Apr 19, 2026
Fischer Black was an iconoclast advocating careful thinking about the vocabulary used in finance. The next step after vocabulary is to consider how what is spoken about comes to be.
Do you have $20 in your pocket? Do you own some shares of stock? How did that happen?
The atoms of finance are holdings, an amount of some instrument owned by a person or company. The amount is an integral multiple of the minimal trading increment for the instrument. It is a mathematical fiction to assume it is a real number and a programming error to represent this as a floating point number. It is rare to find a mathematician who can give a rigorous definition of real numbers and even rarer to find a computer programmer who understand the IEEE 754 floating point standard.
We collect some mathematical facts from a naive perspective.
We use the naive defininton of a set as a collection of elements that are members of a set. We write x\in A to indicate x is an element belonging to set A. The set with no elements is the emptyset \emptyset. Two sets are equal if they have the same members. The set A is a subset of B, A\subseteq B, if every element of A also belongs to B and is a strict subset of B, A\subset B, if A\subseteq B and A\not=B.
Given two sets, the intersection is the collection of elements belonging to both sets, A\cap B = \{x\mid x\in A\text{ and }x\in B\}.
The union is the collection of elements belonging to either set, A\cup B = \{x\mid x\in A\text{ or }x\in B\}.
The set difference is A\setminus B = \{x\mid x\in A\text{ and }x\not\in B\}. If A\supseteq B then A\setminus B is the complement of B with respect to A.
Exericise. If A\supseteq B show (A\setminus B)\cap B = \emptyset and (A\setminus B)\cup B = A.
The cartesian product of two sets is the collection of ordered pairs from each set. The set \{a,b\} is equal to the set \{b,a\} since they have the same elements. The ordered pair (a,b) is equal to (c,d) if and only if a = c and b = d. In particular, (a,b)\not=(b,a). Define (a,b) to be \{\{a\},\{a,b\}\}.
Exercise. Show this is an ordered pair.
Hint: If \{\{a\},\{a,b\}\} = \{\{c\},\{c,d\}\} then \{a\} = \{c\} and \{a,b\} = \{c, d\}.
The set exponential of two set is the collection of functions from one set to the other.
Bertrand Russell blew Frege’s defintion out of the water.
complement, intersection, union, disjoint union, cartesian product, set exponentional.
What is a type? A set together with operations. F-algebra.
What is the difference between a security and an instrument?