"In outer space, there's no up or down."
"In physics, there's nothing called decceleration."
"In relativity, time is the fourth dimension."
Your parents etc. probably taught these things to you as a child, and you might've wondered at the time why that's true. Why can't we just define a direction called "up" in space? Why can't we just define decceleration as negative acceleration (or rather, acceleration in the opposite direction as motion)? Why do we count time as the fourth dimension -- why can't, I don't know, temperature be the fourth dimension?
If you're a child and your parents aren't telling you things like this, please call child protective services immediately. These factoids are incredibly important for any human being worthy of the name to internalise -- they are a special case of the general principle of symmetry, or more specifically: stuff should be defined in terms of its behaviour.
It's not that you can't define a general up or down in space, it's that you really, really shouldn't. It would serve no purpose, and would break the symmetry of space. There is no reason you should hold a specific property of the Earth as fundamental to your study of some physical phenomena in "space". Any facts that you derive must be abstracted to work in any co-ordinate system.
In the other two examples, it's a bit more subtle, as there really are specific physical phenomena associated with decceleration (e.g. harmonic motion), and time does have some special properties distinguishing it from space. Nonetheless, the mental classification is important.
This notion is fundamental to any academic discipline. Unfortunately, it seems that there is no push towards abstraction in the social sciences -- e.g. in economics, where you see a dozen different words for "externality", and a lot of definitions seem to be on entirely social terms.
Insights on interesting things.
Showing posts with label education. Show all posts
Showing posts with label education. Show all posts
"Calculus-based physics"
I dislike this whole “non-calculus physics”/”calculus physics” distinction created in schools, because it degrades mathematics to some kind of a weird tool used in physics.
Physics is just the study of the mathematical stuff we do observe — every physical system is a mathematical system on a fundamental level, which for pedagogical purposes and stuff, we often approximate with other mathematical systems (e.g. modelling stuff as rigid bodies, not considering the motion of every single particle within an extended body, neglecting gravity in particle physics, etc.). So of course you will find math being “used” in physics, because physics is mathematics!
Physics uses math in the same way that mathematics uses math — like how you “use” differentiability in defining lie groups, or how you “use” calculus and linear algebra in differential geometry, or how you “use” matrices in describing linear transformations, or whatever. Neither the physics, nor the mathematics should be classified or segregated by what mathematical methods, or “math” is used in describing or defining it.
You shouldn’t divide physics as “calculus-based” and “non-calculus” for the same reason you don’t divide it into “partial fractions-based” and “non-partial fractions”, or “elementary algebraic” and “non-elementary algebra”, or at a little higher level, “differential geometry-based” and “non-differential geometry-based”.
Use whatever tools you have to use! The point of physics is to describe what we observe — aka the universe — as efficiently and conveniently as possible, not to do elementary calculus.
There are other, more sensible ways to divide physics — experimental, theoretical and phenomenology — “mathematical physics”, which is basically physics done with as much rigor as you find in the mathematics literature, so you ensure everything you know about physics is consistent and stuff (the physics exists), you know what your underlying assumptions/axioms/postulates (that you must verify empirically) are, etc. — you could define it as “symmetry-based physics” and “non-symmetry based physics”, where the good physics is symmetry-based and the bad physics isn’t, but since Einstein, all physics is symmetry-based, so this is irrelevant today.
Physics is just the study of the mathematical stuff we do observe — every physical system is a mathematical system on a fundamental level, which for pedagogical purposes and stuff, we often approximate with other mathematical systems (e.g. modelling stuff as rigid bodies, not considering the motion of every single particle within an extended body, neglecting gravity in particle physics, etc.). So of course you will find math being “used” in physics, because physics is mathematics!
Physics uses math in the same way that mathematics uses math — like how you “use” differentiability in defining lie groups, or how you “use” calculus and linear algebra in differential geometry, or how you “use” matrices in describing linear transformations, or whatever. Neither the physics, nor the mathematics should be classified or segregated by what mathematical methods, or “math” is used in describing or defining it.
You shouldn’t divide physics as “calculus-based” and “non-calculus” for the same reason you don’t divide it into “partial fractions-based” and “non-partial fractions”, or “elementary algebraic” and “non-elementary algebra”, or at a little higher level, “differential geometry-based” and “non-differential geometry-based”.
Use whatever tools you have to use! The point of physics is to describe what we observe — aka the universe — as efficiently and conveniently as possible, not to do elementary calculus.
There are other, more sensible ways to divide physics — experimental, theoretical and phenomenology — “mathematical physics”, which is basically physics done with as much rigor as you find in the mathematics literature, so you ensure everything you know about physics is consistent and stuff (the physics exists), you know what your underlying assumptions/axioms/postulates (that you must verify empirically) are, etc. — you could define it as “symmetry-based physics” and “non-symmetry based physics”, where the good physics is symmetry-based and the bad physics isn’t, but since Einstein, all physics is symmetry-based, so this is irrelevant today.
Why are calculus and linear algebra taught early?
Linear algebra and function theory are related — you can construct plenty of accurate analogies here, like functions and vectors, linear transforms and integral transforms, etc. In addition, the elementary techniques of calculus allow you to talk about non-linear transformations in a pretty nice manner — e.g. the Jacobian matrix as a change-of-basis matrix for non-linear co-ordinate transformations.
In general, calculus is just a special case and a “constructivist” kind of way of understanding the much deeper mathematical field of analysis. The calculus of variations, basic complex analysis, matrix calculus, etc. are other examples of this. It’s taught, despite its non-fundamental nature, not only because it locally linearises things with infinitesimals, allowing us to study non-linear things, e.g. in differential geometry, but also because a lot of its results are special cases of purer results in advanced mathematics. Some elementary examples: the chain rule, a special case of a change-in-basis-variables/the Jacobian matrix; the fundamental theorem of calculus and Stokes’ theorem, special cases of the generalised Stokes’ theorem in differential geometry.
Linear algebra is taught for similar reasons — it introduces you to a lot of things in algebra, much like how calculus introduces you to a lot of things in analysis. Together, they also introduce you to a lot of things in geometry — largely because the “ideas” behind the two allow us to describe a lot of things in a linear way — completing the algebra-analysis-geometry trinity.
In general, calculus is just a special case and a “constructivist” kind of way of understanding the much deeper mathematical field of analysis. The calculus of variations, basic complex analysis, matrix calculus, etc. are other examples of this. It’s taught, despite its non-fundamental nature, not only because it locally linearises things with infinitesimals, allowing us to study non-linear things, e.g. in differential geometry, but also because a lot of its results are special cases of purer results in advanced mathematics. Some elementary examples: the chain rule, a special case of a change-in-basis-variables/the Jacobian matrix; the fundamental theorem of calculus and Stokes’ theorem, special cases of the generalised Stokes’ theorem in differential geometry.
Linear algebra is taught for similar reasons — it introduces you to a lot of things in algebra, much like how calculus introduces you to a lot of things in analysis. Together, they also introduce you to a lot of things in geometry — largely because the “ideas” behind the two allow us to describe a lot of things in a linear way — completing the algebra-analysis-geometry trinity.
Subscribe to:
Posts (Atom)