# What is the ontological stance of formalists on mathematical objects?

Are modern proponents of formalism associated with an ontoglogical opinion regarding numbers?

If they view mathematics as the process of manipulating string according to agreed upon rules, there is no initial need for semantics. In a way you don't need the notion of the number three to do some work, which involved the expression "xxx". And you can make comparison rules like

'consider "xxx,xxxx", which holds "xxx" and "xxxx" and then remove one symbol "x" on the left and the right hand side iteratively and choose left or right according to which has no symbols left first'.

Some points why I have a problem seeing an answer: On the wikipedia page, for example, it says

"According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they aren't "about" anything at all. They are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics)."

I'm not sure if this statement should be interpreted in such a way that the mere possibility of, for example, considering the number of symbols in a string and the possibility of abstraction of comparing two string lengths, should imply that there is some existence and truth to 3 and 4 and then 3<4 respecively.

The observation that, in the last line, here in this question I can only tell you what I'm talking about by introducing new symbols and strings "3", "4" and "3<4" obscures or amplifies the problem: If one interacts with other people, when can one really talk about the abstract concept, if I need to denote them by strings, or words, sounds or other signals anyway. The longer one talks about it, the more the any possitive onthological assumptions seems unnecessary. And I could kind of translate the problem to my own thought as well. After all, am I comunicating here, or do I write text on a computer, which only, maybe, gets read by other people later. That language always has to get expressed (or already is in a state of an expression) is a case for formalism being enough to talk about.

But if it's the case that formalists don't feel the need to talk about existence of other things than their strings (i.e. if there is "nothing real" to any number etc.), is for example ultrafinitism considered to be a radical position. They are talked about as having a very strong restricting point of view, but if there are other mathematicans/logicans/philosophers around, which discard the concept altogether, that is people whos position is totally vertical to the believers position, then ultrafinitism is not particlularly radical, one one step down in the hierarchy ideas that deny certain things.