I’m wondering about the different types of intuitions in physics and mathematics.
What I remember from prepa (two years after high school where we did the full undergraduate program of maths and physics) was that some people had maths intuition (like me) and some had physics intuition (not me). That’s how I recall it, but thinking back on it, there were different types of maths intuitions, which correlated very differently with physics intuition. I had algebra intuition, which means I could often see the way to go about algebraic problems, whereas I didn’t have analysis intuition, which was about variations and measures and dynamics. And analysis intuition correlated strongly with physical intuition.
It’s also interesting that all your examples of physicist using informal mathematical reasoning successfully ended up being formalized through analysis.
This observation makes me wonder if there are different forms of “informal mathematical reasoning” underlying these intuitions, and how relevant each one is to alignment.
An algebra/discrete maths intuition which is about how to combine parts into bigger stuff and reversely how to split stuff into parts, as well as the underlying structure and stuffs like generators. (Note that “the deep theory of addition” discussed recently is probably there)
An analysis/physics intuition which is about movement and how a system reacts to different changes.
Also the distinction becomes fuzzy because there’s a lot of tricks which allow one to use a type of intuition to study the objects of the other type (things like analytic methods and inequalities in discrete maths, let’s say, or algebraic geometry). Although maybe this is just evidence that people tend to have one sort of intuition, and want to find way of applying it at everything.
Interestingly, I have better algebra intuition than analysis intuition, within math, and my physics intuition almost feels more closely related to algebra (especially “how to split stuff into parts”) than analysis to me.
Although there’s another thing which is sort of both algebra and analysis, which is looking at a structure in some limit and figuring out what other structure it looks like. (Lie groups/algebras come to mind.)
I’m wondering about the different types of intuitions in physics and mathematics.
What I remember from prepa (two years after high school where we did the full undergraduate program of maths and physics) was that some people had maths intuition (like me) and some had physics intuition (not me). That’s how I recall it, but thinking back on it, there were different types of maths intuitions, which correlated very differently with physics intuition. I had algebra intuition, which means I could often see the way to go about algebraic problems, whereas I didn’t have analysis intuition, which was about variations and measures and dynamics. And analysis intuition correlated strongly with physical intuition.
It’s also interesting that all your examples of physicist using informal mathematical reasoning successfully ended up being formalized through analysis.
This observation makes me wonder if there are different forms of “informal mathematical reasoning” underlying these intuitions, and how relevant each one is to alignment.
An algebra/discrete maths intuition which is about how to combine parts into bigger stuff and reversely how to split stuff into parts, as well as the underlying structure and stuffs like generators. (Note that “the deep theory of addition” discussed recently is probably there)
An analysis/physics intuition which is about movement and how a system reacts to different changes.
Also the distinction becomes fuzzy because there’s a lot of tricks which allow one to use a type of intuition to study the objects of the other type (things like analytic methods and inequalities in discrete maths, let’s say, or algebraic geometry). Although maybe this is just evidence that people tend to have one sort of intuition, and want to find way of applying it at everything.
Interestingly, I have better algebra intuition than analysis intuition, within math, and my physics intuition almost feels more closely related to algebra (especially “how to split stuff into parts”) than analysis to me.
Although there’s another thing which is sort of both algebra and analysis, which is looking at a structure in some limit and figuring out what other structure it looks like. (Lie groups/algebras come to mind.)
Have you read What Should A Professional Mathematician Know? The relevant bits are in the last two sections.