Serious mathematicians ask:
What is a function?
2021 James B. Wilson, Colorado State University
\(\lambda\)
\(SKI\)-combinators
\((x,y),(x,z)\in F\Rightarrow y=z\)
I learned this in Set Theory
(Was my teach wrong?)
Popular when learning Set Theory
A function \(f:A\to B\) is a subset \(f\subset A\times B\) where
- Domain \(\forall x\in A,\exists y\in B, (x,y)\in f\)
- Well-defined \(\forall (x,y),(x,z)\in f,(y=z)\)
So \((x,y)\in f\) means \(y\) is unique to \(x\) can we call it the output denoted \[f(x)=y.\]
Can't start with functions as sets
- What is \(A\times B\)?
- Is it \(\{(a,b)\mid a\in A, b\in B\}\)? So what is \((a,b)\)?
- \((a,b)\) is gadget you can get both \(a\) and \(b\) from, i.e. there is a function \((a,b)\mapsto a\)...
- Circular to define functions as subsets of \(A\times B\)
- Axiom of Specification: all \(\{x\mid S(x)\}\) are sets. Set Theory axioms assume functions already exist.
- Cardinality, functors, and more are functions but by Russell's paradox their domains are not sets.
Two views on first functions
Replace variables
\(f(x)=x+2\)
If \(P\) and \(P\Rightarrow Q\) then \(Q\)
In \(\lambda\)-notation
\(\lambda x.2x\)
Compose atomic functions
CMP AX BX
DIV CX DX
JMP RX
println(toString(541));
In combinator notation
\(S(BBS)(KK)\)
Replace variables
Church (1930) \(\lambda\)-calculus
- Laws of substitution
Compose atomic functions
Schoenfinkel (1920)
Curry (1927) Combinators
- Three primitive functions, Identity \(I\), Constant \(K\), and Strong composition \(S\)
Where do we go from here
- Choose a side: substitution or primitive functions.
- Define reductions: a means to evaluate functions to an outcome.
- Prove reductions terminate in the same answer, Its Normal Form, though some will refuse.
- Add types to make it clear which class of functions have normal forms and that the theory is consistent.
References
- Hindley-Seldin Lambda-Calculus and Combinators, Cambridge, 2008
Serious Math Asks: What is a function?
By James Wilson
Serious Math Asks: What is a function?
When we substitutes values into variables we seem to know intuitively that it makes sense. But with simple constant and identity functions we can soon run into paradoxes. This points at the need to define a theory of substitution known today as lambda-calculus.
