system
Given the following post, respond as if you're commenting on Reddit:
human
What are numbers?

We use them everyday, but what are they really?
gpt
This is probably still the most important question in the field and it continues to draw the most attention. 

As with any area of philosophy there are many different views on this question, and how you feel about it often depends on your general epistemology. I'll briefly go over a few of the most popular views held by philosophers today, but as well I'll touch on some of the older views.

 "Full blooded Platonism" is classically described as the belief that mathematical objects: exist, are abstract objects, and are independent of human thought. This view has been with us for a long time, primarily because it has more than a few appealing qualities. It is intuitive, it describes how classical mathematicians work, it provides an answer to the "impredicative" definition problem, it explains why certain axioms that seem very distant from the physical world "force themselves" as true upon us. Despite its explanatory ability it also has a big problem, the classic problem of "access", if the objects are abstract, how is it that humans living in a spatio-temporal world understand them? This view is not particularly common these days, but Kurt Godel comes very close to it. Even though he was a logician, his papers on Cantor's continuum problem and others were very influential in philosophical terms.

 Scientific realists following Quine are often described as "reluctant platonists", they believe that because of the reliance of science on the language of mathematics that there must be some sense that mathematical objects (sets in particular) are real. In particular, Quine viewed mathematics as a part of a larger, conceptual scheme  and treated sets like electrons, base objects that are required to do science. The fact that this conceptual scheme works (as science does), is considered pragmatic grounds for accepting the existence of all these objects. It is the confirmation of the entire scheme itself as it related to explaining empirical sensations that Quine is concerned with, not the individual parts. The problem with this philosophy, is that there is a lot of hand-waving about what sets are. Putnam and Benacerraf argue in "Quine and Godel", that in essence Quine's view looks very similar to traditional Platonism in many respects and suffers from similar problems.

Nominalism is many different views, but the key idea is to deny the existence of abstract objects and to try and base mathematics upon some concrete foundations. Some people view mathematical objects as "fictional objects" and we simply assume they exist for the purpose of playing some sort of game. This has many of the same deficiencies as formalism does with regards to how mathematics as an activity seems to operate. Others view mathematical objects as empirical objects, traditionally objects based on human psychology, but some hold that the objects are structural properties of the universe. The difficulties with the first, "psychologism" are based on the idea that human psychology is not "objective" and it creates some sort of mathematical relativism (see Frege and Husserl for their still relevant criticisms). The idea that mathematics is based upon some structure of the universe suffers primarily from the fact that it is very hard to imagine how the infinite, set theory and other extremely abstract fields relate to the physical universe. There is also the question of how exactly we are learning about these objects, in some sense, this question is almost as hard as the one posed by abstract objects.

Formalism is generally identified with the work of Hilbert, and the basic idea was to reduce mathematics to a formal, proven using finite methods, human created, language. In a very important sense it died with Godel's incompleteness theorem. The ultimate goal was to provide a system that could A) be shown to be logically consistent, B) contained all the theorems we associate with mathematics. This goal failed, but different types of formalism have emerged although none of which is anywhere near as popular as Hilbert's form once was. Generally speaking, the biggest problem to new systems is that the axioms do not seem to be content-free, some axioms seem more credible than others, some formal systems seem more likely than others to describe mathematics. We don't even choose axioms on mathematical fertility, the axiom of constructability would resolve many problems but it is not considered to be "true" by mathematicians.