Re: The seven step-Mathematical preliminaries
Brian Tenneson skrev:
How do you know that there is no biggest number? Have you examined all
the natural numbers? How do you prove that there is no biggest number?
In my opinion those are excellent questions. I will attempt to answer
them. The intended audience of my answer is everyone, so please forgive
me if I say something you already know.
Firstly, no one has or can examine all the natural numbers. By that I
mean no human. Maybe there is an omniscient machine (or a maximally
knowledgeable in some paraconsistent way) who can examine all numbers
but that is definitely putting the cart before the horse.
Secondly, the question boils down to a difference in philosophy:
mathematicians would say that it is not necessary to examine all natural
numbers. The mathematician would argue that it suffices to examine all
essential properties of natural numbers, rather than all natural numbers.
There are a variety of equivalent ways to define a natural number but
the essential features of natural numbers are that
(a) there is an ordering on the set of natural numbers, a well
ordering. To say a set is well ordered entails that every =nonempty=
subset of it has a least element.
(b) the set of natural numbers has a least element (note that it is
customary to either say 0 is this least element or say 1 is this least
element--in some sense it does not matter what the starting point is)
(c) every natural number has a natural number successor. By successor
of a natural number, I mean anything for which the well ordering always
places the successor as larger than the predecessor.
Then the set of natural numbers, N, is the set containing the least
element (0 or 1) and every successor of the least element, and only
successors of the least element.
There is nothing wrong with a proof by contradiction but I think a
forward proof might just be more convincing.
Consider the following statement:
Whenever S is a subset of N, S has a largest element if, and only if,
the complement of S has a least element.
By complement of S, I mean the set of all elements of N that are not
elements of S.
Before I give a longer argument, would you agree that statement is
true? One can actually be arbitrarily explicit: M is the largest
element of S if, and only if, the successor of M is the least element of
the compliment of S.
I do not agree that statement is true. Because if you call the Biggest
natural number B, then you can describe N as = {1, 2, 3, ..., B}. If
you take the complement of N you will get the empty set. This set have
no least element, but still N has a biggest element.
In your statement you are presupposing that N has no biggest element,
and from that axiom you can trivially deduce that there is no biggest
element.
--
Torgny Tholerus
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Re: The seven step-Mathematical preliminaries
If you are ultrafinitist then by definition the set N does not
exist... (nor any infinite set countably or not).
If you pose the assumption of a biggest number for N, you come to a
contradiction because you use the successor operation which cannot
admit a biggest number.(because N is well ordered any successor is
strictly bigger and the successor operation is always valid *by
definition of the operation*)
So either the set N does not exists in which case it makes no sense to
talk about the biggest number in N, or the set N does indeed exists
and it makes no sense to talk about the biggest number in N (while it
makes sense to talk about a number which is strictly bigger than any
natural number).
To come back to the proof by contradiction you gave, the assumption
(2) which is that BIGGEST+1 is in N, is completely defined by the mere
existence of BIGGEST. If BIGGEST exists and well defined it entails
that BIGGEST+1 is not in N (but this invalidate the successor
operation and hence the mere existence of N). If BIGGEST in contrary
does not exist (as such, means it is not the biggest) then BIGGEST+1
is in N by definition of N.
Regards,
Quentin
2009/6/4 Torgny Tholerus tor...@dsv.su.se:
Brian Tenneson skrev:
How do you know that there is no biggest number? Have you examined all
the natural numbers? How do you prove that there is no biggest number?
In my opinion those are excellent questions. I will attempt to answer
them. The intended audience of my answer is everyone, so please forgive
me if I say something you already know.
Firstly, no one has or can examine all the natural numbers. By that I
mean no human. Maybe there is an omniscient machine (or a maximally
knowledgeable in some paraconsistent way) who can examine all numbers
but that is definitely putting the cart before the horse.
Secondly, the question boils down to a difference in philosophy:
mathematicians would say that it is not necessary to examine all natural
numbers. The mathematician would argue that it suffices to examine all
essential properties of natural numbers, rather than all natural numbers.
There are a variety of equivalent ways to define a natural number but
the essential features of natural numbers are that
(a) there is an ordering on the set of natural numbers, a well
ordering. To say a set is well ordered entails that every =nonempty=
subset of it has a least element.
(b) the set of natural numbers has a least element (note that it is
customary to either say 0 is this least element or say 1 is this least
element--in some sense it does not matter what the starting point is)
(c) every natural number has a natural number successor. By successor
of a natural number, I mean anything for which the well ordering always
places the successor as larger than the predecessor.
Then the set of natural numbers, N, is the set containing the least
element (0 or 1) and every successor of the least element, and only
successors of the least element.
There is nothing wrong with a proof by contradiction but I think a
forward proof might just be more convincing.
Consider the following statement:
Whenever S is a subset of N, S has a largest element if, and only if,
the complement of S has a least element.
By complement of S, I mean the set of all elements of N that are not
elements of S.
Before I give a longer argument, would you agree that statement is
true? One can actually be arbitrarily explicit: M is the largest
element of S if, and only if, the successor of M is the least element of
the compliment of S.
I do not agree that statement is true. Because if you call the Biggest
natural number B, then you can describe N as = {1, 2, 3, ..., B}. If
you take the complement of N you will get the empty set. This set have
no least element, but still N has a biggest element.
In your statement you are presupposing that N has no biggest element,
and from that axiom you can trivially deduce that there is no biggest
element.
--
Torgny Tholerus
--
All those moments will be lost in time, like tears in rain.
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Re: Cognitive Theoretic Model of the Universe
Russell: Maybe you might be interested in gfortran(http://gcc.gnu.org/wiki/ GFortran)? Ronald On Jun 2, 6:38 pm, russell standish li...@hpcoders.com.au wrote: On Tue, Jun 02, 2009 at 07:45:22AM -0700, ronaldheld wrote: Bruno: Since I program in Fortran, I am uncertain how to interpret things. Ronald Maybe if he said Fortran IV or Fortran 66, it might have made the point clearer. I know guys who still program in Fortran 66. The rest of us have moved on ... Fortran 95 is not a bad language to program in for instance - and 2003 has some interesting features, although I don't know of any freely available compilers. Personally, I went C++ in the early 90s because g++ was available and the equivalent for Fortran 90 was not (gfortran or g95 arrived by about 2000 IIRC). -- ---- Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 hpco...@hpcoders.com.au Australia http://www.hpcoders.com.au ---- --~--~-~--~~~---~--~~ 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-Mathematical preliminaries 2
On Thu Jun 4 1:15 , Bruno Marchal sent:
Very good answer, Kim,
Just a few comments. and then the sequel.
Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
3, ...}?
No idea what square-root(2) means. When I said I was innumerate I wasn't
kidding! I
could of course look
it up or ask my mathematics teacher friends but I just know your explanation
will make
theirs seem trite.
