View source for SurrealNumbers


Surreal numbers have relative values.  The set of surreals is an impossible infinity but the universe itself may be conceived of as a dynamical subset if integer surreals missing a zero point reference.

http://en.wikipedia.org/wiki/Surreal_number  [[IntegerEnergy]]  [[RelativeIntegerDifferenceCalculus]]
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Imam Tashdid ul Alam wrote:
> *NEW) 8) INFORMATION PHYSICIST: will specify what can be known and
> what cannot with the presence of a black hole or otherwise and
> reformulate quantum mechanics by placing surreal numbers in the
> ground floor and having fractals as elements of the governing
> equation
>
> For Jim, I wonder if your imagination got tickled when you read
> about surreal numbers, and if you haven't already, I am sure you
> should pay a visit...

Imam Tashdid ul Alam further wrote:
Listen guys, knowing from nill that the group has readers that do
not speak, I am almost forces to clarify something: INTERNET IS NOT
A SERIOUS PLACE. Have fun. Write whatever you want to. NO-ONE, I
emphasize, NO-ONE cares what you do on internet. So be free, and be
in a light mood, like I always am talking to 'somoy'. Jim, has
anyone told you yet that somoy means time in Bangla?

About surreal numbers...anyone serious in physics must understand
that my descriptions of the possible directions are meant to be fun
only, and they are highly sarcastic. Surreal numbers are totally non-
constructive stuff. No one sane should heed what I say now.

You see Jim, John Conway invented surreal numbers, having the
simplest laws possible, and that "mathematicizes" our intuition that
everything is made up of some sort of binary stuff (everything
cannot be built up from UNARY stuff, because there's no way to tell
which is which====). On-Off, Left-Right, { 1, -1 } things like that.
====
They are so intuitive yet so hard to describe...what do we exactly
mean by "everything can be analyzed"...

A gentle introduction. Conway said something like --- these are so
general that I will simply call them numbers --- see "On Numbers And
Games" by Conway...and that was horrible. So Don Knuth (the most
famous computer scientist in the world, author of "The Art Of
Computer Programming", Standford professor, and yeah, creator of [[TeX]]
and Metafont) changed the name to surreal numbers and Conway liked
that. Great minds in general appreciate each other.

By the way, [[TeX]] is pronounced "tech", because that is exactly the
greek word 	au epsilon chi that the English word tech (as in
technology) is derived from. Surprizingly, it means art, more
appropriately, craft.

Back to surreal numbers. The most horrible things about surreal
numbers is that they are ordered. Everybody knows complex numbers
are not ordered (cannot be ordered as an "ordered field"), so they
are out of the scope.

Conway defined them as pair of "sets". { L || R } is a surreal
numbers, where each of L and R are sets of, ahem, ...err.. surreal
numbers. The condition of such a pair of sets to be a surreal number
is ... simple ... each member of L must be less that or equal to
each member of R.

Harry Gonshor took another approach, designating a surreal numebr by
a sequence of + or - signs. So one surreal number could look like ++-
------+-+--+++-----. The relationship of the two "representations"
of surreal number comes much later than you think.

So, Conway says { -2, -3 || 4, 2 } is a surreal number. Note: we
haven't really got any instruction yet as to why the natural numbers
can be considered surreal numbers.

Now the most amazing revelation that mathematicians take for
granted...these "sets" or "sequences" can be infinite...

So you see, { -1 || 0, 1, 2, 3, .... } is a surreal number alright.
What does it MEAN for a number to be surreal...it simply means it
could be a real number, or it could be "high above" it.

There are rules for deciding the sum and product of two surreal
numbers. Believe me, they are complicated. In the end you get a
number epsilon, { 0 || 1, 1/2, 1/4, 1/3, }. See, this number is
certainly bigger than zero. But it is smaller than any real number====
====
[[can you see why?]]

And guess what, ... epsilon omega <code> 1, where omega </code> { 1, 2, 3,
4, ... || }, or in other words, infinity==== The space, denotes the
====
empty set, of course. And there are marvelous numbers like omega to
the power of omega and yet another square root of omega, SQUARE
ROOT OF INFINITY======
======

If that doesn't interest you, what can I say, I should say more then.

In the "neighborhood" of each real number there are swarms of
surreal numbers lurking around. You can project the real number out
of it. This of course means you can reinterpret the infinitesimals
of calculus in terms of surreal numbers==== Unfortunately, the path is
====
not fully explored, and a more promising candidate in that direction
is the hyperreal number system. Not related to surreal system.

