| Topic: |
Science > Physics |
| User: |
"Robert Clark" |
| Date: |
10 Apr 2007 12:42:33 PM |
| Object: |
What will be the next paradigm shift in Math? |
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
Bob Clark
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| User: "Dirk Van de moortel" |
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| Title: Re: What will be the next paradigm shift in Math? |
10 Apr 2007 12:57:39 PM |
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"Robert Clark" <rgregoryclark@yahoo.com> wrote in message news:1176226953.801593.212730@p77g2000hsh.googlegroups.com...
Be aware that Paradigm Shifts only occur in the marketing
business.
Dirk Vdm
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| User: "Herman Rubin" |
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| Title: Re: What will be the next paradigm shift in Math? |
10 Apr 2007 02:47:58 PM |
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In article <1176226953.801593.212730@p77g2000hsh.googlegroups.com>,
Robert Clark <rgregoryclark@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place.
When an addition was made, many things had to be checked.
Complex numbers made it possible to extend the space of
solutions, just as negative numbers had done before.
Newton's infinitesimals CANNOT be added to the usual model.
In the 19th century, Bolzano, Cauchy, and others showed what
needed to be used to do what could be done in calculus
precisely. In the 20th century, using the Skolem-Lowenheim
Theorem, Robinson showed that infinitesimals could be added,
but NOT as Newton envisaged them. Adding infinitesimals
does not enable doing anything about the "ordinary" numbers
which was not done before, but it may give another way of
looking at it.
Infinite sets had been in mathematics for more than 2000 years
before Cantor showed that there was more than one size of
infinity. He only demonstrated what was already there; he
no more invented infinite sets than Euclid invented primes.
Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
The shifts which need to be explained at the high school
or even earlier level are mainly there. Set algebra is at
the elementary level, but logic and set theory (as the
mathematician uses the term) are not there. Even induction
is no longer there. And the analysis of Bolzano and Cauchy,
the group theory from the 18th and 19th century, the abstract
algebra of the 19th and early 20th century, good measure and
probability, all can get down to the high school level, but
not if arithmetic is seen as the basis of mathematics, and not
if teachers who cannot understand the ABSTRACT concepts and
teach them as concepts by themselves, and not as generalizations
of calculational observations, are put in charge of teaching.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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| User: "quasi" |
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| Title: Re: What will be the next paradigm shift in Math? |
10 Apr 2007 07:49:10 PM |
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On 10 Apr 2007 10:42:33 -0700, "Robert Clark"
<rgregoryclark@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
Bob Clark
I don't think the basic math up through calculus will change much.
It's been pretty stable since the modernizations of the 1960s.
I think there will be some changes in how the material is presented.
As knowledge of computer algebra systems becomes more standard, more
courses will make use of them.
At the high school level, there may be a return to teaching of sets,
logic, and proofs at an introductory level.
At the undergraduate level for math majors, there are some courses
which are currently gaining ground and which will probably become more
standard. For example:
Modern Geometry (with possible ties to elementary aspects of Group
Theory, Model Theory, Algebraic Geometry, Differential Geometry).
Geometry used to be big, then fell out of favor. Now it's coming back.
Dynamical Systems (assuming a first course in Differential Equations).
Algorithmic Math (analysis of algorithms together with the underlying
mathematics).
Stochastic Methods (courses emphasizing simulations as a way to solve
problems in various areas of math).
But the above are not really paradigm shifts. In fact, I don't think
there will be a paradigm shift. Once a math is done right, that's it.
Most of the math up through the undergraduate level has reached that
level of maturity. The 1960s saw the last major overhaul. Will it ever
dramatically change? Maybe, but I think it's good for at least another
1000 years.
quasi
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| User: "Traveler" |
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| Title: Re: What will be the next paradigm shift in Math? |
11 Apr 2007 07:44:25 AM |
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On 10 Apr 2007 10:42:33 -0700, "Robert Clark"
<rgregoryclark@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
The next paradigm shift will come when mathematicians and physicists
realize that continuity (infinite divisibility) is a farce. Everything
is discrete. Why? Because continuity leads to an infinite regress. The
very fact that we have no trouble doing math on our digital computers
(the ultimate discrete machines) should be a clue. In a discrete
universe, only discrete positions exist and things like lines, curves,
circles, surfaces are all abstract concepts. Once we realize that we
never had a continuous math to start with, the old debate between
Euclidian and non-Euclidian geometries about whether or not parallel
lines meet will become pointless since both wrongly assume the
existence of continuous structures. One man's opinion, of course.