Well thanks. The square root of 2 is a number x, such that x*x (x times x, x
multiplied by
itself) gives 2.For example, the square root of 4 is 2, because 2*2 is 4. The
square root of
9 is 3, because 3*3 is 9. Her by square root I mean the positive square root,
because we
will see (more later that soon) that numbers can have negative square root, but
please
forget this. At this stage, with this definition, you can guess that the square
root of 2
cannot be a natural number. 1*1 = 1, and 2*2 = 4, and it would be astonishing
that x
could be bigger than 2. So if there is number x such that x*x is 2, we can
guess that such
a x cannot be a natural number, that is an element of {0, 1, 2, 3 ...}, and the
answer of
exercise 4 is no. The square root of two will reappear recurrently, but more
in examples,
than in the sequence of notions which are strictly needed for UDA-7.
OK - I find this quite mind-blowing; probably because I now understand it for
the first
time in my life. So how did it get this rather ridiculous name of square
root? What's it
called in French?
(snip)
=== Intension and extension
Before defining intersection, union and the notion of subset, I would like to
come back
on the ways we can define some specific sets.
In the case of finite and little set we have seen that we can define them by
exhaustion.
This means we can give an explicit complete description of all element of the
set. Example. A = {0, 1, 2, 77, 98, 5}
When the set is still finite and too big, or if we are lazy, we can sometimes
define the set
by quasi exhaustion. This means we describe enough elements of the set in a
manner
which, by requiring some good will and some imagination, we can estimate having
define
the set.
Example. B = {3, 6, 9, 12, ... 99}. We can understand in this case that we
meant the set of
multiple of the number three, below 100.
A fortiori, when a set in not finite, that is, when the set is infinite, we
have to use either
quasi-exhaustion, or we have to use some sentence or phrase or proposition
describing
the elements of the set.
Definition. I will say that a set is defined IN EXTENSIO, or simply, in
extension, when it is
defined in exhaustion or quasi-exhaustion. I will say that a set is defined IN
INTENSIO, or
simply in intension, with an s, when it is defined by a sentence explaining
the typical
attribute of the elements.
Example: Let A be the set {2, 4, 6, 8, 10, ... 100}. We can easily define A in
intension: A
= the set of numbers which are even and smaller than 100. Mathematicians will
condense
this by the following:
A = {x such that x is even and smaller than 100} = {x ⎮ x is even x
special character, abbreviating such that, and I hope it goes through the
mail.
Just an upright line? It comes through as that. I can't seem to get this symbol
happening so I will
use such that
If not I will use such that, or s.t., or things like that.The expression
{x ⎮ x is even} is
literally read as: the set of objects x, (or number x if we are in a context
where we talk
about numbers) such that x is even.
Exercise 1: Could you define in intension the following infinite set C = {101,
103, 105,
...}C = ?
C = {x such that x is odd and x 101}
Exercise 2: I will say that a natural number is a multiple of 4 if it can be
written as 4*y,
for some y. For example 0 is a multiple of 4, (0 = 4*0), but also 28, 400, 404,
... Could
you define in extension the following set D = {x ⎮ x 10 and x is a multiple
of 4}?
D = 4*x where x = 0 but also { 1, 2, 3, 4, 8 }
I now realise I am doomed for the next set of exercises because I cannot get to
the special
symbols required (yet). As I am adding Internet Phone to my system, I am
currently using an
ancient Mac without the correct symbol pallette while somebody spends a few
days to flip a single
switch...as soon as I can get back to my regular machine I will complete the
rest.
In the meantime I am enjoying the N+1 disagreement - how refreshing it is to
see that classical
mathematics remains somewhat controversial!
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Re: The seven step-Mathematical preliminaries
Quentin Anciaux skrev: If you are ultrafinitist then by definition the set N does not exist... (nor any infinite set countably or not). All sets are finite. It it (logically) impossible to construct an infinite set. You can construct the set N of all natural numbers. But that set must be finite. What the set N contains depends on how you have defined natural number. If you pose the assumption of a biggest number for N, you come to a contradiction because you use the successor operation which cannot admit a biggest number.(because N is well ordered any successor is strictly bigger and the successor operation is always valid *by definition of the operation*) You have to define the successor operation. And to do that you have to define the definition set for that operation. So first you have to define the set N of natural numbers. And from that you can define the successor operator. The value set of the successor operator will be a new set, that contains one more element than the set N of natural numbers. This new element is BIGGEST+1, that is strictly bigger than all natural numbers. -- Torgny Tholerus So either the set N does not exists in which case it makes no sense to talk about the biggest number in N, or the set N does indeed exists and it makes no sense to talk about the biggest number in N (while it makes sense to talk about a number which is strictly bigger than any natural number). To come back to the proof by contradiction you gave, the assumption (2) which is that BIGGEST+1 is in N, is completely defined by the mere existence of BIGGEST. If BIGGEST exists and well defined it entails that BIGGEST+1 is not in N (but this invalidate the successor operation and hence the mere existence of N). If BIGGEST in contrary does not exist (as such, means it is not the biggest) then BIGGEST+1 is in N by definition of N. Regards, Quentin --~--~-~--~~~---~--~~ 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-Mathematical preliminaries
This is a denial of the axiom of infinity. I think a foundational set theorist might agree that it is impossible to -construct- an infinite set from scratch which is why they use the axiom of infinity. People are free to deny axioms, of course, though the result will not be like ZFC set theory. The denial of axiom of foundation is one I've come across; I've never met anyone who denies the axiom of infinity. For me it is strange that the following statement is false: every natural number has a natural number successor. To me it seems quite arbitrary for the ultrafinitist's statement: every natural number has a natural number successor UNTIL we reach some natural number which does not have a natural number successor. I'm left wondering what the largest ultrafinist's number is. Torgny Tholerus wrote: Quentin Anciaux skrev: If you are ultrafinitist then by definition the set N does not exist... (nor any infinite set countably or not). All sets are finite. It it (logically) impossible to construct an infinite set. You can construct the set N of all natural numbers. But that set must be finite. What the set N contains depends on how you have defined natural number. If you pose the assumption of a biggest number for N, you come to a contradiction because you use the successor operation which cannot admit a biggest number.(because N is well ordered any successor is strictly bigger and the successor operation is always valid *by definition of the operation*) You have to define the successor operation. And to do that you have to define the definition set for that operation. So first you have to define the set N of natural numbers. And from that you can define the successor operator. The value set of the successor operator will be a new set, that contains one more element than the set N of natural numbers. This new element is BIGGEST+1, that is strictly bigger than all natural numbers. --~--~-~--~~~---~--~~ 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-Mathematical preliminaries
Brian Tenneson skrev: This is a denial of the axiom of infinity. I think a foundational set theorist might agree that it is impossible to -construct- an infinite set from scratch which is why they use the axiom of infinity. People are free to deny axioms, of course, though the result will not be like ZFC set theory. The denial of axiom of foundation is one I've come across; I've never met anyone who denies the axiom of infinity. For me it is strange that the following statement is false: every natural number has a natural number successor. To me it seems quite arbitrary for the ultrafinitist's statement: every natural number has a natural number successor UNTIL we reach some natural number which does not have a natural number successor. I'm left wondering what the largest ultrafinist's number is. It is impossible to lock a box, and quickly throw the key inside the box before you lock it. It is impossible to create a set and put the set itself inside the set, i.e. no set can contain itself. It is impossible to create a set where the successor of every element is inside the set, there must always be an element where the successor of that element is outside the set. What the largest number is depends on how you define natural number. One possible definition is that N contains all explicit numbers expressed by a human being, or will be expressed by a human being in the future. Amongst all those explicit numbers there will be one that is the largest. But this largest number is not an explicit number. -- Torgny Tholerus --~--~-~--~~~---~--~~ 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: Consciousness is information?