Anyway, surreal numbers are the biggest ordered field possible====
====
Biggest in the sense that every other ordered field must be a
subfield of surreals====
====

Now back to our binary intuition. The first surreal number is { || }.
Because empty set is a set you can ascribe any property...there is
nothing inside to contradict the given property==== So you see, every
====
member of the left hand empty set is smaller than or equal to every
member of the right hand empty set==== MIRACULOUS!
====

It is denoted 0. 1 = { 0 || }. If you write it down, it will look
like { { || } || }. If you choose to write down the empty set and O,
it's like { { O || O } || O }. But enough of this. -1 = { || 0 }.
The "choice" of the names come from the definition of addition,
which I will not go into. But know this, when definining new number
systems, you must DEFINE your operations, instead of having
meaningful operations ready made.

2 <code> { 0, 1 || }. And 1/2 </code> { 0 || 1 }. 1/2 sits in the middle of 0 and
1, we all know that, but how is it THE number between 0 and 1? Here
we use the fact that this 1/2 added to itself gives the
aforementioned 1. And the definition of multiplication is consistent
with that two.

Now imagine a tree. At the root, sits 0 { || }. Second level has two
things, 1, { 0 || }, and -1 { || 0 }. Third level is like { 1, 0 || } =
{ 1 || } <code> 2, { 0 || 1 } </code> 1/2, { || -1 } <code> { || -1, 0 } </code> -2, and { -1
|-
| 0 } = -1/2. Amazing, if you think about it. The tree grows day by
day, filling up every number of the for p/(2^q). It is the
infinite'th day you get numbers like 1/3, sqrt(2) and e. Funny.

Gonshor choose a left turn to be a -, a right turn to be a +. so
zero is a sequence of +'s and -'s alright, it is the EMPTY sequence.
1 is +, -1 is -, 1/2 is +-, -1/2 is -+, -2 is --.

Happy?
-------------------------------------------------------------------
Jim Wrote:
> --- In somoy@yahoogroups.com, &quot;Imam Tashdid ul Alam&quot;
> uchchwhash@y... wrote:
>...
> No one sane should heed what I say now.

Sanity is not all that much fun from what I have heard.

> You see Jim, John Conway invented surreal numbers, having the
> simplest laws possible, ...

In the 70's Knuth and Conway were gods to us.  But surreals were even too
insane for me as they have been employed.>

> ....
> Back to surreal numbers. The most horrible things about surreal
> numbers is that they are ordered. Everybody knows complex numbers  are
> not ordered (cannot be ordered as an "ordered field"), so
> they  are out of the scope.
>

Complex numbers are 2 dimensional, two orderings.  It's nice how the
complex numbers emerge naturally in mathematics, but not essential.>
The one thing I like about surreals is they are ordered, and orderings are
all that I can determine to have existence.> ...

> Harry Gonshor took another approach, designating a surreal numebr by  a
> sequence of + or - signs. So one surreal number could look like ++-
> ------+-+--+++-----. ...

This is very cool

>
> ... In the end you get a
> number epsilon, { 0 || 1, 1/2, 1/4, 1/3, }. See, this number is
> certainly bigger than zero. But it is smaller than any real number====
====
> [[can you see why?]]

errr. you mean smaller than all rational numbers as you can, in theory,
similarly construct a an irrational real smaller than all rational
numbers.  But this is where math transcends reality and where my love of
both real and surreal numbers ends.>

> ...
>
> If that doesn't interest you, what can I say, I should say more then.
>

I am interested only in the subset of numbers that are manifest with
respect to physics, the others are just mind games.  All the reals do not
exist just because we say so.  Numbers that are not generated by some
algorithm are not manifest.  Cantor's proof of the existence transfinite
numbers is self-referential and insolvable by Godel.>

Real numbers are all not real-- manifest numbers are.  Manifest numbers
include the rational numbers and the irrationals which can be generated by
some algorithm.  Since the countable set of all Turing machines or
theorems of the predicate calculus are countable, and represent all
algorithms, then the manifest numbers are also countable.>

> ...
> It is the
> infinite'th day you get numbers like 1/3, sqrt(2) and e. Funny.
>

This may really have a physical interpretation in the quantum.  1/3 and
sqrt(2) may not be manifest at the bottom layer, only nature will tell.>

> Gonshor choose a left turn to be a -, a right turn to be a +. so
> zero is a sequence of +'s and -'s alright, it is the EMPTY sequence.  1
> is +, -1 is -, 1/2 is +-, -1/2 is -+, -2 is --.
>
> Happy?
>

Indeed====  I am so very glad you did not make me explain surreals.  You did
====
an excellent job and enlightened me to the potential importance of
Gonshor's work.  Great stuff====  ...definately not junk with respect to
====
information physics.