Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]

From	Marko Rauhamaa <marko@pacujo.net>
Newsgroups	comp.lang.python
Subject	Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Date	2015-07-23 11:24 +0300
Organization	A noiseless patient Spider
Message-ID	<87si8fgu2x.fsf@elektro.pacujo.net> (permalink)
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Steven D'Aprano <steve@pearwood.info>:

> I think that we can equally choose the natural numbers to be
> axiomatic, or sets to be axiomatic and derive natural numbers from
> them. Neither is more correct than the other.

Mathematicians quit trying to define what natural numbers mean and just
chose a standard enumerable sequence as *the* set of natural numbers.
That's analogous to physicists defining the meter as a particular rod in
a vault in Paris. So not even the length of the rod but the rod itself.

To modern mathematicians, the concept "three" simply means the fourth
element in the standard enumeration. When mathematicians need to use
natural numbers to count, they have to escape to predicate logic. For
example, to express that a natural number n has precisely two divisors,
you have to say,

   ∃x∈N ∃y∈N ¬x=y ∧ x|n ∧ y|n

(which avoids counting) or:

   ∃B∈P({x∈N : x|n}×2)
     (∀x∈{x∈N : x|n} ∃y∈2 ((x,y)∈B ∧ ∀z∈2 (x,z)∈B→y=z) ∧
      ∀x∈2 ∃y∈{x∈N : x|n} ((y,x)∈B ∧ ∀z∈{x∈N : x|n} (z,x)∈B→y=z))

This latter clumsy expression captures the notion of counting. However,
in programming terms, you could say counting is not first-class in
mathematics. Counting is done as a "macro" if you will.

If the logicians had managed to define natural numbers the way they
wanted, counting would be first class and simple:

