Re: The seven step-Mathematical preliminaries
On Sat, Jun 06, 2009 at 10:22:11AM -0700, Brent Meeker wrote:
I wonder if anyone has tried work with a theory of finite numbers: where
BIGGEST+1=BIGGEST or BIGGEST+1=-BIGGEST as in some computers?
Brent
The numbers {0,...,p-1} with p prime, and addition and multiplication
given modulo p (ie
a plus b = (a+b) mod p
a times b = (ab) mod p
)
is an interesting mathematical object known as a finite field (or
Galois field) -
http://en.wikipedia.org/wiki/Finite_field
Interesting examples of infinite fields are those quite familiar to
you: rational, real and complex numbers.
It might make sense for Torgny to work with a Galois field for some
large but unnamed prime :)
Cheers
--
Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 hpco...@hpcoders.com.au
Australiahttp://www.hpcoders.com.au
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Re: The seven step-Mathematical preliminaries 2
Marty,
On 07 Jun 2009, at 02:03, Brent Meeker wrote:
m.a. wrote:
*Okay, so is it true to say that things written in EXTENSION are
never
in formula style but are translated into formulas when we put them
into INTENSION form? You can see that my difficulty with math
arises from an inability to master even the simplest definitions.
marty a.*
It's not that technical. I could define the set of books on my
shelf by
giving a list of titles: The Comprehensible Cosmos, Set Theory and
It's Philosophy, Overshoot, Quintessence. That would be a
definition by extension. Or I could point to them in succession and
say, That and that and that and that. which would be a definition by
ostension. Or I could just say, The books on my shelf. which is a
definition by intension. An intensional definition is a descriptive
phrase with an implicit variable, which in logic you might write as:
The
set of things x such that x is a book and x is on my shelf.
This is a good point. A set is just a collection of objects seen as a
whole.
A definition in extension of a set is just a listing, finite or
infinite, of its elements.
Like in A = {1, 3, 5}, or B = {2, 4, 6, 8, 10, ...}.
A definition in intension of a set consists in giving the typical
defining property of the elements of the set.
Like in C= the set of odd numbers which are smaller than 6. Or D =
the set of even numbers.
In this case you see that A is the same set as C? And B is the same
set as D.
Now in mathematics we often use abbreviation. So, for example, instead
of saying: the set of even numbers, we will write
{x such-that x is even}.
OK?
Bruno
Suppose,
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries 2
Thank you, Brent, This is quite clear. Hopefully I can apply it as clearly to Bruno's examples.marty a. - Original Message - From: Brent Meeker meeke...@dslextreme.com To: everything-list@googlegroups.com Sent: Saturday, June 06, 2009 8:03 PM Subject: Re: The seven step-Mathematical preliminaries 2 m.a. wrote: *Okay, so is it true to say that things written in EXTENSION are never in formula style but are translated into formulas when we put them into INTENSION form? You can see that my difficulty with math arises from an inability to master even the simplest definitions. marty a.* It's not that technical. I could define the set of books on my shelf by giving a list of titles: The Comprehensible Cosmos, Set Theory and It's Philosophy, Overshoot, Quintessence. That would be a definition by extension. Or I could point to them in succession and say, That and that and that and that. which would be a definition by ostension. Or I could just say, The books on my shelf. which is a definition by intension. An intensional definition is a descriptive phrase with an implicit variable, which in logic you might write as: The set of things x such that x is a book and x is on my shelf. 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 2
Bruno,
Yes, this seems very clear and will be helpful to refer back to if
necessary. m.a.
- Original Message -
From: Bruno Marchal marc...@ulb.ac.be
To: everything-list@googlegroups.com
Sent: Sunday, June 07, 2009 4:33 AM
Subject: Re: The seven step-Mathematical preliminaries 2
Marty,
On 07 Jun 2009, at 02:03, Brent Meeker wrote:
m.a. wrote:
*Okay, so is it true to say that things written in EXTENSION are
never
in formula style but are translated into formulas when we put them
into INTENSION form? You can see that my difficulty with math
arises from an inability to master even the simplest definitions.
marty a.*
It's not that technical. I could define the set of books on my
shelf by
giving a list of titles: The Comprehensible Cosmos, Set Theory and
It's Philosophy, Overshoot, Quintessence. That would be a
definition by extension. Or I could point to them in succession and
say, That and that and that and that. which would be a definition by
ostension. Or I could just say, The books on my shelf. which is a
definition by intension. An intensional definition is a descriptive
phrase with an implicit variable, which in logic you might write as:
The
set of things x such that x is a book and x is on my shelf.
This is a good point. A set is just a collection of objects seen as a
whole.
A definition in extension of a set is just a listing, finite or
infinite, of its elements.
Like in A = {1, 3, 5}, or B = {2, 4, 6, 8, 10, ...}.
A definition in intension of a set consists in giving the typical
defining property of the elements of the set.
Like in C= the set of odd numbers which are smaller than 6. Or D =
the set of even numbers.
In this case you see that A is the same set as C? And B is the same
set as D.
Now in mathematics we often use abbreviation. So, for example, instead
of saying: the set of even numbers, we will write
{x such-that x is even}.
OK?
