Paul Dirac’s father insisted that he become an engineer. So Paul speedran his education and graduated with a first-class electrical engineering degree at nineteen. But after failing to land a job, he went back to school to study math and physics. Over the next decade, he crushed it:

• Wrote the equation that describes the electron

• Founded quantum electrodynamics

• Predicted antimatter

• Discovered Fermi-Dirac statistics (which every transistor depends on, btw)

• Took Newton’s old chair at Cambridge

• Won the Nobel Prize at thirty-one

Not bad for an unemployable electrical engineer.

In light of his academic success, it’s tempting to see his engineering era as nothing more than an unnecessary side quest. One wonders how anyone could picture the shy and “slovenly” young Paul in a factory. Wouldn’t he have been better off studying math all along?

Surprisingly, no.

Although Dirac never became an engineer, his training as one gave him an unexpected edge as a mathematician and physicist. You can see this most clearly in three habits that he never dropped.

1) Tolerating imperfection

Before studying engineering, Dirac had a mathematician’s appetite for exactness. An equation was either right or it was wrong. After seeing the inevitable rounding and safety margins that emerged naturally while working with circuits and rotors, Dirac started to see math as a tool to get accurate (enough) answers for the problem at hand. He studied fellow engineer and mathematician Oliver Heaviside and his step function, which made it easy to study the pulses of currents through circuits. No one really understood them, though, including Heaviside. Mathematicians mocked the function because it lacked rigor. Heaviside didn’t care, summarizing the engineer’s perspective rhetorically: “Shall I refuse my dinner because I do not fully understand the process of digestion?”

Paul always sided with the engineers on this issue.

“I owe a lot to my engineering training because it [taught] me to tolerate approximations. Previously to that I thought…one should just concentrate on exact equations all the time. Then I got the idea that in the actual world all our equations are only approximate. We must just tend to greater and greater accuracy. In spite of the equations being approximate, they can be beautiful.”

After pivoting to academics, Dirac proved unusually comfortable dropping unnecessary terms and trusting his instincts, which his singular-minded math peers couldn’t do. Paul had a sense of freedom in a field that overemphasized rigor for its own sake.

Another clear example of this approximation at work is his delta function, which returned zero everywhere except at zero, where it’s infinitely large. Mathematicians claimed that no such function could exist, and they were technically right. But it worked, which was enough for both Dirac and Heaviside. Both were happy to use a method that worked and let the math catch up later. (It took another two decades before Laurent Schwartz’s theory of distributions put the delta function on solid theoretical footing.)

2) Visualizing reality

Engineering taught Dirac to turn a blind eye to the lure of perfection, but it also taught him to see. During high school and undergrad, he practiced geometric drawing, which taught him to visualize complicated objects in motion from different perspectives. He considered cylinders and cones in motion, and learned to see in his mind’s eye in ways that he never could’ve if he confined himself to static algebra.

These drawings primed him to learn projective geometry, a study of perspective that encouraged geometric over algebraic approaches. It challenges students to think not just about distance between points, but their relationships at different angles. Most found this difficult. Dirac ate it up.

Projective geometry I found a most fascinating subject. One could get quite powerful results — theorems about straight lines and conics intersecting each other — just from elementary arguments about 1 to 1 correspondence. That appealed to me very much. All my work since then has been very much of a geometrical nature, rather than of an algebraic nature.
— Dirac (AHQP interview, 1 April 1962)

Physicist Freeman Dyson noticed Dirac’s visual skill. Dyson marveled at how he could seemingly conjure laws of nature from pure thought. His discoveries were like “exquisitely carved marble statues, falling out of the sky, one after another,” Freeman said. Wolfgang Pauli might’ve had the same impression. His joke that “There is no God, and Dirac is his prophet” was as much a quip about Dirac’s atheism as it was a compliment about his visual imagination. I can’t help but think that the visual practice from his engineering education helped him probe nature more effectively.

Seeing a problem clearly is half the battle. The other is ensuring it stays clear after you write it down.

