Re: Some comments on The Mathematical Universe
I found a paper that might be of interest to those interested in Tegmark's work. http://arxiv.org/abs/0904.0867 Abstract I discuss some problems related to extreme mathematical realism, focusing on a recently proposed shut-up-and-calculate approach to physics (arXiv:0704.0646 http://arxiv.org/abs/0704.0646, arXiv:0709.4024 http://arxiv.org/abs/0709.4024). I offer arguments for a moderate alternative, the essence of which lies in the acceptance that mathematics is (at least in part) a human construction, and discuss concrete consequences of this--at first sight purely philosophical--difference in point of view. -Brian --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: The seven step series
On 20 Jul 2009, at 15:34, m.a. wrote:
Bruno,
I don't know about Kim, but I'm ready to push on. I'm
waiting for the answer to problem 2) see below. And could you please
retstate that problem as I'm not sure which one it is? Thanks,
marty a.
Let us see:
1) What is common between the set of all subsets of a set with n
elements, and the set of all finite sequences of 0 and 1 of
length
n.
2) What is common between the set of all subsets of N, and the
set of
all infinite sequences of 0 and 1.
Let me restate by introducing a definition (made precise later). The
cardinal of a set S is the number of elements in the set S.
The cardinal of { } = 0. All singletons have cardinal one. All pairs,
or doubletons, have cardinal two.
Problem 1 has been solved. They have the same cardinal, or if you
prefer, they have the same number of elements. The set of all subsets
of a set with n elements has the same number of elements than the set
of all strings of length n.
Let us write B_n for the sets of binary strings of length n. So,
B_0 = { }
B_1 = {0, 1}
B_2 = {00, 01, 10, 11}
B_3 = {000, 010, 100, 110, 001, 011, 101, 111}
We have seen, without counting, that the cardinal of the powerset of a
set with cardinal n is the same as the cardinal of B_n.
And then we have seen that such cardinal was given by 2^n.
You can see this directly by seeing that adding an element in a set,
double the number of subset, due to the dichotomic choice in creating
a subset placing or not placing the new element in the subset.
Likewise with the strings. If you have already all strings of length
n, you get all the strings of length n+1, by doubling them and adding
zero or one correspondingly.
This is also illustrated by the iterated self-duplication W, M. Mister
X is cut and paste in two rooms containing each a box, in which there
is a paper with zero on it, in room W, and 1 on it in room M. After
the experience, the 'Mister X' coming out from room W wrote 0 in his
diary, and the 'Mister X' coming out from room M wrote 1 in his diary.
And then they redo each, the experiment. The Mister-X with-0-in-his-
diary redoes it, and gives a Mister-X with-0-in-his-diary coming out
from room W, and adding 0 in its diary and a Mister-X with-0-in-his-
diary coming out from room M, adding 1 in its diary: they have the
stories
00
01,
and then the Mister-X-coming-from room W, and with 1 written in the
diary, similarly redoes the experiment, and this gives two more
Misters X, having written in their diaries
10
11.
Obviously the iteration of the self-duplication, gives as result
2x2x2x2x...x2 number of Mister X. If those four Mister X duplicate
again, there will be 8 of them, with each of those guys having an
element of B_3 written in his diary. OK.
And we have seen that a powerset of a set with n elements can but put
in a nice correspondence with B_n.
For example: The powerset of {a, b}, that is {{ }, {a}, {b}, {a, b}},
has the following nice correspondence
00 .. { }
01 .. {a}
10 ...{b}
11 ...{a, b}
Each 0 and 1 corresponding to the answer to the yes/no questions 'is a
in the subset?', is 'b in the subset?'.
Such a nice correspondence between two sets is called a BIJECTION, and
will be defined later. What we have seen, thus, is that there is a
bijection between the powerset of set with n elements, and the set of
binary strings B_n.
And the second question?
What is common between the subsets of N, and the set of infinite
binary sequences. An infinite binary sequence is a infinite sequences
of 0 and 1.
For example: 000..., with only zero is such a sequence. It
could be the first person story of 'the mister X who comes always from
room W. Or, if the zero and one represents the result of the fair coin
throw experiment, it could be the result of the infinitely unlucky
guy: he always get the head.
Another one quite similar is ..., the
infinitely lucky guy.
A more 'typical' would be 11001001110110101010001000...
(except that this very one *is* typical, it is PI written in binary;
PI = 11. 00100...).
