Notes on variables

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.

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