Louis Savain
Physics From the Bible
Shaking the Foundations of Physics:
http://www.rebelscience.org/Seraphim/Physics.htm
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| User: "translogi" |
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| Title: Re: What will be the next paradigm shift in Math? |
11 Apr 2007 09:16:41 AM |
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Interesting tread,
I just want to put something philosophical in the had.
What is a paradigm shift?
I was discussing with my philosophy lecturer if Russell, with the
"Principa mathematica" or Frege, Peano or somebody else around that
time with their publications caused a paradigm shift in Logic.
Syllogistic logic still exists, Barbara Celarent and that ilk are
still thought, and true. can we still call it a paradigm shift?
Inductive logic is a bit out of fashion. But seems to be making a
comeback.
what logic means did change, from descriptive (the law of thought) to
normative (how we should try to think).
But was it a Paradigm shift?
As counter argument i gave that appels before Newton also fell to the
ground.
(according to Aristotle to their normal position)
Do we therefore say there was no Newtonian paradigm shift?
I recently found (and bought cheapely) an second hand logic book from
190?, (Jevons, Logic)
Boole, Frege and Russell aren't even in the bibliography.
De Morgan and Lewis Carroll are only in some footnotes.
Lots of writing if we need O propositions.
De Morgans symbolism isn't there.
and so on.
Can we say this book is from before the paradigm shift?
Was it still used in 1920?
If we don't know what a paradig,m shift is, how can we be sure some
did happen?
Or what the next one will bring?
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| User: "Androcles" |
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| Title: Re: What will be the next paradigm shift in Math? |
12 Apr 2007 03:09:55 AM |
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"translogi" <wilemien@googlemail.com> wrote in message =
news:1176301001.659298.158600@q75g2000hsh.googlegroups.com...
Interesting tread,
I just want to put something philosophical in the had.
=20
What is a paradigm shift?
paradigm
One entry found for paradigm.=20
Main Entry: paradigm=20
Pronunciation: 'per-&-"dIm, 'pa-r&- also -"dim
Function: noun
Etymology: Late Latin paradigma, from Greek paradeigma, from =
paradeiknynai to show side by side, from para- + deiknynai to show -- =
more at DICTION
1 : EXAMPLE, PATTERN; especially : an outstandingly clear or typical =
example or archetype
2 : an example of a conjugation or declension showing a word in all its =
inflectional forms
3 : a philosophical and theoretical framework of a scientific school or =
discipline within which theories, laws, and generalizations and the =
experiments performed in support of them are formulated; broadly : a =
philosophical or theoretical framework of any kind=20
Fact: The Michelson-Morley experiment has a null result.
This is because :
a) time dilates and lengths contract according to the framework of =
special relativity. "light is always propagated in empty space with a =
definite velocity c which is independent of the state of motion of the =
emitting body."- Einstein.
or
b) the velocity of light is source dependent.=20
"But the ray moves relatively to the initial point of k, when measured =
in the stationary system, with the velocity c-v" - Einstein.
=20
"We are to admit no more causes of natural things than such as are both =
true and sufficient to explain their appearances." -- Sir Isaac Newton
b) is a paradigm shift from a).
"Everything should be as psychotic as possible, but not simpler."- =
Einstein.
Once trained, some pathetic minds are incapable of a paradigm shift.=20
Here's another paradigm shift:
http://www.androcles01.pwp.blueyonder.co.uk/Algol/Algol.htm
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| User: "Timothy Golden BandTechnology.com" |
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| Title: Re: What will be the next paradigm shift in Math? |
11 Apr 2007 08:14:14 PM |
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On Apr 11, 8:44 am, Traveler <trave...@nospam.net> wrote:
On 10 Apr 2007 10:42:33 -0700, "Robert Clark"
<rgregorycl...@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
The next paradigm shift will come when mathematicians and physicists
realize that continuity (infinite divisibility) is a farce. Everything
is discrete. Why? Because continuity leads to an infinite regress. The
very fact that we have no trouble doing math on our digital computers
(the ultimate discrete machines) should be a clue. In a discrete
universe, only discrete positions exist and things like lines, curves,
circles, surfaces are all abstract concepts. Once we realize that we
never had a continuous math to start with, the old debate between
Euclidian and non-Euclidian geometries about whether or not parallel
lines meet will become pointless since both wrongly assume the
existence of continuous structures. One man's opinion, of course.
