Alright, i'm trying to do three things at once, and I'm almost succeeding.
The first thing is learn Ruby, the second thing is learn Surreal Numbers,
and the third is to make a Ruby class for Surreal numbers. My problem
is this: part of the definition of a surreal number is pretty much a
comparison with nil. So how would one go about this? Should I write a <=>
and mixin Comparable? What else should I include to make this easier?? Any
help & suggestions are most welcome!!
And for those of you who are interested and want more info on surreal
numbers, here are some links...
A possibly unhelpful suggestion about nil <=> y and y <=> nil: does the nil
for Surreal *have* to be the Ruby nil of NilClass?
I think it could be (as you say, write Nil#<=> and mixin Comparable) and I
guess that it's unlikely that a SurrealNumbers class would be used with
anything else?? (But you can never be sure: another of my lecturers (see
Semi-OT below) was Ian Stewart, and in one of his 1990s (sort of) popular
books on modern mathematics he says non-standard arithmetic has been used to
devise better ways of representing images using pixels (or something like
that): basically work out the theory using "finite" "infinite" integers,
then use the results to make a practical algorithm by changing a "finite"
"infinite" integer to a large finite integer.)
So if you wanted to avoid possible clashes with other code which expects
(nil <=> other) to raise an exception you could set up
class Surreal::SurrealNil
# define appropriate methods
end
Surreal::Nil = Surreal::SurrealNil.new
Nil = Surreal::Nil # maybe
You can do:
class Surreal::SurrealNil < NilClass
but then there isn't Surreal::SurrealNil.new, presumably because there isn't
NilClass.new
I'd be interested to see what you come up with, because periodically I try
to really understand NonStandard Analysis, and the NonStandard Reals are a
subset of the Surreals.
*** Semi-OT: I followed up some links from your links, and found a name I
recognized as the lecturer who gave my first (or at least one of my first)
lectures in mathematics at the University of Warwick in October 1973, a one
term course on the Foundations of Mathematics. (Basically set theory using
Paul Halmos's Naive Set Theory.) I knew he became very interested in
mathematical education some time after I'd graduated, but I didn't know that
he was also interested in "intuitive" concepts of infinity. Following up
links and trying to find out more about David O Tall's "super-real" numbers
I found: Infinitesimals in Modern Mathematics (section 3.2)
A less ambitious but much more accessible approach to defining
infinitesimals is one by David Tall from the University of Warwick. His
motivation was to create a system which was more intuitive for students and
to make Calculus concepts easier to grasp. The simplicity of his approach is
very appealing, as it quickly gets to the use of infinitesimals without the
large construction found *R's construction. ... David Tall Research Papers David Tall - Limits, Infinitesimals and Infinities
and in particular this delightful conversation about infinity between David
Tall and his seven year old son: http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001l-childs-infinity.pdf
···
On Fri, Dec 24, 2010 at 3:45 AM, serialhex <serialhex@gmail.com> wrote:
Alright, i'm trying to do three things at once, and I'm almost succeeding.
The first thing is learn Ruby, the second thing is learn Surreal Numbers,
and the third is to make a Ruby class for Surreal numbers. My problem
is this: part of the definition of a surreal number is pretty much a
comparison with nil. So how would one go about this? Should I write a <=>
and mixin Comparable? What else should I include to make this easier?? Any
help & suggestions are most welcome!!
Colin, your amazing insight has led me to programming greatness!!!
...ok, mabye not so much, but i have (mostly) solved the problem using the
Delegate class, heres the relevant code:
···
##
require 'delegate'
class SurrealNil < DelegateClass(NilClass)
include Comparable
def <=> other
return -1
end
def initialize
super nil
end
end
##
it returns -1 all the time so no matter what you compare it against it's
less than that (i mean, sereously, an empty set is WAAAAYYYYYYY less than
neg infinity, cause with neg infinity you still have SOMETHING right?)
so while the rest of the project is FAR from finished, at least this part is
completed. thanks for the help!!
-hex
p.s. i'll post a link to the source once it's in a form that's not
embarrasing!
On Fri, Dec 24, 2010 at 5:40 PM, Colin Bartlett <colinb2r@googlemail.com>wrote:
On Fri, Dec 24, 2010 at 3:45 AM, serialhex <serialhex@gmail.com> wrote:
> Alright, i'm trying to do three things at once, and I'm almost
succeeding.
> The first thing is learn Ruby, the second thing is learn Surreal
Numbers,
> and the third is to make a Ruby class for Surreal numbers. My
problem
> is this: part of the definition of a surreal number is pretty much a
> comparison with nil. So how would one go about this? Should I write a
<=>
> and mixin Comparable? What else should I include to make this easier??
