The Unreasonable Effectiveness of Mathematics for Everything
A review of a classic math and physics paper, with reflections on life, the universe and everything. But mainly life.
In this essay, we review seminal work on how come mathematics works so well in the natural sciences, extending the idea to look at how our own lives are shaped by complex factors with which we may be unfamiliar. Making the case that becoming more familiar with basic concepts originating in nonlinear science helps us get more of what we desire and require, we review those concepts and discuss how they might apply in our day-to-day lives. Follow-up readings are linked at the end, along with references.
So that’s pretty unreasonable. We made up a number “just because”. We said it, and it seemed to bring things into existence. We don’t need the math for a lot of things because our brains do math intuitively for a lot of things. You throw a ball and it lands in the basket, or you miss. There’s no time to do the calculations formally — some people have really great aim, and you can train aim — and it’s be a different game of basketball if all the shots were calculated first. I guess then every shot would go in, and whoever went first would win? Fully deterministic sports is a bit of an oxymoron.
However, for more difficult, improbable events — math is needed. Math is tantamount to magic, because with some energy source — electricity seems to be a big hit nowadays but there are others (nuclear forces, dark energy, gravity, and so on) — you can make amazing stuff happen!
Entanglement is a quantum mechanical truth — particles formed together in the right way remain eternally bound, until one of them is perturbed–then the exact same thing happens to it’s twin, a billion years late and a billion light years away. No one knows why reality is this way but experiments have proven it to be true.
You can create particles on a space station, beam them to earth, do something to one of them and observe the result instantaneously, hundreds of miles away. Note that it doesn’t matter if it’s one inch or one lightyear — there is no delay. From a regular logic point of view, it is as if they are in the same place at the same time, even if they are far away. (One wonders if there is space entanglement, such that different places can be the same at different times).
There’s another form of entanglement, the same phenomenon but occurring everywhere, all at once — and not just with special particles. OK, it’s a little fantastical; you’ll have to use your imagination.
At the most basic level, space is chunky, quantized. This is the fundamental observation in modern physics, that energy, matter, space, gravity, everything (consciousness?) has a granular nature to it, and you can’t get smaller than that. It’s closely related to the Uncertainty Principle (Heisenberg) which shows us that you can’t know everything about anything, on a quantum level. Within that granularity is uncertainty–if you measure the mass of an atom, you will introduce error into your precision about speed. If you measure the speed accurately, you will throw off knowing where it with more accuracy. It’s not a matter of better measuring instruments–that helps, but it appears there is irreducible fuzziness to reality.
And for space, that minimum quantum size is the Planck length. It’s tiny, but space is in little chunks. At the finest level, physicists call it “quantum foam”. Imagine bath suds. There are myriad little curved surfaces making up all those bubbles and where they touch each other, sharing part of a wall. It’s a bit geometrical and bit organic.
OK, so space is made up of tiny chunks which are inherently uncertain on a tiny scale, but of course predictable in a lot of ways on the macro level, mainly stable otherwise we wouldn’t have three dimensional space to live within. Tiny chunks. Quantum foam.
Why is that important? Because of that uncertainty, on the quantum level there are fluctuations in possibility. The fluctuations are big enough that there might be particles which spring into existence for an instant, always in pairs–a photon and an anti-photon. It’s weird, but (use your imagination) they net out to zero–but they could exist, within the limits of the Uncertainty Principle. And this fluctuation in possibility, in probability, is happening all over space, from small to large areas of space.
What does it mean? OK… stay with it… it means that space is highly quantum mechanically tangled with itself. Entanglement knits space together. If you break the entanglement, the physics changes such that those areas of space no longer interact with each other. As with the original two entangled particles, once you make an observation they are no longer entangled, and you have extracted the information from that system.
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I think this is important because our imagination is at the heart of math, and it is whatever happens when we come up with a new idea, put it together, and test it, that makes the future. As a kid, you build a bridge without math. You use your physical intuition. Maybe it holds; maybe it collapses a but but still works, maybe it falls into the stream. You learn over time what works and what does not work, and you can learn this quite well without ever writing anything down. The math is in your head. You may or may not have proper explanations for why something works, but the physical world (and the social world in different ways) follows rules. Now later, maybe you are in physics class and you go over how bridges, work, F=MA, compression and tension, working out the forces and levers so that you know the bridge will work, as long as no one has changed the rules.
