Re: The seven step-Mathematical preliminaries
2009/6/13 Torgny Tholerus tor...@dsv.su.se:
Jesse Mazer skrev:
Date: Fri, 12 Jun 2009 18:40:14 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
It is, as I said above, for me and all other humans to understand what
you are talking about. It is also for to be able to decide what
deductions or conclusions or proofs that are legal or illegal.
Well, most humans who think about mathematics can understand
rule-based definitions like 0 is a whole number, and N is a whole
number if it's equal to some other whole number plus one--you seem to
be the lone exception.
As for being able to decide what deductions or conclusions or proofs
that are legal or illegal, how exactly would writing out all the
members of the universe solve that? For example, I actually write
out all the numbers from 0 to 1,038,712 and say that they are members
of the universe I want to talk about. But if I write out some axioms
used to prove various propositions about these numbers, they are still
going to be in the form of general *rules* with abstract variables
like x and y (where these variables stand for arbitrary numbers in the
set), no? Or do you also insist that instead of writing axioms and
making deductions, we also spell out in advance every proposition that
shall be deemed true? In that case there is no room at all for
mathematicians to make deductions or write proofs, all of math
would just consist of looking at the pre-established list of true
propositions and checking if the proposition in question is on there.
What do you think about the following deduction? Is it legal or illegal?
---
Define the set A of all sets as:
For all x holds that x belongs to A if and only if x is a set.
This is an general rule saying that for some particular symbol-string x
you can always tell if x belongs to A or not. Most humans who think
about mathematics can understand this rule-based definition. This rule
holds for all and every object, without exceptions.
So this rule also holds for A itself. We can always substitute A for
x. Then we will get:
A belongs to A if and only if A is a set.
And we know that A is a set. So from this we can deduce:
A beongs to A.
---
Quentin, what do you think? Is this deduction legal or illegal?
It depends if you allow a set to be part of itselft or not.
If you accept, that a set can be part of itself, it makes your
deduction legal regarding the rules. If you don't then the statement
is illegal regarding the rules (it violates the rule saying that a set
can't contains itself, which means that A in this system is not a set
thus all the reasoning in *that system* is false.
Choosing one rule or the other tells nothing about the rule itself
unless you can find a contradiction by choosing one or the other.
Regards,
Quentin
But I can't see why a set as I understand it cannot be part of
itself... {1,2,3} is included in {1,2,3} is true, what is the exact
problem with that statement ? (written differently all elements of the
set A are elements of the set B === A is included in B, here as A and
B are the same A is included in A.
--
Torgny Tholerus
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Re: The seven step-Mathematical preliminaries
Quentin Anciaux skrev: 2009/6/13 Torgny Tholerus tor...@dsv.su.se: What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. This is an general rule saying that for some particular symbol-string x you can always tell if x belongs to A or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for A itself. We can always substitute A for x. Then we will get: A belongs to A if and only if A is a set. And we know that A is a set. So from this we can deduce: A beongs to A. --- Quentin, what do you think? Is this deduction legal or illegal? It depends if you allow a set to be part of itselft or not. If you accept, that a set can be part of itself, it makes your deduction legal regarding the rules. OK, if we accept that a set can be part of itself, what do you think about the following deduction? Is it legal or illegal? --- Define the set B of all sets that do not belong to itself as: For all x holds that x belongs to B if and only if x does not belong to x. This is an general rule saying that for some particular symbol-string x you can always tell if x belongs to B or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for B itself. We can always substitute B for x. Then we will get: B belongs to B if and only if B does not belong to B. --- Quentin, what do you think? Is this deduction legal or illegal? -- 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
Well it is illegal regarding the rules meaning with these rules set B does not exist as defined. 2009/6/13 Torgny Tholerus tor...@dsv.su.se: Quentin Anciaux skrev: 2009/6/13 Torgny Tholerus tor...@dsv.su.se: What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. This is an general rule saying that for some particular symbol-string x you can always tell if x belongs to A or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for A itself. We can always substitute A for x. Then we will get: A belongs to A if and only if A is a set. And we know that A is a set. So from this we can deduce: A beongs to A. --- Quentin, what do you think? Is this deduction legal or illegal? It depends if you allow a set to be part of itselft or not. If you accept, that a set can be part of itself, it makes your deduction legal regarding the rules. OK, if we accept that a set can be part of itself, what do you think about the following deduction? Is it legal or illegal? --- Define the set B of all sets that do not belong to itself as: For all x holds that x belongs to B if and only if x does not belong to x. This is an general rule saying that for some particular symbol-string x you can always tell if x belongs to B or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for B itself. We can always substitute B for x. Then we will get: B belongs to B if and only if B does not belong to B. --- Quentin, what do you think? Is this deduction legal or illegal? -- Torgny Tholerus -- All those moments will be lost in time, like tears in rain. --~--~-~--~~~---~--~~ 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: Sat, 13 Jun 2009 11:05:22 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Date: Fri, 12 Jun 2009 18:40:14 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries It is, as I said above, for me and all other humans to understand what you are talking about. It is also for to be able to decide what deductions or conclusions or proofs that are legal or illegal. Well, most humans who think about mathematics can understand rule-based definitions like 0 is a whole number, and N is a whole number if it's equal to some other whole number plus one--you seem to be the lone exception. As for being able to decide what deductions or conclusions or proofs that are legal or illegal, how exactly would writing out all the members of the universe solve that? For example, I actually write out all the numbers from 0 to 1,038,712 and say that they are members of the universe I want to talk about. But if I write out some axioms used to prove various propositions about these numbers, they are still going to be in the form of general *rules* with abstract variables