The true essence of Montessori education is for even young children to learn advanced mathematics through their senses, such as the Pythagorean theorem and exponents. Beyond that, however, lies “Synergetics,” a new mathematical system that discards Cartesian coordinates and Platonic solids. In this episode, we discussed the evolution of mathematics education itself, in a way that is more scientific than anywhere else.
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. Works as a software developer and a weekend farmer. Father of two.
Psychoarithmetic, Psychogeometry, and Beyond: Synergetics
Hiro
There’s something called “synergetics.” What is it? In Montessori education, we do “sensorial mathematics,” right? Synergetics is what lies at the end of the road if you keep doing sensorial mathematics without stopping.
Yati
I see. Maria Montessori wrote in her books Psychoarithmetic and Psychogeometry about how to let 7-year-olds experience high-school-level math through the senses. So synergetics is what comes after that.
Hiro
Exactly. Even if you start with Montessori’s sensorial math, conventional education makes you stop and switch to math that doesn’t use the senses. But if you keep going with sensorial math, you reach synergetics, which is cosmic mathematics. Most of the time, though, people stop using their senses midway and start doing human mathematics. That’s why so many people find math difficult.
Yati
We talked about cosmic math versus human math in the 6 to 12 math-education feature, too. Recently, I was shocked to realize that our oldest daughter, who was raised with Montessori education, says the same thing as Buckminster Fuller, the founder of synergetics. When someone explains something to her, she asks, “Is that something you experienced yourself?” In other words, she refuses rote memorization or blindly accepting theories. She refuses to the point of crying.
Hiro
Exactly. Things no one has experienced don’t exist in the universe. Human math defines things that don’t exist in the universe, then twists logic around arbitrarily to produce answers.
Yati
That’s a striking perspective. I wouldn’t have noticed without being told. I never questioned it, but it’s true.
Hiro
People think math is about scribbling away with paper and pen, or chalk on a blackboard. But that’s human math. We need to evolve from there to cosmic mathematics, synergetics, which uses the senses. We need to advance to sophisticated math that uses the senses.
Yati
To evolve, I want to understand synergetics better. First of all, what does the word mean? I know “synergy” is in there. Synergy means multiple elements interacting to produce greater effects than they would alone, a “synergistic effect.”
Hiro
Synergetics is translated as “the geometry of the universe.” It means we advance our understanding of the universe by asking why it’s designed the way it is. Specifically, you understand the universe through its structures and patterns. You might think other disciplines do the same, but the difference is that you understand it through the senses. That’s why research is done through modeling. Modeling means using wooden sticks and thread, strong carbon-fiber wire, to build something called a tensegrity. That’s “a structure composed of tension and compression,” known as a “tensegrity structure.”
Yati
Tensegrity, a word combining “tension” and “integrity.”
Hiro
Right. In the universe, tensegrity structures are the only things that qualify as structures. Everything else isn’t a structure. This may be shocking, but most people’s understanding of “structure” is shallow, so they build buildings that depend on the ground. Because they depend on the ground, when an earthquake hits, they crumble to pieces. In space, the standard is mobile things like spacecraft that don’t depend on gravity. If something can’t exist stably in space, it can’t be called a “structure” in cosmic terms.
Yati
It’s difficult, but I’m starting to understand. By “structure” here, you mean something cosmically self-supporting, something that won’t break even if you take it into outer space or to another planet. There’s talk that microorganisms have tensegrity structures, and that’s why the panspermia hypothesis, the idea that Earth’s first life came from space, is possible. Microorganisms can travel through space alive because they have tensegrity structures. By the same logic, anything that maintains its structure in space has a tensegrity structure. Does that include atoms?
Hiro
Yes. Fuller said atoms are tensegrity structures. That’s why there are 92 naturally occurring elements. Since synergetics is the geometry of the universe, it can show through calculation why there are exactly 92 natural elements.
Yati
I see. Tensegrity structures are also found in cells and skeletal systems in nature, because that’s the structure the universe has adopted. I’ve looked at various examples of tensegrity structures and tried to understand them in my head, but I couldn’t grasp it. This is something you have to actually build to understand. In the Montessori international course, I learned that at the secondary level, which covers ages 12 to 18, students build real bridges and sheds. Through building a bridge that real people will walk across, they learn math, physics, art, whatever is needed. When you start thinking about building a bridge that absolutely won’t collapse in an earthquake, you naturally arrive at tensegrity. If you only do paper-and-pen human math, detached from the universe and the senses, you might never get that idea.