The link with the subsets of N? It is really the same as above, except
that we extend the idea on the infinite.
A subset of N, that is, a set included in N, is entirely determined
by the answer to the following questions:
Is 0 in the subset?
Is 1 in the subset?
Is 2 in the subset?
Is 3 in the subset?
etc.
You will tell me that nobody can answer an infinity of questions. I
will answer that in many situation we can.
Let us take a simple subset of N, the set {3, 4, 7}. It seems to me we
can answer to the infinite set of corresponding question:
Is 0 in the subset?NO
Is 1 in the subset?NO
Is 2 in the subset? NO
Is 3 in the subset? YES
Is 4 in the subset?YES
Is 5 in the subset?NO
Is 6 in the subset? NO
Is 7 in
Re: Some comments on The Mathematical Universe
Exercise: criticize the following papers mentioned below in the light of the discovery of the universal machine and its main consequences from incompleteness to first person indeterminacy. Think of the identity thesis. To be sure Tegmark is less wrong than Jannes. Solution: search in the archive of this list where I have already explained this, or use directly UDA, or wait for what will (perhaps) follow. I should send some of my papers on arXiv, but up to now, only logicians understand the whole trick, so I have to better appreciated what physicians don't understand in logic, before making a version free of references to mathematical logical baggage. Logicians are not interested in mind, nor really matter, and physicians are still naïve on the link consciousness/reality, I would say. To be sure Tegmark is closer than most physicists except perhaps Wheeler. Also, Tegmarks' argument for mathematicalism is invalid (even with strong non-comp axioms). But I prefer to help you to understand this by yourself through the understanding of what a universal machine is, than trying a direct argument. According of the part of UDA (or perhaps AUDA) you understand, you can already see the weakness of such direct mathematical approach. Note that comp makes physics much more fundamental, and separate it much clearly from possible geograpies. Above all comp does not eliminate the person, which Tegmark is still doing: the frog view is not yet a first person view, in the comp sense. Interesting stuff, still. Thanks for the references. Bruno On 20 Jul 2009, at 19:44, Brian Tenneson wrote: I found a paper that might be of interest to those interested in Tegmark's work. http://arxiv.org/abs/0904.0867 Abstract I discuss some problems related to extreme mathematical realism, focusing on a recently proposed shut-up-and-calculate approach to physics (arXiv:0704.0646, arXiv:0709.4024). I offer arguments for a moderate alternative, the essence of which lies in the acceptance that mathematics is (at least in part) a human construction, and discuss concrete consequences of this--at first sight purely philosophical--difference in point of view. -Brian 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 -~--~~~~--~~--~--~---
Re: Some comments on The Mathematical Universe
Comments below. Bruno Marchal wrote: Exercise: criticize the following papers mentioned below in the light of the discovery of the universal machine and its main consequences from incompleteness to first person indeterminacy. Think of the identity thesis. To be sure Tegmark is less wrong than Jannes. Solution: search in the archive of this list where I have already explained this, or use directly UDA, or wait for what will (perhaps) follow. I should send some of my papers on arXiv, but up to now, only logicians understand the whole trick, so I have to better appreciated what physicians don't understand in logic, before making a version free of references to mathematical logical baggage. Logicians are not interested in mind, nor really matter, and physicians are still naïve on the link consciousness/reality, I would say. To be sure Tegmark is closer than most physicists except perhaps Wheeler. Also, Tegmarks' argument for mathematicalism is invalid (even with strong non-comp axioms). But I prefer to help you to understand this by yourself through the understanding of what a universal machine is, than trying a direct argument. *I need to get a better grasp on what a universal machine is, yes. I am interested in finding out how Tegmark's argument for mathematicalism is invalid, especially since I'm using it to motivate my research.* According of the part of UDA (or perhaps AUDA) you understand, you can already see the weakness of such direct mathematical approach. Note that comp makes physics much more fundamental, and separate it much clearly from possible geograpies. Above all comp does not eliminate the person, which Tegmark is still doing: the frog view is not yet a first person view, in the comp sense. Interesting stuff, still. Thanks for the references. *I'll have to think more on Jannes' paper. As I basically resting the motivation of my research on the correctness of ERH implies MUH, I'm trying to formulate a good refutation to his paper.* --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Some comments on The Mathematical Universe
