# Do mathematical objects exist after their definition? [duplicate]

Suppose we have a system with a set of axioms. Now we begined to define new terms. E.g. in maths we have a particular set of axioms and then we define what a function is. But do all the functions automatically start to exist? For example, did the exponential function came into existence because we defined the term function? Or it was already there and we just give a name of what function is so we can say that exponential function is a member of the set of functions?

Example Consider now the statement "All functions are even". Where does the "all functions" refer? All the functions that exist? If no function exists then the term "all functions" fail to refer. So should we prove existence before we assign a truth value to that statement?

I have read also the post Do numbers exist independently from observers?

But I am asking why in a given system with axioms the the objects we assign properties are already exist so we have not the problem of "fail to refer".

• For platonists functions exist regardless of any definitions, for fictionalists they do not exist before or after derfining, just like Tolkien's Sauron. So "all functions" does fail to refer, but that does not preclude proving theorems about them any more than it precludes talking about Sauron's ring. Either way, defining does not do anything ontologically, only epistemically. Commented Jul 28, 2020 at 12:27
• As the word "axiom" comes up 0 times in the question philosophy.stackexchange.com/questions/451/… and answers to it, I would humbly argue that we are dealing with a different question, here.
– user14511
Commented Jul 28, 2020 at 12:43
• Also duplicate of this previous post by you: How definition relates to abstract/concrete objects? Commented Jul 28, 2020 at 12:53
• See alos the post How do philosophers formally characterise mathematical objects? Commented Jul 28, 2020 at 12:55
• @MauroALLEGRANZA I have both read them. But the main question which I summarized in the last sentece can't be answered from these 2 posts. Commented Jul 28, 2020 at 13:01