Bruno
Suppose,
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries
On Sat, Jun 6, 2009 at 4:20 PM, Jesse Mazer laserma...@hotmail.com wrote: Date: Sat, 6 Jun 2009 21:17:03 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: [[[ Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed.]]] Why humans, specifically? What if an alien could conceptualize a larger number? For that matter, since you deny any special role to consciousness, why should it have anything to do with the conceptualizations of beings with brains? A volume of space isn't normally said to conceptualize the number of atoms contained in that volume, but why should that number be any less real than the largest number that's been conceptualized by a biological brain? *JohnM:* *Jesse, * *you don't have to go out to 'aliens', just eliminate the format possible as of 2009. Our un-alien species is well capable of learning (compare to 2000BC) and whatever is restricted today as 'impossible' may be everyday's bread after tomorrow. You are absolutely right - even as of today. * *Especially in your next reply-par below.* Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. You can only create a finite number of proposition with finite length during your lifetime. Just like the number of natural numbers are unlimited but finite, so are the possible propositions unlimited but finte. But you said earlier that as long as we admit only a finite collection of numbers, we can prove the consistency of mathematics involving only those numbers. Well, how can we prove that? If we only show that all the propositions we have generated to date are consistent, how do we know the next proposition we generate won't involve an inconsistency? Presumably you are implicitly suggesting there should be some upper limit on the number of propositions about the numbers as well as on the numbers themselves, but if you define this limit in terms of how many a human could generate in their lifetime, we get back to problems like what if some other being (genetically engineered humans, say) would have a longer lifetime, or what if we built a computer that generated propositions much faster than a human could and checked their consistency automatically, etc. or about my question about whether you have any philosophical/logical argument for saying all sets must be finite, My philosophical argument is about the mening of the word all. To be able to use that word, you must associate it with a value set. What's a value set? And why do you say we must associate it in this way? Do you have a philosophical argument for this must, or is it just an edict that reflects your personal aesthetic preferences? Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. *I* certainly don't define numbers in terms of any specific mapping between numbers and objects in the universe, it seems like a rather strange notion--shall we have arguments over whether the number 113485 should be associated with this specific shoelace or this specific kangaroo? One of the first thing kids learn about number is that if you count some collection of objects, it doesn't matter what order you count them in, the final number you get will be the same regardless of the order (i.e. it doesn't matter which you point to when you say 1 and which you point to when you say 2, as long as you point to each object exactly once). Also, am I understanding correctly in thinking you don't believe there can be
Re: The seven step-Mathematical preliminaries 2
*Bruno et. al., Good news! I have discovered that the math symbols copy faithfully here in my Thunderbird email.* *Henceforth, I will open all list letters here. Please refresh my memory for the following symbols:* * 1. The ***?** *is called_and means__ 2. The***?** *is called___*_*and means__ 3. The ***? is called__and means ** -* Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, June 03, 2009 1:15 PM Subject: Re: The seven step-Mathematical preliminaries 2 ? ? A = ? ? B = A ? ? = B ? ? = N ? ? = B ? ? = ? ? B = ? ? ? = ? ? ? = * --~--~-~--~~~---~--~~ 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
Bravo Thunderbird!
On 07 Jun 2009, at 18:39, m.a. wrote:
Bruno et. al.,
Good news! I have discovered that the math
symbols copy faithfully here in my Thunderbird email. Henceforth, I
will open all list letters here. Please refresh my memory for the
following symbols:
1. The ∅ is called___THE EMPTY SET_and means__THE SET
WITH NO ELEMENTS
The empty set described in extension: { }
The empty set described in intension. Well, let me think. The set of
french which are bigger than 42 km tall.
A cynical definition would be: the set of honest politicians.
A mathematical one: the set of x such that x is different from x.
It is just the set which has no elements. It is empty.
2. The∪ is calledUNION__and means: A ∪ B__= {x
such-that x belongs to A or x belongs to B};
A u B is the set obtained by doing the union of A and B.
3. The ∩ is called_INTERSECTIONand means__A ∩ B__=
{x such-that x belongs to A andr x belongs to B}; A u B is the
set obtained by doing the intersection of A and B. It is the set of
elements which are in both A and B._
Examples:
{1, 2, 3} ∩ {2, 4, 3} = {2, 3}
{1, 2, 3} u {2, 4, 3} = {1, 2, 3, 4}
{1, 2, 3} ∩ {4, 5, 6} = ∅
{1, 2, 3} u {4, 5, 6} = {1, 2, 3, 4, 5, 6}
OK?
Bruno
- Original Message -
From: Bruno Marchal
To: everything-list@googlegroups.com
Sent: Wednesday, June 03, 2009 1:15 PM
Subject: Re: The seven step-Mathematical preliminaries 2
∅ ∪ A =
∅ ∪ B =
A ∪ ∅ =
B ∪ ∅ =
N ∩ ∅ =
B ∩ ∅ =
∅ ∩ B =
∅ ∩ ∅ =
∅ ∪ ∅ =
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries 2
Marty, Kim,
I realize that, now, the message I have just sent does not have the
right symbols. Apparently my computer does not understand the
Thunderbird!
From now on I will use capital words for the mathematical symbols.
And I will write mathematical expression in bold.
For examples:
{1, 2, 3} INTERSECTION {2, 4, 3} = {2, 3}
{1, 2, 3} UNION {2, 4, 3} = {1, 2, 3, 4}
{1, 2, 3} INTERSECTION {4, 5, 6} = EMPTY
{1, 2, 3} UNION {4, 5, 6} = {1, 2, 3, 4, 5, 6}
All right? Mathematics will get a FORTRAN look but this is not
important, OK? It is just the look. I will do a summary of what we
have seen so far.
With those notions you should be able to invent exercises by yourself.
Invent simple sets and compute their union, and intersection.
Remenber that the goal consists in building a mathematical shortcut
toward a thorugh understanding of step seven. In particular the goal
will be to get an idea of a computation is, and what is the difference
between a mathemarical computation and a mathematical description of a
computation. It helps for the step 8 too.
Marty, have a nice holiday,
Kim, ah ah ... we have two weeks to digest what has been said so far
(which is not enormous), OK?
Bruno
http://iridia.ulb.ac.be/~marchal/
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