**DIALECTICS of MODELING **

Based on a talk given by Al Vilcius at York University, November 1996.

This discussion gives my personal view of mathematics in industry, from the perspective of a person in industry, and keeping it not too technical. I have worked in the Finance Industry for over 20 years now, but still think of myself as a retread algebraist.

So what can a mathematician do in industry?

Indeed the landscape has change quite dramatically over the last 20 years. At first, being a mathematician in industry was a liability; it was a fact that needed to be played, even concealed at times, and certainly required justification and "selling" at all times. However, it was possible to sell this situation, and slowly it started to become a real asset.

Rather than just reviewing a list of job opportunities, or a list of formulas required, I will try to present a little more philosophical view.

I regard mathematics as __an extension of the natural language that
increases our power to make inference__. This may be taken almost at the
level of a slogan that should trigger an enormous amount of concepts in support
of the process of solving problems.

We generally accomplish this through modeling: I'll explain with a picture:

I consider this picture to represent a framework for applying math to "real world" problems

of course we can iterate this through several steps

example: pricing a derivative security

..................... some discussion of derivatives: options, first option, ....

- abstract market factors as variables in a stochastic process, and
- interpret expectations as price

see "Optional Math is not optional" Notices AMS for a start

So where is the math?

and where is the philosophy?

thesis, antithesis, synthesis

What sort of math might be useful in industry?

**statistics** - but do we actually understand variation?

i.e. reasoning with uncertainty to draw conclusions

not just computational, but creative e.g. risk neutral measures

so real analysis and measure theory are important prerequisites

Economists are finding good applications for **dynamical systems** and
**game theory** also

Even though I tend to have a non-standard view, I still believe in the basics

** logic** - ability to make inference in different "worlds"
e.g. quantum worlds

** number theory** - cryptography

**category theory** - computer science, or maybe to really understand
variation

The gateway to mathematical applications in industry for me has been through risk management

however, I have recently made another change, from Investment Banking to electronic commerce: "Smart Card" IC Technology

new challenges of quantifying risk are leading me into simulation modeling, including AI concepts - also "data mining" is a hot topic!

** the essence is pattern recognition**

I suppose I have always been prone to hero worship

but my heroes have not been entertainers or sports stars

rather they have been mathematicians, like John von Neumann

Another one of my heroes was a math professor I had as an undergraduate who greatly influenced my life

I once asked him: what do I really need to know to be a good mathematician ?

I expected a list of topics or even theorems

but rather he said: two things you must know

- how to count
- how to recognize patterns

That gave me the connection of mathematics to life within the framework of
the **dialectic**

and the motivation to study the structural side

at least to me, that is what mathematics is all about.

and that is what I wanted to leave you with today.

thank you very much ................ Al Vilcius

**Talk given Nov. 5, 1996 Vari Hall Lecture Theatre "A", 5:30
pm**

Ref: Prof. Norm Purzitsky, Organizer