On 21 Jul 2009, at 00:22, Brian Tenneson wrote: Comments below. Bruno Marchal wrote: Exercise: criticize the following papers mentioned below in the light of the discovery of the universal machine and its main consequences from incompleteness to first person indeterminacy. Think of the identity thesis. To be sure Tegmark is less wrong than Jannes. Solution: search in the archive of this list where I have already explained this, or use directly UDA, or wait for what will (perhaps) follow. I should send some of my papers on arXiv, but up to now, only logicians understand the whole trick, so I have to better appreciated what physicians don't understand in logic, before making a version free of references to mathematical logical baggage. Logicians are not interested in mind, nor really matter, and physicians are still naïve on the link consciousness/reality, I would say. To be sure Tegmark is closer than most physicists except perhaps Wheeler. Also, Tegmarks' argument for mathematicalism is invalid (even with strong non-comp axioms). But I prefer to help you to understand this by yourself through the understanding of what a universal machine is, than trying a direct argument. I need to get a better grasp on what a universal machine is, yes. I am interested in finding out how Tegmark's argument for mathematicalism is invalid, especially since I'm using it to motivate my research. At least you are aware that a mathematicalism à-la Tegmark needs a rather sophisticated universal structure, but if we assume even very weak version of comp, the universal machine provides that structure, or that structure has to be reducible as an invariant for a set of effective transformation of that machine. We can come back on this. I may be wrong also. According of the part of UDA (or perhaps AUDA) you understand, you can already see the weakness of such direct mathematical approach. Note that comp makes physics much more fundamental, and separate it much clearly from possible geograpies. Above all comp does not eliminate the person, which Tegmark is still doing: the frog view is not yet a first person view, in the comp sense. Interesting stuff, still. Thanks for the references. I'll have to think more on Jannes' paper. As I basically resting the motivation of my research on the correctness of ERH implies MUH, I'm trying to formulate a good refutation to his paper. OK, nice. My main critics is that they seem not be aware of the consciousness/ reality problem. They are using an identify thesis which is not allowed by comp. The UD argument shows exactly that. It is build to show that if we keep consciousness, eventually, physics is even more fundamental than physicist imagine. The physical world(s) is(are) not just a 'sufficiently rich' part of math, it is somehow the border of the ignorance of any (Löbian) universal machine which introspects itself. This connects in some way all part 'sufficiently rich' part of math. It explains also the non communicable part of what we can be conscious of, including physical sensations (as modalities related to self-references). 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 -~--~~~~--~~--~--~---
Re: Dreams and Machines
Brent, I intend to reply more directly to your post soon, as I think there's a lot to be said in response. But in the meantime: So I just finished reading David Deutsch's The Fabric of Reality, and I'm curious what you (Brent, Bruno, and anyone else) make of the following passage at the end of chapter 10, The Nature of Mathematics. The first paragraph is at least partly applicable to Brent's recent post, and the second seems relevant to Bruno's last response. It makes one wonder what other darkly esoteric abstractions may stalk the abyssal depths of Platonia??? The passage: Mathematical entities are part of the fabric of reality because they are complex and autonomous. The sort of reality they form is in some ways like the realm of abstractions envisaged by Plato or Penrose: although they are by definition intangible, they exist objectively and have properties that are independent of the laws of physics. However, it is physics that allows us to gain knowledge of this realm. And it imposes stringent constraints. Whereas everything in the physical reality is comprehensible, the comprehensible mathematical truths are precisely the infinitesimal minority which happen to correspond exactly to some physical truth - like the fact that if certain symbols made of ink on paper are manipulated in certain ways, certain other symbols appear. That is, they are the truths that can be rendered in virtual reality. We have no choice but to assume that the incomprehensible mathematical entities are real too, because they appear inextricably in our explanations of the comprehensible ones. There are physical objects - such as fingers, computers and brains - whose behaviour can model that of certain abstract objects. In this way the fabric of physical reality provides us with a window on the world of abstractions. It is a very narrow window and gives us only a limited range of perspectives. Some of the structures that we see out there, such as the natural numbers or the rules of inference of classical logic, seem to be important or 'fundamental' to the abstract world, in the same way as deep laws of nature are fundamental to the physical world. But that could be a misleading appearance. For what we are really seeing is only that some abstract structures are fundamental to our understanding of abstractions. We have no reason to suppose that those structures are objectively significant in the abstract world. It is merely that some abstract entities are nearer and more easily visible from our window than others. --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