Louis Savain
Physics From the Bible
Shaking the Foundations of Physics:http://www.rebelscience.org/Seraphim/Physics.htm
The discrete paradigm has a basic challenge in rotation. If everything
merely translated through space without rotation we might get some
sense of things jarring position at an infinitessimal scale, but as
soon as an object in this structure rotates say five degrees then we
are left with a failed resolution. Is this too literal of an
interpretation to the discrete models? We observe discrete matter in a
continuous space. Could a multidimensional discrete model be the
answer? If the yield is continuous 3D then you'd have a physics model.
PDNKS?
-Tim
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| User: "Timothy Golden BandTechnology.com" |
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| Title: Re: What will be the next paradigm shift in Math? |
11 Apr 2007 08:15:55 PM |
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On Apr 11, 8:44 am, Traveler <trave...@nospam.net> wrote:
On 10 Apr 2007 10:42:33 -0700, "Robert Clark"
<rgregorycl...@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
The next paradigm shift will come when mathematicians and physicists
realize that continuity (infinite divisibility) is a farce. Everything
is discrete. Why? Because continuity leads to an infinite regress. The
very fact that we have no trouble doing math on our digital computers
(the ultimate discrete machines) should be a clue. In a discrete
universe, only discrete positions exist and things like lines, curves,
circles, surfaces are all abstract concepts. Once we realize that we
never had a continuous math to start with, the old debate between
Euclidian and non-Euclidian geometries about whether or not parallel
lines meet will become pointless since both wrongly assume the
existence of continuous structures. One man's opinion, of course.
Louis Savain
Physics From the Bible
Shaking the Foundations of Physics:http://www.rebelscience.org/Seraphim/Physics.htm
The discrete paradigm has a basic challenge in rotation. If everything
merely translated through space without rotation we might get some
sense of things jarring position at an infinitessimal scale, but as
soon as an object in this structure rotates say five degrees then we
are left with a failed resolution. Is this too literal of an
interpretation to the discrete models? We observe discrete matter in a
continuous space. Could a multidimensional discrete model be the
answer? If the yield is continuous 3D then you'd have a physics model.
PDNKS?
-Tim
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| User: "Shubee" |
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| Title: Re: What will be the next paradigm shift in Math? |
10 Apr 2007 08:31:19 PM |
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On Apr 10, 10:42 am, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
I predict that a mathematician's version of special relativity will
become a standard subject of study in high school algebra classes.
http://www.everythingimportant.org/relativity/special.pdf
Currently, it is a well-known and uncontested fact that bright college
students in physics are confused by standard university instruction in
Einstein's concept of time.
http://arxiv.org/ftp/physics/papers/0207/0207109.pdf
Shubee
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| User: "david petry" |
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| Title: Re: What will be the next paradigm shift in Math? |
10 Apr 2007 08:19:04 PM |
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On Apr 10, 10:42 am, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit?
The adoption of reality checks into the foundations of mathematics
will very possibly be the next big paradigm shift. Mathematics in the
twentieth century went well beyond what could be reasonably considered
to be "reality".
It's the computer revolution which will pave the way for this paradigm
shift. As the computer revolution progresses, people will become
increasingly comfortable with the idea that the world of computation
is a very real world, and that mathematics is the study of the
phenomena observable in that world. We can think of the computer as
the mathematicians' microscope, and then mathematics is the study of
what we observe when we look through that microscope. All of the
mathematics that is potentially applicable in technology - i.e. all of
mathematics that could reasonably be called "real" - fits within that
paradigm. Cantorian set theory doesn't.
It's a good bet that the current older generation of mathematicians
will never accept this paradigm shift, but they will eventually die
off and be replaced by people who have grown up with computers and
will enthusiastically embrace the change.
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| User: "Bob Kolker" |
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| Title: Re: What will be the next paradigm shift in Math? |
10 Apr 2007 09:01:14 PM |
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david petry wrote:
On Apr 10, 10:42 am, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit?
The adoption of reality checks into the foundations of mathematics
will very possibly be the next big paradigm shift. Mathematics in the
twentieth century went well beyond what could be reasonably considered
to be "reality".
