On 09 Jun 2015, at 07:21, meekerdb wrote:
On 6/8/2015 7:30 PM, LizR wrote:
On 9 June 2015 at 14:00, meekerdb <meeke...@verizon.net> wrote:
On 6/8/2015 4:16 PM, LizR wrote:
On 9 June 2015 at 05:31, meekerdb <meeke...@verizon.net> wrote:
On 6/8/2015 1:03 AM, Bruno Marchal wrote:
or that maths exists independently of mathematicians.
That even just arithmetical truth is independent of
mathematician. This is important because everyone agree with any
axiomatic of the numbers, but that is not the case for analysis,
real numbers, etc.
Everyone agrees on ZFC in the same sense. So does that make set
theory and its consequences real?
Reality isn't defined by what everyone agrees on.
Tell it to Bruno, I was just following him.
If it was then the religious majority throughout history would have
been right.
What makes ZFC (or whatever) real, or not, is whether it kicks back.
Mathematics doesn't kick back - except metaphorically.
Are you claiming an alien in another galaxy wouldn't find that
arithmetic works?
No. Is that what you mean by "kicks back"?
I'm not making any metaphysical claims about the status of maths,
merely saying that most mathematicians would, I think, agree that
two people working independently can make the same mathematical
discovery by different routes, and that some maths has real-world
applications, and that when it does, it works.
Arithmetic is a hard example to discuss because it is so simple and
probably even hardwired into our thinking by evolution (crows can
supposedly add and subtract up to six), but it's not really so
inevitable as it seems. In order to count you have to discern
distinct objects and group them in imagination into a whole: So you
count the players on a college football team (U.S.) and you get
105. Then you count the number on the basketball team of the same
school, 35, and you add them to the football team you get 140 - but
that may well be wrong. Of course you will say that's just a
misapplication; but that's the point, that arithmetic is an
abstraction that is invented to apply to certain cases and it is no
more "out there" than other aspects of language. I agree that it's
hard to imagine an intelligent species that doesn't perceive
discrete countable objects and didn't invent arithmetic to describe
them; maybe some plasma being on the surface of the the Sun that
thinks only in continua.
We need the natural number to just define computationalism, Church
thesis, etc. Once you believe in different natural numbers, then you
must explain them, and see if the existence of your notion is
threatening comp, and how. If not, you can imagine anything to avoid
any consequences of any theory.
(But I'm not sure how much kicking back you need from something,
maybe being independently discoverable and working isn't enough?)
Is it something that was invented, and could equally well have
been invented differently, or was it discovered as a result of
following a chain of logical reasoning from certain axioms?
I'd say ZFC and arithmetic were both invented and then an
axiomatization was invented for each of them. I'm not sure what
"invented differently" means?...getting to the same axiomatization
by a different historical path? Or inventing something similar,
but not identical, as ZF is different from ZFC.
It means that two people starting from the same axioms and using
the same system of logic came up with two different results (and
neither made a mistake).
That would mean either the axiom system was inconsistent or there
was a mistake in logic. Note that Graham Priest has written several
books on para-consistent logics, ones in which there can be
contradictions but don't support ex falso quodlibet.
That can be useful in AI, and for natural language. But not in QED,
string theory or theoretical computer science.
A rocket using water instead of hydrogen gas will not work. That does
not refute that rockets can work.
Bruno
If within a given system A always leads to B, then it's reasonable
to say B is discovered - like, for example, a certain endgame in
chess leading to a particular set of possible conclusions.
?? At first reading I thought you meant A logically implies B, which
means B is implicit in A. And so I thought the example was a chess
endgame in which every move is forced (except resignation), A
would be the board position and B the sequence of endgame moves.
But then you say B is a set of possible conclusions. Since chess is
a finite game the starting position already leads to a set of
possible conclusions.
But if within a system A can lead to B, C, D etc then it's
reasonable to say it's invented,
So does the fact that Peano arithmetic lead to many different
theorems mean it's invented? Does the fact that it's incomplete and
can have infinitely many new axioms added to it mean it's invented?
I don't think your criterion for distinguishing invented from
discovered reflects common usage.
like a competition to finish (within the grammatical system of
English) a poem that begins "And now the end is near..."
"And so I face the final curtain
My friend I'll say it clear
I'll state my case of which I'm certain"
Brent
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