Montessori mathematics education is excellent. To make this excellent mathematics education even better, it is necessary to make mathematics cosmically correct. To achieve this, we must discard the Cartesian coordinate system, which fundamentally supports the mathematical system we are accustomed to.
We’d love to hear your thoughts. Feel free to share your comments!
Profile
Yati Obara
Editor-in-Chief, Scientific Montessori
Based in Japan, born in 1989. CEO of Motherhand and Co-Director of the nonprofit think tank Polymath Research. Holds an M.Eng. and is an AMI-certified teacher. Focused on Montessori developmental theory and AI. Mother of two.
Hiro Obara
Publisher, Scientific Montessori
Based in Japan, born in 1990. CEO of StudyX and Co-Director of the nonprofit think tank Polymath Research. Works as a software developer and game designer. Father of two.
The children begin to use an entirely new coordinate system.
Yati
I want to talk about “Why learn mathematics through discovery?” But first, I’d like to define mathematics. Galileo Galilei famously said, “the universe is written in the language of mathematics.” What do you think?
Hiro
Mathematics can be defined as “the study of structure and pattern.”
Yati
Does that mean it describes the universe?
Hiro
Essentially, what we call “mathematics” is the correspondence of the universe’s structures and patterns to symbols. If you don’t map the universe to numerals, numbers, and mathematical symbols, but instead just speak in ordinary language and turn it into a story, it becomes mythology.
Yati
Is “the mathematics that humanity has developed” the same as “the mathematics that describes the universe”?
Hiro
There are two kinds of mathematics. One is human mathematics, the mathematics we learn in education. The other is cosmic mathematics, the mathematics the universe has. The history of human mathematics is a history of corrections to human mathematics. We hypothesize “Maybe there’s a law like this” and verify it, but eventually it turns out to be wrong and gets corrected by later generations.
Yati
Like relativity theory to Newtonian mechanics.
Hiro
That’s physics. For a pure mathematics example, Euclidean geometry to non-Euclidean geometry.
Yati
Right.
Hiro
The history of mathematical development is mostly like this: first you have some illusion, like “Isn’t this the pattern?” Then the next generation says, “Wait, that’s wrong.” It gets corrected. Like Gauss or Euler. But some people hit the right answer from the start, like Pythagoras, discovering that triangles have certain properties. The Pythagorean theorem. For other people, the next generation, they can’t deny it. It’s too useful. Even ordinary children can calculate area with base times height divided by 2. But we couldn’t use it with confidence if Pythagoras or earlier generations hadn’t discovered the law, or if no one had verified it. However, even things used that way for a long time might still have errors. That’s both the fascination and the terror of mathematics.
Yati
That’s true. Is there a pattern to when corrections happen?
Hiro
Usually corrections come through technological development. What I mean is, for example, when you want to see far into the universe, you build large telescopes. Space telescopes. First you have to think about how to launch them. You need mathematics and physics for the launch. Then you also need technology for seeing farther, and that uses mathematics and physics too. And then how to handle the data you get, what patterns to find from it. You need to answer propositions like whether the universe is round.
Yati
The shape of the universe was actually solved using mathematics and physics with the Poincaré conjecture.
Hiro
Right. And there are various unsolved problems that need proof, like Fermat’s Last Theorem.
Yati
Like verifying whether the proof of the ABC conjecture is correct.
Hiro
Right. And to solve these more efficiently, new mathematics is sometimes needed. And that new mathematics is actually undiscovered cosmic mathematical principles.
Yati
Like Edward Witten doing new mathematics while researching superstring theory.
Hiro
Here’s a spoiler for readers. That “coordinate system” you learned in education is called “Cartesian coordinates” because Descartes created it, and it’s very convenient. When drawing graphs, you say X axis, Y axis for 2 axes. For 3 axes, you add Z axis. This Cartesian coordinate system is not correct cosmically.
Yati
That’s a “What?!” moment. [laughs]
Hiro
It doesn’t work in the universe.
Yati
Since Earth is a sphere too, you can’t express it. When you go to the edge, you have to come out the other side.
Hiro
Right, even on Earth we’re approximating calculations quite a lot to get answers. We’re forcing it, quite a lot actually. GPS calculations, for example. Approximating curves with straight lines, using the squeeze theorem and limits to cleverly handle complex differentiation. The calculations are incredibly difficult.
Yati
Right.
Hiro
Since computers were invented, it’s fine if the computational load is enormous. Or rather, computers were developed because of that computational load.
Yati
Right. [laughs]
Hiro
Being able to do all sorts of things with computers is a really great thing. But really, if there’s a way to reduce computational load, that would be better.
