Wise Fools

Posted on Tue 12 August 2025 in education

emerson

Holy and Wise: The mythology and contributions of the fool in mathematics and physics

On August 8th, the NY Times published a long essay about Allan Brooks who fell under a dellusion that he had "had discovered a novel mathematical formula, one that could take down the internet and power inventions like a force-field vest and a levitation beam." His story of struggle and low self-esteem and encouragement to delusion by openAI's chatGPT really touched me as a student of mathematics. I sincerely hope that people will not draw the wrong the conclusions from it. I currently teach Python programming to formerly incarcerated students and in the past have taught remedial math to college students.

Here are five articles from popular sources that consider the possibility that human thought or creativity will cease because of AI:

  1. New York Times – "A.I. May Not Take Your Job, but It Could Change It" – Discusses concerns about AI reducing the scope for human creativity and independent thought. https://www.nytimes.com/2023/03/20/technology/ai-chatgpt-jobs.html

  2. The Atlantic – "The End of Human Art? An AI Wins an Art Competition" – Raises questions about whether AI creativity displaces or erodes human artistic thinking. https://www.theatlantic.com/technology/archive/2022/09/ai-generated-artwork-wins-state-fair-competition/671413/

  3. The Guardian – "Will AI Spell the End of Creativity as We Know It?" – Explores whether AI could fundamentally replace human creative processes. https://www.theguardian.com/technology/2023/apr/09/will-ai-spell-the-end-of-creativity-as-we-know-it

  4. Washington Post – "Is ChatGPT Coming for Human Creativity?" – Looks at fears that language models could take over the act of original thought. https://www.washingtonpost.com/technology/2023/02/20/chatgpt-ai-creativity/

  5. New Yorker – "Will Artificial Intelligence Replace Human Writers?" – Frames the discussion in terms of literary creativity and intellectual labor. https://www.newyorker.com/news/the-weekend-essay/will-artificial-intelligence-replace-human-writers

Here are three which specifically call for the end of mathematics:

  1. "Will A.I. Kill Math?" – The New Yorker Explores whether AI’s ability to generate proofs and solve problems could make traditional human mathematical discovery obsolete.
    https://www.newyorker.com/tech/annals-of-technology/will-ai-kill-math

  2. "The End of Math as We Know It?" – Quanta Magazine Looks at AI systems that can find proofs far beyond human reach and what this means for the future of the field.
    https://www.quantamagazine.org/the-end-of-math-as-we-know-it-20231005/

  3. "A.I. May Not Kill Us, but It Will Break Our Brains" – The New York Times Discusses the potential erosion of human problem-solving skills, including in mathematics.
    https://www.nytimes.com/2023/09/29/opinion/ai-brain-effect.html

Of course, I will argue that these fears are overblown. More importantly this article, I hope will convince you that we all start out as fools. I hope to give you a list of practices you can do, wherever you are (including prison) to do to help bring yourself out of the mud and darkness of mathematical ignorance.

Assuming nothing else, keep your research ideas in a notebook. Next, realize that for people to take you seriously and for you to guide the fruits of any discovery, you will have to master many areas of mathemtics. This will take time and great effort but it is possible.

Becoming Numerate

In fact, there was a time -not that long ago -when many believed that only a small slice of humanity could ever be literate. In the Middle Ages, and even into the 18th century, some assumed there was a natural ceiling — maybe 20 or 30 percent of people could ever read and write well enough to participate in intellectual life. History proved them wrong.

With mass education, cheap books, and a belief that literacy was for everyone, reading and writing went from elite privilege to a near-universal skill. Not everyone arrived there at the same time. So you did not go to an elite private school for primary school addition and subtraction. So you learned Calculus in a detention center. So what?

I think about that history when I think about mathematics -and about anyone who has ever been told they aren't "a math person." Maybe you don’t have a degree. Maybe you never finished high school or got a GED. Maybe formal education feels like a door that closed a long time ago. But none of that means you can't learn mathematics deeply, contribute to the field, and be taken seriously by professional mathematicians.

You need to start with whatever tools you have -free online courses, open textbooks, public lectures, even formal proof assistants like Lean 4 to help check your reasoning. Over time, you can join conversations with researchers, share your work, get feedback, improve, and build the kind of mathematical writing that earns respect in serious venues. I know this is a lot to take in. This article is going to give you a list of free resources and practices to begin this journey and not stop until you are ready to reveal your thoughts and have them seriously considered.

