Math for a Linguist

 Imagine that you are 5 and you struggle with reading, but of course, you must learn how to read in order to pass your first grade class.  Would you:

            1.    Read lots of books, struggling but improving, even if slowly.

            2.    Read 1 or 2 books, struggling but improving even more slowly.

            3.    Give up, because you’re just not good at reading and you never will be, because people are just born to be readers or they’re not.

 Now imagine that you’re in high school, and you’re struggling with math.  Would you:

            1.    Practice lots of math, even doing practice work not addressed within the required work in the class itself.

            2.    Barely skate by the required math, and do just enough to pass.

            3.    Give up, because you’re either born with innate math ability, or you’re not, and you’re just not.

I bring up this thought experiment because I find that most people I have met would acknowledge that options one or two in the reading scenario are appropriate, but tend to think that math knowledge is innate, and thus find option 3 appropriate in the math scenario. I realize that this thought experiment isn’t perfect – lots of people who have learning disabilities will find fault with my implicit assertion that effort is “all you need” in order to become proficient at math or reading.  Obviously, there are a lot of factors that affect a person’s skills, but the thought experiment is meant to illustrate a belief that many people hold to be a fact: you can either do math, or you can’t. That should be reevaluated.

If you think about it, Math is also a language! Math is the formal language of Nature. Look at the language basics.

            1.    Alphabets, the basic definition of sounds

            2.    Words, collection of alphabets that represent something of meaning

            3.    Sentences, collection of words that convey a thought.

            4.    Paragraphs, collection of sentences assembled to elaborate a single thought.

            5.    Articles, stories, books, etc., – advanced expression of thoughts.

Now look at the Math basics.

            1.    Numbers, the fundamental building blocks of counting, are the alphabets of Math.

            2.    Arithmetic is about manipulating numbers (addition, multiplication, etc.,)

            3.    Algebra finds patterns between numbers. a2 + b2 = c2 is a famous relationship, describing the sides of a right triangle. Algebra finds entire sets of numbers – if you know a and b, you can find c.

            4.    Calculus finds patterns between equations. You can see how one equation (circumference = 2 * pi * r) relates to a another one (area = pi * r2)

            5.    Many advanced Math concepts that express nature such as the Newtonian equations, Maxwell’s equation, Schrodinger’s equations all express the behavior of nature that is hard to express with other languages.

For example, the “wave or particle - duality of light” debate exists among us only because of the inadequate expressiveness of our languages. When written as a Math equation, nature expresses itself as a continuous function with smooth and unambiguous transition of state from particle nature to wave nature of light.

It is hard to express nature in a language other than Math. Einstein said “Pure math is, in its way, the poetry of logical ideas.”

There is another beautiful language – language of Music. It uses symbols and notations to build symphonies and incredible compositions that melt all living beings. The greatest music moves us in a deep way. It engages thought and emotion; it expresses itself through the physical sensation of sound.

Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of Mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds. Math is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms. It is an art – the most intellectual and classical of the arts.

Philosophy is written in this grand book - I mean the universe - which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it.

- GALIELO GALILEI, Il Saggiatore (1623)