Proof, by Joseph Caputo '07
Or, a mathematician shows you how you do math all the time without even knowing it, to achieve certain outcomes and keep your life in order. And why the journey matters most. And how to order pizza 1,000 different ways. (Photograph by Peter Howard).
Assume statement P means “it is raining,” while Q translates into “I will bring an umbrella.” Logically, the following statements are all equivalent:
if P, then Q: “It’s raining, so I will bring an umbrella.”
Q is necessary if P: “If it’s raining, I have to bring an umbrella.”
not Q unless P: “I won’t bring an umbrella unless it’s raining.”
and if not Q, then not P: “It’s not raining, so forget the umbrella.”
However, the truth of Q alone does not reveal any information about the truth of P.
Make sense? It should. It’s a common decision we face more often than we’d like, and our brains are wired to make such logical assessments. Our Ps and Qs pair, vary, permute and pilot us towards conclusions.
This innate ability to achieve certain outcomes through reason is apparent in all intelligent organisms. In the human mind, illogical pairs provide fuel for the imagination and keep us from making bad choices. What if I brought my umbrella while it wasn’t raining? How silly! This same process allows us to ponder what graduate school to choose, house to buy or bestseller to write.
Here’s the catch-this is math. After using logic to decide what belt to wear in the morning, professional (and avocational) mathematicians condense this process into a written symbolic form that provides a powerful tool to study and understand the world of numbers and ideas they’ve created.
Some millennia before SLC faculty member Dan King and his colleague, Joseph Woolfson, started teaching math at the College, Euclid, a Greek mathematician from 300 B.C.E., introduced one method to establish mathematical truth: the axiomatic approach. “Take a statement to prove,” says King, “like, all triangles have an angle sum equal to 180 degrees [known as the Triangle Sum Postulate]. You can then prove twenty other things from that statement.” Axioms, according to King, are statements assumed to be true, accepted without evidence and expressed in undefined terms like point, line, or intersect. Two or more axioms are joined and manipulated using logic in order to deduce the truth of another statement, just like our rainy day Ps and Qs. Acceptable modes of logical reasoning are called syllogisms. Whether through use of syllogisms or other forms of argumentation (like mathematical induction or proof by contradiction), “if we prove a statement, we classify it as true and label it a theorem,” King says. A proof is the ultimate convincing argument acknowledging the truth of a statement.
“A proof, mathematically, absolutely, without a doubt, provides certainty to a statement under consideration,” King explains. Writing a proof is like baking a cake from scratch. We begin with the basic ingredients (in this case, axioms, undefined terms and previously proven theorems), mix them together, add logic (syllogisms, etc.) and see what happens. “Math is pure creativity,” King says. “How do you get from A to B? There may be only one right answer—a popular complaint about mathematics—but how do you get to that answer and what path should you take? “This is where creativity enters the picture. When I’m conceiving a proof, I feel like I’m involved in an intricate puzzle. I’ve got all the pieces (or at least most of them), so how do I arrange them?” [Editor’s note: take a break-and a look at the sidebar.]
Just like a game of dominos, lining up the statements for that one satisfying outcome is the excitement for mathematicians. According to King, “The goal is the final sentence of the proof, but the joy is the search.” This is where college-level math shows its true colors: the rote memorization of formulas characteristic of high school math class is gone, replaced by mathematics that values mistakes and requires creativity. “A wrong turn can be extremely valuable. It can reveal a completely unexpected path to the prize. The ability to make and recognize good errors is the sign of an insightful mathematician.”
King, his gleeful grin stretching from ear to ear, speaks of math the way a teenager does a high school crush. This is what he loves. The more he talks about it, the more enthusiastic he gets. “The key to writing proofs is being able to ask the right questions, and being able to be caught up in play. ‘To get from A to B... well, what happens if I consider this? What happens if I take this route?’ To move from one true statement, you have to be somewhat fearless, boldly playful, and willing to accept what comes across your plate. There have been times in history that mathematicians were on the verge of making great discoveries but thought they erred because the places they were reaching with their ideas were unfathomable to them.”
In math, victory may be the desired outcome, but what everyone talks about is, well, how you played the game. “A talented mathematician is someone who has developed a capacity for distinguishing a good option from a dead end among the many paths available. Just like chess players: they don’t think thirty steps in advance, but rather have an intuitive feel about which current move will lead them along a more profitable path.” Mathematicians desire only to create elegant proofs. Maybe you recall the image of Matt Damon scribbling wildly on a blackboard in the film Good Will Hunting.
The real ideal is a proof that can be done quickly, with no excessive details and without requiring complicated mathematics.
If mathematicians have been doing proofs since Euclid, can there be anything left to prove? Absolutely, King assures us. “Mathematics is always extending itself. Every statement proved to be true only produces another question or conjecture.”
Now that planetary movements have been tracked and the oceans mapped, many mathematicians have switched from the natural world to grabbling with imaginary numbers and other abstract creations of the mind.
King is a testament to the unending range of mathematics. With the words of philosophers snuggling next to textbooks on his bookshelf, “I’m not your average mathematician,” he smiles. “I love exploring the social life of mathematics and how it extends to other areas of mathematical pursuit such as philosophy, psychology, literature and political science.” But King prides himself on being a teacher first and mathematician second. He teaches math from a discussion-based interdisciplinary approach that is ideal for Sarah Lawrence, raising new possibilities for students who never envisioned doing proofs in college.
“An interest in math is a gift I received at a very young age, a gift I continue to cherish. And when you love something deeply it always comes with a compulsion to share it.”
Quod erat demonstratum.