Maybe you will realize that you haven't broken the internet or invented a levitation beam but you will gain a deep appreciation for those that have done great work and one day you might be cited in someone else's great advance.

I would argue the idea that "most people can't do math" is not a law of nature. It is a cultural assumption -one that can be overturned just as surely as the old ceiling on literacy. The tools now exist to make advanced mathematics learnable by you and I will show you how to use your access to gain materials, develope mentors and expect more of yourself. All this, so you can refine your ideas, discover the gaps in your reasoning without becoming so discouraged that you quit.

It is my dream that if we do that together the next generation of mathematicians will include people who were never supposed to be there including you and me -and we'll make discoveries we can't yet imagine.

The tools

The first thing you need is instruction and tools. If you are in prison or you love someone in I will give you instruction to get Kahn Academy Lite. For everyone else the first step should be Kahn Academy. I first used the 15 years ago when it was a bunch of random links on a page. They will meet you wherever you are and give you feedback. You can even learn Python on it.

  1. OpenStax – Precalculus

  2. CK-12 Foundation – Precalculus Concepts

  3. College of the Redwoods – Precalculus Text

  4. OpenMathReference – Precalculus Problems with Solutions

  5. Stitz-Zeager Open Source Precalculus

For Calculus, the standard text is Spivak Calculus. You can get it used on Ebay for $40. The problem is it does not have worked solutions but you can find them if you search Math Stack Exchange or GitHub.

I would start with:

Michael Corral — Elementary Calculus (Early Transcendentals) Thorough single-variable text with a decent amount of theory; free, with answers/hints to odd problems provided by the author (easier than Spivak but solid).

and move to:

Keisler — Elementary Calculus: An Infinitesimal Approach A beautiful, rigorous take on single-variable calculus; very Spivak-adjacent in depth and style. Freely hosted; many chapters include answers or hints (full teacher’s solutions aren’t generally public).

Ancillary skills

Now if you want to be taken seriously, you need a few more skills. You need to learn LaTeX. This will allow you to write and format you theories, questions and homework. The best introduction I know is the Not so short introduction to LaTeX. Write you homework up in LaTeX. It will be painful and slow. Your TA's will thank you.

You also need to learn Python. Again the best introduction I know is Python the Hard Way. There will be math problems that call for computation. This makes you useful in any group.

Finally, to make sure you are not being fed a delusion by AI chatbot you need to learn use a proof assistant. It is really just a compiler realy that makes sure you are not kidding yourself because it will only follow the rules of logic. I am not going to kid you, this is 10x harder than writing proofs in LaTeX and not generally part of undergraduate math education yet.

Here are the resources you won't find anywhere else yet

  1. Formalized Euclidean Geometry by Alex Kontorovitch

  2. There is one university where Lean 4 was taught as an integral part of a course on proofs, Fordham University. It was taught by Heather Macbeth who is now on a research sabbatical to Imperial College. I would not count on her returning. Here is a link to Heather Macbeth's: The Mechanics of Proof

  3. At Stanford, the Computer Science Department sponsored an equally innovative course on the connection between formal verification and theorem proving. This is the link to Leni Aniva's:

I don't care what your background is. If you can write your proof and verify it in Lean4. You are almost ready to ask for feedback.

You need at least one mentor. This is someone who has published a math research paper in the last five years. We do not teach students how to judge researchers.

How to judge a professor's research for an outsider

The h-index is a widely used metric for evaluating mathematics researchers by balancing two aspects of scholarly impact: productivity (number of papers) and influence (number of citations). A mathematician has an h-index of h if they have h papers each cited at least h times. This makes it more robust than simply counting total citations, which can be skewed by one blockbuster paper, or total papers, which ignores whether anyone read them. While imperfect, it does encourage a long-term pattern of substantive contributions rather than short bursts of activity or purely incremental work.

That said, the h-index has clear limitations. It favors researchers in larger or faster-moving subfields where citation rates are higher, and it underestimates those working in niche but foundational areas where the audience — and thus citation count — is naturally smaller. It also accumulates over time, which can make younger but high-potential mathematicians look less accomplished compared to senior figures. Moreover, it says nothing about originality, depth, teaching impact, or the beauty of results — qualities mathematicians themselves often value most.