Formal mathematics was never about reality. It is about abstract
contstucts and propositions that follow from underlying assumptions.
It's a good bet that the current older generation of mathematicians
will never accept this paradigm shift, but they will eventually die
off and be replaced by people who have grown up with computers and
will enthusiastically embrace the change.
And mathematics will become the calculation of grocery bills. In short,
it will be trivialized.
Bob Kolker
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| User: "Newberry" |
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| Title: Re: What will be the next paradigm shift in Math? |
14 Apr 2007 10:09:08 AM |
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On Apr 10, 10:42 am, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering
What paradigm shift are you considering?
I want them to be explainable
on the high school or undergraduate level.
Bob Clark
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| User: "" |
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| Title: Re: What will be the next paradigm shift in Math? |
18 May 2007 02:43:24 PM |
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On Apr 10, 12:42 pm, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers.
Part of the reason for that was that people regarded "actual numbers"
as things you could crank out digits for in a single digit stream.
This also put negatives under the shadow, because of its apparent need
for extraneous markers too.
Well, let's fix that problem. First negatives.
Digits: + = +1, 0 = -1, - = -1. Base 3.
Done.
Method 2: Digits: P = -4, E = -3, S = -2, C = -1, O = 0, D = 1, Z = 2,
3 = 3, 4 = 4
Base 9.
Done, again.
Next, complex numbers.
Digits: P = -4, E = -3, S = -2, C = -1, O = 0, D = 1, Z = 2, 3 = 3, 4
= 4
Base 3i.
Done.
Method 2: Digits, F = -1 + i, A = i, 7 = 1 + i, C = -1, O = 0, D = 1,
L = -1 - i, U = -i, J = -1 + i
Base 3.
Done, again.
There. Now that should finally settle the remaining loose ends.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit?
Whatever it is, a bunch of reactionaries will pop up and say "that's
meaningless abstraction" or "we can already do it this way, and that's
unnecessary" -- particularly when it starts to intrude on their petty
domains and turf, in the process of sweeping over all the fields of
math.
So, I guess that already answers the question. Category theory.
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| User: "Marcus" |
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| Title: Re: What will be the next paradigm shift in Math? |
11 Apr 2007 11:00:27 AM |
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On Apr 10, 1:42 pm, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
Bob Clark
I suppose by paradigm shift you mean a new concept that will
fundamentally change the way we look at mathematics. These things are
of course difficult to predict in our present state of mind (A quote
of Einstein's, although used in a different context, comes to mind,
"We can't solve problems by using the same kind of thinking we used
when we created them.")
It seems to me that these types of shifts occur typically due to some
long-standing problem that can't be solved. Even though the concept is
new and there is initial resistance to it, it eventually is accepted
because it solves the problem.
There are some very fundamental problems that it seems are not well
understood. The one area that comes to my mind is geometry. There are
a number of problems that are very simple to state and visualize, yet
their proofs seem extremeley complicated and very computationally
intensive. The 4 color map problem (solved by Appel and Haken) and the
sphere packing problem (solved by Hale) come to mind. A humorous
anecedote about the latter proof illustrates what I'm trying to say:
Hale started getting letters from farmers (supposedly) who told him,
"Ok, now you've solved how to pack oranges, can you do asparagus
next?" Even though there is no axiom that simple problems have to have
simple solutions, I think there is a feeling that there may be a
"better" way of solving them.
MB
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| User: "boson boss" |
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| Title: Re: What will be the next paradigm shift in Math? |
12 Apr 2007 05:59:20 AM |
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I suppose by paradigm shift you mean a new concept that will
fundamentally change the way we look at mathematics. These things are
of course difficult to predict in our present state of mind (A quote
of Einstein's, although used in a different context, comes to mind,
"We can't solve problems by using the same kind of thinking we used
when we created them.")
Physical facts become ridiculous little things on which constructs
beyond comprehension are made.
It seems to me that these types of shifts occur typically due to some
long-standing problem that can't be solved. Even though the concept is
new and there is initial resistance to it, it eventually is accepted
because it solves the problem.
Math for physics. I don't believe there is a reality of pure math --
and I claim it as a fact of greater reality! ...There could be
millions of pages long proofs for stuff in pure math. Its a wrong kind
of infinity.
There are some very fundamental problems that it seems are not well
understood. The one area that comes to my mind is geometry.