Hi Jesse, On 01 May 2009, at 19:36, Jesse Mazer wrote: I found a paper on the Mandelbrot set and computability, I understand very little but maybe Bruno would be able to follow it: http://arxiv.org/abs/cs.CC/0604003 The same author has a shorter outline or slides for a presentation on this subject at http://www.cs.swan.ac.uk/cie06/files/d37/PHP_MandelbrotCiE2006Swansea_Jul2006.pdf and at the end he asks the question If M (Mandelbrot set) not Q- computable, can the Halting Problem be reduced to determining membership of (intersection of M and Q^2), i.e. how powerful a 'hypercomputer' is the Mandelbrot set? I believe Q^2 here just refers to the set of all possible pairs of rational numbers. Maybe by reducing the Halting Problem he means that for any Turing machine + input, there might be some rule that would translate it into a pair of rational numbers such that the computation will halt iff the pair is included in the Mandelbrot set? Whatever he means, it sounds like he's saying it's an open question... Jesse On Thu, Apr 30, 2009 at 10:35 AM, Bruno Marchal marc...@ulb.ac.be wrote: The mathematical Universal Dovetailer, the splashed universal Turing Machine, the rational Mandelbrot set, or any creative sets in the sense of Emil Post, does all computations. Really all, with Church thesis. This is a theorem in math. The rock? Show me just the 30 first steps of a computation of square-root(2). ... Bruno, I am interested about your statement regarding the Mandelbrot set implementing all computations, could you elaborate on this? So, indeed the conjecture I made on the Mandelbrot Set concerns the decidability-on-the-rationals of the set M intersected with QXQ. And it is indeed still an open problem. Actually my question is the creativity (in the sense of Post) of M, and this would mean that you can reduce the halting problem of any Turing machine into a problem of membership of a rational complex number a+bi (a, b, in Q) to M. There would be one fixed algorithm transforming any computable problem on N into such a membership problem. If the solution is positive, then the Mandelbrot Set would be a compact representation of a Universal Dovetailing. Also, this would entail the existence of interesting relationship between classical computability theory and the theory of Chaos on the reals. The universality in chaos phenomenon (Feigenbaum) would be related to the Turing Universality. Also, each of us would be, in a sense, distributed densely on the boundary of M, and each little Mandelbrot would represent the third person projection view of each of our first person plenitude. That would be cute, mainly for the pedagogy of the UD, but also, it would made it possible to borrow mathematical tools from chaos theory theory for the pursue of deriving physics from numbers. Not everything is clear for me in Potgieter paper, probably a result of my incomptence, but it is very interesting. Thanks for the link. Did I give you the link of the last, impressive M-zoom by phaumann? Look at it with the high quality option + full screen, if you are patient enough. Love it! http://www.youtube.com/watch?v=x6DD1k4BAUgfeature=channel_page 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: The seven step-Mathematical preliminaries
Torgny Tholerus wrote: Brian Tenneson skrev: This is a denial of the axiom of infinity. I think a foundational set theorist might agree that it is impossible to -construct- an infinite set from scratch which is why they use the axiom of infinity. People are free to deny axioms, of course, though the result will not be like ZFC set theory. The denial of axiom of foundation is one I've come across; I've never met anyone who denies the axiom of infinity. For me it is strange that the following statement is false: every natural number has a natural number successor. To me it seems quite arbitrary for the ultrafinitist's statement: every natural number has a natural number successor UNTIL we reach some natural number which does not have a natural number successor. I'm left wondering what the largest ultrafinist's number is. It is impossible to lock a box, and quickly throw the key inside the box before you lock it. I disagree. It is impossible to create a set and put the set itself inside the set, i.e. no set can contain itself. No one here is suggesting that you can with regards to natural numbers. It is impossible to create a set where the successor of every element is inside the set, there must always be an element where the successor of that element is outside the set. I disagree. Can you prove this? Once again, I think the debate ultimately is about whether or not to adopt the axiom of infinity. I think everyone can agree without that axiom, you cannot build or construct an infinite set. There's nothing right or wrong with adopting any axioms. What results is either interesting or not, relevant or not. What the largest number is depends on how you define natural number. One possible definition is that N contains all explicit numbers expressed by a human being, or will be expressed by a human being in the future. Amongst all those explicit numbers there will be one that is the largest. But this largest number is not an explicit number. This raises a deeper question which is this: is mathematics dependent on humanity or is mathematics independent of humanity? I wonder what would happen to that human being who finally expresses the largest number in the future. What happens to him when he wakes up the next day and considers adding one to yesterday's number? --~--~-~--~~~---~--~~ 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: Consciousness is information?
