A problem I noticed with if-then-ism in the philosophy of mathematics

Question

In the philosophy of mathematics, if-then-ism is the view that mathematical assertions of existence, like the statement that there exist numbers which are their own squares, should, strictly speaking, be prefaced by "If numbers exist, then there exist...". However, I have just noticed a problem with this view. For someone who believes that mathematical objects don't exist, they would believe something like, "If numbers exist, then 2+2=5", since they believe the antecedent is false. So, how does one deal with this problem? Have any philosophers talked about this problem, and suggested possible solutions? Of course, this is not a problem for someone like me who believes mathematical entities like sets and numbers exist. I am merely asking how someone who does not believe in mathematical entities would deal with this conundrum.

Answer 1

Originally, if-then-ism was an approach to the philosophy of mathematics defended by Bert Russell. Russell held that mathematical truths are necessarily true, but that no existential statement is necessarily true, i.e. nothing exists of necessity. Combining these two things is problematic, because mathematical theorems may take the form, "There exists a number such that...". So if this is a mathematical truth it ought to be necessarily true.

Russell was not denying the existence of abstract objects such as numbers, but he didn't consider that the existence of numbers was necessarily true. His preferred approach was to consider that mathematics is concerned only with implication relations. He even defines mathematics this way:

"Pure Mathematics is the class of all propositions of the form 'p implies q', where p and q are propositions containing one or more variables, the same in two propositions, and neither p nor q contains any constants except logical constants." - Bertrand Russell, The Principles of Mathematics, chapter 1, §1.

Russell's if-then-ism is part of his project of logicism under which all of mathematics is supposed to follow from a bunch of axioms. Hence the necessity of any mathematical theorem is conditional in nature. It is not that theorems are necessarily true, but that theorems follow by necessity from the axioms. The axioms are universal, not existential, and Russell did not consider the necessity of universals to be problematic.

Logicism mostly died out, but others picked up the idea of if-then-ism and deployed it within a structuralist approach to the philosophy of mathematics. Under this approach, mathematical theorems are understood to be statements that hold universally when we quantify over relevant formal systems. So, e.g. "there exists a number zero" would be understood as something like: for any formal system that is capable of interpreting arithmetic and that has models of the Peano axioms, there exists a numeral within that system that plays the role of zero.

Some structuralists use this approach to deny the real existence of abstract objects, perhaps treating them as fictions, while others are quite happy to regard them as real. The SEP article describes quite a few different options. But if-then-ism isn't a matter of saying, "if numbers exist..." but "if you have a formal system that is capable of..." Structuralists who prefer the eliminative approach and don't accept the real existence of numbers and other abstract objects are not rejecting Peano's axioms.

There are no models of Peano arithmetic under which 2+2=5, at least under the standard interpretation. So that sentence would not be a problem for a structuralist.

Answer 2

Using a form natural deduction, here is an example how formal mathematics works. Here, we start with two axioms that define 0, n and s. From these axioms, you construct, i.e. infer the existence of 1 and 2 using various rules of inference (universal specification, detachment, existential specification, splitting a conjunction and substitution):