This page does not represent the most current semester of this course; it is present merely as an archive.
In mathematics, we discuss and name values. In (imperative) programming, we discuss and name entities.
7 is a value. James E. Ryan is an entity.
Let x and y both refer to the number 7. Let Jim Ryan
and UVA’s President
both refer to James E. Ryan.
If I make x a little bigger, x is no longer 7. It might be 7.1 or 8 or the like, but it can’t be bigger than it was and still be 7. This is because 7 is a value: an unchanging Platonic ideal. It is impossible, by definition, to make 7 itself bigger.
If I make Jim Ryan
bigger, perhaps by feeding him huge amounts of fatty foods and preventing him from exercising, he’s still James E. Ryan. This is because James E. Ryan is an entity: something that retains its identity even if details about it change. He can get bigger, smaller, richer, poorer, older, wiser, ill, well: none of this changes his identity.
Because x and y refer to a value, changing x means changing which value x refers to and thus has no impact on y. Because Jim Ryan
and UVA’s President
refer to an entity, changing that entity changes the entity both refer to: if I give UVA’s President some money, Jim Ryan’s gets richer.
This is a mathematics course. We discuss mathematical topics, meaning we discuss values. In that context, you should know the following:
Whether distinct writings describe the same value or not is determined by the properties of the value type. For example, 3+4
and 7
represent the same integer. However, 3+4
and 7
represent different arithmetic expressions.
In programing, x might be 2 one time you look at it and 3 the next. That can’t happen in math; if x is 2, then x is 2.
In programming, the name x can appear in several scopes to refer to different variables. This can also happen in mathematics; in each problem x might refer to a different value. However, within a single problem x cannot change.
We will learn about quantifiers in this class, which will add another kind of mathematical scope beyond just the in this problem
-kind.
We’ll learn about sets in this class. Sets are values in discrete mathematics, just like numbers are: unchangeable Platonic ideals. Most programming languages have a datatype they call a set which is not a value but an entity: changeable objects with durable identity.
The fact that this class refers to mathematical sets while other classes refer to mutable data structure sets can be confusing, so we’ll try to point out in math, it’s not mutable
from time to time. Know that these assertions are trying to contrast math from programming, not one math concept from another. Everything in this class (sets, sequences, strings, relations, functions, sums, permutations, logarithms, …) is a mathematical construct and thus a value, not an entity.
Even once we get to mathematical models of changing systems (state machines in this course, others in subsequent courses) we’ll still use unchanging values to describe them, talking about an immutable sequence of immutable states as a mathematical model of a moving, changing machine.