Problem Solvers and Theorizers

by Gian-Carlo Rota 

From Indiscrete Thoughts 

Birkhäuser (Boston, 1997),  pp. 45-46. *

Mathematicians can be 

subdivided into two types:

problem solvers and theorizers.

Most mathematicians 

are a mixture of the two

although it is easy to find

extreme examples of both types.

To the problem solver,

the supreme achievement

is the solution to a problem

that has been given up as hopeless.

It matters little that 

the solution may be clumsy; 

all that counts is that it should be 

the first and that the proof is correct.

Once the problem solver

finds the solution, 

he will permanently lose 

interest in it, and will listen 

to new and simplified proofs

with an air of condescension 

suffused with boredom.

The problem solver 

is a conservative at heart.

For him, mathematics consists

of a sequence of challenges to be met,

an obstacle course of problems.

The mathematical concepts 

required to state mathematical problems

are tacitly assumed to be eternal and immutable.

Mathematical exposition 

is regarded as an inferior undertaking.

New theories are viewed with deep suspicion,

as intruders who must prove their worth

by posing challenging problems

before they can gain attention.

The problem solver resents generalizations,

especially those that may succeed in trivializing 

the solution of one of his problems.

The problem solver is the role model

for budding young mathematicians.

When we describe to the public

the conquests of mathematics,

our shinning heroes are the problem solvers.

To the theorizer, 

the supreme achievement in mathematics

is a theory that sheds sudden light

on some incomprehensible phenomenon.

Success in mathematics does not lie

in solving problems but in their trivialization.

The moment of glory comes 

with the discovery of a new theory

that does not solve any of the old problems

but renders them irrelevant.

The theorizer is a revolutionary at heart.

Mathematical concepts received from the past

are regarded as imperfect instances

of more general ones yet to be discovered.

Mathematical exposition is considered 

a more difficult undertaking

than mathematical research.

To the theorizer, 

the only mathematics

that will survive

are the definitions.

Great definitions

are what mathematics

contributes to the world.

Theorems are tolerated as a necessary evil

since they play a supporting role

-or rather, as the theorizer

will reluctantly admit, an essential role-

in the understanding of definitions.

Theorizers often have trouble

being recognized

by the community of mathematicians.

Their consolation is the certainty,

which may or may not be borne 

out by history, that their theories

will survive long after

the problems of the day

have been forgotten.

If I were a space engineer

looking for a mathematician 

to help me send a rocket into space,

I would choose a problem solver.

But if I were looking for a mathematician

to give a good education to my child,

I would unhesitatingly prefer a theorizer.

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* Also reproduced in

Proving Darwin - Making Biology Mathematical

Gregory Chaitin

Pantheon Books (New York, 2012). pp. xi-xiii