31–40 of 68 posts
Earlier quoted context omitted.
Speaking about math: I'm not even sure what the meaning of polynomial is. E.g. x^2+3x+1. Each component is measured in different units. Is it intuitive?
Such a polynomial only makes sense when x is a dimensionless quantity, or alternatively when the coefficients have appropriate dimensions to compensate.
Earlier quoted context omitted.
Such a polynomial only makes sense when x is a dimensionless quantity, or alternatively when the coefficients have appropriate dimensions to compensate.
The definition of a polynomial has nothing to do with units of measurement. The coefficients come from a ring and a ring has nothing g to do with units of measurement.
Why do we extrapolate 4d based on 3D volume, and not surface area? For 2D->3D we use area. Why would we then use volume in 3D->4D when surface area is an option?
In dimensions 4 through infinity we quickly come to a problem. Do we come up with unique names to refer to amount of space enclosed by a unit sphere? Mathematicians have decided to just use the word volume. We rely on context to make it clear what the dimension is. Similarly we use the word n-sphere to refer to the collection of points in n-dimensional space that are exactly 1 unit from the origin.
Surface area of an n-sphere is used and studied.
Earlier quoted context omitted.
The definition of a polynomial has nothing to do with units of measurement. The coefficients come from a ring and a ring has nothing g to do with units of measurement.
That’s fine as a pure mathematics perspective, but in physics dimensions come into play often, and one often has polynomials whose domain and coefficients come “tagged” with particular dimensions.
The intuition for operating with polynomials is best obtained by not worrying about units.
from the article: The volume of the unit d-sphere goes to 0 as d grows! A high dimensional unit sphere encloses almost no volume! Maybe I'm nitpicking, but this interpretation is not accurate IMO. Volume of N-dimensional sphere is measured is different units than that of (N-1)-dimensional sphere. E.g. one is m^5, and another is m^4. Comparing values measured in different units is not the best idea. It would be better…
Earlier quoted context omitted.
That’s fine as a pure mathematics perspective, but in physics dimensions come into play often, and one often has polynomials whose domain and coefficients come “tagged” with particular dimensions.
OP did say, “speaking about math...”. The operations on polynomials and dealing with polynomials doesn’t have anything to do with units of measurement. It may be the case that when used in some areas and in some contexts that the units matter but I think it clouds issues to bring them up. The intuition for operating with polynomials is best obtained by not worrying about units.
Earlier quoted context omitted.
That's what I thought reading that too. In general I don't think it's a good idea for maths people to just forget about units ("because it is for physics")
Speaking about math: I'm not even sure what the meaning of polynomial is. E.g. x^2+3x+1. Each component is measured in different units. Is it intuitive?
Earlier quoted context omitted.
Such a polynomial only makes sense when x is a dimensionless quantity, or alternatively when the coefficients have appropriate dimensions to compensate.
I admit that I have no intuition of a product of 2 dimensionless quantities :)
Earlier quoted context omitted.
OP did say, “speaking about math...”. The operations on polynomials and dealing with polynomials doesn’t have anything to do with units of measurement. It may be the case that when used in some areas and in some contexts that the units matter but I think it clouds issues to bring them up. The intuition for operating with polynomials is best obtained by not worrying about units.
I think we’re not really disagreeing about anything. In a pure math context, x would be, as you say, a quantity in which dimensions play no role.