Differential Equations Govern the Future

Mathematics is the language used to quantify anything in science.  Still, some aspect of the tool we call math is utilized in nearly every field from business, to sales, and of course, engineering.

The field of Mathematics may be divided into several branches like calculus, linear algebra, and statistics.  While all of these subjects are fascinating, let us focus on calculus, which can be summarized as "the study of change."  If no facet of the universe ever changed, we would have no use for calculus, but then again, such a universe could not support life altogether (life cannot exist without chemical reactions).

At its heart, calculus focuses on functions, which are equations describing how variables are related to one another.  The simplest kind of function consists of two variables; one variable is dependent, the other is independent.  If a taxi driver charges a customer $1.50 per minute, then the cost function for riding in the taxi would be C = 1.5t, where C is in dollars and t is in minutes.  Here, time is an independent variable, and cost is a dependent variable (as the cost depends on the duration of the cab ride).  In other words, cost is a function of time, or C = f (t).

The simple taxi function given above relates cost to time, but functions can describe the relationships between other dependent variables and time.  We could, for example, consider the temperature of a hot cup of coffee.  One can imagine that the coffee's temperature value would decrease as time goes on until it reaches the room's air temperature.  The key difference between the taxi function and the coffee function, is that the function for the ride in the taxi was specified by the taxi driver.  No one specified the temperature function of the coffee.  The coffee's temperature function is the result of a differential equation, which is also known as a governing equation - a law of nature, and in the case of the coffee, thermodynamics.

You can Count on Asimov

Isaac Asimov is on a short list of my favourite science authors.  The list has two names on it: Arthur C. Clarke and Isaac Asimov.  Both write excellent science fiction (Asimov's "I Robot" and Clarke's "Fountain's of Paradise" are my personal favourites) and both write excellent topical science articles and essays.  While Orson Scott Card writes some compelling sci-fi (Ender's Game is probably my favourite novel of all time), he is not a "popularizer" of science - it is rare to find an author that is skilled in both fiction and non-fiction.

Asimov may have been the most prolific author ever, having published upwards of four hundred pieces.  His direct style in story-telling and, at times, redundant style of communicating in non-fiction does not appeal to everyone, but it does appeal to me.

Asimov's largest volume of work involves the communication of science, but physics in particular.  I just finished reading "Asimov on Numbers," which involves mathematics, but really focuses on, well, numbers.  It is a collection of essays written over a period of many years, beginning in 1959.  Although all articles are between four and five decades old, they have aged well.  The only instances where the book feels old is when population and financial figures are discussed, as the absolute values of both have inflated significantly in the ensuing years.

I learned a lot in reading these essays, mostly about the history of mathematics (very fascinating) and the Earth's geography (which, as it turns out, can be described so well numerically).

The Love Function

Typically, my semesters are spent teaching Physics courses.  The past two semesters have been a little bit different for me, with some Mathematics courses thrown into the mix.  A few weeks ago, in my Calculus course, one student, we will call “Jimmy”, asked if math could be used for something useful.

I explained that math was the language of science, and science affects us every day.  Also, business requires math to predict where it will go based on where it has been and other factors.  The social sciences require a deep understanding of statistics to make sense of our lives.

Jimmy said that business and science are good, but not all that important to him at this stage of his life.  “Can math help us find love?” he asked.  What an excellent question.