Notes on variables

things to talk about eventually:

• $p\leftrightarrow q$ vs $p\equiv q$.

• Propositional functions: view them as $f:\{T,F{\}}^n\to \{T,F\}$? The tautologies are all $f$ such that for any $x\in \{T,F{\}}^n$, we have $f\left(x\right)=T$?

  Now, for some $x\in \{T,F{\}}^n$, what do we make of the statement $f\left(x\right)$? Is it true or false?

• bound variables in expressions like ${\int }_a^{b}f\left(x\right)dx$ and ${\sum }_{i=0}^{n}f\left(i\right)$ vs in ones like $\exists xP\left(x\right)$

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

  Tao’s example: take $f$ defined by $f(x,y)=(x^2,y^2)$. Then $\frac{\partial f}{\partial x}=(2x,0)$, whereas $\frac{d}{dx}f(x,x)=(2x,2x)$.

  May 2022: I discussed this example more recently here.

• transfer pdf notes

• what do we make of statements like “when $x=3$…”?

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

• “as $x$ gets larger…”

• “$a_n\to \infty$ as $n\to 5$”

• $ax+b=0$

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 $O\left(n\right)$ (which is imprecise, because it doesn’t specify what the input variable is; is $n$ the parameter or a constant?) instead of “$O\left(f\right)$, where $f\left(n\right)=n$” or “$O(\lambda n.n)$”.

Similarly, when dealing with Laplace transforms, it seems common to write both $L\left\{f\right(t\left)\right\}=F\left(s\right)$ and $L\left\{f\right\}=F\left(s\right)$; but $s$ is not present on the left hand side of either denotation! To be pedantic, we would need to write $L\left\{f\right\}=F$ or $L\left\{f\right\}\left(s\right)=F\left(s\right)$ or $L\{\lambda t.f(t\left)\right\}\left(s\right)=F\left(s\right)$.

In differential equations, it also seems common to write something like $$y^{\prime \prime }+p\left(t\right)y^{\prime }+q\left(t\right)y=f\left(t\right)$$ but $y$ is a function depending on $t$, so shouldn’t it instead be the following? $$y^{\prime \prime }\left(t\right)+p\left(t\right)y^{\prime }\left(t\right)+q\left(t\right)y\left(t\right)=f\left(t\right)$$ Or more simply $$y^{\prime \prime }+p{y}^{\prime }+qy=f$$

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