Any
> help & suggestions are most welcome!!
>
A possibly unhelpful suggestion about nil <=> y and y <=> nil: does the nil
for Surreal *have* to be the Ruby nil of NilClass?
I think it could be (as you say, write Nil#<=> and mixin Comparable) and I
guess that it's unlikely that a SurrealNumbers class would be used with
anything else?? (But you can never be sure: another of my lecturers (see
Semi-OT below) was Ian Stewart, and in one of his 1990s (sort of) popular
books on modern mathematics he says non-standard arithmetic has been used
to
devise better ways of representing images using pixels (or something like
that): basically work out the theory using "finite" "infinite" integers,
then use the results to make a practical algorithm by changing a "finite"
"infinite" integer to a large finite integer.)
So if you wanted to avoid possible clashes with other code which expects
(nil <=> other) to raise an exception you could set up
class Surreal::SurrealNil
# define appropriate methods
end
Surreal::Nil = Surreal::SurrealNil.new
Nil = Surreal::Nil # maybe
You can do:
class Surreal::SurrealNil < NilClass
but then there isn't Surreal::SurrealNil.new, presumably because there
isn't
NilClass.new
I'd be interested to see what you come up with, because periodically I try
to really understand NonStandard Analysis, and the NonStandard Reals are a
subset of the Surreals.
*** Semi-OT: I followed up some links from your links, and found a name I
recognized as the lecturer who gave my first (or at least one of my first)
lectures in mathematics at the University of Warwick in October 1973, a one
term course on the Foundations of Mathematics. (Basically set theory using
Paul Halmos's Naive Set Theory.) I knew he became very interested in
mathematical education some time after I'd graduated, but I didn't know
that
he was also interested in "intuitive" concepts of infinity. Following up
links and trying to find out more about David O Tall's "super-real" numbers
I found: Infinitesimals in Modern Mathematics (section 3.2)
A less ambitious but much more accessible approach to defining
infinitesimals is one by David Tall from the University of Warwick. His
motivation was to create a system which was more intuitive for students and
to make Calculus concepts easier to grasp. The simplicity of his approach
is
very appealing, as it quickly gets to the use of infinitesimals without the
large construction found *R's construction. ... David Tall Research Papers David Tall - Limits, Infinitesimals and Infinities
and in particular this delightful conversation about infinity between David
Tall and his seven year old son: http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001l-childs-infinity.pdf
On Sat, Dec 25, 2010 at 2:34 AM, serialhex <serialhex@gmail.com> wrote:
Colin, your amazing insight has led me to programming greatness!!!
...ok, mabye not so much, but i have (mostly) solved the problem using the
Delegate class, heres the relevant code:
##
require 'delegate'
class SurrealNil < DelegateClass(NilClass)
include Comparable
def <=> other
return -1
end
def initialize
super nil
end
end
##
it returns -1 all the time so no matter what you compare it against it's
less than that (i mean, sereously, an empty set is WAAAAYYYYYYY less than
neg infinity, cause with neg infinity you still have SOMETHING right?)
so while the rest of the project is FAR from finished, at least this part is
completed. thanks for the help!!
hey robert, thanks for the great article, i'll keep that stuff in mind as
i'm writing the rest of this class!
hex
···
On Mon, Dec 27, 2010 at 5:46 AM, Robert Klemme <shortcutter@googlemail.com>wrote:
On Sat, Dec 25, 2010 at 2:34 AM, serialhex <serialhex@gmail.com> wrote:
> Colin, your amazing insight has led me to programming greatness!!!
>
> ...ok, mabye not so much, but i have (mostly) solved the problem using
the
> Delegate class, heres the relevant code:
>
> ##
> require 'delegate'
>
> class SurrealNil < DelegateClass(NilClass)
> include Comparable
> def <=> other
> return -1
> end
> def initialize
> super nil
> end
> end
> ##
>
> it returns -1 all the time so no matter what you compare it against it's
> less than that (i mean, sereously, an empty set is WAAAAYYYYYYY less than
> neg infinity, cause with neg infinity you still have SOMETHING right?)
>
> so while the rest of the project is FAR from finished, at least this part
is
> completed. thanks for the help!!
If you need additional input I have written about numeric classes in our
blog
hey robert, thanks for the great article, i'll keep that stuff in mind as
i'm writing the rest of this class!
hex
Colin, your amazing insight has led me to programming greatness!!!
...ok, mabye not so much, but i have (mostly) solved the problem using
the
Delegate class, heres the relevant code:
##
require 'delegate'
class SurrealNil< DelegateClass(NilClass)
include Comparable
def<=> other
return -1
end
def initialize
super nil
end
end
##
it returns -1 all the time so no matter what you compare it against it's
less than that (i mean, sereously, an empty set is WAAAAYYYYYYY less than
neg infinity, cause with neg infinity you still have SOMETHING right?)
so while the rest of the project is FAR from finished, at least this part
is
completed. thanks for the help!!
