On 12/17/2018 8:38 PM, Jason Resch wrote:
On Mon, Dec 17, 2018 at 12:26 PM Brent Meeker <[email protected]
<mailto:[email protected]>> wrote:
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.
Are they?
Two is a mathematical object.
One of the properties of two is the number of primes it separates.
For example "3 and 5", "5 and 7", etc.
If mathematical objects are completely defined by their axioms, then
shouldn't this property be defined and known for two? Yet we don't
even know the answer to this question, we don't know if it is infinite
or finite. It might even be that no proof exists under the axioms we
currently use.
A fair point. Although that means there may be no fact of the matter.
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.
Was the physical universe meaningless before Newton?
The universe was defined as all that existed. A question about the
physical universe was probably meaningless to Thag the caveman.
Brent
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