Maths doesn't matter, says scientist

Way back during my PhD, a friend with some strong transhumanist leanings asked me why I'd want to study the real universe if a more interesting simulated one were available. Heck, if I could live inside a simulation, surely I would immediately opt out of reality and become immortal ?

Well ! I was stumped for an answer. I knew I didn't want to study some silly simulation, but I couldn't for the life of me say why - it just felt wrong. Hours later I realised what bothered me was simple : a simulation isn't real. Leaving aside the immortality aspect, If I want to study how the universe works, I need to study the real universe. Oh, simulations might give me some clues, but ultimately it's observations that matter.

This article is a very nice companion piece to that one a few days ago about how physical theories sometimes can't be tweaked, but must be completely revamped. For most situations, the Einsteinian notion of spacetime makes only minor differences in its numerical predictions, but its conception is radically different.

He [Kepler] noticed that there were six planets total, if you included Earth but not Earth's Moon. He also noticed that mathematically, there were only five Platonic solids: five mathematical objects whose faces are all equal-sided polygons. By drawing a sphere inside and outside each one, he could "nest" them in a way that fit the planetary orbits extremely well: better than anything Copernicus had done. It was a brilliant, beautiful mathematical model, and arguably the first attempt at constructing what we might call "an elegant Universe" today. 

But observationally, it failed. It failed to even be as good as the ancient Ptolemaic model with its epicycles, equants and deferents. It was a brilliant idea, and the first attempt to argue — from pure mathematics alone — how the Universe ought to be. But it just didn't work.

Mathematics is a language : whether a sentence is grammatically correct or not doesn't tell you anything about whether it's true. "The dishwasher went cave diving with a well-endowed elephant who was also an exploding pink fairy" makes perfect sense, but nothing like that has ever happened in all of history, and it never will. Mathematics provides a toolkit. It's well worth rummaging around in and banging some stuff together, but just because something seems plausible doesn't mean it has any bearing on reality.

What came next was a stroke of genius that would define Kepler's legacy. He took his beautiful, elegant, compelling model that disagreed with observations, and threw it away. Instead, he went and dove into the data to find what types of orbits would match how the planets actually moved, and came away with a set of scientific (not mathematical) conclusions. The key advance that happened is that science needed to be based in observables and measurables, and that any theory needed to confront itself with those notions. Without it, progress would be impossible.

This idea came up again and again throughout history, as new mathematical inventions and discoveries empowered us with new tools to attempt to describe physical systems. But each time, it wasn't simply that new mathematics told us how the Universe worked. Instead, new observations told us that something beyond our currently understood physics was required, and pure mathematics alone was insufficient to get us there.

Yes, but surely of more importance is that there must be some conceptual framework behind the resulting theory. You can and should use equations to get a handle on understanding what might be happening. But a pure equation, backed up by endless observations but without any physical mechanism proposed to explain it, is disappointingly impotent. It doesn't convey any real understanding or meaning. It's just a bunch of accurate numbers dressed up in fancy clothing.

In some ways, it's a lesson that every physics student learns the first time they calculate the trajectory of an object thrown into the air. How far does it go? Where does it land? How long does it spend in the air? When you solve the mathematical equations — Newton's equations of motion — that govern these objects, you don't get "the answer." You get two answers; that's what the mathematics gives you. 

But in reality, there's only one object. It only follows one trajectory, landing in one location at one specific time. Which answer corresponds to reality? Mathematics won't tell you. For that, you need to understand the particulars of the physics problem in question, as only that will tell you which answer has a physical meaning behind it. Mathematics will get you very far in this world, but it won't get you everything. Without a confrontation with reality, you cannot hope to understand the physical Universe.

There's some hints here as to what we really mean by "physics". Clearly it's not just applied maths, or a matter of cobbling together a bunch of equations, chucking them at the observational data and seeing what sticks. Kepler created a precise mathematical description of planetary orbits, but it was Newton (and later Einstein) who provided an explanation. That's the interesting and worthy bit, in my opinion. How things happen is the heart of it.

But does this actually help ? Perhaps not. As in the previous article, such mechanisms may ultimately only ever be descriptions, not a true account of the way things are. This, then, provides an answer to the question of what the Ultimate Question actually is. And it's very simple : what the hell is going on ?

No, The Universe Is Not Purely Mathematical In Nature

At the frontiers of theoretical physics, many of the most popular ideas have one thing in common: they begin from a mathematical framework that seeks to explain more things than our currently prevailing theories do. Our current frameworks for General Relativity and Quantum Field Theory are great for what they do, but they don't do everything.