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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 which 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 begun in the first place), a function would remain a mathematical object and a mountain would never become one, irrespective of the actions of humans.

  • 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 at 12:12
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The standard ontological categorization rests on 2 distinctions :

(1) concrete/ abstract entity

(2) particular / universal

Most textbokks on ontology will tell you that mathematical objects are (1) abstract (2) particulars.

They are absract since they cannot be located in space/time nor have any causal link with anything.

They are particulars ( not universals) since they cannot be instanciated ( unlike "humanity" or " yellowness").

A group of 5 apples is not a mathematical object. It is a concrete entity ( a mereological sum) that can be associated to the set :

{ Apple 1, Apple 2, Apple 3, Apple 4, Apple 5}.

This set , though abstract, is not yet a mathematical object, but it can , in turn, be associated ( put in 1-1 correspondance) with the mathematical object :

{ 0, 1,2,3,4 }

which is a set, and is considered to be identical to " the number 5".

The number 5 is said to be the cardinal number of the set

{ Apple 1, Apple 2, Apple 3, Apple 4, Apple 5}.

But this last set is not an "instance" of the number 5.

  • "This set , though abstract, is not yet a mathematical object" ...if sets in and of themselves are not mathematical objects, then apparently mathematics studies non-mathematical objects all of the time. – H Walters Jan 3 at 13:39
  • Maybe I should have said : this collection is not a set. Not all collections count as sets. In standard set theory, no " concrete" object such as an apple can be an element of a set. You may have a look at : karagila.org/wp-content/uploads/2016/01/ests-wh.pdf – Saint James Jan 3 at 14:12
  • But in order to count the apples, I have to consider them as a set and establish a 1-1 correspondance between the set of apples and the cardinal number of this set. Mathematicians do not consider that kind of " for application purpose" sets. – Saint James Jan 3 at 14:15
  • "In standard set theory, no " concrete" object such as an apple can be an element of a set." ...not sure that's true. In standard set theory a set is a collection of distinct elements; so the only requirement on elements is that they be distinct. I can't derive from this that only non-concrete elements count; and see no reason why e.g. {apple 1, apple 2, {}, {apple 1}, unicorn 1, 3, 4} is not a perfectly valid set. – H Walters Jan 3 at 15:15
  • At least, the ZFC set theory seems to be an appleless universe en.wikipedia.org/wiki/Von_Neumann_universe / plato.stanford.edu/entries/set-theory/#UniVAllSet - There might be other versions of set theory in which a concrete object can be a member of a set. I'd be intrested in amathematician's answer to this question. – Saint James Jan 3 at 17:07

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