Nerds assemble, maths/physics/philosophy question

Hello Hive mind,

the kind of thing the keeps me awake at night are things like this (and I’ve tried googling)

I was thinking about the Lorenz Attractor and how each time it goes round it follows a similar but infinitesimally different path from the paths it has been on before and these infinitesimal differences can make huge difference to what happens next. But the Lorenz Attractor is a mathematical model executed on a computer and in maths numbers are infinitely divisible and reality isn’t.

Now this might not be a problem for the Lorenz Attractor, it is after all, only a mathematical model and models are not reality and models always breakdown. But isn’t it a problem that our basic conception of mathematics doesn’t do what reality does?

obviously I’m over thinking things, but is there anyone who can help other than telling me to go and get a phd in something (I just about scraped a B in A – level maths).

thanks

What is the question?

Errrrrr… .

But isn’t it a problem that our basic conception of mathematics doesn’t do what reality does?

I guess its due to the degree of actual accuracy/context needed for a given calculation.

For example we can say water boils at 100c. It’s close enough for 99.9% of people, who are reletivley close to sea level…. however if you are sat on top of Everest, where it boils at a much measurably lower tempeature, you calculations are going to be all messed up… never mind trying to boil water on other planets, etc.

what reality does

Do you mean what reality does or what we can measure?

Reality isn’t infinitely divisible?

Reality isn’t infinitely divisible?

Why not?

42

Why not

Quantum mechanics innit

42

But isn’t it a problem that our basic conception of mathematics doesn’t do what reality does?

I wouldn’t call the Lorenz Attractor our basic conception.  Some bits of maths appear to reflect reality as well as we can measure it, other bits of maths don’t appear to have much to do with the reality we can measure.

But that is to ignore the thorny problem of whether maths is, at a fundamental level, part of reality that we uncover, or whether it is something we make up. That is a mangled explanation, but it is a question that philosophers and theoretical physicists have lively debates about.

in maths numbers are infinitely divisible and reality isn’t

“Impossible to step into the same river twice” – Heraclitus

Now this might not be a problem for the Lorenz Attractor, it is after all, only a mathematical model and models are not reality and models always breakdown. But isn’t it a problem that our basic conception of mathematics doesn’t do what reality does?

The Lorentz Attractor is based on a simplified model of a ‘real’ system. Ergo, it cannot truly reflect such a system, only approximate it.

Maybe the universe runs along mathematical rules, maybe it doesn’t.

the thorny problem of whether maths is, at a fundamental level, part of reality that we uncover, or whether it is something we make up.

Great (3 part)  Hannah Fry docu here.. don’t ask me any more, it melts my brain, lol!

This is why mathematicians and physicists don’t necessarily agree.

The Lorenz attractor is (ish) a set of solutions to some equations – don’t read too much into it. What’s interesting is the equations are a simple description of a system that is chaotic, where chaotic has a very specific meaning: something that looks random but certain aspects are predictable in certain ways.

There are a lot of freakish effects to do with chaos, but the real thing to consider is what do we mean by ‘freakish’? The universe is what it is, it does not care one bit if we find some bits of it weirder than others. That’s our problem. We have a language to describe parts of it, but like all languages it doesn’t exactly match the concepts it’s attempting to describe. With maths though, we can add more bits of language or redefine them if we want.

Some numbers are a bit tricky to represent in computers, what with computers being finite and numbers being…

The universe is under no obligation to make sense to us.

The lorenz attractor doesn’t exist anywhere in reality, but “approximately chaotic” systems seem to exist very commonly.

I say “approximately chaotic” in scare quotes because chaos is a concept that exists in a theoretical unchanging world where time is unending. In reality the chaos of weather systems will end with the death of the sun etc. But to all intents and purposes, weather is chaotic, as are various predator-prey systems, multi-body planetary orbits, etc etc. The mathematical models are only an approximation of reality but can still provide useful insights and even forecasts.

Did they remember to take away the first number they thought of?

More mushrooms are required.

You can translate the theory of the single deterministic number of the Lorentz attactor to a probability, then it works in the real world.

Is your concern that our computational systems can’t determine the exact solution, because of floating point precision? Maths generally finds exact, analytical equations to describe problems. Sometimes these are approximated, but they remain analytical, and can sometimes have exact solutions (e.g. by rearranging an equation). Some equations when you try to find a solution, you have to try numerical solutions, which usually means re-writing the equation, typically in a differential form, which adds another layer of approximation. And yes then the initial conditions or boundary conditions or the computer precision can affect the answer you get if the equation is sensitive to them (e.g. mathematically chaotic). That’s why ensembles are run for things we know are sensitive to this (weather / climate / flood forecasting models).

But the governing, analytical equations remain exact, in the terms/case that the equation was derived for. And I would argue most equations even with approximations work well enough to understand the reality in which we live, within what they were designed / developed for.

We have other ways of viewing problems, than just space-time. In fluid dynamics, the detail of flow through a channel can be extremely complex, but in most situations you can take the mean flow, and variability from that mean, and have a good estimate and predictor of flow, without having to solve the full equations. Most engineering simplifies equations to a set of parameters that can be fit to a situation. Many applications use phase-space rather than time (think of a predator-prey relationship and plot population of foxes and rabbits against each other, rather than in time). Or as quantum physics is mentioned, stochastics. The point is the model/equation is an exact explanation of something, but only within the realms of what is was derived for (scale, accuracy and knowns).

One thing to keep in mind with models is that they are a simplification of something more complex. A good model helps our understanding by identifying the essential elements and ignoring the less relevant things. There’s an old saying, “All models are wrong but some are useful.”

Modelling the earth as flat is useful if you are building a house. It’s not useful if you are navigating around the world.

Newtonian physics are very useful if you are modelling the speed of a racing car around a circuit. They are not adequate if you are dealing with relativistic velocities.

I’d say that in general it’s freaky how good physics is at the simple stuff

By simple i mean the system is simple. So quantum machine can predict what will happen in simple systems with astonishing precision.

But once you have even slightly complicated system then it’s really hard to predict behaviours . Things like the three body problem

Newtonian physics are very useful if you are modelling the speed of a racing car around a circuit. They are not adequate if you are dealing with relativistic velocities.

I agreed and i think that’s worth dwelling on. It’s a personal hang up of mine when people say things like “Einstein proved Newton wrong”. Setting it as conflict. I’d say Einstein extended our models of mechanics. In most real works situations they make the same predictions

More mushrooms are required.

Or fractals!

http://iacopoapps.appspot.com/hopalongwebgl/