Sometimes I have too damn much imagination. What? How can you have too much imagination, isn't that a precious, rare commodity? Not when you're doing mathematics, at least the low-level math that I am currently plowing through. The trigonometry that I'm doing, whether it's solving triangles or finding the components of a vector, demand that the information about any situation be reduced purely to numbers: positions on a graph, distances between the points, measures of angles, and the sines or cosines or tangents of those angles. Anything else is irrelevant. Reductionism rules.

But trigonometry takes place in a scene, a world, a universe. And the universe is full of sense-details, which are endlessly distracting to a would-be reductionist. In this situation, I envy those intellectual souls (mathematical physicists, perhaps) who are naturally oblivious to the world and who find reductionism and abstraction easy. Doing word problems, I find myself involved in problems in ways which aren't mathematical at all. Let's take the beginning of a problem from Schaum's Outlines, the "Red Spine" book:

From a boat sailing due north at 16.5 km/h, a wrecked ship K and an observation tower T are observed in a line due east…

The mathematics student will plot these on a graph or do some other thing which is appropriately reductionist. But when I read this, I am on the boat sailing due north on the calm blue-grey water, into a golden afternoon sky. To the east is the mysterious wreckage of that old ship. How had it gotten wrecked? What had been lost? And further east is the red and white iron openwork of the observation tower, pale in the coastal haze. I smell the sea-brine and feel the wind as our ship steams along at the 16.5 km/h pace. Where are we going? The waves are low, yet I feel the ship's motion under my feet. I take a breath of fresh salt air…

But all I need to do is find the distances the words and data are asking for. A word problem puts me into a little world, whether it's the world of a trigonometric seascape or a chemistry lab or a car accelerating along a highway. Like the synesthetic colors of numbers I talked about some time ago, these irrelevant yet distracting sense-associations accompany most word problems for me.

I thought perhaps this was only a problem for me. Maybe it is a "girl thing," that females are supposedly more concerned with "holistic" sense-data than men are. Or maybe, more likely, it is an "artist thing." I am so used to depicting scenes and creating realistic visual pictures that I can't turn off the artistic imagination when I am doing math problems.

But I am not alone. This phenomenon of getting lost in the details has been documented in an excellent book by Sheila Tobias called OVERCOMING MATH ANXIETY. This book was one of the first ones I turned to at the very beginning of my math journey, as I faced the ordeal of recapitulating painful and difficult parts of my childhood and youth. On page 125, in a chapter discussing different styles of thinking and problem-solving, Tobias cites a (probably autobiographical) passage by Philip Roth, about "Nathan, a sickly and feverish young boy, whose father tried to sharpen his mind by giving him arithmetic problems to solve." The boy, instead of seeing the problem as a simple exercise in arithmetic, turns the details of the problems' situations (a merchant discounting unsold merchandise, a lumberjack buying lengths of chain) into intriguing and poignant stories. Roth continues: "My father … was disheartened to find me intrigued by fantasies and irrelevant details of geography and personality and intention, instead of the simple beauty of the arithmetic solution.…" Author Tobias consolingly comments: "(To distinguish)…"mathematical" and "nonmathematical" minds is to miss what they represent: two beacons in a continuum of human curiosity in search of meaning."

Hmmmm, two beacons. A ship is passing at 16 kilometers/hour on a northwesterly course between two beacons… For Roth, born to be a storyteller, the math problems became stories. For me, accustomed to painting pictures, the math problems become landscapes. Yet, like that steady ship passing by the wreck and the tower, I am determined to sail the sea of mathematics and direct myself in the vector-world of physics. But what do I do when I am faced with the first line of this trigonometry problem in the Barron's book:

You are lost on an endless, flat plain.

I cannot help but feel a sense of surrealistic dread, from which mathematics and physics, two pure abstract beacons, can rescue me.

Posted at 2:11 am | link