If you need additional input I have written about numeric classes in our
blog
I can't imagine where you got the idea that a nul is less than minus infinity. Mathematically, a null is basically undefined as in dividing by zero. I have no idea what your use of this is, but mathematically you are in the wrong pew.
Everett L.(Rett) Williams II
···
On Mon, Dec 27, 2010 at 5:46 AM, Robert Klemme > <shortcutter@googlemail.com>wrote:
On Sat, Dec 25, 2010 at 2:34 AM, serialhex<serialhex@gmail.com> wrote:
I can't imagine where you got the idea that a nul is less than minus
infinity. Mathematically, a null is basically undefined as in dividing by
zero. I have no idea what your use of this is, but mathematically you are in
the wrong pew.
Everett L.(Rett) Williams II
where i get that from is this:
when you look at a graph of 1/x lim(+x->0) +infinity; lim(-x->0) -infinity;
(that's also what i learned in my calculus courses) so 1/0 (kind of) has a
value depending on which direction you approach 0 from.
null on the other hand - as you said - is undefined. and when you compare
null to ANY value - even neg infinity - you still have a value on one hand
and nothing on the other. and while it sucks, having -infinity dollars you
have SOME value (to bill collectors) and having nil dollars you have no
value (to anyone). that example is kind of bad cause nil == 0, but that's
not my point. if you read some of the papers i provided in the first post
(esp. this one: http://www.tondering.dk/main/index.php/surreal-numbers\) part of the
definition of a surreal number is a comparison with nil. the first surreal
number { nil | nil } is defined as 0, and the second generation { { nil |
nil } | nil } is defined as 1 and { nil | { nil | nil } } is -1 ( commonly
written simply as { | }; { 0 | } and { | 0 }. and { 0 | 0 } is not
well-defined in the definition of surreal numbers )
so while it might not make sense to most people (indeed, i'm working on
stretching my brain around all this) this is the definition i'm working
with, and i'm not going to argue with math phd's
as for the utility of it all? surreal numbers are commonly used in game
theory. though the reason why i'm building this class is not for utility,
but as an exercise in 'can i do this?' there are some problems that may
come up later i'm going to have to find some way to deal with ( sets of
infinite objects being one of them, surreal numbers being cool enough that
it gives meaning to things like infinity + 6 and infinity/2 ).
so i hope i answered your question and maybe taught you something.
hex
p.s. i still might be mostly wrong, but again, i'm trusting the articles i'm
reading. maybe i should have one of those cool math-like people look at
this?
A pedantic and cautious (meaning I'm reasonably sure about what I
write below but I'm not 100% confident) person with a maths degree
from just over 30 years ago (so out of date, and I've forgotten a
lot!) writes:
...
it returns -1 all the time so no matter what you compare it against it's
less than that (i mean, sereously, an empty set is WAAAAYYYYYYY less
than neg infinity, cause with neg infinity you still have SOMETHING right?)
...
I can't imagine where you got the idea that a nul is less than minus
infinity. Mathematically, a null is basically undefined as in dividing by
zero. I have no idea what your use of this is, but mathematically you are in
the wrong pew.
1. In this case I think serialhex's "<=>" may not always be returning
the appropriate value for "comparisons" with nil (see below).
2. But for the general case, I remain to be convinced that a null is
*necessarily* undefined: depending on what one means by "null" in a
particular context, then it might make sense to compare something with
that. But if "a null is basically undefined" is restricting the use of
null to stuff that is undefined in a particular context, then
(cautiously) I agree!
3. The statement by serialhex that
"an empty set is WAAAAYYYYYYY less than neg infinity"
did make me wonder if it was correct in the specific context of
Surreal numbers, and I have been looking at the articles. The basis
for comparisons is (in one formulation):
x <= y if and only if
(a) there are no values xLv in the Left set xL of x for which y <= xLv;
and
(b) there are no values yRv in the Right set yR of y for which yR <= x.
If the set xL is empty, then (a) is "vacuously" true; similarly, if
the set yR is empty, then (b) is "vacuously" true.
But the empty set itself is *not* (as I understand) it a Surreal
number, so strictly speaking I think (cautiously!) comparisons are not
with the empty set as such but with pairs of sets (L and R) of which
none or one or both of a pair may be the empty set.
