Higher Algebras to describe Basic Math
amzoti
I am trying to teach the engineers on my team a lesson about looking
for more details as one can never know anything about any topic - so I
am looking for examples to demonstrate my point from various sources.
Even though they are not mathematicians, I thought there may be some
clever examples to share with them.
This example would start out with something as simple as:
1 + 1 = 2
We could then go the Peano Axioms and show a higher algebra that can
be used to describe this identity.
Can someone provide a logical ordering of higher algebras that can be
used to explain these items (I am more of a number theory and
numerical guy - so excuse me if this is an ill-posed question or
exercise)?
For example, are there set-theoretic models, first or second order
models, category theory, algebraic geometry or other systems that can
be used to describe this rather simple identity?
The goal is to show them that no matter what you have learned about a
simple thing, there is always an infinite amount more you could learn.
Hope this is clear and appreciate any insights.
Thanks ~A
Robert Israel
> Hi All,
>
> I am trying to teach the engineers on my team a lesson about looking
> for more details as one can never know anything about any topic - so I
> am looking for examples to demonstrate my point from various sources.
>
> Even though they are not mathematicians, I thought there may be some
> clever examples to share with them.
>
> This example would start out with something as simple as:
>
> 1 + 1 = 2
>
> We could then go the Peano Axioms and show a higher algebra that can
> be used to describe this identity.
From what I know of engineers, this sort of thing is not likely to impress
them favourably. Their focus is on solving the problem at hand as efficiently
as possible, not on generalization for its own sake.
> Can someone provide a logical ordering of higher algebras that can be
> used to explain these items (I am more of a number theory and
> numerical guy - so excuse me if this is an ill-posed question or
> exercise)?
>
> For example, are there set-theoretic models, first or second order
> models, category theory, algebraic geometry or other systems that can
> be used to describe this rather simple identity?
>
> The goal is to show them that no matter what you have learned about a
> simple thing, there is always an infinite amount more you could learn.
But the lesson they would take away with them is: mathematicians are adept
at making simple things complicated. Maybe a better idea would be to show
them examples where mathematical ideas make difficult problems easy to
solve. For example, perhaps, using Gaussian integers to solve problems
involving the sum of two squares.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
scaaahu
The following is a true story,
An unemployeed mathematician went to a computer programming job
interview,(He was unemployeed because he did not get his tenure).
While waiting in the office building lobby, he read a category theory
book. His future boss came out and saw him reading a book. During the
interview, the boss asked what book he was reading about. He showed
the book. The boss asked what category theory is about. He started to
talk about classes, functors, etc. The boss said "Ah, so you are an OO
expert?" He said, "Ya, sort of." He was hired.
William Elliot
> The following is a true story,
>
> An unemployeed mathematician went to a computer programming job
> interview,(He was unemployeed because he did not get his tenure).
> While waiting in the office building lobby, he read a category theory
> book. His future boss came out and saw him reading a book. During the
> interview, the boss asked what book he was reading about. He showed
> the book. The boss asked what category theory is about. He started to
> talk about classes, functors, etc. The boss said "Ah, so you are an OO
> expert?" He said, "Ya, sort of." He was hired.
>
What's OO?
Oppt
scaaahu
Object Oriented programming.
Frederick Williams
> This example would start out with something as simple as:
>
> 1 + 1 = 2
>
> We could then go the Peano Axioms and show a higher algebra that can
> be used to describe this identity.
What does 2 have to do with PA?
--
Needle, nardle, noo.
FredJeffries
Mathematics made Difficult by Carl E. Linderholm
http://books.google.com/books?id=C3jwAAAAMAAJ
Herman Rubin
> amzoti <amz...@gmail.com> writes:
>> Hi All,
>> I am trying to teach the engineers on my team a lesson about looking
>> for more details as one can never know anything about any topic - so I
>> am looking for examples to demonstrate my point from various sources.
>> Even though they are not mathematicians, I thought there may be some
>> clever examples to share with them.
>> This example would start out with something as simple as:
>> 1 + 1 = 2
This is NOT as simple as it sounds. Where do we get the
meaning of what is represented by the four different characters
in that equation?
>> We could then go the Peano Axioms and show a higher algebra that can
>> be used to describe this identity.
We should START with the Peano postulates, and use a
different initial representation for teaching. I came
up with representing the number n with 0 followed by
n strokes. Since we want, at some point, to introduce
the usual base representation, it is convenient for many
reasons to have 0 explicitly.
The explanation from the Peano postulates is SIMPLE;
it is the usual treatment which is difficult and
mysterious.
I forgot to say that algebra has to come first. Here
is a complete description of what is basic in algebra:
A variable is a temporary name for anything.
The same operation applied to equal entities
produces equal results.
