Agreed.
There is an interesting shared concept that I hope does not get lost,
a sort of "join" between the continuous and discrete. That is the
limit, and the epsilon/delta argument.
It is generally placed in the Continuous realm and thus most students
do not see how it is used in the Discrete. But the limit of a
sequence is just one example of its use in the discrete.
http://en.wikipedia.org/wiki/Limit_of_a_sequence
http://en.wikipedia.org/wiki/Cauchy_sequence
-- Owen
On Jun 30, 2009, at 7:19 AM, ERIC P. CHARLES wrote:
While much of the conversation below is steeped in issues I only
peripherally understand, from a pedagogical perspective I am in
complete agreement with Benjamin. A basic understanding of
probability and statistics is more likely to be achieved by
students, and would be more useful in most of their lives than a
basic understanding of calculus. Calculus is a big stumbling block
even for many students who enjoyed the math before that. I'm not
sure how the high school curriculum would change to accommodate the
new agenda, but I'd be really interested in finding out.
Eric
On Mon, Jun 29, 2009 11:19 PM, Owen Densmore <[email protected]>
wrote:
Hi joe.
> However, I don't understand your comment that math notation is the
> roman
> numerals of our times. Which branch of math do you have in mind?
> Certainly
> not calculus, where, as you know, we use Leibniz's elegant notation.
The core problem is the clash between two cultures: not the humanities
vs the sciences, but that between mathematics and computing. Or more
precisely, between mathematics and algorithms.
This is a large topic: it includes the lack of good mathematical
languages (like APL of old, and J today)
http://www.jsoftware.com/
http://www.jsoftware.com/jwiki/Guides/Getting%20Started
..which bridge the gap between symbolic computing and MN. It also
refers to the impossibility of parsing mathematics .. it is ill-
defined as a language. I.e. AB may mean A * B or the single variable
named AB.
It extends to the "Asymptotic Assumption" made by many mathematicians
when a discrete problem is more easily solved by converting to
continuous. (Reminds one of ABM vs Math modeling) Knuth has a good
discussion on this in his book Concrete Mathematics (CON=Continuous,
Crete=Discrete). Basically he makes the case that, although the leap
is reasonable at some point, it generally is taken too quickly.
So Roman Numerals == notational roadblock. MN is not only is
impossible to parse (and apply semantics to), it does not include any
notion of "scripting" .. i.e. pseudo-code.
> I also don't follow your comment about discrete versus continuous.
> Among theoretical computer scientists, people who want to understand
> the power of the computer and questions such as P vs NP study
discrete
> problems whereas people like me who want to solve problems
> coming from, say, physics or computational finance think about
> solving continuous problems such as path integration.
See above on Asymptotic Assumption and MN vs scripting. Certainly
computing, intrinsically discrete, provides wonderful approximations
to continuous problems.
Interestingly enough, the Sage system:
http://www.sagemath.org/
.. was originated by mathematicians who *required* open source so that
their theorems could be solved knowing the system on which they were
built. Sage is the first system I know of that has variable
declarations of Ring, Field, and so on. What would happen if Euclid
were propriatorey and only the results, not proofs were public
knowledge?
Computer use by mathematicians remind me of Statistics use by social
scientists. Often the techniques are used without understanding the
domain within which they are valid. If nothing else, the power law
distribution made many of us run back to see if our assumptions were
reasonable. Economics has fallen prey to this, the Black–Scholes
model apparently assumed a Gaussian where a fatter tail was needed.
This rant is a long one, but the summary is simple enough: Mathematics
and Computing/Algorithms need to be reconciled. Modern MN needs
(minor) changes to be at least machine readable. Computing languages
for mathematics need to bridge the gap between pseudo-code and
symbolics. APL/J are close.
How about a beer or glass of wine over this fascinating topic!
-- Owen
On Jun 29, 2009, at 8:19 PM, Joseph Traub wrote:
> Owen,
>
> I find nothing to argue with in Benjamin's talk. He says that
students
> studying economics, science, engineering, or math should learn
> calculus
> but that it may not be needed by other students who should study
> probability and statistics.
>
> However, I don't understand your comment that math notation is the
> roman
> numerals of our times. Which branch of math do you have in mind?
> Certainly
> not calculus, where, as you know, we use Leibniz's elegant notation.
>
> I also don't follow your comment about discrete versus continuous.
> Among theoretical computer scientists, people who want to understand
> the power of the computer and questions such as P vs NP study
discrete
> problems whereas people like me who want to solve problems
> coming from, say, physics or computational finance think about
> solving continuous problems such as path integration.
>
> Best, Joe
> <>
>
> Joseph F. Traub, Edwin Howard Armstrong Professor of Computer
> Science
> and External Professor, Santa Fe Institute
>
> [email protected] http://www.cs.columbia.edu/~traub
>
> Phone: (212) 939-7013 Messages: (212) 939-7000
Fax: (212)
> 666-0140
>
> Columbia University
> Computer Science Department
> 1214 Amsterdam Avenue, MC0401
> New York, NY 10027
> USA
>
> Administrative Assistant: Sophie Majewski
> [email protected] (212)939-7023
>
>
> **************************************************************
>
> From: Owen Densmore <[email protected]>
> Date: June 29, 2009 12:07:14 PM MDT
> To: The Friday Morning Applied Complexity Coffee Group
<[email protected]
> >,
> General topics & issues <[email protected]>
> Subject: [FRIAM] Arthur Benjamin's formula for changing math
> education |
> Video on TED.com
> Reply-To: The Friday Morning Applied Complexity Coffee Group
> <[email protected]>
>
> This is kinda cool and less than 3 minutes long!
>
http://www.ted.com/talks/arthur_benjamin_s_formula_for_changing_math_education.h
> tml
>
> The thesis is a different spin on my claim that modern Math Notation
> (MN) is
> the roman numerals of our times. Arthur Benjamin clearly explains
> that statistics and probability should be the "pinnacle" of our
> basic math
> education, not calculus. His reasoning includes the discrete vs
> continuous
> argument that resonates with my MN vs Algorithm (or MN vs script)
> concern,
> which I'd love to see resolved in a parsable reworking of MN.
>
> -- Owen
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
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============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
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Eric Charles
Professional Student and
Assistant Professor of Psychology
Penn State University
Altoona, PA 16601
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