In Tøndering's second chapter he does define comparisons between sets
of Surreal numbers, and then states:
"It follows from the definition above that
EmptySet <= b
is always true, regardless of the value of b; and so is
EmptySet > b, etc"
One needs to be very careful: I think (cautiously) that the definition
of Order in the Wikipedia article is correct, but that the "friendly"
explanations in brackets after the proper parts of the definition are
(probably/possibly!) either misleading or wrong.
···
On Tue, Dec 28, 2010 at 10:37 AM, Everett L Williams II <rett@classicnet.net> wrote:
On Sat, Dec 25, 2010 at 2:34 AM, serialhex<serialhex@gmail.com> wrote:
I can't imagine where you got the idea that a nul is less than minus
infinity. Mathematically, a null is basically undefined as in dividing by
zero. I have no idea what your use of this is, but mathematically you are in
the wrong pew.
Everett L.(Rett) Williams II
where i get that from is this:
when you look at a graph of 1/x lim(+x->0) +infinity; lim(-x->0) -infinity;
(that's also what i learned in my calculus courses) so 1/0 (kind of) has a
value depending on which direction you approach 0 from.
null on the other hand - as you said - is undefined. and when you compare
null to ANY value - even neg infinity - you still have a value on one hand
and nothing on the other. and while it sucks, having -infinity dollars you
have SOME value (to bill collectors) and having nil dollars you have no
value (to anyone). that example is kind of bad cause nil == 0, but that's
not my point. if you read some of the papers i provided in the first post
(esp. this one: http://www.tondering.dk/main/index.php/surreal-numbers\) part of the
definition of a surreal number is a comparison with nil. the first surreal
number { nil | nil } is defined as 0, and the second generation { { nil |
nil } | nil } is defined as 1 and { nil | { nil | nil } } is -1 ( commonly
written simply as { | }; { 0 | } and { | 0 }. and { 0 | 0 } is not
well-defined in the definition of surreal numbers )
so while it might not make sense to most people (indeed, i'm working on
stretching my brain around all this) this is the definition i'm working
with, and i'm not going to argue with math phd's
as for the utility of it all? surreal numbers are commonly used in game
theory. though the reason why i'm building this class is not for utility,
but as an exercise in 'can i do this?' there are some problems that may
come up later i'm going to have to find some way to deal with ( sets of
infinite objects being one of them, surreal numbers being cool enough that
it gives meaning to things like infinity + 6 and infinity/2 ).
so i hope i answered your question and maybe taught you something.
hex
p.s. i still might be mostly wrong, but again, i'm trusting the articles i'm
reading. maybe i should have one of those cool math-like people look at
this?
I haven't the time nor energy to get through all of that, but I can clearly state that your explication of division by Zero is just wrong. Undefined is just that...undefined. You can approach by any means that you wish, but it does not change the nature of the beast. You might note that, as much as physicists and cosmologists deal with very, very large numbers, they still dislike infitities in equations, because they often lead to mathematical and sometime physical black holes. All computer logic is finite and deterministic, so you may model infinities on a computer, as metadata, but you cannot do any calculation that includes them.
You can indeed do some calculations, the fact that there are infinite
natural numbers does not mean you can't add some of them in a
computer. Certainly not all of them, but some.
In that sense arithmetic with infinite quantities is no different. You
can represent infinites or infinitessimals just fine and define
operators on them, just the way you do with natural numbers. In a
computer you are always modelling, you also model N.
It is common that formalisms in Set Theory start with the empty set,
because the axioms give you that one. That's serialhex's nil in
Conway's classic book on the subject. That's how the naturals are
modelled in Set Theory also, ∅ is 0, {∅} is 1, and n = n ∪ {n}. You
only have the empty set and operations like union or the power set to
build upon.
···
On Wed, Dec 29, 2010 at 1:59 PM, Everett L Williams II <rett@classicnet.net> wrote:
I haven't the time nor energy to get through all of that, but I can clearly
state that your explication of division by Zero is just wrong. Undefined is
just that...undefined. You can approach by any means that you wish, but it
does not change the nature of the beast. You might note that, as much as
physicists and cosmologists deal with very, very large numbers, they still
dislike infitities in equations, because they often lead to mathematical and
sometime physical black holes. All computer logic is finite and
deterministic, so you may model infinities on a computer, as metadata, but
you cannot do any calculation that includes them.
I haven't the time nor energy to get through all of that, but I can clearly
state that your explication of division by Zero is just wrong. Undefined is
just that...undefined. You can approach by any means that you wish, but it
does not change the nature of the beast. You might note that, as much as
physicists and cosmologists deal with very, very large numbers, they still
dislike infitities in equations, because they often lead to mathematical and
sometime physical black holes. All computer logic is finite and
deterministic, so you may model infinities on a computer, as metadata, but
you cannot do any calculation that includes them.