Hiro
Exactly. In Montessori education, we use the senses, so even a 5-year-old can discover how to prove the Pythagorean theorem while playing. Even young children can do advanced mathematics. Synergetics is what comes after that.
Yati
Fuller’s discussion starts with the stability of triangles. In Montessori education, there’s a material called the constructive triangles, which teaches sensorially that all shapes are made of triangles. Isn’t that related?
Hiro
That’s part of the flawed human math. Plane geometry doesn’t exist in the universe. In the real world, the interior angles of a triangle don’t add up to 180 degrees. You can experience this by drawing a triangle on a globe and measuring the angles. If you look at your own body or the parts of any organism, you’ll see that straight lines don’t exist in the universe, only curves. We’re just approximating everything with straight lines.
Yati
I see. We need to evolve to geometry that can actually be experienced. So solid geometry is also wrong from a cosmic-math perspective. We need to remake the teaching materials using tensegrity solids.
Hiro
Yes. We should use vector equilibrium, where angles are flexible. Platonic solids were used in ancient Greece, 2,500 years ago. Using them is hopelessly outdated. People tend to think of shapes and solids as fixed, but rigid solids like concrete are actually very fragile. They collapse under their own weight. You can see that if you keep building a tower taller and taller. What’s the tallest building in the world right now? Doesn’t matter. But if you look at Tokyo Tower or Tokyo Skytree, the famous tall towers in Tokyo, they’re built from combinations of triangles. Squares wouldn’t work; they’d buckle partway up. Tensegrity structures are used in the Tokyo Skytree, too. People focus too much on things that are solid and stationary. In reality, everything is much more dynamic. Tensegrity structures are constantly pulling on each other, always in motion.
Yati
So if we made everything from housing to everything else with tensegrity structures, they’d be self-supporting, wouldn’t collapse in earthquakes, and would use fewer materials, benefits all around. But because education teaches outdated math, everyone ends up thinking and building structures that depend on the ground.
Hiro
Exactly. Spoiler alert: from cells to galaxies, everything uses the same tensegrity structure. Once you realize the universe itself is made of tensegrity structures, studying their properties lets you connect quantum theory and general relativity, understand what gravity is, understand what time is, and explain quantum entanglement and wave-particle duality.
Yati
That’s a lot of information! [laughs] If you could do that, it would be a massive breakthrough. Now I understand why Polymath School is so focused on new mathematics.
Hiro
Yes, we are. Yesterday, I came up with a new interpretation of the quantum double-slit experiment based on tensegrity structures, and I’m going to share it with the kids soon.
Yati
That’s fascinating. It might be another scientific prediction. [laughs]
Hiro
So it’s self-evident that the universe has a tensegrity structure. Calculations based on other premises lead to breakdowns and contradictions. But if you assume tensegrity, there are no contradictions. That said, modern physics is making progress. Loop quantum gravity, for example, is approaching tensegrity. Witten, from superstring theory, says things like, “I’m doing 22nd-century mathematics.” Cute, right?
Yati
So Polymath School is doing 30th-century mathematics?
Hiro
No, ours is eternal mathematics. Cosmic math. [laughs]
Yati
[laughs] Having been steeped in conventional math and physics, I’m very interested in bridging human math and cosmic math.
Hiro
Then work hard on bridging them.
Yati
To incorporate this into the six to twelve environment, isn’t it just about making things? Synergetics is basically about considering how everything in the universe interacts and pulls on each other, and then building the most efficient, sturdy, waste-free things, right?
Hiro
No, no. You’re underestimating it. It’s about “understanding the universe correctly.”
Yati
We covered this in the work feature, that’s humanity’s mission. When we understand the universe correctly, we can become more efficient, help nature, and support the evolution of the universe.
Hiro
Right. This isn’t about the individual level. It’s about collectives or even the cosmic scale. The universe itself has consciousness. As part of that consciousness, the universe reaching self-awareness is what advancing cosmic understanding means.
