Math and Art Without the Filters

Barry Mazur is author of "Imagining Numbers: Particularly the Square Root of Minus Fifteen" (Farrar, Straus, Giroux, 2003).

We are surrounded by experts. Often we cannot live without them.

We are adept at seeking out second and third opinions. We carry guidebooks and listen to audio tours as we walk through museums. The omniscient Internet is at our fingertips. How difficult it is, nowadays, to think about anything without relying on some external authority.

But independence from authority still lies at the core of a few modes of thought. Encounters with art and encounters with mathematics -- even the simplest unscary math -- can be exhilarating, for that reason: In experiencing the impact of a work of art, or understanding a piece of mathematics, you are -- or at least you can be -- entirely on your own, with no authority in sight.

Here is what I mean by the type of independence of thought that simple math offers, perhaps requires.


Everyone who serves on a jury gets the same basic lecture from the judge so he or she can distinguish between evidence, inference and hearsay.

If John told Jane that Jim stole a doughnut, and Jane reports this on the witness stand, then she provides evidence that John said something, but her testimony about Jim is only hearsay.

Now compare these two assertions, one in history, the other in math:

(1) George Washington was the first president of the United States;

(2) If you add up the consecutive odd numbers starting with 1 you get perfect squares:

1 + 3 = 4 = 2 x 2

1 + 3 + 5 = 9 = 3 x 3

No matter how many primary sources, archival materials, Google searches or histories we collect about George Washington, when you or I say anything about his presidency, our testimony is, as any good trial judge would explain, mere hearsay. Our assertions rest, necessarily, on outside authority.

Now consider the statement about the sum of odd numbers that is the first sentence of Leonardo of Pisa’s “The Book of Squares,” published in 1225: “I thought about the origin of all square numbers and discovered that they arise out of the increasing sequence of odd numbers.”

I could simply take this on his authority: “If Leonardo of Pisa says it’s true, that’s good enough for me.”

Or I could do something partial like check out Leonardo’s sequence in a bunch of cases.

1+ 3 = 4 = 2 x 2

1+ 3+ 5 = 9 = 3 x 3

1+ 3+ 5 + 7 = 16 = 4 x 4

1+ 3+ 5 +7 + 9 = 25 = 5 x 5

Then I could mutter, “Didn’t I tell you Leonardo of Pisa was trustworthy?” and leave it at that. Either way, I still do not know why it’s so or whether it is always so.


There is another option: I can cast aside Leonardo’s authority and confront this intellectual material (the sum of consecutive odd numbers starting with 1 is a perfect square) head-on, alone. Moreover, I have some reasonable hope of understanding, alone, that it is true, and why it is true. For a hint about this, gaze at the following diagram.

Then, if I can understand why it is true, when I make the assertion that the sum of consecutive odd numbers starting with 1 is a perfect square I’m doing it more as an eyewitness and not merely offering hearsay testimony.

Just as math requests that we be eyewitnesses to it, art insists on it.

When, for instance, we consider some lines of poetry -- like Theodore Roethke’s “Snail, snail glister me forward / Bird, soft-sigh me home, / Worm, be with me. / This is my hard time.” -- authorities of various stripes can bring us closer to them, but in the end, the gurus, the learned scholars, the biographers and the grammarians must get out of our way as we experience these lines.

Art and mathematics invite us to leave our authoritative guides behind and ultimately to do the essential work on our own.