The Mathematics of the Future
PHPBABY3
passing of time. If we observe the development of mathematics since
the beginning of time, we see that man is constantly generalizing the
existing mathematics. How? By introducing mathematical objects that
take on a required value when none is avaiulable. For example, there
was no X such that X*X=-1, so i the imaginary number was introduced.
Note the use of a double reference to X. A single reference is
typically easily solved e.g. X+1=2
Thus we need to track the generality of mathematics to see how it has
been expanded.
PHPBABY3
We will see this double reference repeatedly and it will be used as a
primitive operation to create new Mathematics. In the mainstream
press this is called "self-reference" - although it really only a
double reference P(x,x) for some relation (or otherwise) P.
When we first become aware of our existance, in the womb, we know
there is ourself and our surroundings. At this point we make our
first decision: Do we stay where we are or do we try to learn about
the surroundings? The conservative values say to favor the security
and stay put. The liberal approach says to explore the surroundings
and learn about an unlimited (as far as we know) universe beyond
ourselves. (For examples, conservatives are more likely than liberals
to fear those from a different origin.)
The conservative approaches produces a finite universe: ourself. The
liberal approach produces an infinite universe. The distinction of
finite-infinite will be made repeatedly as a 2nd. operation for
creating new Masthematics.
If we leave the womb, we see a universe of things. But there is a
heterogeneous mixture in the universe. So we need to make the
distinction between and among these different things. For this we
invent the natural numbers.
The natural numbers are not associated with the things (as is commony
believed) but rather the distinctions that delimit things. This is
why there is constant confusion and disagreement over whether 0 is to
be included. In reality, 0 is the distinction that occured when we
noticed something other than ourselves. 1 is the distinction between
our noticing (exploring) everything and the change that delimits the
first thing from the rest of the universe. So 0 is at a higher level
of abstraction, associated with the universe, while 1 is the delimiter
between the first thing and the rest.
The naturlal numbers are the delimiters that divide everything into
things. Did they exist already? No - they must be different from all
existing things to avoid ambiguity as delimiters.
We begin to develop mathematics from observing the physical world. We
see that we can take 3 piles of something and combine any two and then
that with the last pile, and we get the same result each time. It is
then predictable.
Not everything is predictable. For example, you cannot make a correct
prediction to answer the simple question "Will your next answer be
no?". But, as we have just seen, there are some things that are
predictable. And so we call those things that are predictable
"science". And that science which does not require the use of any
input (our 5 senses) we call "Mathematics".
[Curiously, man seems only able to invent mathematics that ultimately
models the physical world, even though, as defined above, Mathematics
can be done without refrerence to the physical world.]
Now, we formalize this predictable property of combining piles as
addition with associativity: (A+(B+C))=((A+B)+C) We define this
relationship of addition as relation ADD(x,y,z) iff x+y=z. And we
note that if I and J are inputs - independent values - any values -
and x are whatever values are needed - dependent values - output,
then:
ADD(I,J,x)
always has exactly one output. It defines a set of exactly one
value. That is the process of addition.
From this point on we try different values for each component in
relations, input I J . . . or output X Y . . . and also quantified
value (eA) for "there exists a value A such that" where A B . . . are
the quantified variables. And so we get:
ADD(I,I,x)
ADD(I,J,x)
ADD(I,X,I)
ADD(I,X,J)
ADD(X,I,I)
ADD(X,I,J)
ADD(I,X,X)
ADD(X,I,X)
ADD(X,X,I)
etc.
Using principle # 1, we introduce a value for x whenever there is
none.
ADD(x,I,J) : Then we say that "x=J-I" and we introduce the negative
numbers, generalizing the natural numbers into the integers (which can
be positive or negative) -1, -2, -3, etc.
ADD(X,X,I) : Then we say that "X=I/2" and we introduce the rational
numbers (fractions.) We need 1/2, 1 1/2, 2 1/2, etc.
ADD(I,I,X) : We say that "X=I*2" and introduce multiplication and the
specific number 2, as MUL(x,y,z) iff x*y=z. Continuing:
MUL(X,X,-1) : We say "X = square root(-1)" and introduce the imaginary
number i.
The task, then, is to continue this process to produce the rest of
current Mathematics, and continue beyond that to create the
Mathematics of the future.