Take for example a bunch of fractals. Its addition, subtraction,
multiplication iterated round and round. But what is "actual math" for
that?
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| User: "Timothy Golden BandTechnology.com" |
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| Title: Re: What will be the next paradigm shift in Math? |
11 Apr 2007 07:29:21 AM |
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On Apr 10, 1:42 pm, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
Bob Clark
The polysign numbers
http://bandtechnology.com/PolySigned/PolySigned.html
seem to be meeting plenty of resistance. My assurance to you that the
construction is valid is meaningless yet I give you such assurance.
That the real number carries two signs and follows the form
s x
where s is sign and x is magnitude brings the possibility of
generalizing sign. As it turns out the three-signed numbers are the
complex numbers with no additional rule making. They follow directly
in a progression using the same laws as the two-signed reals. Taking
the polysign numbers as fundamental implies that they define
dimension. All of these arguments come from the simplest symmetry:
Sum over s ( s x ) = 0
where s is sign and x is magnitude. At two(P2) this law brings:
- x + x = 0
which is readily believable though it is not ordinarily used in the
definition of the real numbers. Yet this is the extensible form of the
compilation of those old rules whereby for P3 we see
- x + x * x = 0
and this hopefully gives you an inkling of dimensionality. This
cancellation type law dictates that for the unidirectional one-signed
numbers
- x = 0
and so zero dimensional. This is how we see time since we have no
freedom or control over it as we do the other dimensions. The
unidirectional zero-dimensional time qualities are unlikely to be
exposed by any other system than one which investigates sign.
The higher sign systems are also interesting. A natural breakpoint
allows the progression to claim congruence with spacetime. The
treatment of sign is beyond the modern advanced mathematician due to
embedding the real number as a fundamental type in all continuum
constructions. This approach leaves sign beneath them though it stares
at them with every concrete instance.
The operations are easy and may resolve sign errors by looking at sign
in a new way. I think it might be easier to teach a grade schooler
this math than to teach people with a degree. Still, the polysign
system carries many consequences that should be of interest even for
the advanced.
The trouble comes because the coherence of the real number which we
have been obeying for some time is challenged by this construction.
The real number is not at all broken but it is built in such a way
that prevents the polysign generalization and that helps explain how
the polysign construction has gone unnoticed. I find it believable
that it was built by some ancients and forgotten in sanitizing to the
real number. Or is this proof of how young mathematics is?
-Tim
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| User: "RogerB" |
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| Title: Re: What will be the next paradigm shift in Math? |
11 Apr 2007 06:52:32 AM |
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On Apr 10, 6:42 pm, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit?
snip
Recognition that REAL numbers are equivalenced from pairs of unsigned
numbers, and ALGEBRAS are equivalenced from multiplication tables for
sets of unsigned coefficients. Most mathematical texts introduce
subtraction, negation, and real numbers without discussion (or
indexing); a few rigorous authors (including B.L.van der Waerden,
History of Algebra) point out that reals are an artefact, introduced
to simplify much of mathematics to the study of "fields". The "Hoop
Algebra" concept, http://library.wolfram.com/infocenter/MathSource/6198/
, unifies most algebras by showing that they are equivalence relations
on multiplication tables for sets of unsigned coefficients or
operators. If the table has r-fold symmetry, it can be r-folded to a
smaller table that multiplies (and defines division for) vectors with
generalised signs. 2-folding the C2, C4 & quaternion groups (and the
16-element octonion table) gives the real, complex, quaternion and
octonion "real algebras without divisors of zero". Clifford,
hypercomplex, wedge, non-commutative, etc algebras result from folding
other tables. In particular, groups with 3-fold symmetry give TERPLEX
algebras that may describe quarks. These algebras replace division-by-
zero by "projection" onto sub-algebras of reduced symmetry. The
confusing state of modern algebra is clarified.
Roger Beresford.
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| User: "John Jones" |
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| Title: Re: What will be the next paradigm shift in Math? |
10 Apr 2007 01:44:49 PM |
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On Apr 10, 6:42=EF=BF=BDpm, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
=A0In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
=A0Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
=A0Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
=A0Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
=A0If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
=A0 =A0 Bob Clark
Complex numbers, infinitesimals, and infinite sets have never been
accepted. They've been taken and used. You can't 'accept' something
that is incoherent even if it puts food on the table.
