Download Dear Charles, - Abstractmath.org

1
Dear Charles,
I saw the article in this month's FOCUS and checked out your site. I
think
it's a great idea. I have written a piece of software for group theory
that
has a lot of the same goals as your site--to make the abstract more
accessible and understandable to first time visitors--although I'm
concentrating only on group theory. You can see it here.
http://groupexplorer.sourceforge.net
So I went to see if there was a place on your site where I could
suggest you
add a link to the URL above. What I found was that on this page
http://www.abstractmath.org/MM/MMGlossaryA.htm
you have a link to a page that is supposed to contain mention of
groups,
rings, and fields
http://www.abstractmath.org/MM/MMAbsAlg.htm
but that this page does not exist.
So my two suggestions are
1. there is a broken link that you may want to fix, and
2. if you feel it would be beneficial on that page to point to
Group
Explorer at the URL above, that would be great.
Thank you, and keep up the good work!
-Nathan Carter
Mathematical Sciences Department
Bentley College
175 Forest St., Morison 319
Waltham, MA 02452
(781) 891-3171
2
Laws of elementary algebra
Addition is a commutative operation (two numbers add to the same thing whichever order
you add them in).
Subtraction is the reverse of addition.
To subtract is the same as to add a negative number:
Example: if 5 + x = 3 then x = − 2.
Multiplication is a commutative operation.
Division is the reverse of multiplication.
To divide is the same as to multiply by a reciprocal:
Exponentiation is not a commutative operation.
Therefore exponentiation has a pair of reverse operations: logarithm and exponentiation with
fractional exponents (e.g. square roots).
Examples: if 3x = 10 then x = log310. If x2 = 10 then x = 101 / 2.
The square roots of negative numbers do not exist in the real number system. (See: complex
number system)
Associative property of addition: (a + b) + c = a + (b + c).
Associative property of multiplication: (ab)c = a(bc).
Distributive property of multiplication with respect to addition: c(a + b) = ca + cb.
Distributive property of exponentiation with respect to multiplication: (ab)c = acbc.
How to combine exponents: abac = ab + c.
Power to a power property of exponents: (ab)c = abc.
Laws of equality
If a = b and b = c, then a = c (transitivity of equality).
a = a (reflexivity of equality).
If a = b then b = a (symmetry of equality).
Other laws
If a = b and c = d then a + c = b + d.
If a = b then a + c = b + c for any c (addition property of equality).
If a = b and c = d then ac = bd.
If a = b then ac = bc for any c (multiplication property of equality).
If two symbols are equal, then one can be substituted for the other at will (substitution
principle).
If a > b and b > c then a > c (transitivity of inequality).
If a > b then a + c > b + c for any c.
If a > b and c > 0 then ac > bc.
If a > b and c < 0 then ac < bc.
3
I recently sent the message quoted below to a colleague in neuroscience. I am sending it to
this list in case some of you know about this sort of thing.
This was triggered by Ed Dubinsky’s last message. I have rewritten these sections as a way
of bringing out the idea of math object more clearly:
http://www.abstractmath.org/MM//MMMathObj.htm
http://www.abstractmath.org/MM//MMImagesMetaphorsFunctions.htm
http://www.abstractmath.org/MM//MMFunctions.htm
http://www.abstractmath.org/MM//MMSetMetaphorImage.htm
Warning: I am still rewriting these. They may contain missing pieces, broken links and even
(gasp) errors.
Charles Wells
QUOTE
I am creating a website (URL below) about the problems of learning abstract math for
postcalculus math students. I want to say SOMETHING about what "concepts" really are. My
point is that the number 42, Sherlock Holmes, a unicorn, and the month of September are all
objects of thought in a certain sense, and they have properties and relationships with other
objects of thought.
1) Is it reasonable to say: abstract and fictional concepts are stored in the brain in much the same
way as concepts of physical objects such as the Washington Monument.
2) Is it in accordance with modern understanding to say that basically each concept corresponds
to a particular bunch of neurons?
3) Do you know of a good website that discusses this sort of thing from the point of view of
neuroscience? I would like to expand on this discussion on my site:
http://www.abstractmath.org/MM//MMMathObj.htm#otherkinds
I don't want to do a presentation on neuroscience on my site, but my idea is that if the student
can believe that a particular concept is implemented physically in the brain in some sense then it
is easier to think about abstract mathematical objects as THINGS you can think about rather than
as something you do. This is a big stumbling block to higher math.
