn't a computer ... but people can make a computer.
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Re: Arithmetic for Cavebear
« Reply #7 on: November 06, 2017, 04:41:17 PM »
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My 9th grade Algebra 2 teacher offerred an automatic "A" to anyone who could prove 1+1=2.  Of course it can't; it's an axiom.

But I tried.  I offerred counting mechanisms.  1+1+2 coins.  Ordinal thinking.  Subtractive thinking (2 coins and remove one).  But he admired my efforts.  He said 10th grade plane geometry was more my style.  And it was.  I understood that from day one. 
Hell, it took Russell and Whitehead, in You are not allowed to view links. Register or Login, some 300 pages to prove that 1+1=2!

So don't feel bad if you couldn't do it so easily.
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"No matter how cynical you become, it's never enough to keep up."
Lily Tomlin

Offline Baruch

Re: Arithmetic for Cavebear
« Reply #8 on: November 06, 2017, 07:34:42 PM »
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Hell, it took Russell and Whitehead, in You are not allowed to view links. Register or Login, some 300 pages to prove that 1+1=2!

So don't feel bad if you couldn't do it so easily.

That was possible, because of the bisection of the angle.  Since trisection is impossible using compass and straight edge (in affine geometry), it would have taken a lot more to prove that 2+1=3 (you need affine geometry plus a metric).  I will prefer my practical approach, thanks.  In number theory, mathematical induction (successor function) is an axiom, it isn't attempted to be proven.

=====
Addition is a function that maps two natural numbers (two elements of N) to another one. It is defined recursively as:

a+0=a,   (1)
a+S(b)=S(a+b).   (2)

For example:

a+1=a+S(0)   by definition
 =S(a+0)      using (2)
 =S(a),      using (1)
a+2=a+S(1)   by definition
 =S(a+1)      using (2)
 =S(S(a))      using a+1=S(a)      
a+3=a+S(2)   by definition
=S(a+2)      using (2)
=S(S(S(a))) using a+2=S(S(a))   
etc.
=====

Basically assuming that ... "a commutative monoid with identity element 0" exists.  Most mathematicians agree that it does, but not all ... those who reject infinity for example.
« Last Edit: November 06, 2017, 07:38:09 PM by Baruch »
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Offline Cavebear

Re: Arithmetic for Cavebear
« Reply #9 on: November 07, 2017, 02:31:41 AM »
I surely wish I had known those arguments in 9th grade in 1965!  And I'll bet you didn't either.  LOL!
Atheist born, atheist bred.  And when I die, atheist dead!

Offline Baruch

Re: Arithmetic for Cavebear
« Reply #10 on: November 07, 2017, 06:58:11 AM »
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I surely wish I had known those arguments in 9th grade in 1965!  And I'll bet you didn't either.  LOL!

True but I was working on the old chestnut of number theory, Fermat's Last Theorem, when I was 18.  I had an insight then, but still can't connect it to the conclusion.  Sigh.  Sometimes I have my brain working on things (multitasking) for decades, and out pops an answer (happened a year ago on a different math problem).  Some people have a lazy unconscious, apparently I put mine to work in the salt mine of math.
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Offline Cavebear

Re: Arithmetic for Cavebear
« Reply #11 on: November 07, 2017, 08:33:11 AM »
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True but I was working on the old chestnut of number theory, Fermat's Last Theorem, when I was 18.  I had an insight then, but still can't connect it to the conclusion.  Sigh.  Sometimes I have my brain working on things (multitasking) for decades, and out pops an answer (happened a year ago on a different math problem).  Some people have a lazy unconscious, apparently I put mine to work in the salt mine of math.

I tend to think in terms of successfully-built objects.  You seem to think in terms of failed maths.  At least I end up with positive items. 
Atheist born, atheist bred.  And when I die, atheist dead!

Offline Baruch

Re: Arithmetic for Cavebear
« Reply #12 on: November 07, 2017, 01:24:24 PM »
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I tend to think in terms of successfully-built objects.  You seem to think in terms of failed maths.  At least I end up with positive items.

Excellent ... such objects are the grist of civilization.  Utopian politics, less so ... all failures.

Some maths isn't failed, it is impossible.  But the gullible are taken in by math and science charlatans.
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Re: Arithmetic for Cavebear
« Reply #13 on: November 07, 2017, 01:43:37 PM »
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Has to do with number theory.  There are numbers that are inaccessible to any algorithm, not just the restrictive compass/rule algorithms.  BTW ... this is actually why real AI is impossible ... approximating real numbers with rational numbers is good enough for engineering, but not good enough for AI.  Real AI requires those non-algorithmic numbers (which is no problem with any analog system).  Analog signals are used to approximate digital numbers in electronics ... but you can't do it the other way around with complete accuracy.  The two sets of numbers are incommensurable (different classes of infinity).

There are numbers that are inaccessible to any algorithm?   Really?
Winner of WitchSabrinas Best Advice Award 2012


We can easily forgive a child who is afraid of the dark; the real
tragedy of life is when men are afraid of the light. -Plato

Offline Cavebear

Re: Arithmetic for Cavebear
« Reply #14 on: November 07, 2017, 01:44:58 PM »
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There are numbers that are inaccessible to any algorithm?   Really?

The last digit of pi?
Atheist born, atheist bred.  And when I die, atheist dead!

 

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