On 03 Jun 2009, at 20:11, Jason Resch wrote: On Fri, May 22, 2009 at 4:37 PM, Bruno Marchal marc...@ulb.ac.be wrote: Do you believe if we create a computer in this physical universe that it could be made conscious, But a computer is never conscious, nor is a brain. Only a person is conscious, and a computer or a brain can only make it possible for a person to be conscious relatively to another computer. So your question is ambiguous. It is not my brain which is conscious, it is me who is conscious. My brain appears to make it possible for my consciousness to manifest itself relatively to you. Remember that we are supposed to no more count on the physical supervenience thesis. It remains locally correct to attribute a consciousness through a brain or a body to a person we judged succesfully implemented locally in some piece of matter (like when we say yes to a doctor). But the piece of matter is not the subject of the consciousness. It is only the abstract person or program who is the subject of consciousness. To say a brain is conscious consists in doing Searle's'mistake when he confused levels of computations in the Chinese room, as well seen already by Hofstadter and Dennett in Mind's I. Thanks for your response, if I understand you correctly, you are saying that if we run a simulation of a mind, we are not creating consciousness, only adding an additional instantiation to a mind which already has an infinity of indeterminable instantiations. Is that right? Yes, you are right. When you implement an emulation of a mind, you are just adding such an instanciation relatively to you. Of course you are not adding anything in Platonia. Does this imply that it is impossible to create a simulation of a mind that finds it lives in an environment without uncertainty? That is correct. If so is it because even if the physical laws in one instantiation may be certain, where some of the infinite number of computations that all instantiate that mind may diverge and in particular which one that mind will find itself in is not knowable? Yes. I will come back on this in the seven step thread. The consequence being that all observers everywhere live in QM-like environments? Absolutely. We can consider that we live in an infinity of computations, but we cannot distinguish them ... until they differentiate sufficiently so that they are in principle distinguishable (like being in Washington or being in Moscow). This entails that below our substitution level what can be observed depends directly on some average on an infinity of computations. The quantum-like aspect of nature is, in that sense, a consequence of digitalism in the cognitive science. The classical, and computational, aspect of physics remains the hard things to derive. 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: Cognitive Theoretic Model of the Universe
Hi Ronald, On 02 Jun 2009, at 16:45, ronaldheld wrote: Bruno: Since I program in Fortran, I am uncertain how to interpret things. I was alluding to old, and less old, disputes again programmers, about which programming language to prefer. It is a version of Church Thesis that all algorithm can be written in FORTRAN. But this does not mean that it is relevant to define an algorithm by a fortran program. I thought this was obvious, and I was using that known confusion to point on a similar confusion in Set Theory, like Langan can be said to perform. In Set Theorist, we still find often the error consisting in defining a mathematical object by a set. I have done that error in my youth. What you can do, indeed, is to *represent* (almost all) mathematical objects by sets. Langan seems to make that mistake. The point is just that we have to distinguish a mathematical object and the representation of that object in some mathematical theory. I will have the opportunity to give a precise example in the 7th thread later. In usual mathematical practice, this mistake is really not important, yet, in logic it is more important to take into account that distinction, and then in cognitive science it is *very* important. Crucial, I would say. The error consisting in identifying consciousness and brain state belongs to that family, for example. To confuse a person and its body belongs to that family of error too. All such error are of the form of the confusion between the Moon and the finger which point to the moon, or the confusion between a map and the territory. I have nothing against the use of FORTRAN. On the contrary I have a big respect for that old venerable high level programming language :) 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: Cognitive Theoretic Model of the Universe
From my understanding of logic, there is made the distinction between objects and descriptions of objects. For example, the relation is less than is considered different from the relation symbol So what you said makes sense. Bruno Marchal wrote: Hi Ronald, On 02 Jun 2009, at 16:45, ronaldheld wrote: Bruno: Since I program in Fortran, I am uncertain how to interpret things. I was alluding to old, and less old, disputes again programmers, about which programming language to prefer. It is a version of Church Thesis that all algorithm can be written in FORTRAN. But this does not mean that it is relevant to define an algorithm by a fortran program. I thought this was obvious, and I was using that known confusion to point on a similar confusion in Set Theory, like Langan can be said to perform. In Set Theorist, we still find often the error consisting in defining a mathematical object by a set. I have done that error in my youth. What you can do, indeed, is to *represent* (almost all) mathematical objects by sets. Langan seems to make that mistake. The point is just that we have to distinguish a mathematical object and the representation of that object in some mathematical theory. I will have the opportunity to give a precise example in the 7th thread later. In usual mathematical practice, this mistake is really not important, yet, in logic it is more important to take into account that distinction, and then in cognitive science it is *very* important. Crucial, I would say. The error consisting in identifying consciousness and brain state belongs to that family, for example. To confuse a person and its body belongs to that family of error too. All such error are of the form of the confusion between the Moon and the finger which point to the moon, or the confusion between a map and the territory. I have nothing against the use of FORTRAN. On the contrary I have a big respect for that old venerable high level programming language :) 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: Consciousness is information?
On Thu, Jun 4, 2009 at 9:29 AM, Bruno Marchal marc...@ulb.ac.be wrote: On 03 Jun 2009, at 20:11, Jason Resch wrote: On Fri, May 22, 2009 at 4:37 PM, Bruno Marchal marc...@ulb.ac.be wrote: Do you believe if we create a computer in this physical universe that it could be made conscious, But a computer is never conscious, nor is a brain. Only a person is conscious, and a computer or a brain can only make it possible for a person to be conscious relatively to another computer. So your question is ambiguous. It is not my brain which is conscious, it is me who is conscious. My brain appears to make it possible for my consciousness to manifest itself relatively to you. Remember that we are supposed to no more count on the physical supervenience thesis. It remains locally correct to attribute a consciousness through a brain or a body to a person we judged succesfully implemented locally in some piece of matter (like when we say yes to a doctor). But the piece of matter is not the subject of the consciousness. It is only the abstract person or program who is the subject of consciousness. To say a brain is conscious consists in doing Searle's'mistake when he confused levels of computations in the Chinese room, as well seen already by Hofstadter and Dennett in Mind's I. Thanks for your response, if I understand you correctly, you are saying that if we run a simulation of a mind, we are not creating consciousness, only adding an additional instantiation to a mind which already has an infinity of indeterminable instantiations. Is that right? Yes, you are right. When you implement an emulation of a mind, you are just adding such an instanciation relatively to you. Of course you are not adding anything in Platonia. But is the computer emulating the mind not also a platonic object? If the computer simulation does not count toward anything then what is the point of saying yes to the doctor, or to pursue mind uploading technology as a method to obtain immortality and escape eternal aging as QM-immortality would predict? Does this imply that it is impossible to create a simulation of a mind that finds it lives in an environment without uncertainty? That is correct. If so is it because even if the physical laws in one instantiation may be certain, where some of the infinite number of computations that all instantiate that mind may diverge and in particular which one that mind will find itself in is not knowable? Yes. I will come back on this in the seven step thread. The consequence being that all observers everywhere live in QM-like environments? Absolutely. We can consider that we live in an infinity of computations, but we cannot distinguish them ... until they differentiate sufficiently so that they are in principle distinguishable (like being in Washington or being in Moscow). This entails that below our substitution level what can be observed depends directly on some average on an infinity of computations. The quantum-like aspect of nature is, in that sense, a consequence of digitalism in the cognitive science. The classical, and computational, aspect of physics remains the hard things to derive. Interesting, I am curious is there some relationship between ones substitution level and where one will find the QM uncertainty? If all observers live in uncertain environments, and it took us this long to discover QM behavior, I imagine for some observers it could be much harder or much easier to find this uncertainty level. What do you think controls how deep one must look to see the QM behavior first hand? I suppose it might also be related to the complexity of one's observer moment; the more information one takes in from the environment and has in memory the lower the level the uncertainty should be. A God like mind that knew the position of every particle in the universe in which it lived might not have any uncertainty, but of course the mind couldn't encode everything about itself... Jason --~--~-~--~~~---~--~~ 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-Mathematical preliminaries 2
Hi Marty,
On 04 Jun 2009, at 01:11, m.a. wrote:
Bruno,
I stopped half-way through because I'm not at all sure of
my answers and would like to have them confirmed or corrected, if
necessary, rather than go on giving wrong answers. marty a.
No problem.
Exercise 1: Could you define in intension the following infinite set
C = {101, 103, 105, ...}
C = ? C={x such that x is odd x 101}
I guess you meant C = {x such that x is odd and x 101}. means
bigger than, and means little than. OK.
Exercise 2: I will say that a natural number is a multiple of 4 if
it can be written as 4*y, for some y. For example 0 is a multiple of
4, (0 = 4*0), but also 28, 400, 404, ... Could you define in
extension the following set D = {x ⎮ x 10x is a multiple of
4}.D=4*x where x = 0 (but also 1,2,3...10)
You cannot write D = 4*x ..., given that D is a set, and 4*x is a
(unknown) number (a multiple of four when x is a natural number).