Why? Sciences are irrelevant in formalisms.
You can indeed do some calculations, the fact that there are infinite
natural numbers does not mean you can't add some of them in a
computer. Certainly not all of them, but some.
In that sense arithmetic with infinite quantities is no different. You
can represent infinites or infinitessimals just fine and define
operators on them, just the way you do with natural numbers. In a
computer you are always modelling, you also model N.
It is common that formalisms in Set Theory start with the empty set,
because the axioms give you that one. That's serialhex's nil in
Conway's classic book on the subject. That's how the naturals are
modelled in Set Theory also, ? is 0, {?} is 1, and n = n ? {n}. You
only have the empty set and operations like union or the power set to
build upon.
You are confusing computer logic and meta-data manipulation. No computer can natively deal with the representation of, much less the calculation of anything that involves infinity, either negative or positive. You certainly can define a set of rules and attempt to create a program that models those rules, but you cannot naatively do any such calculation. Computers are, by definition, finitie and deterministic, and there is not room here to explain exactly what that means, but there is plenty of information spread all over the internet on the subject. Let me take a small stab at an example. Given to finitie numbers whose sum is within the capacity of the instructions of a computer, I can add those two numbers and get a third number. Anything beyond that is modeled and entirely dependent on my logic rather than the logic of the computer. So, you can declare that infinity plus 6 has meaning and you can declare what that meaning is, providing a routine that will decode your expression of infinity and then follow your instructions for creating whatever you have defined as infinity plus 6, but there is no native instruction, even in floating point, that can impinge on the correctness or the calculation of the answer. It is entirely dependent on the meta-logic and meta-data that you have provided. Even extended precision math libraries can break a large number down into segments and then use the native facilities of the computer in a logically and mathematically valid process that leads to arithmetically correct answers, but infinity cannot be represented in any nat8ive form within a computer.
If you look up infinity on the wiki, you will find pages upon pages of various means of manipulating infinities, and yours may be the latest. When I have the time and energy, I will look, but it is hard to get excited about the umpty-unth attempt.
Everett L.(Rett) Williams II
···
On Wed, Dec 29, 2010 at 1:59 PM, Everett L Williams II > <rett@classicnet.net> wrote:
My reply addressed a couple of points of your post:
1. If we are programming symbolic mathematics, we are doing
mathematics. The convenience or lack thereof of such and such concept
for scientists doesn't matter in discussing whether something can be
given a well-defined formal meaning.
2. Computations in computers: from a formal point of view I disagree,
but do not want to enter into that. If by metadata you mean eg
programs versus CPU registers, and if you agree that we can represent
something infinite like the set of quadratic polynomials in on
variable in Z, then we agree in this point. Not its members, but the
set and its rules, akin to how we represent Z in C.
···
On Wed, Dec 29, 2010 at 8:06 PM, Everett L Williams II <rett@classicnet.net> wrote:
You are confusing computer logic and meta-data manipulation. No computer can
natively deal with the representation of, much less the calculation of
anything that involves infinity, either negative or positive. You certainly
can define a set of rules and attempt to create a program that models those
rules, but you cannot naatively do any such calculation. Computers are, by
definition, finitie and deterministic, and there is not room here to explain
exactly what that means, but there is plenty of information spread all over
the internet on the subject. Let me take a small stab at an example. Given
to finitie numbers whose sum is within the capacity of the instructions of a
computer, I can add those two numbers and get a third number. Anything
beyond that is modeled and entirely dependent on my logic rather than the
logic of the computer. So, you can declare that infinity plus 6 has meaning
and you can declare what that meaning is, providing a routine that will
decode your expression of infinity and then follow your instructions for
creating whatever you have defined as infinity plus 6, but there is no
native instruction, even in floating point, that can impinge on the
correctness or the calculation of the answer. It is entirely dependent on
the meta-logic and meta-data that you have provided. Even extended precision
math libraries can break a large number down into segments and then use the
native facilities of the computer in a logically and mathematically valid
process that leads to arithmetically correct answers, but infinity cannot be
represented in any nat8ive form within a computer.
If you look up infinity on the wiki, you will find pages upon pages of
various means of manipulating infinities, and yours may be the latest. When
I have the time and energy, I will look, but it is hard to get excited about
the umpty-unth attempt.