3) Mastering symbols

In 1939, sixteen years into his physics career, Dirac submitted a three-page paper titled “A New Notation for Quantum Mechanics.” In it, he argues that the standard notation’s mix of linear and coordinate operations “do not fit together naturally” and “give rise to an awkward jump in the flow of one’s thoughts.” Then he proposes “bra-ket” notation. A quantum state was a ket, |ψ⟩; its corresponding dual was a bra, ⟨φ|; and their pairing as ⟨φ|ψ⟩ — a bracket. Instead of forcing physicists to mentally switch between the abstract and concrete, this “unification of ideas” allowed them to stay in one symbolic system. No wonder why, eighty-seven years later, physicists and quantum-computing engineers still use it.

When asked how he could manipulate symbols so easily, Dirac mentions how he had to calculate the breaking stress of structures as part of his general engineering training, which forced him to get familiar with new symbols.

He refers to this training again when considering how he came up with delta functions, a generalized way to model mass and load, which he says were inspired by electrical circuits and stress diagrams.

I think it was probably that sort of training that first gave me the idea of a delta function because when you think of loads in engineering structures, sometimes you have a distributed load and sometimes you have a concentrated load at the point. Well, it’s essentially the same whether you have a concentrated load or a distributed load but you use somewhat different equations in the two cases. Essentially it’s only to unify these two things which sort of led to the delta function.
— Dirac (Kuhn AIP interview)

Dirac’s symbol work feels like a side quest born out of pure irritation, like Linus inventing git in a few weeks so he could keep his kernel repo organized. If Dirac were a SWE, he’d have created his own mathematical programming language. Instead, he created new symbols that unified two ways of reasoning about quantum mechanics.

This work stemmed from a deep instinct for abstraction that he maintained throughout his career. His delta function unified concentrated and distributed loads mathematically; his notation unified abstract states and coordinates; his physics unified quantum mechanics with special relativity. He never became a structural engineer, but the guy couldn’t help building bridges.

Takeaway

Much like my liberal arts degree, Paul’s engineering training didn’t land him a job. But it did give him something more durable: pragmatism. He used this intuition to approximate without apology, visualize relationships, and invent better abstractions when the others got in his way. His skill at combining engineering with theory might’ve been why his discoveries were so unexpectedly elegant.

If you’re a software engineer worried about what’s left once agents write all the code, I hope Dirac’s career is encouraging. Frameworks go out of style. Languages die. Job listings disappear. Your coding skills might not be as necessary as anticipated, but the sensibilities you developed while learning them will stay with you for a lifetime.

Further Exploring

Essays: Paul Dirac’s papers. Scanned and organized into a GitHub repo.

Blog: When simplicity isn’t enough (me). What SWEs can learn from the Dirac equation.

Book: The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius (Farmelo).

Book: A history of mathematical notations (Cajori). More symbol stories.

Audio: Basic Beliefs and Prejudices in Physics (Dirac). 58-minute Nobel speech.

Video: Why is there antimatter? (Veritasium). A meaty yet approachable explanation of Dirac’s equation.

Kobe is beloved for his consistency, obsession, and attention to detail. Each of his teammates has a story about how he stayed in the gym longer, studied film closer, and continued evolving later into his career.

Yet things didn’t look so promising at the beginning. Despite entering the league at 18 with plenty of raw talent, he sat on the bench for most of his rookie year. He lacked the physical dominance of his teammate Shaq, who could neglect his diet and free throws and still become a hall-of-famer. Kobe also suffered many career-threatening injuries. Yet he always did the unglamorous work to get back on the court. As his athleticism waned later in his career, he repurposed his game around tempo, angles, and passing.

Although less storied, Richard Hamming (1915-1998) wasn’t so different.

After getting his PhD in Math at 27, he joined the Los Alamos crew on the Manhattan Project. But instead of doing the fission work, he was told to manage the IBM accounting machines they used to run bomb simulations. He described himself as a “stooge,” a “janitor of science.”

Then he got to Bell Labs, where he was again given the role of a glorified IT admin in their computer center. He only got to use the machines he managed when the big shots were gone on the weekend.

