When Math Invents Reality: Imaginary Numbers &

Every week I write about tools that exist only as math before they touch biology. A CRISPR guide RNA gets designed computationally — algorithms predicting which 20-letter sequence will dock cleanly onto a gene — before anyone in a lab has confirmed it works. AlphaFold predicted protein structures that took experimental biologists decades to chase down. The math runs ahead and biology scrambles to catch up. So when Neil deGrasse Tyson sat down with comedian Chuck Nice on a recent episode of StarTalk to walk through the history of mathematical concepts that had "no connection to reality, to an objective world," I was maybe more primed than the average viewer to find it interesting. And maybe more primed to find it a little unsettling.

The episode opens with a history of math as a series of reluctant inventions. Nobody sat down one morning and decided to dream up negative numbers for fun. Tyson's argument — clean, satisfying, told through a running bit about owing someone chickens — is that math has almost always followed a practical need. You need to track a debt you can't pay? You need negative numbers. You need to split something? You need fractions. You need to represent nothing? Zero, which has its own complicated biography: while the concept has South Asian origins (the mathematician Brahmagupta formalized it in 7th century CE India), placeholder zeros also developed independently in Babylonian and Mayan mathematics. Tyson notes that zero traveled through the Islamic Golden Age and became formalized in what we now call Arabic numerals — though that name credit, as he points out, arguably belongs to India. The point is that the idea migrated because it was useful, not because it was beautiful.

This is the rhythm of the episode: need creates math, math waits around, reality eventually shows up to collect. The sharpest example is non-Euclidean geometry. Euclid's rules — triangles always add up to 180°, parallel lines never meet — held for millennia because they described flat surfaces, which is mostly what humans needed to measure. Then mathematicians asked what happens on a curved surface, and suddenly the rules broke. On a sphere, you can draw a triangle with three 90° angles — one vertex at the North Pole, two on the equator — and the angles sum to 270°. (And that's not even the ceiling: spherical triangle angle sums can approach 540°, depending on how large the triangle gets relative to the sphere. The 270° example is just the tidy one.) On a saddle-shaped, negatively curved surface, triangles sum to less than 180° and parallel lines diverge. Whole self-consistent geometric systems, none of which looked like they described anything real.

And then Einstein wandered in. Tyson's version of this is delightful — Einstein as a man who showed up to a math party and said, essentially, watch this. He showed that matter and energy curve spacetime, and the geometry you need to describe that curvature is exactly the non-Euclidean kind that mathematicians had been developing with no particular destination in mind. "That was math kind of waiting for something to do," Tyson says. It's a good line. It also describes most of the computational biology I cover.

The imaginary number section of the episode is where Tyson earns his keep as a science communicator. The square root of a negative number has no answer in the real numbers — multiply any real number by itself and you get something positive or zero, never negative. So mathematicians defined an answer: the square root of -1 is i, the imaginary unit. Tyson flags the naming as a disaster. "It's a most unfortunate name," he says. "There's no less real than negative numbers. In a corporeal world, you could not possess a negative number." He's right — imaginary is a PR problem, not a mathematical one. In the complex plane, imaginary numbers are just another axis, another coordinate direction. A number like 5 + 3i isn't weird; it just lives in two dimensions instead of one.

And they're genuinely essential. Electrical engineers use complex numbers to represent the phase relationships between current and voltage in AC circuits — the way the two oscillate out of sync in ways that a single real number literally cannot encode. The math isn't decorative. Remove it and your circuit analysis collapses.

So far, so triumphant. Math invents things. Reality shows up later. Everyone wins.

The final example is where I started chewing on this differently.

When physicists developed the equations describing black holes — the collapse of spacetime into a singularity — a squared term appeared, as squared terms do. Take the square root: two solutions. One gives you a black hole. The other gives you its mathematical opposite: a white hole. Where a black hole pulls everything in and lets nothing out, a white hole would push everything out and let nothing in. Tyson describes it clearly: "Thus was born the notion of a white hole." A valid solution to the Einstein field equations under time-reversal symmetry. Mathematically coherent. Physically... nowhere to be found. Astronomers have looked. Calculated what a white hole would look like — very bright, very small, very intense. Searched the universe. Nothing.

White holes are a real solution to real equations. But the physics community is genuinely divided on what that means — many treat them as mathematical artifacts, objects that fall out of the algebra but don't correspond to anything the universe actually builds. They're not mandatory, just permitted. Tyson is careful here, leaving it as an open question: "maybe the space folds itself so much that the white hole exists on another plane altogether — possibly." That "possibly" is doing a lot of work, and it should.

This is the moment the episode made me a little nervous, and I think it's worth sitting in that feeling longer than Tyson and Nice do.

I cover this dynamic constantly in a different domain. Polygenic risk scores — mathematical models that claim to predict your likelihood of developing a disease based on thousands of genetic variants — are increasingly being used clinically before we have solid evidence that acting on them improves outcomes. The math is real. The models are internally consistent. But we haven't always confirmed which solutions are physically relevant and which are the negative-coordinate-behind-the-cliff variety. Speculative models get deployed in drug development, in genetic screening, in disease risk communication, and we find out which solutions mattered after people have already made decisions based on them.

I'm not saying Tyson is wrong to find the white hole beautiful. I find it beautiful too. The idea that the equations for black holes contain their own mirror image — and that we might someday find it, or find out why it can't exist — is genuinely thrilling. "It's a fun dance," Tyson says at the end, "between what the math is giving you and what we look for in the universe. And it continues to this day."

Yes. It does. And the dance is mostly wonderful. What I'd add — what my beat keeps teaching me — is that we're often already dancing with a partner we haven't confirmed is in the room. Sometimes it's AlphaFold, and the partner turns out to be real and revolutionary. Sometimes it's a risk score model, and the partner is a ghost, and patients are making irreversible decisions based on its weight and warmth.

The math running ahead of reality is one of the most beautiful features of human cognition. The question I can't quite let go of is how we decide, in the gap between equation and confirmation, how much to trust the answer we haven't verified yet.

By Mei Zhang, Biotech & Genetics Reporter, Buzzrag