Scalar done right

Almost three years also , I posted an entry titled Scalar Done Wrong.  The entry has received a large number of comments, and 16,793 reads.  I have since tried to get scalars right.  I have just attached today’s version of my notes as a PDF file to the original entry.

A collection of things is called a scalar set if the things are additive to one another, scalable by numbers, and proportional to one another.  Each thing in the collection is called a scalar.  We will define the words “additive”, “scalable” and “proportional” with care.

Scalars in two worlds.  Scalars are fundamental building blocks of many models of the world.  In physics, scalars model length, time, mass, charge, energy, and entropy.  In economics, scalars model money, commodity, and labor.  In politics, scalars model population, votes, and polls.  In social networks, scalars model friends, posts, and likes.

Scalars are also fundamental building blocks of a virtual world, called linear algebra.  Being linear is being scalars.  Scalars build vectors.  Scalars and vectors then build everything else in linear algebra: linear maps, linear forms, bilinear forms, quadratic forms, inner products, and tensors.  These concepts are the characters that play in the virtual world of linear algebra.

Give us scalars.  We build two worlds.

Happiness is not a scalar. We have some notion of various states of happiness.  But these states of happiness are not additive, not scalable, and not proportional to one another.  States of happiness do not form a scalar set.

Indeed, many things in the real world are not scalars.  Love is not scalar.  Color is not scalar.  Temperature is not scalar.  Linear algebra can only represent part of the real world.  Such is the limitation of linear algebra.  To learn a subject is to learn its interior, as well as its boundary.

The boundary moves.  One subject morphs to another.  The known morphs to the unknown.  We know how temperature relates to two scalars, energy and entropy.  We can also relate color in some way to scalars.  Much of science is a program to relate things in the world to scalars.  And we do not know how far we can push this program.  It is conceivable that one day we will relate happiness to scalars.