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| User: "boson boss" |
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| Title: Re: What will be the next paradigm shift in Math? |
10 Apr 2007 04:33:20 PM |
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On Apr 10, 7:42 pm, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
Bob Clark
The death of math and birth of invention.
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| User: "boson boss" |
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| Title: Re: What will be the next paradigm shift in Math? |
10 Apr 2007 04:46:02 PM |
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On Apr 10, 11:33 pm, "boson boss" <junker...@gmail.com> wrote:
On Apr 10, 7:42 pm, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
Bob Clark
The death of math and birth of invention.
Also, the algorithm could get a definition, Random numeric
decomposition, Resemblance, Occurrence. :-))
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| User: "" |
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| Title: Re: What will be the next paradigm shift in Math? |
10 Apr 2007 11:34:55 PM |
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On Apr 11, 3:42 am, "Robert Clark" <rgregorycl...@yahoo.com> wrote:
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit?
It will be of a geometric nature.
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| User: "Nam D. Nguyen" |
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| Title: Re: What will be the next paradigm shift in Math? |
11 Apr 2007 12:53:57 AM |
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Robert Clark wrote:
In reading Mario Livio's "The Equation That Couldn't Be Solved", I
was interested to see how complex numbers were originally viewed with
suspicion but gradually became accepted as actual numbers. They of
course are now taught routinely in high school math classes.
Newton's idea of infinitesimals was also initially distrusted before
becoming an indispensable part of modern mathematics which also is now
taught in high school math. Such also was and is the case with
Cantor's theory of different classes of infinity.
Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Three paradigm-shifts have come to my mind:
a) *Formally* recognize the limitation of finite understanding
of infinite non-recursion.
b) *Formally* recognize the relativity of logical reasoning
c) *Formally* recognize the fact that, in term of formal expressions,
certain natural complexities could span a long spectrum of many
different order-logics and/or could span many levels of introspection.
Also, I'm looking for ideas that will become routinely discussed in
for example high school math classes or at least in undergraduate
mathematics.
If you'll notice complex numbers, infinitesimals, and infinite sets
were being discussed by the experts in Math on their introduction
before becoming accepted and common place. Then on that basis the
upcoming paradigm shifts in Math might be some things that are being
discussed by the experts in certain rarefied fields of Math but have
not filtered down to the general mathematics community. But at least
for the paradigm shifts I'm considering I want them to be explainable
on the high school or undergraduate level.
Bob Clark
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| User: "Jack Campin - bogus address" |
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| Title: Re: What will be the next paradigm shift in Math? |
11 Apr 2007 03:54:52 AM |
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Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
Maybe the biggest new area of research is "quantum" mathematics of
various sorts. It may stay forever a solution looking for a problem,
but if anybody does figure out something you can do with it, there's
a lot there already to work with.
============== j-c ====== @ ====== purr . demon . co . uk ==============
Jack Campin: 11 Third St, Newtongrange EH22 4PU, Scotland | tel 0131 660 4760
<http://www.purr.demon.co.uk/jack/> for CD-ROMs and free | fax 0870 0554 975
stuff: Scottish music, food intolerance, & Mac logic fonts | mob 07800 739 557
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| User: "Dr. V I Plankenstein" |
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| Title: Re: What will be the next paradigm shift in Math? |
11 Apr 2007 07:01:12 AM |
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Anyone have any ideas of what will be the next paradigmic shift in
Math that will become routine in the mathematicians toolkit? I know
there are many ideas of the next revolutionary ideas in physics,
chemistry, biochemistry and the other sciences, but I'm speaking
strictly of Mathematics here.
One major change might be more social, as math may be regarded as a martial
art for the mind, it would be great to see more people doing math for
recreation instead of smoking drugs or engaging in other frivolous
activities. If people can get away from the idea that math is merely an
instrument of torture, that it can be a really fascinating and enjoyable way
to spend one's hours, people's lives can become enriched not by the
contraptions and trappings of material goods but by the aquisition of
"abstract goods" and appreciation of beauty.
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| User: "boson boss" |
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| Title: Re: What will be the next paradigm shift in Math? |
11 Apr 2007 06:50:51 AM |
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If its not in few lines it has no basic sobriety.
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| User: "Gib Bogle" |
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| Title: Re: What will be the next paradigm shift in Math? |
12 Apr 2007 02:39:01 AM |
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Surrogate factoring.
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