On 12/12/2018 3:19 AM, Bruno Marchal wrote:
On 10 Dec 2018, at 20:26, Brent Meeker <[email protected]
<mailto:[email protected]>> wrote:
On 12/9/2018 11:38 PM, Philip Thrift wrote:
On Sunday, December 9, 2018 at 8:43:59 PM UTC-6, Jason wrote:
On Sun, Dec 9, 2018 at 2:02 PM Philip Thrift <[email protected]
<javascript:>> wrote:
On Sunday, December 9, 2018 at 9:36:39 AM UTC-6, Jason wrote:
On Sun, Dec 9, 2018 at 2:53 AM Philip Thrift
<[email protected]> wrote:
On Saturday, December 8, 2018 at 2:27:45 PM UTC-6,
Jason wrote:
I think truth is primitive.
Jason
As a matter of linguistics (and philosophy), *truth*
and *matter* are linked:
"As a matter of fact, ..."
"The truth of the matter is ..."
"It matters that ..."
...
[ https://www.etymonline.com/word/matter
<https://www.etymonline.com/word/matter> ]
I agree they are linked. Though matter may be a few
steps removed from truth. Perhaps one way to interpret
the link more directly is thusly:
There is an equation whose every solution (where the
equation happens to be */true/*, e.g. is satisfied when
it has certain values assigned to its variables) maps
its variables to states of the time evolution of the
wave function of our universe. You might say that we
(literally not figuratively) live within such an
equation. That its truth reifies what we call matter.
But I think truth plays an even more fundamental roll
than this. e.g. because the following statement is
*/true/* "two has a successor" then there exists a
successor to 2 distinct from any previous number.
Similarly, the */truth/* of "9 is not prime" implies the
existence of a factor of 9 besides 1 and 9.
Jason
Schopenhauer 's view: "A judgment has /material
truth/ if its concepts are based on intuitive
perceptions that are generated from sensations. If a
judgment has its reason (ground) in another
judgment, its truth is called logical or formal. If
a judgment, of, for example, pure mathematics or
pure science, is based on the forms (space, time,
causality) of intuitive, empirical knowledge, then
the judgment has transcendental truth."
[ https://en.wikipedia.org/wiki/Truth
<https://en.wikipedia.org/wiki/Truth> ]
I guess I am referring to transcend truth here. Truth
concerning the integers is sufficient to yield the
universe, matter, and all that we see around us.
Jason
In my view there is basically just *material* (from matter)
truth and *linguistic* (from language) truth.
[
https://codicalist.wordpress.com/2018/06/18/to-tell-the-truth/
<https://codicalist.wordpress.com/2018/06/18/to-tell-the-truth/>
]
Relations and functions are linguistic: relational type
theory (RTT) , functional type theory (FTT) languages.
Numbers are also linguistic beings, the (fictional) semantic
objects of Peano arithmetic (PA).
Numbers can be "materialized" via /nominalization /(cf.
Hartry Field, refs. in [
https://en.wikipedia.org/wiki/Hartry_Field
<https://en.wikipedia.org/wiki/Hartry_Field> ]).
Assuming the primacy of matter assumes more and explains less,
than assuming the primacy of arithmetical truth.
Jason
In today's era of mathematics, Joel David Hamkins (@JDHamkins
<https://twitter.com/JDHamkins>) has shown there is a "multiverse"
of truths:
*The set-theoretic multiverse*
[ https://arxiv.org/abs/1108.4223 ]
/The multiverse view in set theory, introduced and argued for in
this article, is the view that there are many distinct concepts of
set, each instantiated in a corresponding set-theoretic universe.
The universe view, in contrast, asserts that there is an absolute
background set concept, with a corresponding absolute set-theoretic
universe in which every set-theoretic question has a definite
answer. The multiverse position, I argue, explains our experience
with the enormous diversity of set-theoretic possibilities, a
phenomenon that challenges the universe view. In particular, I argue
that the continuum hypothesis is settled on the multiverse view by
our extensive knowledge about how it behaves in the multiverse, and
as a result it can no longer be settled in the manner formerly hoped
for.
/
/
/
/
/
What this means is that for mathematics (a language category), truth
depends on the language.
I think Hamkins could say the same thing in French. His example of
the continuum hypothesis just says that by adding as axioms different
undecidable propositions we get different sets of theorems. He
doesn't use the word "truth" and I think with good reason. Theorems
in mathematics aren't "true" in any normal sense of the word. What
is true is that the axioms imply the theorem...given the rules of
inference.
By incompleteness truth is bigger than what *any* consistent can
prove. Of course you can say that the meaning of
[Exyz(x^3 + y^3 + z^3 = 33) v ~ (Exyz(x^3 + y^3 + z^3 = 33))
Is true because it is an substitution instance of the tautology A v ~A.
Which is exactly what I'd say; and is the only way that you can know it
is true.
But you can also reflect on this, and believe it is true, because
either such numbers do exist, or they don’t.
The phrase following "because" is just rephrasing A v ~A.
But "Exyz(x^3 + y^3 + z^3 = 33) “ is an open problem. Such open
problem would not make sense if you define truth by proof.
Many mathematicians would say you have */discovered/* it. But you could
only discover it by finding a proof I don't define truth by proof. You
can prove what is false just by choice of axioms. So having a proof
doesn't prove the conclusion is true; it only shows that it is true that
the conclusion follows from the premises.
Recently I have proven that TOT (the set of Gödel numbers of the codes
of the total computable function is not recursively enumerable. That
proofs would have no meaning if truth = proof. Intuitionism is
recovered in Mechanism as the particular first person solipsistic view
([]p & p, S4Grz1).
Many mathematicians would say you have */discovered/* it. But you could
only discover it by finding a proof.
In mathematical logic, we study since long the difference between
proof and truth (satisfied by a reality).
Yes, you create a whole theology around not all truths are provable.
But you ignore that what is false is also provable. Provable is only
relative to axioms.
Brent
Bruno
PS I will use the word “reality" instead of “model", as most
physicisst continue to use “model” for theories.
Brent
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