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Where Sufficient Reason Isn’t Enough

Frank Morgan - Photo by Jeff Bauer of CitcoSpeaking at the MAA Carriage House on June 16, Judith Grabiner (Pitzer College) noted the meeting space’s abundance of parallel lines. She drew her audience’s attention to the everywhere equidistant outlines of the wood paneling and the beams traversing the ceiling to hit the wall at equal angles. 

Remember the theorem from high school geometry about how two lines are parallel if and only if, when they are cut by a transversal, corresponding angles are equal?

“If you counted how many times that theorem is in this room you’d be here all night,” Grabiner quipped. “This is the kind of room you’d design if you wanted to brainwash people to believe that space is Euclidean.”

In “Space: Where Sufficient Reason Isn’t Enough,” Grabiner argued that geometry subtly influences everything from art and architecture to physics, philosophy, and how people see and think about the world.

The World of Sufficient Reason

Even before displaying Euclid’s five postulates, Grabiner observed that the Greek mathematician’s axiomatic approach to geometry shaped conceptions of proof, truth, and certainty for centuries—and not just in mathematics.

Spinoza capped a supposed proof of God’s existence with “Q.E.D.” Newton called his laws of motion “axioms.” Thomas Jefferson gave the Declaration of Independence the flavor of a logical argument in the Euclidean style, complete with “self-evident” truths and a conclusion introduced by “therefore.”

Grabiner described the 18th-century world—the world on the eve of the discovery (or, if you prefer, invention) of non-Euclidean geometry—as “the world of sufficient reason.”

Dating back at least to Archimedes, the principle of sufficient reason states that, for everything that is, there is a reason why it happens as it does and not otherwise.

Archimedes used sufficient reason to argue that a lever with equal weights at equal distances from the fulcrum must balance. Giordano Bruno claimed that the principle implies the infinitude of space. (There’s no reason for it to stop at any particular place, after all.) No less a mathematical figure than Lagrange even appealed to the principle of sufficient reason in an attempted proof of Euclid’s infamously non-intuitive fifth postulate.

The world of sufficient reason was symmetric, balanced, and based on self-evident and necessary truths, Grabiner said, a world embedded in Euclidean space—a notion reinforced by art and architecture—and susceptible to rational interrogation.

Geometry stood, in other words, as humanity’s bedrock. “There are no sects in geometry,” said Voltaire. “One doesn’t say, ‘I’m a Euclidean.’ ‘I’m an Archimedean.’ Demonstrate the truth, and the whole world will be of your opinion.”

The Paradigm Shifts

But then Gauss, Lobachevsky, and János Bolyai came along and—Grabiner quoted Morris Kline—“knocked geometry off its pedestal.” 

Their breakthrough, said Grabiner, was recognizing that the seemingly absurd implications of negating Euclid’s fifth postulate are not absurd at all, but rather “truths in some alternative, counterintuitive reality.”

Recognition of this reality necessitated a paradigm shift.

“Before non-Euclidean geometry the laws of space and motion implied an infinite space whose properties were always the same, so we knew what was infinitely far away just as well as we knew the geometry in this room,” said W. K. Clifford. “Lobachevsky has taken this away from us.”

As intellectuals grappled with the new multiplicity of geometries, Poincaré declared that no geometry is more true than any other, just more convenient.

Hermann von Helmholtz used convex mirrors to argue that we can, contrary to Immanuel Kant, order our perceptions in a non-Euclidean space. 

(Doubtful of your ability to do this? Spend some time looking into your car mirror. “There’s a warning on it, isn’t there?” Grabiner asked her Carriage House audience. “What it means is, ‘Warning: The space you see in this mirror is not Euclidean!’”)

Spanish philosopher José Ortega y Gasset used the advent of non-Euclidean geometry to highlight the inferiority of provincialism to more broad-minded outlooks. For Ortega, Euclidean geometry was an unwarranted extrapolation to the whole universe of what was locally observed. Einstein’s relativity, on the other hand, relied on the alternative geometry of Riemann and promoted a harmonious multiplicity of all points of view. 

Ortega drew an analogy between mathematics and society: “There is a Chinese perspective that is fully as justified as the Western,” he said.

Western culture was eventually infiltrated by non-Euclidean geometry, Grabiner pointed out. From Man Ray’s helical “Lampshade” to the hyperbolic paraboloid of a roof on London’s Olympic Velodrome, non-Euclidean geometric objects are, increasingly, in the public eye.

And the mindset has changed along with the art and architecture. 

“Euclidean geometry and the principle of sufficient reason came to mean that reason can figure out the whole universe, and it’s symmetric and it’s stable and it’s uniform and there’s a reason for everything and everybody who studies it will come to agree,” Grabiner said at the close of her talk. “I trust you see that this is not the world we live in now.” —Katharine Merow

Grabiner’s talk was part of the Distinguished Lecture Series funded by the National Security Agency.

 

 

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