UNQUOTE
4
Charles
>
> There is some really good stuff here.
>
> One silly point: if delta = min {1, eps/5} then delta < or = 1 (line 4 of
> your explanation).
>
> One big point. There is a major research text by Malcolm Swan
> "Collaborative learning in Mathematics" (2006) ISBN 1-86201-311-X
>
> summarising much theory but based on research with 16-19 year olds.
> He establishes great significance for learning, in social conflict
> relating to mathematical concepts.
>
> In higher education we are desperately short of evidence of habitual
> misconceptions. When we have got it, it is the teachers who need to know
> how to use it.
>
> Bob Burn
>
> On Wed, 31 January, 2007 1:53 am, Charles Wells wrote:
> > I have extensively revised the chapter of abstractmath.org called Doing
> > Math at
>>
> > http://www.abstractmath.org/MM//MMDoingMath.htm
>>
> > and would appreciate comments and suggestions.
>>
> > I have also added a short section on the lack of standardization of
> > definitions of math terms and symbols at
>>
> > http://www.abstractmath.org/MM//MMDefs.htm#nostandardization
>>
> > Charles Wells
>>
>
>
> -> Bob Burn
> Research Fellow, Exeter University
> Sunnyside
> Barrack Road
> Exeter EX2 6AB
> 01392-430028
5
Author: David W. Cantrell
Subject: Re: Nonstandardization in mathematical terminology
Charles Wells <[email protected]> wrote:
> I have just added a short section to abstractmath.org about the lack
of a
> standarizing body for definitions of mathematical words and symbols,
at
>
> http://www.abstractmath.org/MM/MMDefs.htm#nostandardization
>
> I would appreciate any comment or other examples of conflicts in
> terminology.
You have already mentioned some that I had in mind, but there are many
more. Here are just a few others:
To some "multivalued function" is an oxymoron.
Related to the above (letting V serve here as an ersatz radical symbol
denoting square root), some would say that V9 is +/-3, but most would
say
that it is only +3, the principal value. Some would say that x^(1/2) is
bivalued for nonzero x, while others would say that it denotes just the
principal value. Etc.
Special functions have many different conventions. Glancing at the
short
Notation sections near the beginnings of the chapters in Spanier &
Oldham's
_An Atlas of Functions_ would give you some idea of the problems. As an
example, consider elliptic integrals. Suppose I've told my audience
that
I'm dealing with the complete elliptic integral of the second kind,
denoted
by E(x). Then what is the value of E(1/2)? According to one convention,
it's 1.467...; according to another, it's 1.350... Both conventions are
common. Mathematica uses the former, Maple the latter. And things get
much
messier if we deal with _in_complete elliptic integrals!
Is 0^0 = 1 or is it undefined?
What is Arccot(-1)? Is it 3pi/4 or -pi/4? It depends. There are two
different common ways to choose the principal value when the argument
is
negative. Of course that choice affects which identities hold, etc.
IIRC,
only sine, cosine and tangent have inverse functions which are
universally
agreed upon. (That's why I try to avoid using inverses of cot, sec and
csc.)
What is sin(90)? I'd say "That's about 0.894 . But you probably meant
to
ask for sin(90 degrees), which equals 1." But there are those, some of
whom
are mathematicians, who would say that there are _two_ sine functions,
one
taking an argument in radians and the other an argument in degrees [so
that
sin(90) = 1 would be correct].
That's all... for now.
David
6
Author: Robert Israel
Subject: Re: Nonstandardization in mathematical terminology
On Jan 31, 5:06 am, "G. A. Edgar" <[email protected]>
wrote:
> In article
> <[email protected]>,
>
> Charles Wells <[email protected]> wrote:
> > I have just added a short section to abstractmath.org about the
lack of a
> > standarizing body for definitions of mathematical words and
symbols, at
>
> >http://www.abstractmath.org/MM/MMDefs.htm#nostandardization
>
> > I would appreciate any comment or other examples of conflicts in
terminology.
>
> > I have also spent some time doing minor and semi-major edits to the
chapter
> > called Doing Math at
>
> >http://www.abstractmath.org/MM/MMDoingMath.htm
>
> > Charles Wells
>
> I remember when we started the class on topological groups, the
> professor (Andrew Gleason) gave some explanations about terminology.
> "Normal" is used both in group theory (normal subgroup) and topology
> (normal space).
In this course we use it only in the group sense.