You are confusing computer logic and meta-data manipulation. No computer can
natively deal with the representation of, much less the calculation of
anything that involves infinity, either negative or positive. You certainly
can define a set of rules and attempt to create a program that models those
rules, but you cannot naatively do any such calculation. Computers are, by
definition, finitie and deterministic, and there is not room here to explain
exactly what that means, but there is plenty of information spread all over
the internet on the subject. Let me take a small stab at an example. Given
to finitie numbers whose sum is within the capacity of the instructions of a
computer, I can add those two numbers and get a third number. Anything
beyond that is modeled and entirely dependent on my logic rather than the
logic of the computer. So, you can declare that infinity plus 6 has meaning
and you can declare what that meaning is, providing a routine that will
decode your expression of infinity and then follow your instructions for
creating whatever you have defined as infinity plus 6, but there is no
native instruction, even in floating point, that can impinge on the
correctness or the calculation of the answer. It is entirely dependent on
the meta-logic and meta-data that you have provided. Even extended precision
math libraries can break a large number down into segments and then use the
native facilities of the computer in a logically and mathematically valid
process that leads to arithmetically correct answers, but infinity cannot be
represented in any nat8ive form within a computer.
If you look up infinity on the wiki, you will find pages upon pages of
various means of manipulating infinities, and yours may be the latest. When
I have the time and energy, I will look, but it is hard to get excited about
the umpty-unth attempt.
My reply addressed a couple of points of your post:
1. If we are programming symbolic mathematics, we are doing
mathematics. The convenience or lack thereof of such and such concept
for scientists doesn't matter in discussing whether something can be
given a well-defined formal meaning.
2. Computations in computers: from a formal point of view I disagree,
but do not want to enter into that. If by metadata you mean eg
programs versus CPU registers, and if you agree that we can represent
something infinite like the set of quadratic polynomials in on
variable in Z, then we agree in this point. Not its members, but the
set and its rules, akin to how we represent Z in C.
I'd go a bit past registers to the total logic of the computer. No instruction in any computer can deal with infinity in any form, either logically or physically. Of course, programs for symbolic manipulation can and have been written, but there is no enforcement or checking related to the logic or hardware of the computer. Unless something is physically wrong, computers, adding binary 1 to binary 1 will get binary 10 every time. Your program, consisting of meta-data and meta-logic is a construct entirely dependent on your definition of all parts of the program. By the way, if you are programming symbolic mathematics, the computer is merely following your algorithm. All elements and properties of that algorithm are external to the computer.
Everett L.(Rett) Williams II
···
On Wed, Dec 29, 2010 at 8:06 PM, Everett L Williams II > <rett@classicnet.net> wrote:
Everett, even in the act of adding two numbers, the computer is following
our instructions & algorithms. The computer doesn't know the difference
between 'add' or 'subtract' or 'exponentiate' or 'do this cool MMX function
thing'. The entire process is an abstraction, so while I agree that it IS
impossible to *literally* do addition or multiplication with infinite
strings of numbers (unless you have some cool sci-fi infinite-computer
thing) you can *figuratively* do multiplication on infinite stings of
numbers. One example as when you use ruby's rational class for numbers like
1/3 or 17/9. Neither of those numbers can be *completely* represented in
the computer as a floating point number, as they go on indefinitely. So
instead they are represented as a fraction and the math on an infinite
string of digits is done by changing the way the computer sees & acts with
the number.
Now, one thing I do know, is that figuring out how to do the whole 'surreal
multiplication with infinite numbers' thing is going to be a pain in my
arse!! I've got some ideas but I'm not sure my programming-fu is up to the
task just yet. But hey, gotta set goals high right?
hex
···
On Thu, Dec 30, 2010 at 7:19 AM, Everett L Williams II <rett@classicnet.net>wrote:
Xavier Noria wrote:
On Wed, Dec 29, 2010 at 8:06 PM, Everett L Williams II >> <rett@classicnet.net> wrote:
You are confusing computer logic and meta-data manipulation. No computer
can
natively deal with the representation of, much less the calculation of
anything that involves infinity, either negative or positive. You
certainly
can define a set of rules and attempt to create a program that models
those
rules, but you cannot naatively do any such calculation. Computers are,
by
definition, finitie and deterministic, and there is not room here to
explain
exactly what that means, but there is plenty of information spread all
over
the internet on the subject. Let me take a small stab at an example.
Given
to finitie numbers whose sum is within the capacity of the instructions
of a
computer, I can add those two numbers and get a third number. Anything
beyond that is modeled and entirely dependent on my logic rather than the
logic of the computer. So, you can declare that infinity plus 6 has
meaning
and you can declare what that meaning is, providing a routine that will
decode your expression of infinity and then follow your instructions for
creating whatever you have defined as infinity plus 6, but there is no
native instruction, even in floating point, that can impinge on the
correctness or the calculation of the answer. It is entirely dependent on
the meta-logic and meta-data that you have provided. Even extended
precision
math libraries can break a large number down into segments and then use
the
native facilities of the computer in a logically and mathematically valid
process that leads to arithmetically correct answers, but infinity cannot
be
represented in any nat8ive form within a computer.
