A square is a rectangle. This is the statement that I found myself explaining -- actually passionately defending -- with my 6-year old (Lizzie) and 8-year-old (Anna Mae) a few nights ago when straying off topic from Lizzie’s math homework. A rectangle has four sides and four right angles. So, any polygon (or using Lizzie’s language: a shape that has no holes and no curvy sides) with exactly four sides and four right angles must be a rectangle. Ergo, a square is rectangle. Ladies and gentleman of the jury, I rest my case.

I know what you’re thinking. No one likes a math nerd (especially one that pretends he’s Matlock) and isn’t this just some technical nuance of language that will only confuse my daughters that are simply learning the properties of basic shapes? Sadly, I have to concede: you’re probably right on all accounts. But, really it’s a lesson that goes deeper than geometric properties. It’s a lesson on mathematical reasoning, on problem solving, on logic - on starting with a few rules or assumptions and building a whole world of conclusions. So when I saw the window of opportunity for spreading my love of logic to my daughters open slightly, I dove through head first.

Tucked away in your head somewhere is the memory of learning about postulates and theorems in high school geometry. Postulates are the basic assumptions which are simply accepted as true. For example, postulate one states that a straight line segment can be drawn joining any two points. This very same postulate was scribbled on some papyrus by Old Man Euclid in 300 BC. (He probably preferred to just be called Euclid back then.) Euclid also included four other postulates and five axioms or “common notions.” The basic truths or assumptions are the seeds from which enormous mathematical forests grew. Using only a few basic assumptions, philosophers and mathematicians were able to conclude a few more things, which allowed them to prove something else, which in turn allowed us to determine another thing, opening the door for the next generation to form the next conclusion, and so on. It’s like a Rube Goldberg machine that just keeps propelling the next act. A marble rolls down a ramp that strikes a domino which falls onto a switch that lights a match that causes air to fill a balloon… Only the mathematical machine never comes to an end - with each stunt becoming more complex than the one before.

If you’ve ever even stacked dominos in a row, you know that each domino is completely reliant on the one that precedes it. The same is true of logic. Place one domino a little out of position and the dominos that follow will remain standing. Introduce faulty logic along your path of mathematical discovery and everything that follows is erroneous. This is why I love math and why I want my girls love to math. For me, it’s never been about algorithms or contrived steps that march you from point A to D. It’s about knowing that you have a bag of tools - tricks really - that if applied accurately and appropriately, allow you to discover your next destination. Mathematicians don’t just solve problems; they uncover opportunities.

Reasoning and logic also teaches us of the importance of knowing that the beliefs that we hold are rooted in some basic assumptions that we each individually hold to be true. Time and time again, we’ve learned that what seems obvious and certain based on our individual perspectives do not fit the perspective of others, the believes of the next generation, or even the facts that we’ve yet to discover.

Galileo and a few other scientists before him had the courage to question that the Earth was the center of the universe. Albert Einstein challenged some of Isaac Newton’s assumptions and showed that of Newton’s Laws of Motion were only approximately correct, falling apart when objects approached the speed of light. Euclid’s very own fifth postulate became quite the controversial topic for nearly 20 centuries. (You know how mathematicians like to find fodder.) The postulate said something like: “At most one line can be drawn through any point not on a given line parallel to the given line in a plane.” There was much debate about whether or not this really needed to be included as a postulate. It wasn’t that mathematicians necessarily thought of it as a concept that didn’t hold true, it’s just that they thought it was really unneeded; that his other basic assumptions (or postulates) basically had him covered. (In modern time, think of it as using unnecessary or redundant lines of code resulting in bloated, slower software.) Well, they were wrong. In fact, the more they tried to prove that it wasn’t needed, the more interesting things became. It was finally around the 19th century that entire branches of geometry - that use a different fifth postulate - became accepted as plausible alternatives giving shape to non-Euclidean geometries like elliptical geometry and hyperbolic geometry.

So, logic teaches us that a square is a rectangle. It also teaches us animals in the air likely have wings and that it’s wise to wear a jacket when we see snow on the ground. But, logic also teaches us that our conclusions are based on some definitions or assumptions. Logic reminds us that in every argument or nearly everything we hold to be true, there are assumptions. Mathematics and logic teaches us not just on how to build on our assumptions to form a stronger argument; it forces us to acknowledge the vulnerabilities of our thoughts and respect the positions of others. And, that is a lesson I want my kids to know.

Reasoning and logic also teaches us of the importance of knowing that the beliefs that we hold are rooted in some basic assumptions that we each individually hold to be true. Time and time again, we’ve learned that what seems obvious and certain based on our individual perspectives do not fit the perspective of others, the believes of the next generation, or even the facts that we’ve yet to discover.

Galileo and a few other scientists before him had the courage to question that the Earth was the center of the universe. Albert Einstein challenged some of Isaac Newton’s assumptions and showed that of Newton’s Laws of Motion were only approximately correct, falling apart when objects approached the speed of light. Euclid’s very own fifth postulate became quite the controversial topic for nearly 20 centuries. (You know how mathematicians like to find fodder.) The postulate said something like: “At most one line can be drawn through any point not on a given line parallel to the given line in a plane.” There was much debate about whether or not this really needed to be included as a postulate. It wasn’t that mathematicians necessarily thought of it as a concept that didn’t hold true, it’s just that they thought it was really unneeded; that his other basic assumptions (or postulates) basically had him covered. (In modern time, think of it as using unnecessary or redundant lines of code resulting in bloated, slower software.) Well, they were wrong. In fact, the more they tried to prove that it wasn’t needed, the more interesting things became. It was finally around the 19th century that entire branches of geometry - that use a different fifth postulate - became accepted as plausible alternatives giving shape to non-Euclidean geometries like elliptical geometry and hyperbolic geometry.

So, logic teaches us that a square is a rectangle. It also teaches us animals in the air likely have wings and that it’s wise to wear a jacket when we see snow on the ground. But, logic also teaches us that our conclusions are based on some definitions or assumptions. Logic reminds us that in every argument or nearly everything we hold to be true, there are assumptions. Mathematics and logic teaches us not just on how to build on our assumptions to form a stronger argument; it forces us to acknowledge the vulnerabilities of our thoughts and respect the positions of others. And, that is a lesson I want my kids to know.

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