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

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From: Bruno Marchal Sent: Monday, February 14, 2011 3:47 AM To: everything-list@googlegroups.com Subject: Re: Belief in PlatoniaHi 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 thereis an 800 Pound Gorilla in the Room that needs to be addressed andit is the nature of the assumptions that we bring into ourmodelizations. Whether the Goldbach conjecture is true or false is aquestion 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 arethe sum of two primes? No, obviously, but is the choice betweenfalsity or truth necessarily sound? Does not there exist adifference between finite and infinite sets such that we can definemeasure and ratios on the finites but not on the infinites. TheGoldbach 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 itstruth by the fact that it is infinite and has the property of anisomorphism between a proper subset of the infinity and its whole.We can distinguish instances of even number from instances ofodd numbers because we can extract from the infinity of Integers afinite subset and use it as a basis for a reasoning, but we have theproblem of induction to deal with. How can we be certain that thesubset that we extracted is typical of any other extractable subset?Additionally, how can we even consider notion of extraction unlessthe infinity that we are operating upon is a subset of an evenlarger infinity or substrate. If we are “just numbers” how can anumber reach back out from its place and operate upon some othernumber. Diagonalization is effectively the compounding ofdimensionality, no?The infinite regress rears it head!I disagree with Ultrafinitism on many grounds (particularly itsrejection of infinities which I believe can be established to existon grounds of the Completeness of Existence) but this is somethingthat has caused debates and even advances in mathematics. Witnessfor example how Brouwer’s intuitionist rejection of the law ofexcluded 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/IntuitionismI find that the Turing thesis makes sense to me, but I am freelyallowing for the premises and tacit assumptions that go with it. Butthere is a difference between the belief in an entity because itsexistence is necessary for some other to exist allowing for a chainof necessitation and the belief that something exists in order tosupport 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: amathematical statement corresponds to a mental construction, and amathematician can assert the truth of a statement only by verifyingthe validity of that construction by intuition. “ and “the claimthat an object with certain properties exists is a claim that anobject with those properties can be constructed. Any mathematicalobject is considered to be a product of a construction of a mind,and therefore, the existence of an object is equivalent to thepossibility of its construction.”If there is a mind necessary for the establishment of truth of amathematical statement and if minds can only be finite this does notpreclude the situation where we have an infinite tower of evenlarger finite minds that can construct (equivalent to their abilityto apprehend a mathematical theorem) ever more complex mathematicalstatements.Why not consider that a mind can be defined partially by theproperty of being able to construct a true statement, thus areciprocal relationship is claimed to exist between minds and thetruths of statements. I see aspects of this in your work and maybe Iam just missing the obvious because of my poor ability to interpretsymbols (dyslexia’s curse), but this does not go so far as you claimabove.We need to better understand the metaphysical underpinnings ofour models before we wander off to far to see what we considered tobe true at the start is contradicted by what we discover far intothe forest of reasonings. We saw this before in Whitehead andHilbert’s work... Are we going to re-explore the path of theScholastics, yet again?Bruno, my dear friend, did you know that the statement of “Evento say "I am not arithmetical realist" is enough to be anarithmetical realist. A real anti-ariothmetical realist cannot evenspaeak about arithmetical realism. You need to be an arithmeticalrealist to make sense of denying it.” follows the exact samereasoning of the transcendental argument of the existence of Godthus 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 thatthe Christian God is the precondition of all human knowledge andexperience, by demonstrating the impossibility of the contrary; inother words, that logic, reason, or morality cannot exist withoutGod. The argument proceeds as follows:[1]Knowledge is possible (or some other statement pertaining to logicor 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 anon-theistic point of view as well... It is this that we ought tomean when we say that we reason from the impossibility of thecontrary. 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 andReformed, 1969], p. 204).Therefore, the TAG differs from Thomistic and Evidentialistarguments, which posit the probable existence of God in order toavoid an infinite regress of causes or motions, to explain life onEarth, and so on. The TAG posits the necessary existence of aparticular conception of God in order for human knowledge andexperience to be possible at all. The TAG argues that, because thetriune God of the Bible, being completely logical, uniform, andgood, exhibits a character in the created order and the creaturesthemselves (especially in humans), human knowledge and experienceare possible. This reasoning implies that all other worldviews (suchas atheism, Buddhism, and Islam), when followed to their logicalconclusions, descend into absurdity, arbitrariness or inconsistency.”You are effectively claiming that my tentative assumption of theexistence of Numbers as existing independent of any mind for thesake of discussion of an argument necessitates that that existenceof number follow independent of my temporary and conditionalapprehensions 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 themin our hands, but we can have models or representations of them justas I can have models of pink unicorns in my mind! Conceivabilityalone does not necessitate existence, or does it?! A parrot can makesounds that can be mistaken for human speech, does this require thatthe parrot be a Realist if he happened to state “I am not aRealist”? The same would apply to Turing Machines that do notinvolve the ability to both generate internal models of themselvesas they compute some algorithm and that the behavior of thoseinternal models can have causal efficacy on the output of the Turingmachine. Mere dovetailing the model of the Machine is insufficient,the supervened system must have a means to act upon the substancethat 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 aboutthings which implies that our concept of a number or numbers pluralis merely a finite simulation of what it is like to be conscious ofnumber. Unless we allow that simulations of a thing can be the thingitself...

`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/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.