Re: [FRIAM] Arthur Benjamin's formula for changing math education | Video on TED.com

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
>
> traub@...          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
> sophie@... (212)939-7023
>
>
> **************************************************************
>
> From: Owen Densmore <owen@...>
> Date: June 29, 2009 12:07:14 PM MDT
> To: The Friday Morning Applied Complexity Coffee Group
<friam@... 
> >,
> General topics & issues <discuss@...>
> Subject: [FRIAM] Arthur Benjamin's formula for changing math  
> education |
> Video on TED.com
> Reply-To: The Friday Morning Applied Complexity Coffee Group
> <friam@...>
>
> 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
>
>
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