Read carefully the problem. I gave the set in intension, and the
exercise consisted in writing the set in extension. Let us translate
in english the definition of the set D = {x ⎮ x 10x is a
multiple of 4}: it means that D is the set of numbers, x, such that x
is little than 10, and x is a multiple of four. So D = {0, 4, 8}.
Indeed 0 is little than 10, and 0 is a multiple of four (because 0 =
4*0), and
4 is little than 10, and 4 is a multiple of four (because 4 = 4*1)
8 is little than 10, and 8 is a multiple of 4 (because 8 = 4*2)
The next mutiple of 4 is 12. It cannot be in the set because 12 is
bigger than 10.
The numbers 1, 2, 3, 5, 7, 9 cannot be in D, because they are not
multiple of 4. You cannot write 1 = 4 * (some natural numbers), nor
can you write 3 or 5, or 7 or 9 = 4 * x with x a natural number.
Example: the set of multiple of 4 is {0, 4, 8, 12, 16, 20, 24, 28, 32,
36, ...}, all have the shape 4*x, with x = to 0, 1, 2, 3, ...
The set of multiple of 5 is {0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50,
55, ...}
Etc.
A ∩ B = {x ⎮ x ∈ A and x ∈ B}.
Example {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}
Similarly, we can directly define the union of two sets A and B,
written A ∪ B in the following way:
A ∪ B = {x ⎮ x ∈ A or x ∈ B}.Here we use the usual
logical or. p or q is suppose to be true if p is true or q is true
(or both are true). It is not the exclusive or.
Example {3, 4, 5, 6, 8} ∪ {5, 6, 7, 9} = {3, 4, 5, 6, 7, 8}.
Question: In the example above, 5,6 were the intersection because
they were the (only) two numbers BOTH groups had in common. But in
this example, 7 is only in the second group yet it is included in
the answer. Please explain.
In the example above (that is {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5,
6}) we were taking the INTERSECTION of the two sets.
But after that, may be too quickly (and I should have made a title
perhaps) I was introducing the UNION of the two sets.
If you read carefully the definition in intension, you should see that
the intersection of A and B is defined with an and. The definition
of union is defined with a or. Do you see that? It is just above in
the quote.
I hope that your computer can distinguish A ∩ B (A intersection B)
and A ∪ B (A union B).
In the union of two sets, you put all the elements of the two sets
together. In the intersection of two sets, you take only those
elements which belongs to the two sets.
It seems you have not seen the difference between intersection and
union. I guess you try to go to much quickly, or that the font of
your computer are too little, or that you have eyesight problems, or
that you have some dyslexia.
Exercice 3.
Let N = {0, 1, 2, 3, ...}
Let A = {x ⎮ x 10}
Let B = {x ⎮ x is even}
Describe in extension (that is: exhaustion or quasi-exhaustion) the
following sets:
N ∪ A = {0,1,2,3...} inter {x inter x10}= {0,1,2,3...9}
N ∪ B = {0,1,2,3} inter {x inter x is even}= {0,2,4,6...}
A ∪ B = {x inter x 10} inter {x inter x is even}= {0,2,4,6,8}
B ∪ A = {x inter x is even} inter {x inter x 10}= {0,2,4,6,8}
All that would be correct if you were searching the intersection, but
∪ is the UNION symbol. (and ∩ is the INTERSECTION symbol).
also you wrote the ⎮ as inter, instead of such that.
N ∩ A = {0,1,2,3...} inter {x inter x10}= {0,1,2,3...9}
B ∩ A = {x inter x is even} inter {x inter x 10}= {0,2,4,6,8}
N ∩ B = {0,1,2,3} inter {x inter x is even}= {0,2,4,6...}
A ∩ B = {x inter x 10} inter {x inter x is even}= {0,2,4,6,8}
All that is correct. Careful you were still using inter in place of
such that. Your last line should be
A ∩ B = {x such that x 10} inter {x such that x is even}=
{0,2,4,6,8}
Exercice 4
Is it true that A ∩ B = B ∩ A, whatever A and B are? yes
Is it true that A ∪ B = B ∪ A, whatever A and B are? yes
Both are correct.
Not bad Marty! Just read carefully. I
Re: The seven step-Mathematical preliminaries 2
On Thu, Jun 4, 2009 at 7:28 AM, kimjo...@ozemail.com.au
kimjo...@ozemail.com.au wrote:
On Thu Jun 4 1:15 , Bruno Marchal sent:
Very good answer, Kim,
Just a few comments. and then the sequel.
Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
3, ...}?
No idea what square-root(2) means. When I said I was innumerate I wasn't
kidding! I
could of course look
it up or ask my mathematics teacher friends but I just know your explanation
will make
theirs seem trite.
Well thanks. The square root of 2 is a number x, such that x*x (x times x, x
multiplied by
itself) gives 2.For example, the square root of 4 is 2, because 2*2 is 4. The
square root of
9 is 3, because 3*3 is 9. Her by square root I mean the positive square
root, because we
will see (more later that soon) that numbers can have negative square root,
but please
forget this. At this stage, with this definition, you can guess that the
square root of 2
cannot be a natural number. 1*1 = 1, and 2*2 = 4, and it would be astonishing
that x
could be bigger than 2. So if there is number x such that x*x is 2, we can
guess that such
a x cannot be a natural number, that is an element of {0, 1, 2, 3 ...}, and
the answer of
exercise 4 is no. The square root of two will reappear recurrently, but
more in examples,
than in the sequence of notions which are strictly needed for UDA-7.
OK - I find this quite mind-blowing; probably because I now understand it for
the first
time in my life. So how did it get this rather ridiculous name of square
root? What's it
called in French?
I don't know what it is called in French, but I can answer the first
part. I remember the day I first figured out where the term came
from.
When you have a number multiplied by itself, the result is called a
square. 3*3 = 9, so 9 is a square. Imagine arranging a set of peas,
if you can arrange them in a square (the four cornered kind) with the
same number of rows as columns, then that number is a square. Some
examples of squares are: 4, 9, 16, 25, 36, 49, 64, 81, see the
pattern? And the roots of those squares are 2, 3, 4, 5, 6, 7, 8,
and 9. The square root is equal to the number of items in a row, or
column when you arrange them in a square.
This is a completely extraneous fact, but one I consider to be very
interesting: Multiply any 4 consecutive positive whole numbers and the
result will always be 1 less than a square number. For example,
5*6*7*8 = 1680, which is 1 less than 1681, which is 41*41. Isn't that
neat?
Jason
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Re: The seven step-Mathematical preliminaries
I've never seen an ultrafinitist definition of the natural numbers.
The usual definition via Peano's axioms obviously rules out there being
a largest number. I would suppose that an ultrafinitist definition of
the natural numbers would be something like seen in a computer (which is
necessarily finite). The successor operation would be defined such that
Successor (Biggest) = 0 or -Biggest.
Brent
Quentin Anciaux wrote:
If you are ultrafinitist then by definition the set N does not
exist... (nor any infinite set countably or not).
