OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Steven D'Aprano
steve at pearwood.info
Wed Jul 22 18:58:39 CEST 2015
On Wed, 22 Jul 2015 11:34 pm, Rustom Mody wrote:
> On Tuesday, July 21, 2015 at 4:09:56 PM UTC+5:30, Steven D'Aprano wrote:
>
>> We have no reason to expect that the natural numbers are anything less
>> than "absolutely fundamental and irreducible" (as the Wikipedia article
>> above puts it). It's remarkable that we can reduce all of mathematics to
>> essentially a single axiom: the concept of the set.
>
> These two statements above contradict each other.
> With the double-negatives and other lint removed they read:
I meant what I said.
> 1. We have reason to expect that the natural numbers are absolutely
> fundamental and irreducible
That's wrong. If we had such a reason, we could state it: "the reason we
expect natural numbers are irreducible is ..." and fill in the blank. But I
don't believe that such a reason exists (or at least, as far as we know).
However, neither do we have any reason to think that they are *not*
irreducible. Hence, we have no reason to think that they are anything but
irreducible.
> 2. We can reduce all of mathematics to essentially a single axiom: the
> concept of the set.
Heh, yes, that is a bit funny. But I don't think it's really a
contradiction, because I don't think that numbers "really are" sets.
The set-theoretic definition of the natural numbers is quite nice, but is it
really simpler than the old fashioned idea of natural numbers as "counting
numbers"? Certainly it shows that we have a one-to-one correspondence from
the natural numbers to sets, building from the empty set alone, and in some
sense sets seem more primitive. But I think that proving that sets actually
are more primitive is another story.
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.
By the way, my comment about "essentially a single axiom" should be read
informally. Formally, you need a whole bunch of axioms:
http://math.stackexchange.com/questions/68659/set-theoretic-construction-of-the-natural-numbers
I stated that the set-theoretic definition was generally accepted, but
that's quite far from saying it is logically proven. The definition comes
from Zermelo–Fraenkel set theory (ZF) plus the Axiom of Choice (ZFC), but
the Axiom of Choice is not uncontroversial. E.g. the Banach-Tarski paradox
follows from AC.
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
Like any other Axiom, the Axiom of Choice is unprovable. You can either
accept it or reject it, and there is also ZF¬C. Jerry Bona jokes:
"The Axiom of Choice is obviously true, the well-ordering principle
obviously false, and who can tell about Zorn's lemma?"
the joke being that all three are mathematically equivalent.
> So are you on the number-side -- Poincare, Brouwer, Heyting...
> Or the set-side -- Cantor, Russel, Hilbert... ??
Yes. I think both points of view are useful, even if only useful in
understanding what the limits of that view are.
>> On Tuesday 21 July 2015 19:10, Marko Rauhamaa wrote:
>> > Our ancestors defined the fingers (or digits) as "the set of numbers."
>> > Modern mathematicians have managed to enhance the definition
>> > quantitatively but not qualitatively.
>>
>> So what?
>>
>> This is not a problem for the use of numbers in science, engineering or
>> mathematics (including computer science, which may be considered a branch
>> of all three). There may be still a few holdouts who hope that Gödel is
>> wrong and Russell's dream of being able to define all of mathematics in
>> terms of logic can be resurrected, but everyone else has moved on, and
>> don't consider it to be "an embarrassment" any more than it is an
>> embarrassment that all of philosophy collapses in a heap when faced with
>> solipsism.
>
> That's a bizarre view.
Really? Which part(s) do you consider bizarre?
(1) That the lack of any fundamental definition of numbers in terms is not a
problem in practice?
(2) That comp sci can be considered a branch of science, engineering and
mathematics?
(3) That there may still be a few people who hope Gödel is wrong?
(4) That everyone else (well, at least everyone else in mathematics, for
some definition of "everyone") has moved on?
(5) That philosophy is unable to cope with the problem of solipsism?
I don't think any of those statements are *bizarre*, i.e. conspicuously or
grossly unconventional or unusual; eccentric; freakish; gonzo; outlandish;
outre; grotesque.
In hindsight, my claim of "everyone else" having moved on is too strong --
there are millions of mathematicians in the world, and I'm sure that you
can find one or two who don't fall into either of the two categories I
gave. That is, they neither wish to dispute Gödel nor do they accept the
irreducibility of numbers as something matter-of-fact and not embarrassing.
So let's just quietly insert an "almost" before "everyone" and move on,
shall we?
> As a subjective view "I dont feel embarrassed by..." its not arguable
> other than to say embarrassment is like thick-skinnedness -- some have
> elephant-skins some have gossamer skins
I don't think any of my comments relies on embarrassment being an objective
state.
> As an objective view its just wrong: Eminent mathematicians have disagreed
> so strongly with each other as to what putative math is kosher and what
> embarrassing that they've sent each other to mental institutions.
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;
(2) Those who understand Gödel and have given up or rejected Russell's
program to define mathematics in terms of pure logic.
> And -- most important of all -- these arguments are at the root of why CS
> 'happened' : http://blog.languager.org/2015/03/cs-history-0.html
Again, this is the past.
> The one reason why this view -- "the embarrassments in math/logic
> foundations are no longer relevant as they were in the 1930s" -- is
> because people think CS is mostly engineering, hardly math. So (the
> argument runs) just as general relativity is irrelevant to
> bridge-building, so also meta-mathematics is to pragmatic CS.
>
> The answer to this view -- unfortunately widely-held -- is the same as
> above: http://blog.languager.org/2015/03/cs-history-0.html
> A knowledge of the history will disabuse of the holder of these
> misunderstandings and misconceptions
I am very aware of the history of both comp sci and mathematics. It's not
that I think history is irrelevant. I think history is very important to
understand where we are, how we got here, and to where we are going.
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.
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."
https://en.wikipedia.org/wiki/Finitism
--
Steven
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