From: Bruno Marchal
Sent: Monday, February 14, 2011 3:47 AM
To: [email protected]
Subject: Re: Belief in Platonia
Do you believe that Goldbach conjecture is either true or false? If you agree
with this, then you accept arithmetical realism, which is enough for the comp
consequences.
Do you believe that Church thesis makes sense? That is enough to say that you
believe in the 'arithmetical platonia'. People needs to be ultrafinitist to
reject the arithmetical platonia.
Personnaly I am a bit skeptical on set realism, because it is hard to define
it, but for the numbers I have never met people who are not realist about them.
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.
Bruno
snip[SPK]
> Don’t we need to be able to define exactly what Life is first, in order
to know whether or not it is Turing Emulable? AFAIK there is no agreed upon
definition of life and the folks that believe that Life is Turing emulable seem
> to do so as a act of faith, given that there is no evidence at all that such
is possible or impossible. Additionally, the existence of the Platonic realm
cannot be established by empirical means nor logical necessity even if some >
famous people wrote papers about it, its existence is mere conjecture. Thus it
too is believed as an act of faith. There has not be a resolution to the debate
between nominalism and universals that I know of, so the jury is still > out on
even the objective existence of numbers.
> I grew up among people with faith like that, except they believed in a
God that would condemn mankind to an existence in a lake of fire for not
accepting blah blah ... until I realized that it was all a power game to
control my > mind. Thus am very leery of beliefs that cannot be justified by
either empirical evidence or logical necessity or some combination of the two.
> One can tentatively accept the existence of some entity for the sake of
an argument and see what the implications are, but to base one’s ontology on
such without very careful deliberation is to engage in the same kind of >
irrationality that we disdain religions fanatics for. I have been holding back
on expressing this thought here, but seriously, we need to step back and
reconsider what we are taking for granted in our “reasonings”.
> I must admit this because I have been advocating for a form of dualism
that would claim that numbers and even information has an objective existence
of sorts but this dualism is not so bold as the dualism that is inherent > in
the belief in Platonia. The Platonic realm is obviously not a physical place
and thus has to be considered as separate from our world of experience. Roger
Penrose seems to be the only person to be up front about this
> aspect of Platonism and he got his books panned for his honest attempt to
defend his claims.
Hi Bruno,
Umm, I did not mean to upset you personally. 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.
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
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.
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]
1.. Knowledge is possible (or some other statement pertaining to logic or
morality).
2.. If there is no god, knowledge is not possible.
3.. 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. This is
Cartesian dualism in pure form! 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 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...
Onward!
Stephen
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