On 14 Feb 2011, at 11:11, Stephen Paul King wrote:
From: Bruno Marchal
Sent: Monday, February 14, 2011 3:47 AM
To: [email protected]
Subject: Re: Belief in Platonia
Hi Bruno,
Umm, I did not mean to upset you personally.
I was not upset. May be too straight. Sorry if I look upset.
I find your ideas to be very interesting and even elegant, but there
is an 800 Pound Gorilla in the Room that needs to be addressed and
it is the nature of the assumptions that we bring into our
modelizations. Whether the Goldbach conjecture is true or false is a
question that needs to have its premise examined.
The question is not "is Goldbach conjecture true, or false", but is
"is Goldbach conjecture true or false"?
If you prefer, the question is "do you accept the third excluded
principle" for the arithmetical proposition, as I do, if only to be
able to use the (classical) Church thesis.
Can we examine all of the even integers to determine if they are
the sum of two primes? No, obviously, but is the choice between
falsity or truth necessarily sound? Does not there exist a
difference between finite and infinite sets such that we can define
measure and ratios on the finites but not on the infinites. The
Goldbach conjecture is a conjecture about an infinite set
?
It is a conjecture on the even numbers.
and thus we may be prevented from proving the decidability of its
truth by the fact that it is infinite and has the property of an
isomorphism between a proper subset of the infinity and its whole.
We can distinguish instances of even number from instances of
odd numbers because we can extract from the infinity of Integers a
finite subset and use it as a basis for a reasoning, but we have the
problem of induction to deal with. How can we be certain that the
subset that we extracted is typical of any other extractable subset?
Additionally, how can we even consider notion of extraction unless
the infinity that we are operating upon is a subset of an even
larger infinity or substrate. If we are “just numbers” how can a
number reach back out from its place and operate upon some other
number. Diagonalization is effectively the compounding of
dimensionality, no?The infinite regress rears it head!
I disagree with Ultrafinitism on many grounds (particularly its
rejection of infinities which I believe can be established to exist
on grounds of the Completeness of Existence) but this is something
that has caused debates and even advances in mathematics. Witness
for example how Brouwer’s intuitionist rejection of the law of
excluded middle lead to Heyting algebras.
But this is a red herring, as far as we limit ourselves on natural
numbers. instead of A v ~A, take ~~(~A v ~~A). But comp is classical,
an intuistionist has to say no to the doctor, because the doctor
cannot prove that he will survive.
http://en.wikipedia.org/wiki/Intuitionism
I find that the Turing thesis makes sense to me, but I am freely
allowing for the premises and tacit assumptions that go with it. But
there is a difference between the belief in an entity because its
existence is necessary for some other to exist allowing for a chain
of necessitation and the belief that something exists in order to
support claim within a theory or model.
Why is the premise of the intuitionist not more sensible, that
“the truth of a mathematical statement is a subjective claim: a
mathematical statement corresponds to a mental construction, and a
mathematician can assert the truth of a statement only by verifying
the validity of that construction by intuition. “ and “the claim
that an object with certain properties exists is a claim that an
object with those properties can be constructed. Any mathematical
object is considered to be a product of a construction of a mind,
and therefore, the existence of an object is equivalent to the
possibility of its construction.”
If there is a mind necessary for the establishment of truth of a
mathematical statement and if minds can only be finite this does not
preclude the situation where we have an infinite tower of even
larger finite minds that can construct (equivalent to their ability
to apprehend a mathematical theorem) ever more complex mathematical
statements.
Why not consider that a mind can be defined partially by the
property of being able to construct a true statement, thus a
reciprocal relationship is claimed to exist between minds and the
truths of statements. I see aspects of this in your work and maybe I
am just missing the obvious because of my poor ability to interpret
symbols (dyslexia’s curse), but this does not go so far as you claim
above.
We need to better understand the metaphysical underpinnings of
our models before we wander off to far to see what we considered to
be true at the start is contradicted by what we discover far into
the forest of reasonings. We saw this before in Whitehead and
Hilbert’s work... Are we going to re-explore the path of the
Scholastics, yet again?
Bruno, my dear friend, did you know that the statement of “Even
to say "I am not arithmetical realist" is enough to be an
arithmetical realist. A real anti-ariothmetical realist cannot even
spaeak about arithmetical realism. You need to be an arithmetical
realist to make sense of denying it.” follows the exact same
reasoning of the transcendental argument of the existence of God
thus showing a clear example of the problem that I spoke of earlier! http://en.wikipedia.org/wiki/Transcendental_argument_for_the_existence_of_God
“The TAG is a transcendental argument that attempts to prove that
the Christian God is the precondition of all human knowledge and
experience, by demonstrating the impossibility of the contrary; in
other words, that logic, reason, or morality cannot exist without
God. The argument proceeds as follows:[1]
Knowledge is possible (or some other statement pertaining to logic
or morality).
If there is no god, knowledge is not possible.
Therefore God exists.
It is similar in form to Descartes' Cogito ergo sum.[2]
Cornelius Van Til likewise wrote:
We must point out that [non-theistic] reasoning itself leads to self-
contradiction, not only from a theistic point of view, but from a
non-theistic point of view as well... It is this that we ought to
mean when we say that we reason from the impossibility of the
contrary. The contrary is impossible only if it is self-
contradictory when operating on the basis of its own assumptions.
—(A Survey of Christian Epistemology [Philadelphia: Presbyterian and
Reformed, 1969], p. 204).
Therefore, the TAG differs from Thomistic and Evidentialist
arguments, which posit the probable existence of God in order to
avoid an infinite regress of causes or motions, to explain life on
Earth, and so on. The TAG posits the necessary existence of a
particular conception of God in order for human knowledge and
experience to be possible at all. The TAG argues that, because the
triune God of the Bible, being completely logical, uniform, and
good, exhibits a character in the created order and the creatures
themselves (especially in humans), human knowledge and experience
are possible. This reasoning implies that all other worldviews (such
as atheism, Buddhism, and Islam), when followed to their logical
conclusions, descend into absurdity, arbitrariness or inconsistency.”
You are effectively claiming that my tentative assumption of the
existence of Numbers as existing independent of any mind for the
sake of discussion of an argument necessitates that that existence
of number follow independent of my temporary and conditional
apprehensions of ideas about numbers.
OK.
This is Cartesian dualism in pure form!
?
(I think you assume some primitive matter here).
We do not hold numbers in our heads any more than we can hold them
in our hands, but we can have models or representations of them just
as I can have models of pink unicorns in my mind! Conceivability
alone does not necessitate existence, or does it?! A parrot can make
sounds that can be mistaken for human speech, does this require that
the parrot be a Realist if he happened to state “I am not a
Realist”? The same would apply to Turing Machines that do not
involve the ability to both generate internal models of themselves
as they compute some algorithm and that the behavior of those
internal models can have causal efficacy on the output of the Turing
machine. Mere dovetailing the model of the Machine is insufficient,
the supervened system must have a means to act upon the substance
that underpins it and this to just escape the solipsistic case!
We have finite minds
Hmm... With comp, the mind is not clearly a finite thing, given that
its perspective is directly related to the whole indeterminacy, itself
infinite.
and thus necessarily can only contemplate finite statements about
things which implies that our concept of a number or numbers plural
is merely a finite simulation of what it is like to be conscious of
number. Unless we allow that simulations of a thing can be the thing
itself...
Well, once we say yes to the doctor, we identify ourself locally with
a relative number (the personal backup). But we are more than that.
The talk can be very confusing here, and that is why at some stage I
use the modal logic and their semantics.
Bruno
http://iridia.ulb.ac.be/~marchal/
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