One does not need to learn lots of algorithms first to
understand the above, and even a child can derive
algorithms from it.
> From what I know of engineers, this sort of thing is not likely to impress
> them favourably. Their focus is on solving the problem at hand as efficiently
> as possible, not on generalization for its own sake.
The general is simpler than the special cases;
Szent-Gyorgi, a famous biologist, stated that explicitly.
Adding conditions is easy; removing them requires the
difficult process of unlearning.
One must understand the problem to solve it. The engineer
who understands the principles of differential calculus has
all the tools needed to formulate differential equations.
I was at an oral exam for a student who had done an excellent
master's project, but he was unable to formulate the differential
equation for the rate at which a spherical ball of salt would
dissolve in water, even with the hint making it obvious.
>> Can someone provide a logical ordering of higher algebras that can be
>> used to explain these items (I am more of a number theory and
>> numerical guy - so excuse me if this is an ill-posed question or
>> exercise)?
The integers are an important special case, some of which
needs to be learned for any higher application. What does
not need to be learned is arithmetic; it can be done by
machine. One complication is that there are many concepts
possessed by the integers, and the basic ones need to be
learned, including how the integers can embedded in rationals,
reals, and complex numbers.
>> For example, are there set-theoretic models, first or second order
>> models, category theory, algebraic geometry or other systems that can
>> be used to describe this rather simple identity?
You mean the equation? If one defines 1 and 2 as above
in the Peano model, the identity is immediate. If one
does it in the cardinal model, likewise. The other "models"
are based on these. The usual axiom of infinity in set
theory is usually a version of a model formed from the
Peano axioms, proved if necessary by the use of ordinal
numbers.
>> The goal is to show them that no matter what you have learned about a
>> simple thing, there is always an infinite amount more you could learn.
Showing what one can get from Peano, this will be obvious.
Going from the axioms to our usual base representation is
by no means obvious, although the steps are simple.
> But the lesson they would take away with them is: mathematicians are adept
> at making simple things complicated. Maybe a better idea would be to show
> them examples where mathematical ideas make difficult problems easy to
> solve. For example, perhaps, using Gaussian integers to solve problems
> involving the sum of two squares.
If one starts with concepts instead of manipulations, it
is quite clear. Analytic function theory, Fourier series
and Fourier transforms, are used, and there are even some
uses of number theory.
Sixty years ago, at a meeting of the American Mathematical
Society, I commented at a session that there was mathematics
which has been applied, and mathematics which has not yet
been applied. But even some of the esoterica can help
understand things which are applied, even if one can get
by whithout them.
--
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
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Herman Jurjus
The OP may also want to take a look at 'Shadows of the Truth' by
Alexandre Borovik (downloadable via his blog:
http://micromath.wordpress.com/ )
--
Cheers,
Herman Jurjus
Frederick Williams
Come to think of it, in some formulations of PA, + and 1 don't appear
either.
--
Needle, nardle, noo.
porky_...@my-deja.com
The common acronym is OOP, not OO.
amzoti
wrote:
I used: http://en.wikipedia.org/wiki/Peano_axioms
I know it is not definitive, but it does discuss the matter of 1, et.
al.
Frederick Williams
>
> I know it is not definitive, but it does discuss the matter of 1, et.
> al.
I find this odd:
5. 0 is a natural number.
6. For every natural number n, S(n) is a natural number.
...
Axioms 5 and 6 define a unary representation of the natural numbers:
the number 1 is S(0), 2 is S(S(0)) (which is also S(1)), and, in
general, any natural number n is S^n(0).
5. & 6. make no mention of 1 and 2. Oh, to be sure one _may_ define 1
to be S(0) and 2 to be S(S(0)) but the definitions aren't in the axioms
as quoted, nor is "+".
Later we read
The Peano axioms can be _augmented_ with the operations
of addition and...
(My emphasis.)
--
Needle, nardle, noo.
Herman Rubin
>> amzoti wrote:
Of course not. Addition is defined, and also notation. We
do not write mathematical entities, but representations of them.
I hope that mathematicians realize this, and it is ntecessary
that others be taugnt this. The representation is not the
concept represented.
The use of the characters "1" and "2" for the corresponding
NUMBERS is the convention associated with the Arabic numerals,
and other representations use other characters.
Axel Vogt
Engineers like practical things. And rough, but solutions.
How about approximating data or functions (over a fixed range)
by (orthogonal) polynomials, show in some cases how to find
that and talk about limitation.
Plus using that for integration. Or similar.
Dan Christensen
pull-down menus of the axioms and rules of logic, set theory and
number theory to construct formal mathematical proofs. In the
tutorial, examples 9 and 10 introduce Peano-like axioms for the
natural numbers. Every example include exercises with hints and full
solutions. Visit my website to download my program.
Dan
Web: http://www.dcproof.com