Chris Angelico wrote:
> On Wed, Nov 5, 2014 at 11:31 PM, Ivan Evstegneev
> <webmailgro...@gmail.com> wrote:
>>>> That's what I'm talking about (asking actually), where do you know it
>>>> from?
>>
>>>>I know it because I've been a programmer for 39 years.
>>
>> I didn't intend to offence anyone here. Just asked a questions ^_^
>
> Don't worry about offending people. Even if you do annoy one or two,
> there'll be plenty of us who know to be patient :) And I don't think
> Larry was actually offended; it's just that some questions don't
> really have easy answers - imagine someone asking a great
> mathematician "But how do you KNOW that 2 + 2 is 4? Where's it written
> down?"... all he can say is "It is".
An ordinary mathematician will say: "Hold up two fingers. Count them, and
you get one, two. Now hold up another two fingers. Count them, and you will
get two again. Hold them together, count the lot, and you get one, two,
three, four. Therefore, 2+2 = 4."
A good mathematician might start with the empty set, ∅ = {}. [Aside: if the
symbol looks like a small box, try changing your font -- it is supposed to
be a circle with a slash through it. Lucinda Typewriter has the glyph
for '\N{EMPTY SET}'.] That empty set represents zero. Take the set of all
empty sets, {∅} = {{}}, which represents one. Now we know how to count:
after any number, represented by some set, the *next* number is represented
by the simplest set containing the previous set.
Having defined counting, the good mathematician can define addition, and go
on to prove that 2+2 = 4. This is, essentially, a proof of Peano Arithmetic
(PA), which one can take as effectively the basic arithmetic of counting
fingers, sheep or sticks.
But a *great* mathematician will say, "Hmmm, actually, we don't *know* that
2+2 equals 4, because we cannot prove that arithmetic is absolutely
consistent. If arithmetic is not consistent, then we might simultaneously
prove that 2+2 = 4 and 2+2 ≠ 4, which is unlikely but not inconceivable."
Fields medallist Vladimir Voevodsky is a great mathematician, and he
apparently believes that the consistency of Peano Arithmetic is still an
open question.
http://m-phi.blogspot.com.au/2011/05/voevodsky-consistency-of-pa-is-open.html
Another way to look at this, not necessarily Voevodsky's approach, is to
note that the existing proofs of PA's consistency are *relative* proofs of
PA. E.g. they rely on the consistency of some other formal system, such as
the Zermelo-Frankel axioms (ZF). If ZF is consistent, so is PA, but we
don't know that ZF is consistent...
--
Steven
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