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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.

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  • It may be true that function would remain mathematical and mountain would never become one, but only because "mathematical" already has a meaning, given to it by... humans. It would make little sense to ask what is or is not mathematical in activities of an alien race with radically different sense organs, intellect and history, or none, as those concepts may not apply to them either. The scope underwent very human historical evolution from very human ways of dealing with objects and patterns, see What makes something mathematics?
    – Conifold
    Jan 3 '20 at 12:12
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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.

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What is considered a mathematical object is ultimately conventional. Numbers, sets, relations, functions, etc. have been used long before any ontologically rigorous definition was available of what a mathematical object is and before any mathematically rigorous definitions of these objects were available.

Investigations by Frege, Russell, Zermelo, Fraenkel and others have, at the beginning of the 20th century, shown that sets may be considered basic mathematical objects in the sense that

  1. most other objects (numbers, relations, functions, etc.) can be defined as special sets, viz. sets with special properties
  2. a lot of mathematics can be reformulated in the axioms of set theory (e.g. the axioms according to Zermelo & Fraenkel)

These axioms can -- together -- be considered as implicitly defining what sets are. This insight is a milestone in the philosophy of mathematics (and ontology, actually). Before a collection of a few axioms was accepted a definition, thick books were written on what sets might be. To no avail.

A drop of bitterness remains, though. It would, from an ontological point of view, have been pretty, if the so set-based mathematics not only worked exclusively well in the natural sciences but if the set of axioms could have been proved consistent. However, as Goedel showed in 1931, this is (I simplify, here) not possible.

Therefore, in my opinion, the best argument for the existence of mathematical objects remains the success of the holistic system of the natural sciences in which these mathematical objects play such a vital part.

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  • I like your explanation. Mathematical objects, those that follow from a few basic defined axioms.
    – framontb
    Jun 2 '20 at 22:54
  • No, your argument is simply wrong. There is absolutely no good argument for the existence of a ZFC universe (i.e. a real-world embedding of a model of ZFC). So far all known mathematical theorems that have concrete real-world interpretation can be expressed and proven in HOA (higher-order arithmetic), which is weaker than BZ (Bounded Zermelo) by far, and BZ is in turn far weaker than bounded ZFC, not to say full ZFC. Furthermore, the bare minimum mathematics that we need to get all known real-world applications is just ACA, far weaker than HOA.
    – user21820
    Aug 1 '20 at 13:14
  • You can, however, fix your argument by being less careless with your conclusion. Namely, you can claim that the natural sciences depend so heavily on the truth of theorems of ACA (suitably interpreted in the real world) that it very well justifies the meaningfulness of ACA. It still not necessarily justifies existence, since for instance there is completely no evidence that there is any physical realization of a halting oracle. Nevertheless, the well-definedness of the halting problem as a concept rather than physical entity is well justified. Conceptual sets beyond ACA? Not so much.
    – user21820
    Aug 1 '20 at 13:19
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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.

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