If you pose the assumption of a biggest number for N, you come to a
contradiction because you use the successor operation which cannot
admit a biggest number.(because N is well ordered any successor is
strictly bigger and the successor operation is always valid *by
definition of the operation*)
So either the set N does not exists in which case it makes no sense to
talk about the biggest number in N, or the set N does indeed exists
and it makes no sense to talk about the biggest number in N (while it
makes sense to talk about a number which is strictly bigger than any
natural number).
To come back to the proof by contradiction you gave, the assumption
(2) which is that BIGGEST+1 is in N, is completely defined by the mere
existence of BIGGEST. If BIGGEST exists and well defined it entails
that BIGGEST+1 is not in N (but this invalidate the successor
operation and hence the mere existence of N). If BIGGEST in contrary
does not exist (as such, means it is not the biggest) then BIGGEST+1
is in N by definition of N.
Regards,
Quentin
2009/6/4 Torgny Tholerus tor...@dsv.su.se:
Brian Tenneson skrev:
How do you know that there is no biggest number? Have you examined all
the natural numbers? How do you prove that there is no biggest number?
In my opinion those are excellent questions. I will attempt to answer
them. The intended audience of my answer is everyone, so please forgive
me if I say something you already know.
Firstly, no one has or can examine all the natural numbers. By that I
mean no human. Maybe there is an omniscient machine (or a maximally
knowledgeable in some paraconsistent way) who can examine all numbers
but that is definitely putting the cart before the horse.
Secondly, the question boils down to a difference in philosophy:
mathematicians would say that it is not necessary to examine all natural
numbers. The mathematician would argue that it suffices to examine all
essential properties of natural numbers, rather than all natural numbers.
There are a variety of equivalent ways to define a natural number but
the essential features of natural numbers are that
(a) there is an ordering on the set of natural numbers, a well
ordering. To say a set is well ordered entails that every =nonempty=
subset of it has a least element.
(b) the set of natural numbers has a least element (note that it is
customary to either say 0 is this least element or say 1 is this least
element--in some sense it does not matter what the starting point is)
(c) every natural number has a natural number successor. By successor
of a natural number, I mean anything for which the well ordering always
places the successor as larger than the predecessor.
Then the set of natural numbers, N, is the set containing the least
element (0 or 1) and every successor of the least element, and only
successors of the least element.
There is nothing wrong with a proof by contradiction but I think a
forward proof might just be more convincing.
Consider the following statement:
Whenever S is a subset of N, S has a largest element if, and only if,
the complement of S has a least element.
By complement of S, I mean the set of all elements of N that are not
elements of S.
Before I give a longer argument, would you agree that statement is
true? One can actually be arbitrarily explicit: M is the largest
element of S if, and only if, the successor of M is the least element of
the compliment of S.
I do not agree that statement is true. Because if you call the Biggest
natural number B, then you can describe N as = {1, 2, 3, ..., B}. If
you take the complement of N you will get the empty set. This set have
no least element, but still N has a biggest element.
In your statement you are presupposing that N has no biggest element,
and from that axiom you can trivially deduce that there is no biggest
element.
--
Torgny Tholerus
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Re: The seven step-Mathematical preliminaries
Torgny Tholerus wrote: Brian Tenneson skrev: This is a denial of the axiom of infinity. I think a foundational set theorist might agree that it is impossible to -construct- an infinite set from scratch which is why they use the axiom of infinity. People are free to deny axioms, of course, though the result will not be like ZFC set theory. The denial of axiom of foundation is one I've come across; I've never met anyone who denies the axiom of infinity. For me it is strange that the following statement is false: every natural number has a natural number successor. To me it seems quite arbitrary for the ultrafinitist's statement: every natural number has a natural number successor UNTIL we reach some natural number which does not have a natural number successor. I'm left wondering what the largest ultrafinist's number is. It is impossible to lock a box, and quickly throw the key inside the box before you lock it. It is impossible to create a set and put the set itself inside the set, i.e. no set can contain itself. It is impossible to create a set where the successor of every element is inside the set, there must always be an element where the successor of that element is outside the set. Depends on how you define successor. Brent What the largest number is depends on how you define natural number. One possible definition is that N contains all explicit numbers expressed by a human being, or will be expressed by a human being in the future. Amongst all those explicit numbers there will be one that is the largest. But this largest number is not an explicit number. --~--~-~--~~~---~--~~ 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: Cognitive Theoretic Model of the Universe
Bruno Marchal wrote: Hi Ronald, On 02 Jun 2009, at 16:45, ronaldheld wrote: Bruno: Since I program in Fortran, I am uncertain how to interpret things. I was alluding to old, and less old, disputes again programmers, about which programming language to prefer. It is a version of Church Thesis that all algorithm can be written in FORTRAN. But this does not mean that it is relevant to define an algorithm by a fortran program. I thought this was obvious, and I was using that known confusion to point on a similar confusion in Set Theory, like Langan can be said to perform. In Set Theorist, we still find often the error consisting in defining a mathematical object by a set. I have done that error in my youth. What you can do, indeed, is to *represent* (almost all) mathematical objects by sets. Langan seems to make that mistake. The point is just that we have to distinguish a mathematical object and the representation of that object in some mathematical theory. Just so I'm sure I understand you; do you mean that, for example, the natural numbers exist in a way that is independent of Peano's axioms and the theorems that can be proven from them. In other words you could add to Peano's axioms something like Goldbach's conjecture and you would still have the same mathematical object? Brent I will have the opportunity to give a precise example in the 7th thread later. In usual mathematical practice, this mistake is really not important, yet, in logic it is more important to take into account that distinction, and then in cognitive science it is *very* important. Crucial, I would say. The error consisting in identifying consciousness and brain state belongs to that family, for example. To confuse a person and its body belongs to that family of error too. All such error are of the form of the confusion between the Moon and the finger which point to the moon, or the confusion between a map and the territory. I have nothing against the use of FORTRAN. On the contrary I have a big respect for that old venerable high level programming language :) 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: The seven step-Mathematical preliminaries