But Hamming wasn’t willing to let that hold him back from doing great work.

His plan was to load the machines up on Friday and come back Monday with a week’s worth of answers. But several weekends in a row, the machine hit an error Friday night, stalled, and gave him nothing.

Hamming wondered whether he could get the machine to not only detect an error, but also locate where it is and flip the bit.

He obsessed over this error-correction problem for six months, working on making the machines more efficient during his off-time. He eventually came upon a novel solution that involved combining what he know about geometry with the binary calculations the computers relied on.

It worked: he got a year’s worth of work done in six months. But then he did the part most engineers skip: he patented it, wrote it up so others at Bell could use it, and eventually wrote entire books on coding theory so everyone else could too. That’s what won him the Turing Award in 1968.

After finally getting recognition, he could’ve easily coasted for the rest of his career. Instead, he quit, saying that he thought that old researchers got in the way of the fresh young ideas. Instead, he reinvented himself as a writer and speaker for the Navy.

And like Kobe, who was known to challenge his lazy teammates, Hamming refused to suffer fools or expect less of himself. He was famous for bringing a sheet of paper to meetings so that he could work on math problems when others started talking in circles. He got kicked out from the chemistry lunch table for challenging the scientists for their lack of ambition.

Kobe never had the size or athleticism as his peers like Shaq or LeBron. Hamming didn’t have the IQ of Shannon or Einstein. But they both refused to let those disadvantages temper their ambition. That’s why I admire them. They’re proof that consistent hard work can outpace raw horsepower in the race that is our careers.

HAMMING’S STATS

Richard Wesley Hamming (1915-1998, 82), born in Chicago, Illinois, on February 11, 1915, and died in Monterey, California, on January 7, 1998.

He earned a B.S. in mathematics from the University of Chicago (1937), an M.A. from the University of Nebraska (1939), and a Ph.D. in mathematics from the University of Illinois at Urbana-Champaign (1942). After a short teaching stretch, he joined the Manhattan Project at Los Alamos in 1945, programming IBM calculating machines for bomb simulations. In 1946 he moved to Bell Labs, where he stayed for 30 years and invented the error-detecting and error-correcting codes that still carry his name. He received the ACM Turing Award in 1968. In 1976 he left Bell for the Naval Postgraduate School in Monterey, where he taught until shortly before his death.

Known for: Hamming codes, Hamming distance, the Hamming window, and the lecture-turned-essay You and Your Research.

KEEP EXPLORING

• Book: The Art of Doing Science and Engineering (PDF)

• Article: You and Your Research

• Video: Hamming codes explained (3Blue1Brown)

• Lectures: Learning to Learn (Hamming at the Naval Postgraduate School)

Bloggers point out how he loved juggling, unicycles, chess, his maze-solving mechanical mouse named Theseus.

He’s portrayed as a pure-hearted genius, a whimsical man-child who just wanted to play with his toys.

I thought the same, until reading what his colleague Richard Hamming said in his “You and Your Research” talk:

When you are famous it is hard to work on small problems. This is what did Shannon in. After information theory, what do you do for an encore? The great scientists often make this error. They fail to continue to plant the little acorns from which the mighty oak trees grow. They try to get the big thing right off. And that isn't the way things go. So that is another reason why you find that when you get early recognition it seems to sterilize you.

Hamming praised Shannon for his genius and contributions. But he knew him well enough to recognize that he lost his edge later in life.

It reminds me of Kobe’s analysis of Shaq. Kobe recognized him as one of the most dominant players in NBA history. But, as his teammate, he also knew him well enough to criticize his work ethic and commitment. (Shaq himself admits that he was too stubborn to diet or work on his free throws.)

Shannon’s Stagnation

It got me curious to learn more about Claude’s production. I found that it did, indeed, go down after Information Theory.

Some people speculated that he was using his clout to get out of doing much public work at all. He still made plenty of gadgets and explored his curiosity (e.g. developing a unified theory on juggling). But he didn’t take the same pains to publish his work like he did earlier in his career.