Yati
That sounds like you’re saying something pretty out there. [laughs]
Hiro
What does it mean in simple terms? It means the universe wakes up. Humans are born, and at first, babies don’t understand who they are or where they are, right? Gradually they come to understand what they are. It’s the same idea. So it becomes religious, philosophical, profound, unavoidably. [laughs]
Yati
I see. First, you realize you’re part of the universe and advance your understanding of it. As this happens across many life forms, since those life forms are also part of the universe, that means the universe is advancing its understanding of itself. The universe is understanding itself. Is that it?
Hiro
Yes. In synergetics, everything is pulling on each other, constantly interrelated. The universe is the same. Every person is connected to someone. No object can exist by itself. Someone said this long ago, around the 6th century BCE.
Yati
Gautama Siddhartha?
Hiro
That was his name before he awakened. By the time he realized it, he was awake. The Buddha. Do you know what summarizes everything the Buddha realized?
Yati
Buddhism.
Hiro
Right. And what is Buddhism? Synergetics. It’s cosmic physics. Our family practices Zen Buddhism, a Japanese tradition of Buddhism, so I don’t know much about other sutras, but if you decode the Heart Sutra, a core text in Mahayana Buddhism, it’s about synergetics. In other words, awakening means realizing the reality of the universe instead of human common sense.
Yati
You’re saying something huge. But it’s true. Buddhism teaches that self and others are the same. You could interpret that as saying we’re all the universe.
Hiro
So when I say “let’s begin synergetics,” I’m saying let’s stop learning human math, conventional school math.
Yati
And realize the true nature of the universe and achieve enlightenment. Polymath School is going to produce one Buddha after another.
Hiro
[laughs] The alternative path is for select people whose senses haven’t been corrupted, so I’m not forcing it. The friction is great.
Yati
I’d like to share something that lets you feel, sensorially, that human math is wrong. Let’s consider Fuller’s frequent claim that “there are no straight lines in the universe.” Straight lines are among the first things you learn in arithmetic and plane geometry. And they don’t exist, meaning you can’t experience them. A straight line is supposed to be perfectly straight and extend infinitely. Try drawing a straight line with paper and pen. You can’t make it perfectly straight. Magnify it, and it’s wavy. And you can’t draw an infinitely extending line. The paper has limits. That’s at the micro level. What about the macro level?
Hiro
Earth orbits the sun and rotates on its axis. So even if you think you’ve drawn a straight line, viewed from afar it spirals around. That’s why straight lines don’t exist in the universe. Human experience is amazing. There’s a Japanese saying, “If you’re in a hurry, take the roundabout way.” That’s literally true. Spiraling around is actually the shortest distance. It looks like you’re going farther away, but you’re actually taking the shortest route. It’s all there in human experiential wisdom.
Yati
I still feel like I’m being tricked. So there really are no straight lines? For example, Fuller often uses great circles, lines passing through the Earth’s center, like meridians or the equator. Aren’t those straight?
Hiro
Those aren’t straight lines. They’re called “geodesic lines,” describing the shortest distance on a sphere. They’re curved because they’re on a spherical surface.
Yati
Right. Then what about a rocket flying through space? Doesn’t it fly in a straight line?
Hiro
If you watch a SpaceX rocket launch, you’ll see, right after liftoff, the rocket curves dramatically. The SpaceX logo represents that curved trajectory.
Yati
I didn’t know that. It’s a beautiful design.
Hiro
Rocket engineers and aerospace professionals probably understand this best. When you launch a rocket from Earth to the Moon, there are no straight lines. Everything is curved, spiraling. Landing is also spiraling, gradually approaching while orbiting. Always rotating.
Yati
What if you push a rock floating in space and let it drift in a straight line by inertia?
Hiro
Even that is bent by the gravitational pull of black holes, nearby celestial bodies, everything in the universe pulling on it. So what you think is a straight line is actually curved.
Yati
What about your line of sight? Aren’t you looking straight at the Sun?
Hiro
Light from the Sun takes 8 minutes to reach us, and during that time Earth rotates, so your line of sight is curved in cosmic terms. So straight lines don’t exist in the universe. Give up. Unlearn. Human math assumes imaginary spaces that don’t exist and does calculations there. It’s like a child imagining a fantasy world and trying to navigate reality with it. Reality isn’t like that, so it doesn’t work in the real world.
Yati
Just thinking about straight lines really brought that home. All right, let’s begin synergetics with the children.
