Variables often treated as blanks to be filled. $$f(x) = x+5$$

becomes

Two simple functions emerge.

First are constant functions: $$\begin{aligned} f(x) & = 5\\ g(y) &= \pi \\ h(u) & = cat \end{aligned}$$

Two simple functions emerge.

Second are the identity functions: $$\begin{aligned} f(x) & = x\\ g(y) & = y \\ h(u) & = u\end{aligned}$$

Two simple functions emerge, and then provoke a protest.

Constant functions $K_c(x)=c$ and the identity function $I(x)=x$ should coexist in peace.

Substitutions needs rules, but...

Making rules will impose limits and concern.

• Will banning some substitutions lead to loosing power, like recursion?
• Will the rules be consistent or lead to their own paradoxes?
• If the rules be consistent, will they be easy to follow?
• Will some substitutions never end?

Alonzo Church give us the answers in the form of his $\lambda$-calculus.

It was soon seen as equivalent to Turing's notion of computation.

In their honors today we teach

• The Turing Machine as the universal computer, and
• $\lambda$-calculus as the universal programming language.