When a visitor asked about an unpublished idea, Shannon joked, “Look, you should see my attic.” His wife added they had boxes full of unfinished papers.

Curiosity’s shadow

This led me to an analysis of curiosity itself, since this was the dominant force that Claude directed his life around.

Curiosity, as opposed to interest, involves both emotion and intellect. It takes a degree of emotional investment to ask uncomfortable questions and wander into the unknown in search for answers. Claude mentioned how he felt irritated when he intuited that a solution wasn’t quite right, when it had superfluous parts or processes.

The problem is when there is no mechanism that forces you to come back to society with your solution.

Claude would follow his curiosity, get the answers, and then never tell anyone about it.

His colleagues would bring problems related to his Information Theory to him and find

1. He already knew about the problem

2. He already solved it for himself

3. He never shared his answers

The way out: simplicity

If you simplify your solution and give it to others, you successfully transmute your curiosity into helpfulness. It completes the hero’s journey, where the protagonists goes, seeks, finds, and returns to his tribe with wisdom and bounty.

This is exactly what Claude did with Information Theory. He simplified the problem space of digital engineering for everyone.

And this is exactly what he STOPPED doing afterwards. He avoided people, solved his own problems, and was reluctant to share his answers.

This isn’t an indictment on Claude’s character. If we all drop a gem like Information Theory, we should all have license to become hermits and retire.

Instead, it’s a cautionary tale on the supposed virtues of curiosity. Every over-developed trait has its shadow side. While heartfelt curiosity can lead to astonishing discovery, it can also tempt one to stay forever in their own world.

Keep exploring

• Oral history: Claude E. Shannon (IEEE History Center / ETHW)

• Article: 10,000 Hours With Claude Shannon: 12 Lessons on Life and Learning From a Genius (Jimmy Soni and Rob Goodman)

• Profile / interview: Anthony Liversidge, Omni profile of Claude Shannon (1987 PDF)

Born in Petoskey, Michigan; raised in Gaylord. B.S. in mathematics and B.S.E. in electrical engineering (University of Michigan, 1936); S.M. in electrical engineering and Ph.D. in mathematics (MIT, 1940). Spent his peak research years at Bell Labs in New Jersey, then taught at MIT; lived later years near Boston (his home was nicknamed Entropy House).

Major accomplishment
In 1948 he published “A Mathematical Theory of Communication,” founding information theory: information as a measurable quantity, the bit as its unit, and hard limits on how much you can send reliably over a noisy channel. That framework underpins digital encoding, compression, wifi, storage, and media playback. Earlier, his MIT master’s thesis showed Boolean algebra could design switching circuits, a foundation of digital electronics. He also did wartime cryptography work at Bell Labs.

Quote

“I’ve always pursued my interests without much regard for financial value or value to the world. I’ve spent lots of time on totally useless things.”

(Claude Shannon, interview with John Horgan, Scientific American, 1990. Source: Scientific American profile.)

Fun fact
At Bell Labs he was remembered for juggling while riding a unicycle through the halls. He also built gadgets like Theseus (a maze-solving mechanical mouse), juggling machines, and a rocket-powered Frisbee.

Takeaways

1. Curiosity is power. It alone can also leave the next breakthrough hidden. Shannon solved problems colleagues later “discovered,” then never published. If you are not dropping history-changing papers, you need a habit that pushes answers out to other people.

2. Treat perfectionism as a vice with a schedule, not a virtue with higher standards. Pick a recurring publish cadence (notes, app updates, short videos, a weekly family email) and a simple KPI so shipping beats polishing forever.

3. Dig deep, but don’t forget to give away whatever you find. Following curiosity only for yourself puts you at the center. The fix is to hand your (imperfect) solution to someone else before it feels finished.

