In the 1800s, it seemed like mathematics was a solved problem. The paradoxes in the field were resolved, and even areas like advanced calculus could be taught consistently and reliably at any school. It was clearly understandable in a way that abstract fields like philosophy weren’t, and it was on its way to solving humanity’s problems. Mathematical work on electromagnetism made modern electrical engineering and power systems possible. New research in algebra created the logical basis for future computer science and digital circuits.

But then new problems appeared. In the early 20th century, mathematicians made discoveries that showed them enough to know how little they really knew. Bertrand Russell showed that at its edges, math fell apart. It couldn’t fully define itself on its own terms without becoming logically inconsistent. He gave the analogy of a small-town barber who shaves everyone who doesn't shave himself; the question is, who shaves the barber—if he shaves himself, he breaks the rule, but if he doesn't shave himself, he must, by the rule, shave himself?

In today’s episode, I’m speaking to Jason Bardi, author of The Great Math War: How Three Brilliant Minds Fought for the Foundations of Mathematics and we explore the story of three competing efforts by mathematicians to resolve this crisis. What do you do if math, the most logical of all sciences, becomes illogical at a certain point? Bertrand Russell thought the problem could be solved with even more logic, we just hadn’t tried hard enough. David Hilbert thought redemption lay in accepting mathematics as a formal game of arbitrary rules, no different from the moves and pieces in chess. And L. E. J. Brouwer argued math is entirely rooted in human intuition—and that math is not based on logic but rather logic is based on math. Set against the backdrop of one of the most turbulent periods of European history (from the late 19th century through World War I, the 1920s, the Great Depression, and the early days of World War II), we look at what happens when rock-solid truths don’t seem so rock solid anymore.

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The Battles of Lexington and Concord in April 1775, known as the "shot heard round the world," marked the first military engagements of the American Revolution. Ralph Waldo Emerson named it that because it launched revolutionary movements in Europe and beyond, marking it as a key moment in the fight for liberty and self-governance. But this moment was global in more ways than inspiring other nations. The quest for independence by the 13 North American colonies against British rule rapidly escalated into a worldwide conflict. The Patriots forged alliances with Britain’s key adversaries—France, Spain, and the Netherlands—securing covert arms supplies initially, which evolved into open warfare by 1779. French and Spanish naval campaigns in the Caribbean diverted British forces from North America to defend valuable sugar colonies, while American privateers disrupted British trade, bolstering the rebel economy. All of this international involvement was promoted by the Founding Fathers, because the Declaration of Independence was translated into French, Spanish, Dutch, and other languages and distributed by them across Europe to garner sympathy and support from nations like France and the Netherlands.

Spain’s separate war against Britain in Florida and South America, alongside French efforts to spark uprisings in British-controlled India, further strained Britain’s ability to quash the rebellion. Post-independence, the consequences rippled globally: Britain and Spain tightened their grip on remaining colonies, Native American tribes faced heightened land encroachments due to the loss of British protections, and enslaved African Americans who fought for Britain, lured by promises of freedom, were relocated to Nova Scotia and later Sierra Leone.

To explore this new framework of the Revolutionary War is today’s guest, Richard Bell, author of “The American Revolution and the Fate of the World.”

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The brain acts in strange ways during wartime. Even in active combat situations, when soldiers are one mistake away from death, many can’t fire on their enemies because their brain is triggering compassion centers against other soldiers. Studies of World War II show that while soldiers were willing to risk death, only 15% to 20% fired their weapons in intense combat, indicating a reluctance to kill. That’s why successful military leaders were able to motivate their soldiers with ideas of unfairness and justice, that their enemies weren’t human to make them better at fighting and killing.

All this goes to show that if you want to understand war, you have to understand how the brain makes sense of it. Does war make all of us retreat to our lizard brain and act on pure instinct – so the only way to win is pumping out manipulative propaganda to the masses and use modern technologies like AI and social media exploit the brain's cognitive vulnerabilities? Well, many nations like Russia and China are already using these to their advantage.

Or can we bring higher thinking to the matter? Is a researcher like Robert Sapolsky right when he argues that we can stop wars by persuading enough people that it is bad and pointless.

Today’s guest is Nicholas Wright, author of “Warhead: How the Brain Shapes War and War Shapes the Brain.” He’s a neuroscientist and advisor to the Pentagon. We explore how our brains respond under pressure and how these instincts can shape everything from battlefield outcomes to boardroom decisions. He argues that while conflict is inevitable, it’s not unmanageable - if we understand how the brain drives fear, trust, aggression, and judgment.

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