Charlie-Boo
Philosopher, Social Commentator, Political Pundit
Jack May
"PHPBABY3" <shyma...@gmail.com> wrote in message
news:1c4ee1cd-7303-4857...@25g2000hsx.googlegroups.com...
Science and engineering use very little mathematics these days because the
problems to be solved are very hard, very large, and very non-linear.
Problems now tend to be solved with intuition (partly helped by math) and
large amounts of computer computation.
Even math these days is using a lot more computing to prove theorems. Its
a different type of math where the proof has to be stated in a way were it
is proved with a large number of cases that are proved or disproved in the
computer. I think, but not sure, that the four color map problem was solved
that way.
PHPBABY3
> "PHPBABY3" <shymath...@gmail.com> wrote in message
>
> news:1c4ee1cd-7303-4857...@25g2000hsx.googlegroups.com...
>
> > How do we determine the future? We observe, model, and simulate the
> > passing of time. If we observe the development of mathematics since
> > the beginning of time, we see that man is constantly generalizing the
> > existing mathematics. How? By introducing mathematical objects that
> > take on a required value when none is avaiulable. For example, there
> > was no X such that X*X=-1, so i the imaginary number was introduced.
>
> > Note the use of a double reference to X. A single reference is
> > typically easily solved e.g. X+1=2
>
> > Thus we need to track the generality of mathematics to see how it has
> > been expanded.
>
> Science and engineering use very little mathematics
Isn't mathematics a branch of science? How can you define science in
such a way that mathematics is not included?
> these days because the
> problems to be solved are very hard, very large, and very non-linear.
What are you counting that is little - as opposed to medium or large?
> Problems now tend to be solved with intuition (partly helped by math) and
> large amounts of computer computation.
You seem to be saying that intuition plus math = solution. But don't
we translate the intuitive into the formal i.e. math? Rather than A +
B = Solution it is A => B => Solution. Not intuition + math =
solution but rather intuition => math => solution?
> Even math these days is using a lot more computing to prove theorems.
You seem to mean more often among users or more users or more
calculations being executed by computers. But that has nothing to do
with what the logical connections are between the parts, or what
principles are needed. If a principle solves a small case then it
will solve a large case. If it works for one user, one use or one
program then it will work for the large cases. That is the nature of
computers: to repeat a process indefinitely.
> Its
> a different type of math where the proof has to be stated in a way were it
> is proved with a large number of cases that are proved or disproved in the
> computer.
How is it qualitatively different? It is just the repeated use of the
system to represent multiple instances of the process.
> I think, but not sure, that the four color map problem was solved
> that way.
One rubic's cube solver can memorize a 6 by 6 by 6 cube by looking at
it, spend the next hour developing the algorithm in his head, and then
blindfolded unscramble the cube with his hands using thousands of
moves over a period of another hour.
Whether computers are necessary (due to the smaller limitations of
humans in terms of number of accurate repeats) is very subjective and
quite irrelevant to the description of the process of creating
mathematics - no?
C-B
Jack May
On Aug 6, 8:12 pm, "Jack May" <jack....@comcast.net> wrote:
> "PHPBABY3" <shymath...@gmail.com> wrote in message
>
>Isn't mathematics a branch of science? How can you define science in
> such a way that mathematics is not included?
Science is defined in multiple ways with math being uses at times but
certainly not always.
Evolution says species evolve mostly by survivable of the fittest. A very
powerful concept with no mathematics being used.
There are exceptions like genetic drift which is mainly random changes of
the characteristics of species.
With a major concept like evolution, math can be used to test simple,
idealistic environments. Setting up and solving mathematics for very
complex real world environments is not a realistic approach. Mathematics
is not sufficiently powerful enough to handle most real world environments.
The next step is to go observe real world environments for example of ants.
With the observations the scientist tries to determine what characteristics
of ant colonies that tend to be in common in the most successful surviving
colonies over time. Determining those characteristics will use intuition
and probably some statistical tools along with graphs and analysis of video
or pictures.
The scientist may then use computer simulation to test theories to see how a
simulated colony compares with the real colonies. The simulations are not
really math in the classic sense but simple numerical descriptions saying
for example how single ants react to other ants and conditions being
experienced by the ant.