   {x∈N : x|n}∈2


Marko

Thread

Re: Should non-security 2.7 bugs be fixed? Devin Jeanpierre <jeanpierreda@gmail.com> - 2015-07-18 19:33 -0700
  Re: Should non-security 2.7 bugs be fixed? Rick Johnson <rantingrickjohnson@gmail.com> - 2015-07-18 19:49 -0700
  Re: Should non-security 2.7 bugs be fixed? Rustom Mody <rustompmody@gmail.com> - 2015-07-18 20:52 -0700
    Re: Should non-security 2.7 bugs be fixed? Rick Johnson <rantingrickjohnson@gmail.com> - 2015-07-18 21:18 -0700
    Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Steven D'Aprano <steve@pearwood.info> - 2015-07-19 14:45 +1000
      Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Chris Angelico <rosuav@gmail.com> - 2015-07-19 15:06 +1000
        Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Marko Rauhamaa <marko@pacujo.net> - 2015-07-19 10:16 +0300
          Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-07-19 00:32 -0700
            Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Marko Rauhamaa <marko@pacujo.net> - 2015-07-19 10:44 +0300
            Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Terry Reedy <tjreedy@udel.edu> - 2015-07-19 19:13 -0400
              Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-07-19 19:02 -0700
      Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Anuradha Laxminarayan <lanuradha@gmail.com> - 2015-07-18 23:25 -0700
      Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Terry Reedy <tjreedy@udel.edu> - 2015-07-19 04:26 -0400
      Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Tim Chase <python.list@tim.thechases.com> - 2015-07-19 07:56 -0500
      Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Chris Angelico <rosuav@gmail.com> - 2015-07-20 04:07 +1000
      Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Tim Chase <python.list@tim.thechases.com> - 2015-07-19 14:55 -0500
      Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Chris Angelico <rosuav@gmail.com> - 2015-07-20 07:16 +1000
        Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] wxjmfauth@gmail.com - 2015-07-20 00:43 -0700
      Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] MRAB <python@mrabarnett.plus.com> - 2015-07-19 23:13 +0100
      Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-07-20 20:30 -0700
        Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Chris Angelico <rosuav@gmail.com> - 2015-07-21 13:43 +1000
          Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-07-20 23:11 -0700
        Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Laura Creighton <lac@openend.se> - 2015-07-21 10:10 +0200
          Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Marko Rauhamaa <marko@pacujo.net> - 2015-07-21 12:10 +0300
            Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Chris Angelico <rosuav@gmail.com> - 2015-07-21 19:18 +1000
              Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Marko Rauhamaa <marko@pacujo.net> - 2015-07-21 13:13 +0300
            Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Mark Lawrence <breamoreboy@yahoo.co.uk> - 2015-07-21 11:34 +0100
            Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2015-07-21 20:39 +1000
              Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Marko Rauhamaa <marko@pacujo.net> - 2015-07-21 13:54 +0300
                Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Marko Rauhamaa <marko@pacujo.net> - 2015-08-09 00:27 +0300
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-08-09 09:29 -0700
              Re:  Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-07-22 06:34 -0700
                OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Steven D'Aprano <steve@pearwood.info> - 2015-07-23 02:58 +1000
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Laura Creighton <lac@openend.se> - 2015-07-22 19:17 +0200
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-07-22 10:49 -0700
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Laura Creighton <lac@openend.se> - 2015-07-22 20:14 +0200
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Mark Lawrence <breamoreboy@yahoo.co.uk> - 2015-07-22 21:59 +0100
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Steven D'Aprano <steve@pearwood.info> - 2015-07-23 03:21 +1000
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Dennis Lee Bieber <wlfraed@ix.netcom.com> - 2015-07-22 21:44 -0400
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Chris Angelico <rosuav@gmail.com> - 2015-07-23 12:00 +1000
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Paul Rubin <no.email@nospam.invalid> - 2015-07-22 10:48 -0700
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-07-22 10:51 -0700
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2015-07-23 15:14 +1000
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-07-22 11:09 -0700
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2015-07-23 15:41 +1000
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Marko Rauhamaa <marko@pacujo.net> - 2015-07-23 23:59 +0300
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Chris Angelico <rosuav@gmail.com> - 2015-07-24 07:03 +1000
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Marko Rauhamaa <marko@pacujo.net> - 2015-07-24 00:29 +0300
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Mark Lawrence <breamoreboy@yahoo.co.uk> - 2015-07-23 22:50 +0100
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Laura Creighton <lac@openend.se> - 2015-07-23 23:52 +0200
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Marko Rauhamaa <marko@pacujo.net> - 2015-07-24 00:59 +0300
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Chris Angelico <rosuav@gmail.com> - 2015-07-24 08:02 +1000
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Chris Angelico <rosuav@gmail.com> - 2015-07-24 08:00 +1000
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] MRAB <python@mrabarnett.plus.com> - 2015-07-23 23:01 +0100
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Laura Creighton <lac@openend.se> - 2015-07-24 00:19 +0200
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Mark Lawrence <breamoreboy@yahoo.co.uk> - 2015-07-23 23:56 +0100
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Grant Edwards <invalid@invalid.invalid> - 2015-07-24 00:07 +0000
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Rick Johnson <rantingrickjohnson@gmail.com> - 2015-07-23 18:40 -0700
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Paul Rubin <no.email@nospam.invalid> - 2015-07-23 19:03 -0700
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Rick Johnson <rantingrickjohnson@gmail.com> - 2015-07-23 20:16 -0700
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Grant Edwards <invalid@invalid.invalid> - 2015-07-24 14:13 +0000
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Paul Rubin <no.email@nospam.invalid> - 2015-07-24 08:45 -0700
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Mark Lawrence <breamoreboy@yahoo.co.uk> - 2015-07-24 16:58 +0100
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-07-23 22:15 -0700
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security   2.7 bugs be fixed?] Gregory Ewing <greg.ewing@canterbury.ac.nz> - 2015-07-23 18:57 +1200
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-07-23 02:12 -0700
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Rustom Mody <rustompmody@gmail.com> - 2015-07-23 05:52 -0700
                Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?] Marko Rauhamaa <marko@pacujo.net> - 2015-07-23 11:24 +0300
        Re: Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?] Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2015-07-21 18:57 +1000
    Re: Should non-security 2.7 bugs be fixed? Terry Reedy <tjreedy@udel.edu> - 2015-07-19 02:44 -0400
    Re: Should non-security 2.7 bugs be fixed? Terry Reedy <tjreedy@udel.edu> - 2015-07-19 05:11 -0400
      Re: Should non-security 2.7 bugs be fixed? Rustom Mody <rustompmody@gmail.com> - 2015-07-19 07:30 -0700
    Re: Should non-security 2.7 bugs be fixed? Mark Lawrence <breamoreboy@yahoo.co.uk> - 2015-07-19 15:00 +0100
  Re: Should non-security 2.7 bugs be fixed? Steven D'Aprano <steve@pearwood.info> - 2015-07-19 14:45 +1000
    Re: Should non-security 2.7 bugs be fixed? Devin Jeanpierre <jeanpierreda@gmail.com> - 2015-07-19 18:20 -0700
      Re: Should non-security 2.7 bugs be fixed? Steven D'Aprano <steve@pearwood.info> - 2015-07-20 13:05 +1000
        Re: Should non-security 2.7 bugs be fixed? Devin Jeanpierre <jeanpierreda@gmail.com> - 2015-07-19 20:41 -0700
    Re: Should non-security 2.7 bugs be fixed? Mark Lawrence <breamoreboy@yahoo.co.uk> - 2015-07-20 02:46 +0100
      Re: Should non-security 2.7 bugs be fixed? Rustom Mody <rustompmody@gmail.com> - 2015-07-19 19:16 -0700
        Re: Should non-security 2.7 bugs be fixed? Chris Angelico <rosuav@gmail.com> - 2015-07-20 12:59 +1000
        Re: Should non-security 2.7 bugs be fixed? Mark Lawrence <breamoreboy@yahoo.co.uk> - 2015-07-20 11:59 +0100
        Re: Should non-security 2.7 bugs be fixed? Rick Johnson <rantingrickjohnson@gmail.com> - 2015-07-20 20:04 -0700
          Re: Should non-security 2.7 bugs be fixed? Rustom Mody <rustompmody@gmail.com> - 2015-07-20 20:15 -0700
            Re: Should non-security 2.7 bugs be fixed? Chris Angelico <rosuav@gmail.com> - 2015-07-21 13:33 +1000
            Re: Should non-security 2.7 bugs be fixed? Terry Reedy <tjreedy@udel.edu> - 2015-07-21 00:45 -0400
          Re: Should non-security 2.7 bugs be fixed? breamoreboy@gmail.com - 2015-07-21 14:22 -0700
            Re: Should non-security 2.7 bugs be fixed? Rick Johnson <rantingrickjohnson@gmail.com> - 2015-07-21 19:07 -0700
              Re: Should non-security 2.7 bugs be fixed? Terry Reedy <tjreedy@udel.edu> - 2015-07-22 02:51 -0400
                Re: Should non-security 2.7 bugs be fixed? Rick Johnson <rantingrickjohnson@gmail.com> - 2015-07-22 16:37 -0700
    Re: Should non-security 2.7 bugs be fixed? Terry Reedy <tjreedy@udel.edu> - 2015-07-20 02:25 -0400
    Re: Should non-security 2.7 bugs be fixed? dieter <dieter@handshake.de> - 2015-07-20 08:58 +0200
    Re: Should non-security 2.7 bugs be fixed? Devin Jeanpierre <jeanpierreda@gmail.com> - 2015-07-20 00:13 -0700

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