This is pretty nifty. However, as implied, the human imagination is limited by our intellectual capacity. We can only remember a bit, we can’t imagine (most of us) an entire building’s blueprint, let alone the schematic for an aircraft, or the details required to launch a rocket or build a particle accelerator or a quantum computer–or a lot of complex human relationships, for that matter.
Then the math is more powerful… it amplifies the imagination by providing a way to remember the process of working it out over time, and recording it in condensed ways e.g. equations, proofs, operations manuals, engineering plans, and so forth. For some things, like the bridge, we’ll know what we’ll find and we know what we need. It will always work, unless there was an error or a change in conditions. If you want a bridge which can withstand earthquakes, you have to imagine that scenario at the outset. You will also build and test models experimentally, and use that data to guide design. You may copy nature (biomimetic) to find solutions we wouldn’t think of. You may use AI to find solutions we wouldn’t think of, now.
For our own lives, this has implications in the present moment. I’ve written about that a bit, in terms of choice here. As with the math we’re discussing here, the present moment has real and imaginary components. What is interesting — critically so in my view — is that the imaginary components of the mind actually correspond to real components. Neurons, the brain, wired together in a mathematically-complex, probabilistic way. Different fields within complexity theory helps us to define and predict, and design, the operations of the brain–network theory, complex adaptive systems, and that like. They again use Complex numbers. But the imagination is not imaginary, it’s real — it’s as real as the blood pumping into the brain, as real as the trillions of synapses, as real as the electrochemical potentials thrumming along fibers, as real as the glial cells withdrawing their flaps to expose binding sites on neurons for learning, as real as the neurotransmitters being released and reuptaken, as real as the genes and proteins being translated, modified and produced.
So it begs the questions. Is anything imaginary? What’s imaginary is real, but not everything we imagine will work in the real world. Some things only work in one’s imagination–a perpetual motion machine, which keeps moving without energy. Aside from possibly some specific crystal lattices, perpetual motion machines seem to be only of the imagination. That’s their only real part, in the mind’s eye, a drawing, a video. But not on your tabletop–if you see one, it means there is a hidden source of energy (often a hidden battery and magnet, giving the system a little kick to make up for energy lost to noise and heat).
The other stuff, you can imagine it and you can make it real. Some of them are really easy. You can imagine and fold a paper airplane. It won’t always be great but there’s a real clear path from imagination to reality. Same goes for a giant cruise ship, though there’s many more folds to make a cruise ship. For other things, we are pretty sure we can do it but it’s super hard. It requires setting up very special conditions, for example the manufacture of very delicate computer components, some nanomachines, some chemical processes, and so on.
Other things–the most interesting ones perhaps, are those predicted by theory–but we don’t know if the theory works. We don’t know if it’s right. The experiment follows the math, as the tech to do the experiment wasn’t around yet when the math was written down. This is true for many classical physics experiments. When you hear “they’ve discovered a new particle”, typically it means the theory predicted it years ago, and then experimental physicists have been building the apparatus to set up the conditions required to see if it will happen. Some events are so rare, even in a particle accelerator, that they have to run the same experiment for many years to get a result.
A lot of human endeavors are like this, but not really. One one hand, they have that “you know it will work component”. You know it should work or could work, if the conditions are there. Many times, different from a careful experimental set-up, we can’t control the conditions.
You want to start a coffee shop in your town? You know it could work, and if you convince yourself you have a solid business plan you know it should work… but with rare exceptions there are no guarantees. You could open up the shop and for reasons which aren’t clear, no one goes in. Or you could open up and then a day later two big brands upon up across the streets. You had a great idea — you picked a fantastic location — but so did they. We don’t always know how much to chalk up to “luck” and how much to wonder if we missed something.
Learning how to make the most of the influence which we do have is crucial. Magical thinking can be misleading, as suggested by evidence that people who rely on “manifesting” tend to get swindled more and make bad investment decisions. Knowing oneself well, and developing great “executive function”, can help us leverage the complex mathematical underpinnings of reality — and of one’s own ways of moving through the world — to land close to where we hope to land, to get to a place of success and satisfaction by our own definitions, and avoid common pitfalls.
Further readings/References
This Is Your Day — How to start the day off on the right foot using the Butterfly Effect
How to Make the Most of the Present Moment in Decision-Making
8 Keys to Effective Self-Governance
Do We Have Free Will? If we do, so what now?
Acclaimed Book: Making Your Crazy Work For You: From Trauma and Isolation to Self-Acceptance and Love
References
Fractal Analysis of Jackson Pollack’s Painting Evolution
Wigner, E. P. (1960). “The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959”. Communications on Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM…13….1W.