The scientist will work with the computer simulation testing ideas on colony
life might actually work until the simulation gets fairly near reality.
There may be some mathematics to derive basic emergent behavior for some ant
characteristics. A full mathematical derivation of a large number of ants
is probably beyond the capabilities of mathematics. There may be some
knowledge thrown in for example of what statistical characteristics are of
large complex systems (power law statistics with power law spectrums) which
come mainly from empirical measurement.
and so it goes as knowledge is built up over time with little or no "real
mathematics" used along the way.
> these days because the
> problems to be solved are very hard, very large, and very non-linear.
>What are you counting that is little - as opposed to medium or large?
Home work and PhD dissertations are usually small problems. Problems are
then at all sizes to all the way up to how will global warming effect the
evolution of everything on earth.
> Problems now tend to be solved with intuition (partly helped by math) and
> large amounts of computer computation.
>You seem to be saying that intuition plus math = solution. But don't
we translate the intuitive into the formal i.e. math? Rather than A +
B = Solution it is A => B => Solution. Not intuition + math =
>solution but rather intuition => math => solution?
Not that simple. Solutions are usually the result of many things combined.
> Even math these days is using a lot more computing to prove theorems.
>You seem to mean more often among users or more users or more
calculations being executed by computers. But that has nothing to do
with what the logical connections are between the parts, or what
principles are needed. If a principle solves a small case then it
>will solve a large case.
Large cases have wildly different characteristics never seen in small cases.
I have worked in the field of large systems like how do you detect an attack
on the Internet itself instead of attacks on the computers on the Internet.
>If it works for one user, one use or one
program then it will work for the large cases. That is the nature of
>computers: to repeat a process indefinitely.
Because of the need to have finite computation times, the models that are
used to understand large cases are much different and much simpler than are
used for small cases. That OK because the experience is that typically only
2 or 3 characteristic of a model have much effect on the behavior of very
large cases like the Internet.
> Its
> a different type of math where the proof has to be stated in a way were it
> is proved with a large number of cases that are proved or disproved in the
> computer.
>How is it qualitatively different? It is just the repeated use of the
> system to represent multiple instances of the process.
> I think, but not sure, that the four color map problem was solved
> that way.
>Whether computers are necessary (due to the smaller limitations of
humans in terms of number of accurate repeats) is very subjective and
quite irrelevant to the description of the process of creating
>mathematics - no?
I think the four color problem required the development of an approach to
automatically generate all (a finite set) of all ways boundaries on a map
can interact with all other boundaries. This would be far different than
scaling up simple rules for a few cases.
PHPBABY3
The problem that I am discussing is not how to use Mathematics to
model evolution, but rather to use Mathematics to model the
"evolution" of Mathematics. This has nothing to do with evolution in
the sense of mankind's evolution from apes or Adam & Eve.
I am talking about what has already been developed in Mathematics and
modeling it. I am not referring to using Mathematics to model new
real-world problems at all. I am starting with what Mathematics has
been developed and is useful, examing the Mathematics itself, and
automating the development of Mathematics.
You are talking about applying Mathematics to new real world
problems. I am talking about applying Mathematics to the successful
development of Mathematics of the past, not the use of Mathematics to
solve addition real-world problems - only the problem of the
developent of the Mathematics that has been developed alreay and been
useful.
C-B
PHPBABY3
Addendum:
This is why your comments make no reference to the procedure that I
have described and advocated. You are talking about what Math is good
for. I am not debating what it is good for, only how the Math that
has been proven useful is developed. As far as what problems in
general Math is good for, that is a different question.
You need to look at the procedure that I have developed for modeling
the development of Mathematics and see if that is in fact accurate. I
give several examples of how existing, classical Mathematics is in
fact synthesized by my algorithm. You need to address whether what it
creates is in fact classical Mathematics, not what the overall
limitations of Mathematics are.
The fact it, the algorithm I give does create the Mathematics of the
past, and continuing it can create the Mathematics of the future. It
would be an amazing coincidence if we find ourselves now at the point
in history where "Mathematics" suddenly has a new meaning and the
future Mathematicians discard the Mathematics developed over the past
2,000+ years.
Charlie-Boo
> > scaling up simple rules for a few cases.- Hide quoted text -
>
> - Show quoted text -