If you look up infinity on the wiki, you will find pages upon pages of
various means of manipulating infinities, and yours may be the latest.
When
I have the time and energy, I will look, but it is hard to get excited
about
the umpty-unth attempt.
My reply addressed a couple of points of your post:
1. If we are programming symbolic mathematics, we are doing
mathematics. The convenience or lack thereof of such and such concept
for scientists doesn't matter in discussing whether something can be
given a well-defined formal meaning.
2. Computations in computers: from a formal point of view I disagree,
but do not want to enter into that. If by metadata you mean eg
programs versus CPU registers, and if you agree that we can represent
something infinite like the set of quadratic polynomials in on
variable in Z, then we agree in this point. Not its members, but the
set and its rules, akin to how we represent Z in C.
I'd go a bit past registers to the total logic of the computer. No
instruction in any computer can deal with infinity in any form, either
logically or physically. Of course, programs for symbolic manipulation can
and have been written, but there is no enforcement or checking related to
the logic or hardware of the computer. Unless something is physically wrong,
computers, adding binary 1 to binary 1 will get binary 10 every time. Your
program, consisting of meta-data and meta-logic is a construct entirely
dependent on your definition of all parts of the program. By the way, if you
are programming symbolic mathematics, the computer is merely following your
algorithm. All elements and properties of that algorithm are external to the
computer.
Everett, even in the act of adding two numbers, the computer is following
our instructions& algorithms. The computer doesn't know the difference
between 'add' or 'subtract' or 'exponentiate' or 'do this cool MMX function
thing'. The entire process is an abstraction, so while I agree that it IS
impossible to *literally* do addition or multiplication with infinite
strings of numbers (unless you have some cool sci-fi infinite-computer
thing) you can *figuratively* do multiplication on infinite stings of
numbers. One example as when you use ruby's rational class for numbers like
1/3 or 17/9. Neither of those numbers can be *completely* represented in
the computer as a floating point number, as they go on indefinitely. So
instead they are represented as a fraction and the math on an infinite
string of digits is done by changing the way the computer sees& acts with
the number.
Now, one thing I do know, is that figuring out how to do the whole 'surreal
multiplication with infinite numbers' thing is going to be a pain in my
arse!! I've got some ideas but I'm not sure my programming-fu is up to the
task just yet. But hey, gotta set goals high right?
hex
Xavier Noria wrote:
You are confusing computer logic and meta-data manipulation. No computer
can
natively deal with the representation of, much less the calculation of
anything that involves infinity, either negative or positive. You
certainly
can define a set of rules and attempt to create a program that models
those
rules, but you cannot naatively do any such calculation. Computers are,
by
definition, finitie and deterministic, and there is not room here to
explain
exactly what that means, but there is plenty of information spread all
over
the internet on the subject. Let me take a small stab at an example.
Given
to finitie numbers whose sum is within the capacity of the instructions
of a
computer, I can add those two numbers and get a third number. Anything
beyond that is modeled and entirely dependent on my logic rather than the
logic of the computer. So, you can declare that infinity plus 6 has
meaning
and you can declare what that meaning is, providing a routine that will
decode your expression of infinity and then follow your instructions for
creating whatever you have defined as infinity plus 6, but there is no
native instruction, even in floating point, that can impinge on the
correctness or the calculation of the answer. It is entirely dependent on
the meta-logic and meta-data that you have provided. Even extended
precision
math libraries can break a large number down into segments and then use
the
native facilities of the computer in a logically and mathematically valid
process that leads to arithmetically correct answers, but infinity cannot
be
represented in any nat8ive form within a computer.
If you look up infinity on the wiki, you will find pages upon pages of
various means of manipulating infinities, and yours may be the latest.
When
I have the time and energy, I will look, but it is hard to get excited
about
the umpty-unth attempt.
My reply addressed a couple of points of your post:
1. If we are programming symbolic mathematics, we are doing
mathematics. The convenience or lack thereof of such and such concept
for scientists doesn't matter in discussing whether something can be
given a well-defined formal meaning.
2. Computations in computers: from a formal point of view I disagree,
but do not want to enter into that. If by metadata you mean eg
programs versus CPU registers, and if you agree that we can represent
something infinite like the set of quadratic polynomials in on
variable in Z, then we agree in this point. Not its members, but the
set and its rules, akin to how we represent Z in C.