Still, as a ranking tool, the h-index is far more defensible than something like the U.S. News & World Report rankings, which often rely on opaque “reputation scores,” outdated surveys, and structural biases that reward wealthy institutions rather than actual scholarly contribution. An h-index is at least grounded in measurable research output and peer acknowledgment, even if it must be interpreted carefully and supplemented with qualitative judgment. In that sense, it can serve as a more honest -if still incomplete -starting point for assessing the standing of a researcher in the mathematical community.

Look at the h-index of anyone you are considering and take there class. Lower h-index but closer physical proximity is preferred. Go to the office hours and TA recitations. Do the reading, ask questions. Do the reading before class and nd come with a list of questions. If the questions are not answered during class bring them to recitation and office hours. Ask your TA's and your professor about their research. See if there are internships and research assistantships available. Many departments have money set aside for undergraduate research.

Speaking of funded research, keep Pathways to Science on your radar even for math, it is a listing of federally funded research opportunities for high school students and beyond.

How could that be, I didn’t even graduate high school?

The tradition of the wise fool goes as far tack as the ancient Greeks. Among many other inventions Greek Poets invented the tragic hubristic hero from Greek drama — someone whose virtues, taken too far, become their undoing — is a clear ancestor to Quixote’s mixture of courage and folly. My cultural background is decidely more mid/basic so I will link to a version of the Broadway standard:

To Dream, The Impossible Dream

And from my own ethnic tradition Tevye the Dairyman (Sholem Aleichem, 1894–1914 stories) grew out of Yiddish storytelling traditions that themselves blended Jewish folklore, rabbinic parables, and the wry, self-deprecating humor of the shtetl. The "poor but dignified man with too many daughters" trope has biblical echoes (think of Job's trials mixed with the wry fatalism of Ecclesiastes) and you also see traces of the European peasant storyteller, whose commentary on life balances resignation with wit. Sholem Aleichem elevated this into a character who, like Don Quixote, insists on interpreting his world through a personal moral framework, even when history and economics make him look foolish. I remember my father, a laborer in the garment industry also bargaining with G-d for a better economic turn. I often cry thinking of him when I watch Zero Mostel sing:

If I were a rich man

In the article Allen Brooks asks in the context of how he could have or could possibly make a contribution

How could that be, I didn’t even graduate high school?

The villain of the story ChatGpt retorts with a laundry list of wise fools and outsiders who made contributions including Michael Faraday, Srinvasa Ramanujan, Leonardo Da Vinci and R. Buckminster Fuller. Well I want to offer an even longer list that I cherish to encourage me when I fall into a depression about my own intellectual inadequacies and career errors only tempered by the warning that you can make a contribution, you just have much more work to do before you call the NSA or the Canadian Centre for Cyber Security or any research mathematician.

The list and the patterns

Niels Henrik Abel

Niels Henrik Abel was a mathematician who lived from 1802 to 1829. He is notable for proving quintic unsolvability and advancing elliptic function theory. Despite being born poor in rural Norway and facing financial hardship, Abel's genius was recognized through his work. Unfortunately, he died before receiving the recognition he deserved. For more information, see Arild Stubhaug's book "Niels Henrik Abel and His Times" (Springer, 2013).

Srinivasa Ramanujan

Srinivasa Ramanujan was a mathematician who lived from 1887 to 1920. He made significant contributions to number theory, modular forms, and infinite series. Despite being self-taught in India and lacking a formal degree, Ramanujan relied on G.H. Hardy's mentorship and went on to make groundbreaking contributions. For more information, see Robert Kanigel's book "The Man Who Knew Infinity" (1991).

Michael Faraday

Michael Faraday was a scientist who lived from 1791 to 1867. He is notable for his work on electromagnetic induction and diamagnetism. Faraday was a bookbinder's apprentice with minimal formal schooling, but he learned through lectures and self-study. For more information, see James Hamilton's book "A Life of Discovery: Michael Faraday, Giant of the Scientific Revolution" (2004).

Oliver Heaviside

Oliver Heaviside was a mathematician and physicist who lived from 1850 to 1925. He is notable for developing operational calculus and reformulating Maxwell's equations. Heaviside left school at 16 and was self-taught, working in near isolation. For more information, see Basil Mahon's book "The Forgotten Genius of Oliver Heaviside" (2009).

