OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Oscar Benjamin
oscar.j.benjamin at gmail.com
Wed Jul 22 19:52:09 CEST 2015
On Wed, 22 Jul 2015 at 18:01 Steven D'Aprano <steve at pearwood.info> wrote:
>
> I think that the critical factor there is that it is all in the past tense.
> Today, I believe, the vast majority of mathematicians fall into two camps:
>
> (1) Those who just use numbers without worrying about defining them in some
> deep or fundamental sense;
>
Probably. I'd say that worrying too much about the true essence of numbers
is just Platonism. Numbers are a construct (a very useful one). There are
many other constructs used within mathematics and there are numerous ways
to connect them or define them in terms of each other. Usually these are
referred to as "connections" or sometimes more formally as "isomorphisms"
and they can be useful but don't need to have any metaphysical meaning.
Conventional mathematics treats the natural numbers as subsets of the
complex numbers and usually treats the complex numbers as the most basic
type of numbers. Exactly how you construct this out of sets is not as
important as the usefulness of this concept when actually trying to use
numbers.
(2) Those who understand Gödel and have given up or rejected Russell's
> program to define mathematics in terms of pure logic.
>
<snip>
> It's that I think that Russell's program is a degenerate research program
> and irrelevant paradigm abandoned by nearly everyone. Not only does Gödel
> prove the impossibility of Russell's attempt to ground mathematics in pure
> logic, but mathematicians have by and large rejected Russell's paradigm as
> irrelevant, like quintessence or aether to physicists, or how many angels
> can dance on the head of a pin to theologians. In simple terms, hardly
> anyone cares how you define numbers, so long as the definition gives you
> arithmetic.
>
Actually in the decades since the incompleteness theorems were published
much of mathematics has simply ignored the problem. Hilbert's idea to
construct everything out of formal systems of axioms and proof rules
continues to be pushed to its limits. This is now a standard approach in
the literature, in textbooks and published papers, and in undergraduate
programs. In contrast Gödel's (Platonist IMO) intuitionist idea of
mathematical proof is ignored.
The thing is that it turns out that even if you can't prove everything then
you can at least prove a lot: Gödel demonstrated the existence of at least
one unprovable theorem. Since we know that there are loads of unproven
theorems and that loads of them continue to be proven all the time we
clearly haven't yet hit any kind of "Gödel limit" that would impede further
progress.
> Quoting Wikipedia:
>
> "In the years following Gödel's theorems, as it became clear that there is
> no hope of proving consistency of mathematics, and with development of
> axiomatic set theories such as Zermelo–Fraenkel set theory and the lack of
> any evidence against its consistency, most mathematicians lost interest in
> the topic."
>
They lost interest in the topic of proving consistency, completeness etc.
They didn't lose interest in creating an explosion of different sets of
axioms and proof systems, studying the limits of each and trying to push as
much of conventional mathematics as possible into grand frameworks.
For a modern example of Hilbert's legacy take a look at these guys:
http://us.metamath.org/mpegif/mmtheorems.html
They've tried to construct all of mathematics out of set theory using a
fully formal (computer verifiable) proof database. They define the natural
numbers as a subset of the complex numbers:
http://us.metamath.org/mpegif/mmtheorems87.html#mm8625s
The complex numbers themselves are defined in terms of the ordinal numbers
which are similar to the natural numbers but have a distinct definition in
terms of sets:
http://us.metamath.org/mpegif/mmtheorems77.html#mm7635s
I think they did it that way because it's just too awkward if the complex
number 1 isn't the same as the natural number 1.
--
Oscar
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