# Notes on variables

• 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.