Undergrad What Are Some Examples of Symbolic Manipulation Not Included in Calculus?

[Jonathan212]

Not satisfied with the following definition of calculus. What is a better definition? More detailed?

1a : a method of computation or calculation in a special notation (as of logic or symbolic logic)
b : the mathematical methods comprising differential and integral calculus —often used with the

https://www.merriam-webster.com/dictionary/calculus

 

[PeroK]

Within mathematics "calculus" is not a well-defined term. It has no definition.

"Calculus", like "geometry" and "algebra" are meta-mathematical terms that describe the subject of mathematics and, therefore, have a certain imprecision and fluidity.

 

[fresh_42]

This definition here should cover most possible meanings:
https://ncatlab.org/nlab/show/calculus

 

[mathwonk]

I'll give this a shot. in a general sense it is true that a "calculus" is just a method of computing or calculating anything. (I believe the root word means pebble or stone, so probably goes back to counting things with a pile of pebbles.) If you mean it in the context of usual college math courses, at least in the US, where it refers specifically to the differential and integral calculus, then differential calculus is a method of approximating functions by linear functions, as well as drawing conclusions about a function based on a knowledge of its best linear approximation. The integral calculus may be described as a method for computing average values of functions, with respect to some concept of size or measure, such as length or area. It also allows the computation of length and area, as well as volume,...

 

[fresh_42]

We use analysis (real, complex, vector) for what in English is calculus, and use calculus only for a certain frame, like the predicate calculus in logic.

 

[HallsofIvy]

I don't think anyone can give an answer that will satisfy you without knowing why you are "not satisfied with the following definition of calculus". What do you think is wrong with it?

 

[Jonathan212]

That it is too vague. The college math definition is a lot better and then we add predicate logic to the same bucket as integrals and differentials and it seems less elegant. Not even sure what predicate logic is and what it is doing in the same branch as integrals.

 

[fresh_42]

Not even sure what predicate logic is and what it is doing in the same branch as integrals.

To narrow calculus to differential and integral calculations is the error here. Analysis is a lot more than just this, but it includes these. On the other hand, calculus means simply a framework for calculations. It doesn't say which kind of calculations. So the implicit meaning differential and integral takes away the many other frames, which makes no sense.

Analysis in everything about functions on $\mathbb{R}^n$ and $\mathbb{C}^n$.

Calculus is a frame for calculations, and as such has nothing to do with analysis unless you identify the two, leaving no room for the many other meanings.

 

[Jonathan212]

Some examples of symbolic manipulation NOT included in calculus?