Not to mention all the other uses of this most overworked word in
mathematics:
normal distribution, normal vector, normal operator (or normal element
of a *-algebra, or normal subset of a *-algebra), normal number,
normal family of functions, normal extension of a field, normal
equations in linear regression,
assorted normal forms... what have I left out?
Robert Israel
Department of Mathematics
University of British Columbia
[email protected]
http://www.math.ubc.ca/~israel
Vancouver, BC, Canada
7
Charles, I`ve not had a chance to look at your interesting chapter
at any
length, but I have a couple of quick comments. I plan to look at it
more
closely later.
In point 4), I would say that every real number has exactly one REAL
cube root. Students might think there is only one cube root.
In 1), you mean never instead of always, don`t you? Olaf.
******************************************************
> I have extensively revised the chapter of abstractmath.org called
Doing
> Math at
>
> http://www.abstractmath.org/MM//MMDoingMath.htm
>
> and would appreciate comments and suggestions.
>
> I have also added a short section on the lack of standardization of
> definitions of math terms and symbols at
>
> http://www.abstractmath.org/MM//MMDefs.htm#nostandardization
>
> Charles Wells
8I have just been shown your latest posting to mathedu. Some years ago I
had
quite a campaign to persuade the mathematical community to do something
about
the lack of standards in Mathematics, but received little support from
other
mathematicians who thought that it would hobble their creativity.
Indeed I was
commissioned by the IEEE to rewrite their dictionary of mathematical
definitions for the P610 initiative, the Dictionary of Computing.
However, just
as it was about to be published the whole project collapsed and so it
never saw
the light of day.
My main concern was that while competent mathematicians can cope with
some
vagueness in definitions, as they automatically interpret statements in
the way
intended by the author, the same is not true for people who may be
using results
without complete understanding of what is going on, and this can be
very
serious, for example, in the development of software. You mention the
lack of
clarity as to whether zero is a natural number or not. Most
mathematicians do
not care that much, but software often fails because a parameter
reduces to
zero and this possibilty has not been allowed for in some subroutine.
More seriously I have found over many years teaching students that one
of the
biggest blockages for them is that they misparse formulae and so what
should be
simple logical derivations are perceived by them as magic and
mysterious
procedures which must be learned parrot fashion if they are to succeed.
Yet
there is no clearly accepted rule. I managed to type exactly the same
formula
into eight different calculators/ computer algebra systems and got
eight
different answers, and I am not talking about rounding errors here. For
example
try typing "=-3^2" into a cell in Excel (without the quotes). Now
create a macro
in excel with the statement "cells(1,1)=-3^2", and run it. Would you
expect to
get the same answer? What is the value of 2^2^3? What is log(10)? Even,
what is
2+3x4? Or 12/2x3?
By the way, I noticed a couple of minor typos in your document. At one
point you
say that a number is even if it is divisible by 21. You also give a
definition
of square root which requires it to be an integer, and then apply it to
the
square root of 2.
I could say more, but you may have given up reading this already, so I
will stop
here.
Hugh
Dr Hugh Porteous
116 Totley Brook Road
Sheffield S17 3QU
UK
Tel: +44(0)114 236 6674
8
Stuff from Number Theory and Equivalence Relation in index
Sets as properties
Sets as labels
Abstract math as a game
How to think about transitive, symm, etc
Talk about arbitrary, natural, artificial
Default structure
PARAMETERS
9
Blog about n-dimensional thinking:
http://jonstraveladventures.blogspot.com/2007/08/how-to-work-with-extra-dimensions.html
Ad SciTalks blog
10
Make new website: Amazing stories about math. Links to abstractmath, or make it part of
abstractmath.
Infinite cardinals peculiar behavior
The Perrin function
E8 check out web
can find out if number is composite without knowing the factors
11
New articles for blog
 English agglutinative ish teria etc for word derivation but ish is more like an inflection?
 scope, let vs assume, hard for grasshoppers, methods for helping this -- links
(“remember x is now = to 5”)
 two-valued force to fit
 Talke about Wikipedia articles -- math as a language, the lang. of math
 foreign languages disappearing in math – I used to read math in French, German, and
(less) Spanish, Russian
 differences in logic between math English and real English – and some philosophers
dispute this
12
13
List of claims about math language.
 scope of different words such as let, assume, define
 symbolic language is different from math English
 changing the meaning in the process of discourse – unusual outside math and
mathematical arguments in science writing?
