“Inspiring and informative…deserves to be widely read.”
—Wall Street Journal

“This fun book offers a philosophical take on number systems and revels in the beauty of math.”
—Science News

Because we have ten fingers, grouping by ten seems natural, but twelve would be better for divisibility, and eight is well suited to repeated halving. Grouping by two, as in binary code, has turned out to have its own remarkable advantages.

Paul Lockhart presents arithmetic not as rote manipulation of numbers—a practical if mundane branch of knowledge best suited for filling out tax forms—but as a fascinating, sometimes surprising intellectual craft that arises from our desire to add, divide, and multiply important things. Passionate and entertaining, Arithmetic invites us to experience the beauty of mathematics through the eyes of a beguiling teacher.

“A nuanced understanding of working with numbers, gently connecting procedures that we once learned by rote with intuitions long since muddled by education…Lockhart presents arithmetic as a pleasurable pastime, and describes it as a craft like knitting.”
—Jonathon Keats, New Scientist

“What are numbers, how did they arise, why did our ancestors invent them, and how did they represent them? They are, after all, one of humankind’s most brilliant inventions, arguably having greater impact on our lives than the wheel. Lockhart recounts their fascinating story…A wonderful book.”
—Keith Devlin, author of Finding Fibonacci

For seven years, Paul Lockhart’s A Mathematician’s Lament enjoyed a samizdat-style popularity in the mathematics underground, before demand prompted its 2009 publication to even wider applause and debate. An impassioned critique of K–12 mathematics education, it outlined how we shortchange students by introducing them to math the wrong way. Here Lockhart offers the positive side of the math education story by showing us how math should be done. Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living.

In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.

Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can “do the math” in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work.

Most scholars who write on social contract and classical natural law perceive an irreconcilable tension between them. Social contract theory is widely considered the political-theoretic concomitant of modern philosophy. Against the regnant view, The Social Contract in the Ruins, argues that all attempts to ground political authority and obligation in agreement alone are logically self-defeating. Political authority and obligation require an antecedent moral ground. But this moral ground cannot be constructed by human agreement or created by sheer will—human or divine. All accounts of morality as constructed or made collapse into self-referential incoherence. Only an uncreated, real good can coherently ground political authority and obligation or the proposition that rightful government depends on the consent of the governed. Government by consent requires classical natural law for its very coherence.