Author Topic: Using the counting numbers ...  (Read 370 times)

Offline Baruch

Using the counting numbers ...
« on: May 28, 2018, 02:57:39 PM »
To know is one thing, to understand is another ...

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This is the axiomatic definition of the counting numbers (starting with zero).

The definition is both logical and implicit.  We usually know them explicitly, thru counting marbles etc (addition and subtraction) and derive zero from subtraction problems (and the negative integers as well.

This is a good brain-up to get your head thinking logical, before tackling something else, like a warmup before playing sports ;-)

The axiom of induction may be necessary to clear up "next", to avoid loops in counting, and give entry to using countably infinite sets.  But it looks a lot different from the first four axioms, just like the parallel postulate in Euclidean geometry seems out of place.

So removing the induction axiom, is like removing the parallel postulate.  It opens up to more general data structures (ones that allow indirect self reference (loops) just as ignoring the parallel postulate opens us up to non-Euclidean geometry.
« Last Edit: May 28, 2018, 03:04:09 PM by Baruch »
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Offline Baruch

Re: Using the counting numbers ...
« Reply #1 on: May 28, 2018, 03:39:28 PM »
You get paradoxes if you try to add/subtract infinite series however ... that is much more tricky than doing finite operations ...

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Yes, even though it is used in QFT, this is a misleading analysis (in the video).  The result of averaging one of the series ... is wrong.  Averages are tricky.  The shifting operation isn't legit, it can't be derived from Peano Axioms (confusion between equality of regular sums vs equality of super-sums).  It is like dividing by zero (which is not a number) and not realizing it.  There are partial sums used to sum infinite series ...

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Infinite series, even with natural numbers with addition defined, requires advanced math.  This is alway why physicists (Numberphile) aren't mathematicians (the second guy).  Like in the physics textbook for undergraduates ... making a derivation ... "at this point a miracle occurs" and "the details are left as a student exercise".  This isn't good pedagogy at all.  One might come to doubt if the author of the textbook can even do that exercise themselves!

Of course the average person isn't a mathematician or a physicist ... but is innumerate.  Hence political-economic behavior is irrational.
« Last Edit: May 28, 2018, 03:54:21 PM by Baruch »
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Offline Cavebear

Re: Using the counting numbers ...
« Reply #2 on: May 29, 2018, 06:19:48 AM »
I kind of like Fibonacci numbers myself...
Atheist born, atheist bred.  And when I die, atheist dead!

Offline Baruch

Re: Using the counting numbers ...
« Reply #3 on: May 29, 2018, 01:42:55 PM »
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I kind of like Fibonacci numbers myself...

I hope you aren't fibbing ;-)
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Re: Using the counting numbers ...
« Reply #4 on: May 29, 2018, 01:47:59 PM »
I like phi, the golden ratio, which is related to the Fibonacci numbers.

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Phi can only be approximated using this, it never equals exactly phi.
« Last Edit: May 29, 2018, 01:50:22 PM by Unbeliever »
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Offline Cavebear

Re: Using the counting numbers ...
« Reply #5 on: May 29, 2018, 01:51:04 PM »
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I hope you aren't fibbing ;-)

Not by the 2, 5, 7, 12 hairs of your chinny-chin-ears!
Atheist born, atheist bred.  And when I die, atheist dead!

 

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