Yati
Mathematical proofs sometimes use computers to brute-force things. Like the four-color map problem. If it could be proven without computers, that would be beautiful.
Hiro
Right. Computers would use less electricity too. Now with AI development, they’re consuming enormous computational resources, and there’s even talk of needing nuclear power. Electricity shortage is a problem. Data centers are running at like 100% capacity, constantly operating. Electricity becomes a problem. Then you’re consuming fossil fuels. Why is this happening? It’s because the mathematics is wrong. The underlying mathematics.
Yati
That’s really true. Whether global environmental problems get solved depends on the evolution of mathematics.
Hiro
Right now we’re calculating in extremely inefficient ways. With the cosmic approach, computational load becomes one-third.
Yati
Is that the power of cosmic mathematics?
Hiro
In other words, because we’re calculating in Cartesian coordinates, computational load is 3 times what it would be cosmically. This is essentially an assumption—people think Cartesian coordinates are always correct. Most people think mathematics has no errors and is always correct. They feel mathematics never changes forever, or that it doesn’t change because it’s been proven.
Yati
So separate from whether something is proven or not, there’s the discussion of whether it’s efficient.
Hiro
Right. We know it can be solved that way. For example, there’s brute force—giving random values, trying lots of random things when solving calculation problems, and if you hit the answer, lucky. Just repeating like “Please let tomorrow be sunny.” That method exists.
Yati
AI is a type of that too. Generating images from random seed values, hoping something good comes out. Or when you want new materials with specific properties, you generate randomly and pick what’s probabilistically likely. Same with shogi moves.
Hiro
If you do that, eventually you’ll solve it. Brute forcing. But if you do that for every problem, it’s game over. In the end, “because it’s proven” isn’t really a reason. The legitimacy of human mathematics is simply how much it matches cosmic mathematics. And the more it matches, the more you can predict the future. This is a bit of self-promotion, but Polymath School, which started in April 2025, defines “intellectual” that way and aims to be “the most intellectual school.”
Yati
Right.
Hiro
“Calculation” layers inferences. Like the Japanese proverb, “When the wind blows, the barrel maker profits.” It’s a saying about chains of unlikely cause and effect. You go through about 10 inference steps from wind blowing to the barrel maker profiting. Mathematics is about becoming able to layer inferences. There’s all sorts of randomness. So people immediately start talking about chaos, saying “We give up, we can’t predict the future.” But including that, you can still produce prediction accuracy. You can express it as probability.
Yati
That’s how AI has been developing too.
Hiro
Weather forecasting too. It keeps evolving, using AI to predict more accurately. This is the Laplace’s demon story—if you could accurately predict the state of all particles, you could always predict the next state.
Yati
The idea that the future is determined.
Hiro
We’re approaching that. In other words, cosmic mathematics is that direction. If you understood all of the universe’s state and principles, you could predict the universe.
Yati
That’s where quantum physics comes in.
Hiro
The idea that things are probabilistically determined, so the future is uncertain. Two states superimposed. But that’s human understanding—it’s not that the universe decided superposition is uncertain.
Yati
This discussion is getting complicated.
Hiro
Simply put, as an efficiency of representing information, superposition is good, so you could say the universe uses it. An interesting point here relates to “The Power of Folktales” from the previous issue. For example, in “The Crane’s Return of a Favor,” a crane secretly weaves cloth disguised as a young woman, and children naturally understand superposition. The crane in the weaving room is in a superposition state of daughter and crane.
Yati
Same with the big wicker basket in “Tongue-Cut Sparrow.”
Hiro
Richard P. Feynman said “If you think you understand quantum mechanics, you don’t understand quantum mechanics.” Elon Musk said “Quantum mechanics is the most difficult.” Jeff Bezos said “I was going to become a physicist but gave up at quantum mechanics.” Meanwhile, in Japan, 5-year-olds readily accept it. Entanglement, the Urashima effect—there must be many more folktales if you look. And it’s not because some great scholar thought them up. They’ve been passed down among farmers since ancient times, and people listen because they’re interesting.
Yati
So children are given all sorts of patterns, and much later, they can discover on their own “Oh, this is that pattern.”
Hiro
“Straw Millionaire” is also the law of conservation of energy. Straw = lotus leaf = miso = sword = millionaire.
Yati
But straw does not equal millionaire, right? Isn’t that mathematically incorrect?
Hiro
That’s being tainted by human mathematics. Buckminster Fuller said mathematics must include time, gravity, and temperature. In cosmic mathematics that includes time, the equality holds.
Yati
It’s hard to escape human mathematics. Please properly teach me about how calculations become one-third without Cartesian coordinates.
Hiro
You’ll discover that yourself. That’s what learning through discovery means.
