Is It Okay to Use Fake Math If It Works?
The Impossible Spike That Worked
In 1927 the physicist Paul Dirac (a quiet, precise 20th‑century Englishman) needed a strange tool. He imagined a function that is zero everywhere except at one single point, where it shoots up to infinity. Its area, though, is exactly 1. Mathematicians told him that such a thing couldn’t really exist. It broke the rules. Yet Dirac used it anyway, because it made his quantum equations gorgeously simple — and the numbers that came out matched the lab perfectly.
This wasn’t the first time scientists had gotten into trouble by using “bad math.” It would not be the last. A deep question runs through the history of physics: when is it okay to use a mathematical trick that works, even if nobody can prove it’s real?
The Old Battle Over Tiny Ghosts
Two hundred years before Dirac, the same fight erupted over infinitesimals — numbers that are smaller than any ordinary positive number, but not zero. Think of a slice of time thinner than a heartbeat yet not just nothing. Isaac Newton and Gottfried Leibniz built their physics with these ghostly numbers. They worked. Planets orbited correctly.
Then George Berkeley, an 18th‑century bishop and philosopher, attacked. In 1734 he called infinitesimals ghosts of departed quantities. He said you can’t multiply something that is zero and get a real answer. For a while, mathematicians were embarrassed. In the 1800s they replaced infinitesimals with the rigorous idea of a limit — a precise way to talk about numbers getting arbitrarily close to some value. That clean, logical foundation became standard. Yet, amazingly, in the 1960s Abraham Robinson showed that infinitesimals could be made fully rigorous too. What looked like sloppy cheating eventually earned a respectable home.
The Perfectionist Who Hated the Spike
Dirac’s spike was the next ghost. John von Neumann, a Hungarian‑born mathematical genius of the early 20th century, couldn’t stand it. He called Dirac’s tool an “improper function with self‑contradictory properties.” In the late 1920s and 1930s he built a completely rigorous version of quantum mechanics. He used a special kind of space called a Hilbert space — a tidy universe of vectors and operators where every operation is perfectly defined. There was no room for undefined spikes.
Von Neumann’s formulation was a masterpiece of logic. But working physicists almost never used it. It was heavy, abstract, and slow. Dirac’s informal framework, with its spike and its elegant bracket notation, let people calculate quickly and see the physics clearly. Most textbooks still follow Dirac, not von Neumann.
The Trick That Got a Promotion
Why would anyone trust a tool that mathematicians call a fiction? Dirac’s own answer was practical: as long as you follow simple rules — for example, keep the spike inside an integral and never ask its value at a point — nothing goes wrong. He also pointed out that you could eventually replace it with a proper, clunky expression, but doing so would hide the clarity.
Over the 1940s and 1950s, mathematicians caught up with Dirac’s intuition. They invented distribution theory (which treats objects like the spike as “maps” from nice test functions to numbers) and then the rigged Hilbert space — a clever structure that surrounds a normal Hilbert space with enough extra room to hold all the useful “improper” things. Suddenly the delta spike, and many other tools, had a perfectly rigorous foundation. Dirac’s pragmatic shortcut had been right all along. Still, working physicists rarely bothered with the fancy new wrapping; they just kept using Dirac’s original, hand‑wavy version because it was faster.
Physics Splits Into Two Worlds
When physicists moved to quantum field theory (QFT), which marries quantum mechanics with Einstein’s relativity, the tension between rigor and pragmatics exploded. Now the systems had infinitely many degrees of freedom — like a sea of vibrating points everywhere in space. Mathematics became ferociously hard.
Two camps formed. The axomatic approach tried to build QFT on a completely secure logical base, using either abstract algebras of local observables (algebraic QFT) or operator‑valued distributions (Wightman’s axiomatic QFT). These theories were clean but almost useless for making predictions about real particles. The Lagrangian approach, on the other hand, used a grab bag of tools: path integrals, Feynman diagrams, and especially renormalization. Renormalization is a recipe for sweeping infinities under the rug — subtracting one huge mess from another to leave a tiny, perfectly accurate number behind. Using it, physicists calculated the electron’s magnetic properties to twelve decimal places. The predictions were spectacularly correct, even though the steps would make a pure mathematician wince.
The Map and the Territory
Philosophers entered this debate over the last twenty years. If you want to understand what quantum fields are really telling us about reality, which version of the theory should you trust? David Wallace argues for the Lagrangian picture, which he calls the conventional approach. He points out that renormalization group methods now show why the messy calculations work without needing a perfectly smooth continuum. Physics likely breaks down at extremely tiny distances where gravity takes over, so worrying about infinite spikes at those scales is unnecessary. The effective, cut‑off‑using theory gives us the right ontology.
Doreen Fraser disagrees. She insists that a true interpretation of QFT must be based on an axiomatic framework, because only a theory that obeys the full symmetry of special relativity (with no artificial distance cutoffs) tells the fundamental story. A useful recipe isn’t the same as a deep account of what exists.
Other thinkers search for middle ground. Constructive QFT tries to build rigorous models that match the successful Lagrangian recipes, while pluralists suggest that different formulations are complementary lenses — like using both a map and a satellite photo to understand a city. The question hasn’t closed.
Why the Mess Still Matters
You probably use “bad math” every week. Your phone’s map app guesses traffic using approximations. Weather forecasts crunch messy data with rules of thumb. Those shortcuts work, and you don’t wait for a perfect theory of the atmosphere before you carry an umbrella. Physics is more serious, but the same dilemma appears: do we demand absolute logical cleanliness, or do we allow ourselves to be guided by tools that simply, stubbornly, work?
Dirac once said that in science beauty is more important than rigor. von Neumann believed the opposite. Both were geniuses. Their standoff lives on in the foundations of our best theory of matter. No one has yet built a theory that is both as immaculate as algebra and as agile as a Lagrangian. Maybe that’s because the universe itself doesn’t care about the difference. Or maybe we just haven’t found the right kind of math yet.
Think about it
- If a team of engineers built a bridge using math that “works” but nobody can fully prove it is safe, would you walk across it? Why or why not?
- Can a scientific theory be completely correct about all the numbers it predicts but still give us the wrong idea about what really exists? Give an example from everyday life.
- When is it smarter to trust a messy shortcut, and when is it smarter to demand a perfect foundation? Think of a situation where you’ve had to choose.