On 04 Jun 2009, at 15:40, Brian Tenneson wrote: This is a denial of the axiom of infinity. I think a foundational set theorist might agree that it is impossible to -construct- an infinite set from scratch which is why they use the axiom of infinity. People are free to deny axioms, of course, though the result will not be like ZFC set theory. The denial of axiom of foundation is one I've come across; I've never met anyone who denies the axiom of infinity. Among mathematicians nobody denies the axiom of infinity, but many philosopher of mathematics are attracted by finitism. But Torgny is ultrafinitist. That is much rare. he denies the existence of natural numbers above some rather putative biggest natural number. For me it is strange that the following statement is false: every natural number has a natural number successor. I thought he would have said this, and accepted that the successor of its N is equal to N+1. Nut in a reply he says that N+1 exists but is not a natural number, which I think should not be consistent. To me it seems quite arbitrary for the ultrafinitist's statement: every natural number has a natural number successor UNTIL we reach some natural number which does not have a natural number successor. I'm left wondering what the largest ultrafinist's number is. It cannot be a constructive object. It is a number which is so big that if you add 1 to it, the everything explodes! I dunno. I still suspect that ultrafinitism in math cannot be consistent, unlike the many variate form of finitism. Comp is arguably a form of finitism at the ontological level, yet an ultra-infinitism, if I can say, at the epistemological level. 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: The seven step-Mathematical preliminaries
Torngy, How many numbers do you think exist between 0 and 1? Certainly not only the ones we define, for then there would be a different quantity of numbers between 1 and 2, or 2 and 3. Jason On Thu, Jun 4, 2009 at 10:27 AM, Torgny Tholerus tor...@dsv.su.se wrote: Brian Tenneson skrev: Torgny Tholerus wrote: It is impossible to create a set where the successor of every element is inside the set, there must always be an element where the successor of that element is outside the set. I disagree. Can you prove this? Once again, I think the debate ultimately is about whether or not to adopt the axiom of infinity. I think everyone can agree without that axiom, you cannot build or construct an infinite set. There's nothing right or wrong with adopting any axioms. What results is either interesting or not, relevant or not. How do you handle the Russell paradox with the set of all sets that does not contain itself? Does that set contain itself or not? My answer is that that set does not contain itself, because no set can contain itself. So the set of all sets that does not contain itself, is the same as the set of all sets. And that set does not contain itself. This set is a set, but it does not contain itself. It is exactly the same with the natural numbers, BIGGEST+1 is a natural number, but it does not belong to the set of all natural numbers. The set of all sets is a set, but it does not belong to the set of all sets. What the largest number is depends on how you define natural number. One possible definition is that N contains all explicit numbers expressed by a human being, or will be expressed by a human being in the future. Amongst all those explicit numbers there will be one that is the largest. But this largest number is not an explicit number. This raises a deeper question which is this: is mathematics dependent on humanity or is mathematics independent of humanity? I wonder what would happen to that human being who finally expresses the largest number in the future. What happens to him when he wakes up the next day and considers adding one to yesterday's number? This is no problem. If he adds one to the explicit number he expressed yesterday, then this new number is an explicit number, and the number expressed yesterday was not the largest number. Both 17 and 17+1 are explicit numbers. -- Torgny Tholerus --~--~-~--~~~---~--~~ 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-Mathematical preliminaries 2
Hi Kim,
On 04 Jun 2009, at 14:28, kimjo...@ozemail.com.au wrote:
OK - I find this quite mind-blowing; probably because I now
understand it for the first
time in my life. So how did it get this rather ridiculous name of
square root? What's it
called in French?
Racine carrée. Literally square root.
It comes from the fact that in elementary geometry the surface or area
of a square which sides have length x, is given by x*x, also written
x^2, which is then called the square of x. Taking the square root
of a number, consists in doing the inverse of taking the square of a
number. It consists in finding the length of a square knowing its area.
Mathematician and especially logician *can* use arbitrary vocabulary.
It is the essence of the axiomatic method in pure mathematics that
what is conveying does not depend on the term which are used. Hilbert
said once that he could have use the term glass of bear instead of
line in his work in geometry.
A = {x such that x is even and smaller than 100} = {x ⎮ x is even
x
special character, abbreviating such that, and I hope it goes
through the mail.
Just an upright line? It comes through as that. I can't seem to get
this symbol happening so I will
use such that
Yes, such that is abbreviated by an upright line. Sometimes also by
a half circle followed by a little line, but I don't find it on my
palette!
If not I will use such that, or s.t., or things like that.The
expression {x ⎮ x is even} is
literally read as: the set of objects x, (or number x if we are in
a context where we talk
about numbers) such that x is even.
Exercise 1: Could you define in intension the following infinite
set C = {101, 103, 105,
...}C = ?
C = {x such that x is odd and x 101}
Correct.
Exercise 2: I will say that a natural number is a multiple of 4 if
it can be written as 4*y,
for some y. For example 0 is a multiple of 4, (0 = 4*0), but also
28, 400, 404, ... Could
you define in extension the following set D = {x ⎮ x 10 and x is
a multiple of 4}?
D = 4*x where x = 0 but also { 1, 2, 3, 4, 8 }
Hmm...
Marty made a similar error. D is a set. May be you wanted to say:
D = {4*x where x = 0 but also { 1, 2, 3, 4, 8 }}. But this does not
make much sense. Even if I try to imagine favorably some meaning, I
would say that it would mean that D is the set of numbers having the
shape 4*x (that is capable of being written as equal to 4*x for some
x), and such that x belongs to {0, 1, 2, 3, 4, 8}.
A proper way to describe that set would be
D = {y such that y = 4x and x belongs-to {0, 1, 2, 3, 4, 8}}.
But that would makes D = {0, 4, 8, 12, 32}.
The set D = {x ⎮ x 10 and x is a multiple of 4} is just, in
english, the set of natural numbers which are little than 10 and which
are a multiple of 4. The only numbers which are little than 10, and
multiple of 4 are the numbers 0, 4, and 8. D = {0, 4, 8}.
I now realise I am doomed for the next set of exercises because I
cannot get to the special
symbols required (yet). As I am adding Internet Phone to my system,
I am currently using an
ancient Mac without the correct symbol pallette while somebody
spends a few days to flip a single
switch...as soon as I can get back to my regular machine I will
complete the rest.
Take it easy. No problem.
In the meantime I am enjoying the N+1 disagreement - how refreshing
it is to see that classical
mathematics remains somewhat controversial!
The term is a bit too strong. It is a bit like if I told you that I
am Napoleon, and you conclude that the question of the death of
Napoleon is still controversial. I exaggerate a little bit to make my
point, but I know only two ultrafinitists *in math*, and I have never
understood what they mean by number, nor did I ever met someone
understanding them.
What makes just a little bit more sense (and I guess that's what
Torgny really is) is being ultrafinitist *in physics*, and being
physicalist. You postulate there is a physical universe, made of a
finite number of particles, occupying a finite volume in space-time,
etc. Everything is finite, including the everything.
Then by using the unintelligible identity thesis (and thus
reintroducing the mind-body problem), you can prevent the comp white
rabbits inflation. Like all form of materialism, this leads to
eliminating the person soon or later (by the unsolvability of the mind-
body problem by finite means). Ultrafinitist physicalism eliminates
also mathematics and all immaterial notions, including all universal
machines. Brrr...
The real question is do *you* think that there is a biggest natural
number? Just tell me at once, because if you really believe that
there is a biggest natural number, I have no more clues at all how you
could believe in any of computer science nor UDA.
