On 12/16/2018 10:59 PM, Jason Resch wrote:
On Mon, Dec 17, 2018 at 12:03 AM Bruce Kellett <[email protected]
<mailto:[email protected]>> wrote:
On Mon, Dec 17, 2018 at 4:30 PM Jason Resch <[email protected]
<mailto:[email protected]>> 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?
You are equivocating on the notion of "truth". You seem to be
claiming that "truth" is encapsulated in the axioms, and yet the
axioms and the given rules of inference do not encapsulate all
"truth".
I think I worded that badly. What I mean is given that truth does not
come from axioms (since they cannot encapsulate all of it), then where
does it come from? Does it have an independent, uncaused,
transcendent existence?
Do you agree that arithmetical truth has an existence
independent of the axiomatic system?
I agree that there are true statements in arithmetic that are not
theorems in any particular axiomatic system. This does not mean
that arithmetic has an existence beyond its definition in terms of
some set of axioms. You cannot go from "true" to "exists", where
"exists" means something more than the existential quantifier over
some set. Confusing the existential quantifier with an ontology is
a common mistake among some classes of mathematicians.
I agree, let us ignore "exists" for now as I think it is distracting
from the current question of whether "true statements are true"
(independent of thinking about them, defining them, uttering them, etc.).
What I am curious to know is how how many of these statements you
agree with:
"2+2 = 4" was true:
1. Before I was born
2. Before humans formalized axioms and found a proof of it
3. Before there were humans
4. Before there was any conscious life in this universe
5. As soon as there were 4 physical things to count
6. Before the big bang / before there were 4 physical things
But "2+2=4" is easily seen as an empirical theory generalized from
experience. How about "Every number has a successor." Then the answers
aren't so easy.
Brent
There are syntactically correct statements in the system
that are not theorems, and neither are their negation
theorems.
Yes.
Godel's theorem merely shows that some of these statements
may be true in a more general system.
So isn't this like scientific theories attempting to better
describe the physical world, with ever more general and more
powerful theories?
Except that physics is not an axiomatic system, and does not
confuse theorems with truth. It is not useful to classify physical
theories as 'true' or 'false',
Isn't this what professors do with physics tests? Ask there students
to prove something or determine what some physical law says should
happen? Then they grade an item as wrong if the answer given was
"false" under the working theory.
I've marked many a physics homework and test in my time and I've never
marked an answer "false".
Brent
--
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.
