The general consensus among mathematical philosophers (or philosophical mathematicians, for that matter) is that mathematics is a priori – independent of experience. However, a flourishing contemporary branch within the philosophy of mathematics is that of structuralism, which I will explain in detail in a little bit. Advocates of the structuralist approach are e.g. Shapiro  and Resnik , who argue how this approach answers a lot of important philosophical questions, including Resnik’s way of answering how mathematics can have applications in the physical world. There is a catch however: he assumes that mathematics is a posteriori – dependent of experience – contrary to popular belief.
Let’s start off by first of all explaining what structuralism is. Structuralists believe that mathematics is the study of patterns, meaning that things like numbers don’t exist independently, but only exist as being positions in a certain pattern; e.g. the number 2 is a position in the arithmetical pattern, consisting of a distinguished element called 0 and the successor relation S. One could argue that patterns would then be subject to the same criticism, that patterns exist independently – which would be both right and wrong, or rather, neither. Let me first of all mention that there are different kinds of structuralists and thus different forms of structuralism; but if we for instance take Resnik, who believes that patterns exist, but doesn’t believe there to be any fact of the matter as to them being objects or entities of any kind.
To justify this seemingly odd claim we turn to an analogous situation, which at the same time is one of the main reasons to prefer the structuralist view over other philosophical views of mathematics. From set theory, we have learned to represent the natural numbers as sets; e.g. the ordinals , and more generally , originated from von Neumann. We can check that this way of interpretation of numbers behave as they should – that is, satisfy the same Peano axioms as regular natural numbers and provide the same theorems. However, if we instead interpret and again, but more generally , originated from Zermelo, then this interpretation would satisfy the exact same Peano axioms and arithmetical theorems.
The question then becomes whether the natural numbers are equal to the von Neumann sets or the Zermelo sets. One could be tempted to say “them both!”, but then a problem occurs, since these “natural numbers” are still sets, so we can ask questions as , to which von Neumann would nod in agreement, but Zermelo would claim it false. Thus the two “kinds” of set-numbers aren’t equal, meaning that they both couldn’t be equal to the “real” natural numbers (since if they were, by transitivity the two set-numbers would be equal, yielding a contradiction). So it seems that the answer can either be arbitrarily chosen, based on properties which has nothing to do with arithmetic at all, or one could deem it an invalid question. Since the first option is not really viable, due to the fact that one of the purposes of philosophy is exactly to answer such questions in a meaningful way. We’re thus seemingly forced to say that both the von Neumann numbers and the Zermelo numbers aren’t “real” natural numbers. More generally, given any set and natural number , the question doesn’t have a truth value; there is no fact of the matter as to whether sets are numbers.
This then leads us back to Resnik’s view that there is no fact of the matter as to patterns being entities of any kind, to which the previous discussion should have brought some meaning into. To round off the last paragraph though, structuralism answers the equality problem by introducing patterns, or structures, and claiming that the relations among the positions in the various patterns only hold in between the positions of the same pattern. Since natural numbers lie within the arithmetical pattern and sets being in the set pattern, then the equality relation simply isn’t viable between elements from both patterns.
There are problems attached to the structuralist view though; one of them being how we even become knowledgeable about patterns. This is where both Resnik and Shapiro answer that we gain knowledge of patterns through pattern recognition in the physical world. We see several red entities and can thus get an understanding of the pattern of red items, and in the same way we can see e.g. the pattern
and gain a knowledge about the “1-to-5” pattern. Furthermore, and this is the point where skeptics of structuralism can protest, patterns can be generated through abstraction. That is, after gaining knowledge about the “1” pattern, the “1-to-2” pattern, the “1-to-3” pattern, the patterns themselves form a pattern, from which structuralists argue that this abstraction leads one to gain knowledge of the entire natural number sequence, despite its infinitude.
But another noticeable thing is that in arguing for Resnik’s view, he assume that mathematics is based on these concrete physical patterns before any abstract patterns can be made, thus making mathematics a posteriori, dependent on experience, going against the views of philosophers such as Kant, Plato, Aristoteles and Descartes, to mention a few. Reasons why to deny that mathematics is a posteriori is because it implies that mathematical knowledge is not universal knowledge, it is not universal truths, which most people would disagree on – but it seems like this is exactly what the structuralists of the “Resnikian kind” are arguing is the actual case.