Mary Somerville

Mary Somerville was a mathematician and astronomer who lived from 1780 to 1872. She is notable for popularizing advanced math and astronomy. Somerville was self-taught in algebra and calculus, and she had no formal degree. She was also a key translator of Laplace's work. For more information, see Kathryn Neeley's book "Mary Somerville: Science, Illumination, and the Female Mind" (2001).

Katherine Johnson

Katherine Johnson was a mathematician who lived from 1918 to 2020. She worked at NASA and made key contributions to orbital mechanics, including calculations for the Mercury and Apollo missions. Johnson faced racial and gender barriers, but she worked her way from a segregated school to become a NASA mathematician. For more information, see Margot Lee Shetterly's book "Hidden Figures" (2016).

Louis de Broglie

Louis de Broglie was a physicist who lived from 1892 to 1987. He is notable for his work on wave-particle duality of electrons. De Broglie began in history, but he shifted to physics with Einstein's endorsement. For more information, see Olivier Darrigol's book "From c-Numbers to q-Numbers" (1992).

Freeman Dyson

Freeman Dyson was a physicist who lived from 1923 to 2020. He is notable for reformulating QED and developing visionary space engineering concepts. Dyson did not have a PhD, but he transitioned from RAF operations work to major theoretical contributions. For more information, see his book "Disturbing the Universe" (1979).

Mary Cartwright

Mary Cartwright was a mathematician who lived from 1900 to 1998. She is notable for her work on nonlinear differential equations and early chaos theory in radar systems. Cartwright overcame gender and institutional barriers, becoming the first woman to earn an Oxford D.Phil in math. For more information, see D. Strick's article "Mary Lucy Cartwright, 17 December 1900 – 3 April 1998" in Biographical Memoirs of Fellows of the Royal Society (2000).

Georg Cantor

Georg Cantor was a mathematician who lived from 1845 to 1918. He is notable for developing set theory and transfinite numbers. Cantor faced intense opposition, and his early work was dismissed as metaphysical. For more information, see Joseph W. Dauben's book "Georg Cantor: His Mathematics and Philosophy of the Infinite" (1979).

Grigori Perelman

Grigori Perelman is a mathematician born in 1966. He is notable for solving the Poincaré conjecture. Perelman declined awards and withdrew from academia, working in isolation. For more information, see Sylvia Nasar and David Gruber's article "Manifold Destiny" in The New Yorker (2006).

Yitang Zhang

Yitang Zhang is a mathematician born in 1955. He is notable for his work on bounded gaps between primes. Zhang spent years in obscurity teaching at a small college before making a groundbreaking result. For more information, see Alec Wilkinson's article "The Pursuit of Beauty" in The New Yorker (2015).

Stephen Wolfram

Stephen Wolfram is a mathematician and computer scientist born in 1959. He is notable for developing cellular automata and creating Mathematica. Wolfram earned his PhD at Caltech at age 20 and later left academia to pursue independent research. For more information, see his book "A New Kind of Science" (2002).

Emmy Noether

Emmy Noether was a mathematician who lived from 1882 to 1935. She is notable for developing Noether's theorem and abstract algebra. Noether was barred from formal positions due to her gender and lectured under male colleagues' names. For more information, see Auguste Dick's book "Emmy Noether: 1882–1935".

Lise Meitner

Lise Meitner was a physicist who lived from 1878 to 1968. She is notable for her co-discovery of nuclear fission. Meitner fled Nazi Germany and was excluded from Nobel recognition awarded to Otto Hahn. For more information, see Ruth Lewin Sime's book "Lise Meitner: A Life in Physics".

Leo Szilard

Leo Szilard was a physicist who lived from 1898 to 1964. He is notable for conceiving the nuclear chain reaction and working on the Manhattan Project. Szilard trained in engineering and moved between disciplines, becoming a key political activist in science policy. For more information, see William Lanouette's book "Genius in the Shadows" (1992).

Robert G. Rudnitsky

Robert G. Rudnitsky is a physicist who has worked at NIST and advised at the U.S. Dept. of State. He transitioned from Political Science to physics, taking a cross-disciplinary path. For more information, see public bios and interviews.