Keep exploring

• Article: Profile of Claude Shannon, Inventor of Information Theory (John Horgan / Scientific American; includes the interview)

• Video: How Claude Shannon Invented The Future (Founders #428, based on A Mind at Play)

• Book: Jimmy Soni and Rob Goodman, A Mind at Play: How Claude Shannon Invented the Information Age

Companion video:

Born in Chicago. B.S. in mathematics (University of Chicago, 1937), M.A. (University of Nebraska, 1939), Ph.D. (University of Illinois, 1942). Ran computing at Los Alamos (1945-1946), spent thirty years at Bell Labs in Murray Hill, New Jersey (1946-1976), then taught at the Naval Postgraduate School in Monterey, California until shortly before his death.

Major accomplishment
Invented the Hamming codes (error-detecting and error-correcting codes) that made reliable digital communication practical, from telephone switches to modems, disks, and satellites. Won the ACM Turing Award in 1968 for numerical methods, automatic coding systems, and those codes. Later distilled what he learned watching Shannon, Feynman, Shockley, and the rest into “You and Your Research,” still one of the clearest guides to doing significant technical work.

Quote

“If you do not work on an important problem, it’s unlikely you’ll do important work. It’s perfectly obvious.”

Fun fact
For years he kept “Great Thoughts Time”: after Friday lunch he would only discuss big questions (Where is the field going? What will computers do to science?). He says dedicating ten percent of the week thinking big helped him avoid wasting time working on the wrong things.

Takeaways

1. Ask the lunch-table question: What are the important problems in your field, and are you working on one of them? If the answer is ‘no,’ why not?

2. After a win, keep planting acorns (ie, working on small problems). Hamming points out that Shannon and the other Nobel-Prize winners later refused to work on the small problems, which prevented them from a great encore.

3. Prepare your mind on purpose. If you commit all your mental energy on a problem, your subconscious will keep working on it from different angles. Also leave the door open enough to hear which problems matter.

Keep exploring

• Article: You and Your Research (1986 Bellcore transcript)

• Video: Hamming, “You and Your Research” (June 6, 1995) (him delivering the talk)

• Book: Richard W. Hamming, The Art of Doing Science and Engineering: Learning to Learn

1874-1964 (90)

Born in West Branch, Iowa; orphaned young and raised partly in Oregon. Geology graduate of Stanford’s pioneer class (1895). Made his fortune as an international mining engineer (Australia, China, London), then lived and worked in DC as a relief organizer, Commerce Secretary, and 31st U.S. President.

Major accomplishment
Turned failing mines into profitable operations, built a fortune on the order of Ford and Edison’s, then applied the same operational skill to humanitarian relief. As head of the Commission for Relief in Belgium in World War I, he organized food for millions under occupation. As president, he pushed major public works (including the Hoover Dam and SF bridge). His engineering career got him to the White House, but his inability to inspire people through the Depression prevented him from staying there for a second term.

Quote

“The great liability of the engineer compared to men of other professions is that his works are out in the open where all can see them. His acts, step by step, are in hard substance. He cannot bury his mistakes in the grave like the doctors. He cannot argue them into thin air or blame the judge like the lawyers. He cannot, like the architects, cover his failures with trees and vines. He cannot, like the politicians, screen his shortcomings by blaming his opponents and hope that the people will forget. The engineer simply cannot deny that he did it. If his works do not work, he is damned”

(Article in “Engineer’s Week”)

Fun fact
He read two hours a night for fun from a young age. When U.S. booksellers gave the White House a 500-book “home library” in 1930, Hoover told a guest: “If they sent those here to educate me, I’m afraid it was too late. I’d read 85% of them before.”

Takeaways

1. Engineering skill can provide money and prestige, but leading machines is very different from leading people. Hoover’s optimizer tendencies failed him when the job required empathy, persuasion, and selling a vision.

2. Identify your likely engineer flaw (pride, stubbornness, perfectionism, or emotional flatness), get a baseline, and track it. (See my YouTube video below for tracking examples.) You are not trying to erase the flaw overnight. You are trying to chip away at it before it’s too late.

3. Write to yourself. Hoover blamed the economy, opponents, and “fear of the common man” after his defeat. He wrote to everyone else except himself. A dated notebook conversation with reality is cheaper insurance than waiting for a failed reelection, a stalled promotion, or flat download numbers to humble you.

