On 12/16/2018 10:46 PM, Jason Resch wrote:
On Mon, Dec 17, 2018 at 12:00 AM Brent Meeker <[email protected]
<mailto:[email protected]>> wrote:
On 12/16/2018 9:30 PM, Jason Resch wrote:
On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett
<[email protected] <mailto:[email protected]>> wrote:
On Mon, Dec 17, 2018 at 1:50 PM Jason Resch
<[email protected] <mailto:[email protected]>> wrote:
On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett
<[email protected] <mailto:[email protected]>> wrote:
Are you claiming that there is an objective
arithmetical realm that is independent of any set of
axioms?
Yes. This is partly why Gödel's result was so shocking,
and so important.
And our axiomatisations are attempts to provide a
theory of this realm? In which case any particular
set of axioms might not be true of "real" mathematics?
It will be either incomplete or inconsistent.
Sorry, but that is silly. The realm of integers is
completely defined by a set of simple axioms -- there
is no arithmetic "reality" beyond this.
The integers can be defined, but no axiomatic system can
prove everything that happens to be true about them.
This fact is not commonly known and appreciated outside
of some esoteric branches of mathematics, but it is the case.
All that this means is that theorems do not encapsulate all
"truth".
Where does truth come from, if not the formalism of the axioms?
Do you agree that arithmetical truth has an existence independent
of the axiomatic system?
No. You are assuming that arithmetic exists apart from axioms
that define it.
I am saying truth about the integers exists independently of any
system of axioms that are capable of defining the integers.
There are true things about arithmetic that are not provable
/within arithmetic/.
It's unclear what you mean by "within arithmetic".
But that is not the same as being independent of the axioms. Some
axioms are necessary to define what is meant by arithmetic.
You need to define what you're talking about before you can talk about
it.
But mathematical objects are completely defined by their axioms. There
is no possibility of ostensive or empirical definition. That's the
strength of mathematics; it's "truths" are independent of reality, they
are part of language.
But in any case, the axioms don't define arithmetical truth, which is
my only point.
No, but they define arithmetic, without which "arithmetical truth" would
be meaningless.
Brent
If they don't, then formalism, nominalism, fictionalism, etc. all
fall, and what is left is platonism.
Jason
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send
an email to [email protected]
<mailto:[email protected]>.
To post to this group, send email to [email protected]
<mailto:[email protected]>.
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
