# Notes on variables

things to talk about eventually:

vs .

Propositional functions: view them as ? The tautologies are all such that for any , we have ?

Now, for some , what do we make of the statement ? Is it true or false?

bound variables in expressions like and vs in ones like

why different results for partial and total derivative? cf. Tao’s analysis book

Tao’s example: take defined by . Then , whereas .

transfer pdf notes

what do we make of statements like “when …”?

How do we make sense of things like the partial derivative, where one variable is “moving” while the others stay “constant”?

“as gets larger…”

“ as ”

Concepts to discuss:

- parameter, arbitrary fixed constant, , flowing letters, meta-variables
- distinction between bound universals and free arbitrary variables
- undefined/unspecified/unknown/undetermined
- known constants vs unknown constants
- undefined constants vs variables
- declaration of variable type
- variable assignment
- here is an example of when variables can lead to confusion.
- temporary (new) constants

Mathematicians and computer scientists are usually not careful with a function versus the output of a function. So for instance when using the big-Oh notation, people will write (which is imprecise, because it doesn’t specify what the input variable is; is the parameter or a constant?) instead of “, where ” or “”.

Similarly, when dealing with Laplace transforms, it seems common to write both and ; but is not present on the left hand side of either denotation! To be pedantic, we would need to write or or .

In differential equations, it also seems common to write something like but is a function depending on , so shouldn’t it instead be the following? Or more simply

## Questions

- What is the difference between a parameter and a variable?
- What is the difference between a variable and a meta-variable?
- In expressions like , is there an essential difference between and the other letters, even when we say that is a variable while and are constants?
- What does it mean to define some variable as a function of another variable ?
- e.g. what does it mean to say something like “as gets larger, also gets larger”?
- Two possible interpretations: (1) Treat as a condition, and look at different possibilities like “if , then ”, and so on. (2) treat as a machine that outputs different things for different inputs. So as is “adjusted”, is affected too.

- e.g. what does it mean to say something like “as gets larger, also gets larger”?