On 14 Feb 2011, at 11:11, Stephen Paul King wrote:


From: Bruno Marchal
Sent: Monday, February 14, 2011 3:47 AM
To: everything-list@googlegroups.com
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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