How do philosophers formally characterise mathematical objects?

Question

In the Stanford Encyclopedia of Philosophy article, 'Platonism in the Philosophy of Mathematics', the following formalisation is given for the existence of a mathematical object:

Existence can be formalized as ‘∃xMx’, where ‘Mx’ abbreviates the predicate ‘x is a mathematical object’ which is true of all and only the objects studied by pure mathematics, such as numbers, sets, and functions.

I am curious as to how these objects are rigorously defined and distinguished from other objects. Clearly there is some natural intuition that suggests a function is a mathematical object but a mountain is not. However, I'm not sure whether the distinction that a function is studied in pure maths whilst a mountain isn't is satisfactory: it seems that, were all humans to stop studying mathematics (or if we never began in the first place), a function would remain a mathematical object and a mountain would never become one, irrespective of the actions of humans.

Answer 1

The SEP is narrowly characterising how mathematics is understood in Platonism. For example, Plotinus in his Enneads writes:

Mathematics, which as a student by nature he will take very easily, will be prescribed to train him in abstract thought and to faith in the unembodied. A moral being by native disposition he must be led to make his virtue perfect. After mathematics, he must be put through a course of dialectics and be made an adept in the science.

This is more or less going on from Plato. And hence Platonism. To focus on the ontological status of mathematics to the exclusion of all else in Platonism is, for example, when examining a statue, to note only the toes and to not consider the statue in its totality.

Answer 2

Whether mathematical objects are independent objects, i.e. are objective, is a matter that seems to be given from the term platonism on the SEP entry. If you have some interest in mathematical objects as human constructs, I would recommend glancing at intuitionist philosophical approach.

One distinction is quite settled: mathematical objects are abstract entities, i.e. they are not in time or space. Frege proposed that a number n is the class of sets that had n elements in them. In some way this is true for the definitions in set theory.

Each number has as a reference a set. There's an axiomatic system that

1. implies the existence an empty set (our first object);
2. outrules urulements (things that are not sets);
3. Enables the construction of infinitely many sets with the empty set.

The empty set is considered zero, because it has no elements. Other natural numbers are constructed from it.

Example of construction A. AXIOM OF PAIR: If you have sets x and y, then the set {x,y} exists.

Well, suppose x=y= ∅ (the empty set). Then {∅,∅} exists, being identical with {∅} (by the extensionality axiom).

(Now we have another set: {∅} ({∅}=1}, which is different from ∅, and is its the successor of zero)

B. Given any number, we may produce its successor. Given any n+1, n+1 is the set made by the elements of n and n itself.

This process builds layers of sets (in terms of quantity or, more correctly, its cardinality) and covers all natural numbers and real numbers.

Not all sets are made with the successor operations, though. Those which are not are called limit cardinals and they define the cardinality of infinities of numbers. These limit cardinals are also infinite by a process of infinetely applying the axiom of parts to limit cardinals, which has proven to establish the difference in quantities between any set and the set of its parts.

So they're infinitely many infinites. If you don't think so, you will arrive in a contradiction similar to that which we find in trying to guess the greatest natural number. For any n, n + 1 is greater. For any limit cardinal, the set of its parts is greater.

So that's how you define numbers in set theory, for example. Mathematics in its entirety is not equivalent to set theory, but set theory covers a great range of topics.