On 7/14/2012 5:52 AM, Bruno Marchal wrote:

On 14 Jul 2012, at 11:16, Evgenii Rudnyi wrote:

On 14.07.2012 11:00 Bruno Marchal said the following:

On 14 Jul 2012, at 10:42, Evgenii Rudnyi wrote:

...


If to speak about your theorem, it is unclear to me, how the first
person view accesses numbers and mathematical objects.

Like a digital machine, which can access numbers encoded in their
memory, through logical gates, and so one. More details are given
currently on the FOAR list, but the idea is simple, with comp our
bodies are statistical first person constructs related to infinities
of number relations, so we access to them a bit like a fish can
access water. The price of this is that we have to abandon
physicalism eventually.

I am not sure if I understand. I would like to have an explanation for a phenomenon, for example

1) I see a cat;

2) I see a piece of paper with 2 + 2 = 4.

Yet, when you start explaining, the phenomenon seems to disappear.

1) I see a cat. This is explained by the fact that your current computational state belongs to an infinity of computations making you singling out some stable patterns that you recognize, by access to your previous experience as being cat. The qualia itself is explained by the fact that when you refer to the cat, you are really referring to yourself (with the implicit hope that it corresponds to some relatively independent pattern), and the math shows that such a self-reference involves some true but non rationally communicable feature. The math explained why, if this justification is correct, machines/numbers will not be entirely satisfied by it, for the first person is not a machine from its own first person view.

2) The same with "2+2=4 written on some paper". It is also a stable pattern in the computations going through your state. Here you might just refer to what you have learned in school, and you might considered that the truth referred by that sentence on a paper is more stable than a cat, but the conscious perception of cat or ink on paper admits the same explanation: some universal number reflect a pattern belonging to almost all computations going through your state. You have to take the first person indeterminacy into account, and keep in mind that your immediate future is determined by an infinity of computations/universal number, going through your actual state. For example, all the Heisenberg matrices computing the state of the galaxy at some description level for some amount of steps. They all provably exist independently of us in a tiny part of elementary arithmetic, and admit at least as many variants as there are possible electron location in their energy level orbitals.

I cannot be sure if this helps you as it relies to some familiarity with the first person indeterminacy and the fact that our comp states are distributed in an infinity of distinct, from a third person pov, computations (existing in arithmetic).

Bruno

Dear Bruno,

Please consider the following: from gowers (mathoverflow.net/users/1459),/Non-principal ultrafilters on ?/, http://mathoverflow.net/questions/15889 (version: 2010-02-20)


4
        

There is a nice class of problems that are equivalent to the existence of a non-principal ultrafilter. One such, if I remember correctly, is the existence of a colouring of the infinite subsets of the natural numbers in such a way that no infinite set has all its infinite subsets of the same colour. The obvious proof is to colour the sets in such a way that if you add or take away a single element then you change its colour. To make this proof work, you define two sets to be equivalent if their symmetric difference is finite, and do the colouring in each equivalence class separately. But to get it started you have to pick a set in each equivalence class, and for that the obvious thing to do is use AC.

But you can in fact do it with a non-principal ultrafilter as follows. Given an infinite subset A, define its counting function f(n) to be the cardinality of the intersection of A with {1,2,...,n}. Then take a non-principal ultrafilter ? and define F(A) to be the limit along ? of(-1)f(n). If you add an element m to A, then f(n) is unchanged up to m and then adds 1 thereafter, so its parity is changed everywhere except on a finite set, which implies that F(A) changes. So F gives you your colouring.

I've never actually thought about the other direction (getting from such a colouring to a non-principal ultrafilter) so I don't know how hard it is. I'm not even 100% sure that it's true, but I'm pretty sure I remember hearing that it was.

link <http://mathoverflow.net/questions/15872/non-principal-ultrafilters-on/15889#15889>|flag|cite
        
answeredFeb 20 2010 at 12:30
<http://mathoverflow.net/users/1459/gowers>
gowers <http://mathoverflow.net/users/1459/gowers>
16.1k?8?73?124


The idea here of a "colouring" is a partitioning that would relate to your measure, I think.


--
Onward!

Stephen

"Nature, to be commanded, must be obeyed."
~ Francis Bacon

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