Notes on variables

Contents

• Questions

things to talk about eventually:

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

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

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

• bound variables in expressions like $\int_a^b f(x)\, dx$ and $\sum_{i=0}^n f(i)$ vs in ones like $\exists x P(x)$

• 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 $\displaystyle \frac{\partial f}{\partial x} = (2x,0)$, whereas $\displaystyle \frac{d}{dx}f(x,x) = (2x,2x)$.

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

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

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

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 $y = f(x)$ as a condition, and look at different possibilities like “if $x = 1$, then $y = f(1)$”, 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.

Tags: logic, math.

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