I'd go a bit past registers to the total logic of the computer. No
instruction in any computer can deal with infinity in any form, either
logically or physically. Of course, programs for symbolic manipulation can
and have been written, but there is no enforcement or checking related to
the logic or hardware of the computer. Unless something is physically wrong,
computers, adding binary 1 to binary 1 will get binary 10 every time. Your
program, consisting of meta-data and meta-logic is a construct entirely
dependent on your definition of all parts of the program. By the way, if you
are programming symbolic mathematics, the computer is merely following your
algorithm. All elements and properties of that algorithm are external to the
computer.
Everett L.(Rett) Williams II
Finite addition, subtraction, multiplication, and division are native functions of all but the simplest computers, whether the computer is self-aware or not, silly as that is to even discuss. What we are talking about here is native instructions as opposed to algorithmic processes. Since I have been programming computers for 44 years, I tend to think that most people, at least in these parts understand the distinction. As they say, if you don't understand it, you can't program it. And, your understanding of it will be what is programmed. Most of us who have been programming for more than five minutes know that we must limit our use of an instruction to the finite limits of the hardware capability. Programmers must constantly be aware that all processes must be bounded, or programs may loop until stopped, a condition not desirable, at least in my experience. There is both a quantitative and a qualitative difference between using the basic capacities of hardware and instruction sets, and overlaying on those extreme sets of logic not amenable to any computation.
Everett L.(Rett) Williams II
···
On Thu, Dec 30, 2010 at 7:19 AM, Everett L Williams II > <rett@classicnet.net>wrote:
On Wed, Dec 29, 2010 at 8:06 PM, Everett L Williams II >>> <rett@classicnet.net> wrote:
But don't you realize you are doing an interpretation?
The computer knows *nothing* about numbers. The computer is a physical
machine, numbers are abstractions *you* are putting onto it.
I could say that for every representable n, any given native 4-bytes
represent the infinite number ∞ + n. By definition (∞ + n) + (∞ + m) =
∞ + (n + m).
There you have it, infinite quantities dealt with in a "finite" computer.
My ∞ + 0 is no different than your vanilla 0 for a blank 4-bytes. The
bytes are just electronics, we are the ones interpreting them in a
convenient and natural, but also arbitrary manner.
···
On Thu, Dec 30, 2010 at 8:30 PM, Everett L Williams II <rett@classicnet.net> wrote:
Finite addition, subtraction, multiplication, and division are native
functions of all but the simplest computers, whether the computer is
self-aware or not, silly as that is to even discuss. What we are talking
about here is native instructions as opposed to algorithmic processes. Since
I have been programming computers for 44 years, I tend to think that most
people, at least in these parts understand the distinction. As they say, if
you don't understand it, you can't program it. And, your understanding of it
will be what is programmed. Most of us who have been programming for more
than five minutes know that we must limit our use of an instruction to the
finite limits of the hardware capability. Programmers must constantly be
aware that all processes must be bounded, or programs may loop until
stopped, a condition not desirable, at least in my experience. There is both
a quantitative and a qualitative difference between using the basic
capacities of hardware and instruction sets, and overlaying on those extreme
sets of logic not amenable to any computation.
On Thu, Dec 30, 2010 at 8:30 PM, Everett L Williams II > <rett@classicnet.net> wrote:
Finite addition, subtraction, multiplication, and division are native
functions of all but the simplest computers, whether the computer is
self-aware or not, silly as that is to even discuss. What we are talking
about here is native instructions as opposed to algorithmic processes. Since
I have been programming computers for 44 years, I tend to think that most
people, at least in these parts understand the distinction. As they say, if
you don't understand it, you can't program it. And, your understanding of it
will be what is programmed. Most of us who have been programming for more
than five minutes know that we must limit our use of an instruction to the
finite limits of the hardware capability. Programmers must constantly be
aware that all processes must be bounded, or programs may loop until
stopped, a condition not desirable, at least in my experience. There is both
a quantitative and a qualitative difference between using the basic
capacities of hardware and instruction sets, and overlaying on those extreme
sets of logic not amenable to any computation.
But don't you realize you are doing an interpretation?
The computer knows *nothing* about numbers. The computer is a physical
machine, numbers are abstractions *you* are putting onto it.
I could say that for every representable n, any given native 4-bytes
represent the infinite number ? + n. By definition (? + n) + (? + m) =
? + (n + m).
There you have it, infinite quantities dealt with in a "finite" computer.
My ? + 0 is no different than your vanilla 0 for a blank 4-bytes. The
bytes are just electronics, we are the ones interpreting them in a
convenient and natural, but also arbitrary manner.
nihilism and solipsism are not much fun to discuss, even in a philosophy forum, one of which this is not. There is a nice one-to-one correspondence to a subset of basic math and boolean logic found in the data storage, manipulation, and instruction set of all but the most trivial modern computer chips. Skip the semantic games and move on.
My problem
My problem