[FOM] Mathematics and Explanations

[Lucas Wiman]

Buckner writes:

 >As an interview question it's always useful to test a candidate's
 >knowledge by asking them to explain the meanings (in English, 
obviously) of
 >the different terms of which the [Black-Scholes] equation is composed.  
This is the only way
 >to understand what the formula really means, and it's very important 
that
 >people do understand it, because it's used in financial risk management,
 >where black boxes are dangerous things.

I know little about the Black-Scholes' equation, but I understand that 
it is the solution of a stochastic differential equation which provides 
the optimal price of an option or future.  I can certainly understand 
asking the candidates to explain what "risk" is, or whatever else is 
found in the equation, but do you ask them to explain precisely what a 
stochastic differential equation is?  Continuous time and the Ito 
calculus?  I doubt it very much.  You ask them to explain precisely what 
the meanings of the variables which they plug in to the equation are, 
but these are financial realities with which they must work every day.  
The mathematical components of the equation sort of sit behind the 
financially relevant portions--and they are quite hard to explain.

 >Btw I've noticed that many trained mathematicians can find this very 
hard.
 >They are excellent at rearranging equations, and seem to have a 
supernatural
 >ability to do this correctly and rapidly.  But they are often hopeless 
at
 >explaining what these formula really are.  By contrast, lawyers, whose 
job
 >it is to frame financial contracts in a way that is both precise and
 >intelligible, do have a knack for translating mathematical ideas into 
plain
 >language.  (But then they are paid several orders of magnitude more than
 >your average graduate mathematician - is there a moral to be drawn?).

To take your example a step further, say you tried to explain the 
Black-Scholes equation to someone who had never heard of terms like 
"risk" "option" "stock" or even "money."  How long would it take you to 
explain it?  I would venture to say a good long while.  Perhaps it 
wouldn't even be possible.  Yet I have no doubt that you understand the 
Black-Scholes equation very well.  The point is that you're able to ask 
these questions because you share certain concepts with the applicant, 
and he or she is able to respond using these shared concepts.  Without 
them, things become much more difficult to explain.  Similarly, 
mathematicians have no difficulty explaining to one another what a 
Boolean algebra, a Diophantine equation or a Banach space is.  The basic 
concepts are shared, but they're very far removed from the experience of 
ordinary people.  This is why mathematicians have so much difficulty 
explaining them.

I suppose that a mathematician's ability to work with mathematics might 
seem "supernatural" if you don't know anything about mathematics.  I 
would imagine that many people would find abilities which you possess 
"supernatural", though it is only because they don't understand them.  
"Any sufficiently advanced technology is indistinguishable from magic."

As for whether lawyers are good at explaining mathematical concepts to 
the lay-person, I've no evidence one way or another.  I would guess that 
the reason lawyers make more money than mathematicians is that their 
work is immediately relevant to business--something that is often not 
the case with mathematicians.  Mathematicians who do work in industry 
often make very high salaries.  Indeed, I read an article yesterday 
about a shortage of good mathematics teachers in Britain because math 
students are being tempted by the high-paying financial services 
industry.

Challenge:  find a lawyer who can explain the Stone duality theorem to 
the mathematical novice.

- Lucas Wiman