Remember that Thorgny pretends also to be a zombie. It has already
Re: Cognitive Theoretic Model of the Universe
On 04 Jun 2009, at 19:28, Brent Meeker wrote: Bruno Marchal wrote: Hi Ronald, On 02 Jun 2009, at 16:45, ronaldheld wrote: Bruno: Since I program in Fortran, I am uncertain how to interpret things. I was alluding to old, and less old, disputes again programmers, about which programming language to prefer. It is a version of Church Thesis that all algorithm can be written in FORTRAN. But this does not mean that it is relevant to define an algorithm by a fortran program. I thought this was obvious, and I was using that known confusion to point on a similar confusion in Set Theory, like Langan can be said to perform. In Set Theorist, we still find often the error consisting in defining a mathematical object by a set. I have done that error in my youth. What you can do, indeed, is to *represent* (almost all) mathematical objects by sets. Langan seems to make that mistake. The point is just that we have to distinguish a mathematical object and the representation of that object in some mathematical theory. Just so I'm sure I understand you; do you mean that, for example, the natural numbers exist in a way that is independent of Peano's axioms Not just the existence of the natural numbers, all the true relations are independent of the Peano Axioms, and of me, ZF, ZFC and you. and the theorems that can be proven from them. A formal theory is just a machine which put a tiny light on those truth. In other words you could add to Peano's axioms something like Goldbach's conjecture and you would still have the same mathematical object? The whole point of logic is to consider the Peano's axioms as a mathematical object itself, which is studied mathematically in the usual informal (yet rigorous and typically mathematica) way. PA, and PA+GOLDBACH are different mathematical objects. They are different theories, or different machines. Now if GOLDBACH is provable by PA, then PA and PA+GOLDBACH shed the same light on the same arithmetical truth. In that case I will identify PA and PA+GOLDBACH, in many contexts, because most of the time I identify a theory with its set of theorems. Like I identify a person with its set of (possible) beliefs. If GOLDBACH is true, but not provable by PA, then PA and PA+GOLDBACH still talk on the same reality, but PA+GOLDBACH will shed more light on it, by proving more theorems on the numbers and numbers relations than PA. I do no more identify them, and they have different set of theorems. If GOLDBACH is false. Well GOLBACH is PI_1, that is, its negation is SIGMA_1, that is, it has the shape it exist a number such that it verify this decidable property. Indeed the negation of Goldbach conjecture is it exists a number bigger than 2 which is not the sum of two primes. This, if true, is verifiable already by the much weaker RA (Robinson arithmetic). So, if GOLDBACH is false PA + GOLDBACH is inconsistent. That is a mathematical object quite different from PA! Here, you would have taken the twin primes conjecture, and things would have been different, and more complex. Note that a theory of set like ZF shed even much more large light on arithmetical truth, (and is still incomplete on arithmetic, by Gödel ...). Incidentally it can be shown that ZF and ZFC, although they shed different light on the mathematical truth in general, does shed exactly the same light on arithmetical truth. They prove the same arithmetical theorems. On the numbers, the axiom of choice add nothing. This is quite unlike the ladder of infinity axioms. I would say it is and will be particularly important to distinguish chatting beings like RA, PA, ZF, ZFC, etc... and what those beings are talking about. 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: The seven step-Mathematical preliminaries
Date: Thu, 4 Jun 2009 15:23:04 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Quentin Anciaux skrev: If you are ultrafinitist then by definition the set N does not exist... (nor any infinite set countably or not). All sets are finite. It it (logically) impossible to construct an infinite set. What do you mean by construct? Do we have to actually write out or otherwise physically embody every element? Why can't we think of a particular set as just a type of rule that, given any possible element, tells you whether or not that element is a member or not? In this case there's no reason the rule couldn't be such that there are an infinite number of possible inputs that the rule would identify as valid members. You can construct the set N of all natural numbers. But that set must be finite. What the set N contains depends on how you have defined natural number. How do *you* define natural number, if not according to the usual recursive rule that 1 is a natural number and that if N is a natural number, N+1 is also a natural number? Hopefully you agree that there can be no finite upper limit on possible inputs you could give this rule that the rule would identify as valid natural numbers? I think your claim would be that simply describing the rule is not a valid way of constructing the set of natural numbers. If so, why *isn't* it valid? *You* may prefer to adopt the rule that we should only be allowed to call something a set if we can actually write out every member, but do you have any argument as to why it's invalid for the rest of us to define sets simply as general rules that decide whether a given input is a member or not? This seems more like an aesthetic preference on your part rather than something you have a compelling philosophical argument for (or at least if you have such an argument you haven't provided it). Jesse --~--~-~--~~~---~--~~ 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: Cognitive Theoretic Model of the Universe
Bruno Marchal wrote: ... Bruno Marchal wrote: The whole point of logic is to consider the Peano's axioms as a mathematical object itself, which is studied mathematically in the usual informal (yet rigorous and typically mathematica) way. PA, and PA+GOLDBACH are different mathematical objects. They are different theories, or different machines. Now if GOLDBACH is provable by PA, then PA and PA+GOLDBACH shed the same light on the same arithmetical truth. In that case I will identify PA and PA+GOLDBACH, in many contexts, because most of the time I identify a theory with its set of theorems. Like I identify a person with its set of (possible) beliefs. If GOLDBACH is *true, but not provable* by PA, then PA and PA+GOLDBACH still talk on the same reality, but PA+GOLDBACH will shed more light on it, by proving more theorems on the numbers and numbers relations than PA. I do no more identify them, and they have different set of theorems. If GOLDBACH is false. Well GOLBACH is PI_1, that is, its negation is SIGMA_1, that is, it has the shape it exist a number such that it verify this decidable property. Indeed the negation of Goldbach conjecture is it exists a number bigger than 2 which is not the sum of two primes. This, if true, is verifiable already by the much weaker RA (Robinson arithmetic). So, if GOLDBACH is false PA + GOLDBACH is inconsistent. That is a mathematical object quite different from PA! So what then is the status of the natural numbers? Are there many different objects in Platonia which we loosely refer to as the natural numbers or is there only one such object and the Goldbach conjecture is either true of false of this object? Here, you would have taken the twin primes conjecture, and things would have been different, and more complex. Because, even if it is false, it cannot be proven false by exhibiting an example? Note that a theory of set like ZF shed even much more large light on arithmetical truth, (and is still incomplete on arithmetic, by Gödel ...). Incidentally it can be shown that ZF and ZFC, although they shed different light on the mathematical truth in general, does shed exactly the same light on arithmetical truth. They prove the same arithmetical theorems. On the numbers, the axiom of choice add nothing. This is quite unlike the ladder of infinity axioms. I would say it is and will be particularly important to distinguish chatting beings like RA, PA, ZF, ZFC, etc... and what those beings are talking about. Bruno Do you mean PA talks about the natural numbers but PA+theorems is a different mathematical object than N? Brent --~--~-~--~~~---~--~~ 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-Mathematical preliminaries
On Jun 4, 2009, at 8:27 AM, Torgny Tholerus wrote: How do you handle the Russell paradox with the set of all sets that does not contain itself? Does that set contain itself or not? My answer is that that set does not contain itself, because no set can contain itself. So the set of all sets that does not contain itself, is the same as the set of all sets. And that set does not contain itself. This set is a set, but it does not contain itself. It is exactly the same with the natural numbers, BIGGEST+1 is a natural number, but it does not belong to the set of all natural numbers. The set of all sets is a set, but it does not belong to the set of all sets. So you're saying that the set of all sets doesn't contain all sets. How is that any less paradoxical than the Russell paradox you're trying to avoid? -- Kory --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
