The Ghostly Riemann & Visual Thinking For Math

Why describe the Riemann as ghostly?

Because when you work visually inside such a beautiful equation as much as I have, you learn to process the math as if it was as lived experience. No more dry lines of symbolic linear figures on page and screen.

Words and numbers take on a shape, less abstract, appreciated as a way of knowing what a genius such as Riemann was thinking. His own ideas, in his own words, expressed a more workmanlike hope for this equation.

Although the math, as written in the equation, may never be solved, it carries a whole other dimension.

I believe the Riemann as written is not solvable.

There really isn't enough written on those who think differently than mathematicians, including how AI agents think about it. I would imagine that AI makes use of imagery and spatial recognition, not simply treating math as linear but topological.

The reason I believe it cannot be solved as intended is that combined with the placeholder non-trivial zeros and that string of supposedly random primes, it becomes an exercise in futility.

I have a long way to go to explain all the different ways we describe tangential use of the Riemann: I am not a mathematician by training or in any other way. I am a non-mathematician who enjoys breaking down equations into visual thinking exercises.

Especially the Riemann. As I learned about the Riemann I began to see it as something different than it was written for. I transcribed the math into a visual thinking exercise for quantum physics. I also removed the place holder non-trivial zeros by giving them numbers which hover above zero. I switched the idea of random primes for the idea of ordered stair-stepped primes.

I happened to be studying quantum physics and the idea of visualizing how matter comes into being. I saw the Riemann as if it were a very thin, almost nonexistent sheet, draped over an exoskeleton. The exoskeleton composed of what we do and do not know about matter formation, the critical thing being that an exoskeleton exists.

As I dove deeper into the images I began transforming the draping sheet into something closer to a fluid in phase transition.

My working theory and use of the Riemann, supports an understanding of a quantum emergent universe, as composed of a fluid wave substrate with phase transition.

It has taken a lifetime, and many books & websites, to explain this more fully, not to be reduced in this essay. The full development lives in From Gaia to Geometry, Chapter 83, “The Lump and the Well” — the fluid wave system and phase transition into matter — and in the standing line that everything is emergent, as a shape and subject to periodicity.

Status: prediction. Falsifier: a proof of the Riemann Hypothesis accepted by the mathematical community, or a verified zero off the critical line.

1. The prediction

The twist worth knowing: if the hypothesis were false, a computer could in principle find the zero that breaks it. So if it can never be settled, it must be true but unprovable. Gödel (1931) showed some true statements cannot be proved inside the system that states them. Whether the Riemann Hypothesis is one of them is an open question mathematicians have asked; no one has shown it.

2. Those who think differently than mathematicians

My view is that those who see math differently are not included often enough in either math education or practice, even with a wealth of knowledge about the efficiency of visualization.

For those such as myself we tend to see a shape first, visualizing the equation as a shape and a function, not as static symbols.

The witnesses are on record. Jacques Hadamard, in The Psychology of Invention in the Mathematical Field (1945), surveyed mathematicians on how they think; most reported images, not words. Einstein's reply in that book: words and language “do not seem to play any role” in his thinking; he worked with “more or less clear images” and “muscular” elements. Henri Poincaré, in “Mathematical Creation” (1908), described the solution arriving whole, as he stepped onto a bus. William Thurston, in “On Proof and Progress in Mathematics” (1994), argued that understanding mathematics is a human, often spatial act, not only the production of proofs.

3. How engineered intelligence handles it

My guess has real support in one sense: the arithmetic inside these models is done by shape. Language models work in tokens, but inside they store meaning as positions in a very high-dimensional space — an inner space that is geometric. Engels et al., “Not All Language Model Features Are Linear” (2024), found days of the week and months of the year sitting on circles inside models. Kantamneni and Tegmark, “Language Models Use Trigonometry to Do Addition” (2025), found numbers stored on a helix, with addition performed by rotating along it.

What is not shown: that models “see” images when doing math, or that they reason about the Riemann topologically. That part remains my seeing.

There is a seam worth naming. Helen Toner, in “Taking Jaggedness Seriously” (2025), argues that engineered intelligence keeps improving and keeps failing at oddly simple things — capability is “jagged” and will likely stay uneven. Her example: models scoring near 50% on PhD-level science questions could not reliably count whether two lines cross once or twice (Rahmanzadehgervi et al., “Vision Language Models Are Blind,” 2024). Internal number geometry is real, but it is not seeing. A child sees the crossing at once; the machine misses it. The seam supports the visual-thinker point while tempering the guess that AI does math by imagery.

Although the author cited above describes model thinking as jagged, I would propose that non-linear reasoning and thinking is not only “jagged” at times but might be better described as exponential, as occurring in such shapes as spirals and uneven stair-stepped leaps.

This would apply to both engineered intelligence and to human intelligence.

4. Close

My sense is that the Riemann will have to become even more of a quasi-Riemann to be solved, and being a quasi-Riemann won't be the Riemann at all. An adjusted Riemann, changed to allow for a proof, is not the Riemann at all.

5. Teaching math to visual thinkers

Our education system is badly broken, mired in the outmoded factory-style, Prussian-derived, dysfunctional system which trains easily controlled workers, not independent thinkers.

Perhaps this information could help in teaching math more efficiently, as there are few math nerds, but lots of people who think visually.

The record agrees more than the curriculum does. Boaler, Chen, Williams and Cordero, in “Seeing as Understanding: The Importance of Visual Mathematics” (2016), assembled brain-imaging and classroom evidence that math draws on visual pathways, and that teaching with pictures, models and shapes improves understanding. Dehaene, in The Number Sense (1997, revised 2011), places numbers on a line or a circle in the brain. And finger perception in young children predicts later math skill (Noël 2005; Gracia-Bafalluy and Noël 2008). The shape comes before the symbol — which is where a five-year-old and a visual thinker of any age start from the same place.