I have three more examples I forgot to mention earlier.
-
Number are often interpreted as operators. If we interpret the number 5 as an operator that stretches a line segment, then the number i is an operator that rotates a line segment in the plane. That is in fact the best way to view it. So it’s common fro numbers to be identified with functions.
-
We have function spaces such as the set of all continuous functions that can be added and multiplied pointwise. With these operations of plus and times, the continuous real-valued functions of a real variable become a commutative ring.
Likewise we have famous function spaces like Banach and Hilbert spaces, in which functions are points, we have inner products, and we can do linear algebra on them.
- There is no distinction between numbers and functions in set theory. I don’t know enough about type theory to know how this is handled.
I didn’t represent an uncountable object with a countable one. There are as many continuous functions (reals to reals) as there are reals. Easy proof. That was my only point.
I’m confused about where you’re coming from. What point are you making? Morally, functions and numbers are different. But in practice we use functions as numbers and numbers as functions all the time.