Adam C. Firestone

Adam C. Firestone is a lawyer, CEO of an internet security company, and holder of multiple patents. In Scrappy but Hapless (2025), he blends history, wit, and startup experience to draw lessons on leadership under pressure, from battlefields to boardrooms.

This doesn't scratch the surface or include all the people I know from the English, Comparitive Literature and even Theater department I know from the CUNY graduate Center in AI centric secuurity.

The patterns you should observe

  1. Self-teaching + small network access — Almost all eventually found a mentor or intellectual circle (e.g., Faraday with Humphry Davy, Ramanujan with Hardy).
  2. Initial outsider status — Many were not taken seriously at first (Heaviside was often dismissed as a crank; Ramanujan's early notebooks baffled reviewers).
  3. Persistence — Most worked for years in obscurity before recognition.
  4. Entry point — Public lectures, correspondence, or publishing in non-elite venues often opened the door.

The conclusion you should draw

  1. If you want to be taken seriously take math seriously. Learn the tools and don't skip steps. Read the books, do the problems, learn LaTeX, learn Python and verify your work with Lean4 or some other proof assistant.

  2. Find a mentor, many outsiders were able to find a mentor who championed their cause. Math and Physics are human endeavors. People who love the problems rather than the prizes or fame attract. Don't waste your time with an miserable person. No one is smart enough to tolerate abuse.

  3. You thought you were right and publicly embarassed yourself. Be upset. Cry. Put down the vape pen. Try again. Learn to do better and never give up.

  4. I am not sure where the phrase "What is a winner? Just another loser who got up one more time". It has been attributed George M. Moore Jr. and Vince Lombardi. A Japanese proverb, "Fall seven times and stand up eight" predates it by several centuries. I don't care, the quote speaks to me and I will keep standing up and I hope you do too.

    1. My next post will be on the construction of the real numbers via Cauchy completion of the rationals and the Carathéodory extension theorem for measures on a $\sigma$-algebra share a deep analogy, which can be elegantly formalized using category theory's universal properties and adjoint functors.

Epilogue: A version of the Hannukah song for all of us math outsiders

Adam Sandler struck a chord for many Jews isolated by distance from large communities when he sang about being the only kid without a Christmas tree. Here is a link to the original:

What follows is my poetic response, sung to the tune of Adam sandler's channukah song:

When you're the only kid in town without a degree to show,
Here's a list of folks who proved you don't need school to glow.
Niels Abel, broke but brilliant, cracked quintics' stubborn code,
Ramanujan, self-taught genius, till Hardy paved the road.
Faraday, from bookbinders' shop, made electric dreams ignite,
Mary Somerville's starry pen taught the cosmos to take flight.
Katherine Johnson's math soared high, sent Apollo to the moon,
With sharp minds and helping hands, they all learned to play the tune.

Louis de Broglie waved at particles, sparked a quantum blaze,
Einstein's faith lit up his path through early doubters' haze.
Freeman Dyson skipped the PhD, lit QED's bright phase,
Found Oppenheimer's open door to guide his early days.
Mary Cartwright swapped her classics for math's chaotic dance,
Radar's pulse, her theory's spark, gave chaos a new chance.
Cantor's infinite sets shook minds, made scholars lose their cool,
Perelman solved Poincaré's riddle, spurning fame's old school.

Yitang Zhang, with prime gaps bound, rewrote the math we know,
Help from friends and open ears helped his long-shot theorem grow.
Emmy Noether's algebra broke barriers none could see,
Hilbert cleared the gate for her to shape physics' destiny.
Lise Meitner split the atom, though Nobel passed her by,
Her fission dreams lit up the world beneath a war-torn sky.
Wolfram left his doctorate for cellular streams to chase,
Szilard's chain reaction sparked a world we couldn't face.
Rudnitsky's physics flair turned NIST into a name,
From poli-sci to science halls, he learned to play the game.

So when they say, "No PhD, no shot," don't buy their test,
But if that's your dream, go chase it – and find a mentor who's the best.
Like Hardy backed Ramanujan, or Hilbert backed Noether's hand,
Like Einstein stood by de Broglie when few could understand.
From number games to cosmic rays, from lab bench to the stars,
Start with grit, a spark of you, and change what's ours.
No degree can block your path, no gate can stop your view,
With heart, a guide, and hustle – let the world catch up to you!

ambition persitence