like x and y (where these variables stand for arbitrary numbers in the set), no? Or do you also insist that instead of writing axioms and making deductions, we also spell out in advance every proposition that shall be deemed true? In that case there is no room at all for mathematicians to make deductions or write proofs, all of math would just consist of looking at the pre-established list of true propositions and checking if the proposition in question is on there. What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. It's well known that if you allow sets to contain themselves, and allow arbitrary rules for what a given set can contain, then you can get contradictions like Russell's paradox (the set of all sets which do not contain themselves). But what relevance does this have to arithmetic? Are you afraid the basic Peano axioms might lead to two propositions which can be derived in finite time from the axioms but which are mutually contradictory? If so I don't see how allowing only a finite collection of numbers actually helps--like I said in an earlier post, the number of propositions that can be proved about a finite set of numbers can still be infinite. I suppose it might be possible to make it finite by disallowing propositions which are created merely by connecting other propositions with the AND or OR logical operators, but it's still the case that if your largest whole number BIGGEST is supposed to be at least as large as some numbers humans have already conceived--say, as large as 10^100--then there is no way we could actually write out all possible propositions about these numbers that follow from some Peano-like axiom system to check manually that no two propositions contradicted each other (do you want to try to calculate 10^100 + A and A + 10^100 for every possible value of A smaller than 10^100 to verify explicitly that they are equal in every case?) So, it seems that unless you want to make your universe of numbers *very* small, you have to rely on some sort of mental model of arithmetic to be confident that you won't get contradictions from the axioms you start from, just like how people are confident in the non-contradictoriness of the Peano axioms based on their mental model of counting discrete objects like marbles (see my comments in the last paragraph of the post at http://www.mail-archive.com/everything-list@googlegroups.com/msg16564.html ). 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: Consciousness is information?
On Apr 24, 4:39 pm, Bruno Marchal marc...@ulb.ac.be wrote: Any content of consciousness can be an illusion. Consciousness itself cannot, because without consciousness there is no more illusion at all. - just catching up with the thread, but I feel compelled to comment that this is beautifully and clearly put. Why does this insight escape so many whose grasp of logic in other respects seems quite adequate? The word 'illusion' is often brandished in a scarily 'eliminative' way, but those who do so seem quite 'unconscious' (ironically) that the subtle knife they wield for this excision is precisely that which they seek to excise! David On 24 Apr 2009, at 06:14, Kelly wrote: On Apr 22, 12:24 pm, Bruno Marchal marc...@ulb.ac.be wrote: So for that to be a plausible scenario we have to say that a person at a particular instant in time can be fully described by some set of data. Not fully. I agree with Brent that you need an interpreter to make that person manifest herself in front of you. A bit like a CD, you will need a player to get the music. It seems to me that consciousness is the self-interpretion of information. David Chalmers has a good line: Experience is information from the inside; physics is information from the outside. First person experience and third person experiment. Glad to hear Chalmers accept this at last. In UDA, inside/outside are perfectly well defined in a pure third person way: inside (first person) = memories annihilated and reconstructed in classical teleportation, outside = the view outside the teleporter. In AUDA I use the old classical definition by Plato in the Theaetetus. I still don't see what an interpreter adds, except to satisfy the intuition that something is happening that produces consciousness. Which I think is an attempt to reintroduce time. I don't think so. The only time needed is the discrete order on the natural numbers. An interpreter is needed to play the role of the person who gives some content to the information handled through his local brain. (For this I need also addition and multiplication). But I don't see any advantage of this view over the idea that conscious states just exist as a type of platonic form (as Brent mentioned earlier). The advantage is that we have the tools to derive physics in a way which is enough precise for testing the comp hypothesis. Physics has became a branch of computer's psychology or theology. At any given instant that I'm awake, I'm conscious of SOMETHING. To predict something, the difficulty is to relate that consciousness to its computational histories. Physics is given by a measure of probability on those comp histories. And I'm conscious of it by virtue of my mental state at that instant. In the materialist view, my mental state is just the state of the particles of my brain at that instant. Which cannot be maintained with the comp hyp. Your consciousness is an abstract type related to all computations going through your current state. But I say that what this really means is that my mental state is just the information represented by the particles of my brain at that instant. And that if you transfer that information to a computer and run a simulation that updates that information appropriately, my consciousness will continue in that computer simulation, regardless of the hardware (digital computer, mechanical computer, massively parallel or single processor, etc) or algorithmic details of that computer simulation. OK. But it is a very special form of information. Consciousness is really the qualia associated to your belief in some reality. It is a bet on self-consistency: it speed up your reaction time relatively to your most probable histories. But, what is information? I think it has nothing to do with physical storage or instantiation. I think it has an existence seperate from that. A platonic existence. And since the information that represents my brain exists platonically, then the information for every possible brain (including variations of my brain) should also exist platonically. You make the same error than those who confuse a universal dovetailer with a counting algorithm or the babel library. The sequence: 0, 1, 2, 3, 4, ... , or 0 1 10 11 100 101 110 111 go through all description of all information, but it lacks the infinitely subtle redundancy contained in the space of all computations (the universal dovetaling). You work in N, succ, you lack addition and multiplication, needed for having a notion of interpreter or universal machine, the key entity capable of giving content to its information structure. This is needed to have a coherent internal interpretation of computerland. Conscious experience is more the content, or the interpretation of that information, made by a person or by a universal