Keep exploring

• Article: Herbert Hoover: Life Before the Presidency (Miller Center)

• Video: Remembering Herbert Hoover and the Commission for Relief in Belgium

• Book: Kenneth Whyte, Hoover: An Extraordinary Life in Extraordinary Times

1910-1989 (died at 79)

Born in London to American parents; raised in Palo Alto, California. B.S. in physics (Caltech, 1932), Ph.D. in physics (MIT, 1936). Spent his peak research years at Bell Labs in New Jersey, then founded Shockley Semiconductor Laboratory in Mountain View, California; later taught at Stanford.

Major accomplishment

Led the Bell Labs group that produced the first working transistor (Bardeen and Brattain built the point-contact device; Shockley invented the junction transistor). Shared the 1956 Nobel Prize in Physics. In 1956 he founded Shockley Semiconductor Laboratory, the first silicon semiconductor company in what became Silicon Valley. Eight of his employees (the “Traitorous Eight”) left to found Fairchild Semiconductor, which later spun out Intel and much of the Valley’s chip industry.

Memorable quote

“It has today occurred to me that an amplifier using semiconductors rather than vacuum is in principle possible.”

(Laboratory notebook, 29 December 1939.)

Fun fact

Was an semipro rock climber with a first-ascend.

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Takeaways

1. Intellect isn’t enough. Shockley got really far on brains alone, then his need to feel superior wrecked the teams, the company, and eventually his reputation. Brilliance can get you really far (like, a Nobel Prize), but it can’t make people like you.

2. When people leave and thrive without you, look inward before you look outward. Shockley blamed the world instead of his behavior. Which direction did you look after your last career change?

3. Keep a simple tracker (mood, habits, or a monthly page). You’ll be amazed how hard it is to believe your own BS when you have something to hold you accountable.

Keep exploring

- Article: Beckman, Shockley and the 60th Anniversary of the Birth of Silicon Valley

- Video: 391 San Antonio Rd. — A Semiconductor Documentary

- Book: Joel N. Shurkin, Broken Genius: The Rise and Fall of William Shockley, Creator of the Electronic Age

1903–1995 (91)

Son of a Bulgarian immigrant. B.S. in electrical engineering (University of Florida, 1925), M.S. in mathematics (Iowa State College, 1926), Ph.D. in theoretical physics (University of Wisconsin, 1930). Professor of mathematics and physics at Iowa State in Ames; later lived and worked in Washington, D.C., then Maryland.

Major accomplishment
With graduate student Clifford Berry, he built the Atanasoff–Berry Computer (ABC) at Iowa State (1939–1942), an electronic, binary, digital machine for solving systems of linear equations. In 1973, the Honeywell v. Sperry Rand trial credited him as the inventor of the electronic digital computer and invalidated the ENIAC patent, putting that architecture in the public domain.

Memorable quote

“During that evening in the Illinois roadhouse, I made four decisions for my computer project… I am now amazed and pleased to find that each of my four decisions relates to structures that are in use in modern computers.”

(From his account of the 1937–38 night he locked in electricity/electronics, binary, regenerative capacitor memory, and direct logical computation.)

Fun fact
Iowa State funded the first ABC prototype with a $650 grant. Atanasoff and Berry built a working “breadboard” model in about one semester.

Takeaways

1. Strive for goodness, not greatness. Atanasoff proves that you by staying humble and focusing on yourself and your work, you can still accomplish remarkable things without turning into a jerk. Do the best work you can without trading away being a decent collaborator, mentor, or friend.

2. When you’re stuck, don’t force the solution. Instead, spend time in the problem space, then write your non-negotiable requirements before you worry about how you’ll build it.

3. Keep a dated log of how your thinking evolves. Atanasoff’s notes later proved he invented the computer. What would your notebook prove about the problem you’re wrestling with now?

Keep exploring

• Article: John Vincent Atanasoff Biography (Iowa State)

• Video: The Forces That Led to the Atanasoff-Berry Electronic Computer (Atanasoff’s own 1980 lecture)

• Book: Jane Smiley, The Man Who Invented the Computer