Mathematics

"Geometry’s mathematical use is abstract and markedly different from its quantitative use, which is concerned more with practical matters. The mathematical use of geometry includes within itself an inner logic that was spotted early in the history of civilization. Euclid lived in Alexandria and was likely born around 325 BC, but we sadly do not know more than this one fact about his life. I speculate that Euclid was less the discoverer of geometric theorems himself and more a brilliant chronicler of the common mathematical knowledge of his contemporaries, who were themselves informed by knowledge passed down from Egyptians, Sumerians, and other civilizations ancient even in Euclid’s time. However, that is beside the point—Euclid is identified the author of The Elements , he was Greek, and that is enough.

I do not think people appreciate just how influential the mathematical uses of geometry have been in the development of rationality. Euclid’s Elements was the most popular textbook ever published and was required reading up until the early 20th century."

"Many have been subjected to the “mathematical use” in their geometry class during secondary education (high school), but sadly most saw this as a form of unusually cruel punishment. Do not worry—that is not what this book is about.
Geometry and mathematical education in general have been falling short for a long time. In 2011 one survey found, '77% of the students seemed to believe that math was not something that could be figured out, or that made sense. It was just a set of procedures and rules to be memorized. This is, of course, exactly the opposite of true.'

Given the failure of stimulating conceptual thinking in students, it is no wonder pre-rational mythic belief has erupted onto the global stage. Democracy is itself a rational creation which requires at least that stage of cognition within its citizens to function effectively.

Euclid showed how geometry can teach us to think clearly. From a few simple axioms, an entire interconnected edifice of logic can be reconstructed in each mind."

"Abraham Lincoln studied and nearly mastered the Six-books of Euclid (geometry) since he was a member of Congress. He began a course of rigid mental discipline with the intent to improve his faculties, especially his powers of logic and language. Hence his fondness for Euclid, which he carried with him on the circuit till he could demonstrate with ease all the propositions in the six books; often studying far into the night, with a candle near his pillow, while his fellow-lawyers, half a dozen in a room, filled the air with interminable snoring."
—William Herndon (1818-1891), law partner and biographer of Abraham Lincoln

The Shadow of the Classroom
I highly recommend re-training yourself in mathematics as an adult—or perhaps, learning it truly for the first time. Many of us left school believing we "weren't math people." If you hated math in school, I invite you to look into that shadow. Math anxiety is often just a form of performance fear that robs the brain of its working memory. Revisiting it as an adult is a form of cognitive exposure therapy. You aren't just learning numbers; you are reclaiming the mental territory that fear once occupied.

Why We Learn the Language of the Logos
We don't learn math for its utility in calculating tips or measuring floorboards; we have machines for that. We learn math because it is the ultimate training ground for the Logos—the rational faculty that allows us to perceive the underlying order of the universe..

The Brutal Honesty of the Ascent
Two years ago, I realized that even with my background in engineering and architecture, I had forgotten the core of what I once knew. I decided to go back to the beginning. I used Khan Academy—a free resource that covers everything from Kindergarten to University—and I was brutally honest with myself. I went back as far as I needed to find the thread. For me, that was Algebra I. Sophie Germain had a key insight:

The Depth of the Adult Mind
While a teenager’s brain might be faster at memorization, the adult mind is deeper. You have a lifetime of lived experience to map these abstract truths onto. You aren't just passing a test; you are revitalizing your left hemisphere and building a 'cognitive reserve' that serves as a bulwark against aging and the possibility of cognitive decline with the passage of time. This ascent is a revitalization of the left hemisphere—the part of the mind that seeks order, clarity, and truth.

The Tremendous View
Reclaiming your mathematical capacity is an act of mental sovereignty. It is an uphill slog, but the sense of accomplishment in making it up or back up that hill is a feeling like no other. Whether you are picking up a thread you lost decades ago or starting a brand new journey, the view from the summit is tremendous.

Whether you were an A+ student or someone who struggled to stay in the classroom, the mountain of math is open to you. KhanAcademy.org and other free resources have democratized this ascent. You just need the courage to find your thread and start climbing. You don't need a degree to speak the language of the universe. You just need the courage to take the first step.

Maybe my dark mode iPad notes below could help you, or at least give you a visual sense of what you're in for if you accept this mission...highly structured thoughts. So I recently got 96% on AP College Calculus BC exam, which is damn good, if I do say so myself. See, I've still got it! Just so you know, I truly doubted whether I'd make it at all when I started the repeat journey 40+ years after my first summit attempt. It’s quite a slog, but the view from the top—well, you’ve just got to see it for yourself. To be clear, this is university-level math, not the rarefied altitude of a professional mathematician. But for an amateur, it’s a hell of a vista.

Middle aged perspective: The thing is—nobody cares in the slightest. You have to let go of that. Still, the view from the mountaintop is a moment in time, frozen in a photograph, like my recent notes below. The real challenge and the pleasure, if one could even call it that, is the climb. The 'Orange' strive-drive is all about that climb. And for a guy with two artificial legs (and one artificial hand), this is my kind of climb. I’ve done a lot of transcending-and-including in my life, but there is a unique novelty in getting down off my Turquoise high-horse and revisiting earlier Tier 1 modes—to see if I can still fully inhabit the most challenging stages. It’s like visiting your self-sufficient parents: it’s great to be there, but eventually, you have to return to your current life.