Is there a mainstream school of thought in mathematics? Some years ago(before I started studying mathematics) I was under the impression that most mathematician "believe" in Mathematical Platonism, however, the more I study the more I feel this is not really the case.
From reading and hearing different points of view about the philosophy of mathematics I see there are certain overlaps between the different schools of thought, e.g., Formalism, Deductivism and Logicism seem to be very close to one another.
Is there currently a mainstream school that most mathematicians follow?
 A: Consider this too short to be a full answer, but too long to be a comment.
As far as I am aware, at least from my own experience, the mindset of a mathematician is always in constant flow. I would be hard-pressed to believe than any single mathematician stuck to a single philosophy throughout their career, and because of this, I see a wide variety in perspectives at any given time.
As Karl points out, "most mathematicians are just doing math". I think this is an excellent point, and mutually agreeable. I'd add to this by saying that the average pure mathematician is more-so enjoying crafting theorems and discovering esoteric objects than thinking about philosophical implications perhaps in the same way an athlete simply enjoys playing the sport, rather than thinking about the politics of the sport.
The not-so-mainstream
That being said, although it's difficult to answer what it means for something to be a mainstream school of thought in mathematics, I think most mathematicians do (perhaps unconsciously) hold very sincere opinions on some aspects. Hence, it may be easier to define what isn't mainstream. I.e; areas that generally have no major consensus. Here are a few examples of schools of thought that I've come across that seem to be in constant turmoil.
1. How "real" are the reals and beyond?
I've bared witness to huge debates on how appropriate the real numbers (and other fields) are, when it comes to both intuition, and their relation to the real world.

*

*Do transcendental numbers really "exist" in the real world the same way rational numbers do?

*Why did the Ancient Greeks put so much emphasis on the constructible numbers alone?

*How do we justify the supertask that is doing arithmetic with one or more irrational numbers?

*Are complex numbers and their extensions such as the quaternions, octonions, etc, just mathematical "hacks", or do they serve a very real basis in the universe's doings.

2. The nature of infinity
This is an area who's controversy is hopefully self-explanatory.

*

*Should we think of infinity as actual infinity or as an unreachable, potential infinity?

*Is there any intuition or credibility towards an infinite series having a value different than it's (potentially divergent) limit, such as in the cases of Ramanujan summations or p-adic systems?

3. What is the best foundation of mathematics?

*

*Is it Peano arithmetic, set theory, or something more general, such as category theory?

*Which axioms are the most intuitive?

*How do we justify the self-recursive nature of metalanguage?

The mainstream
In the most general sense, there are a few schools of thought that seem to be very agreeable, when it comes to how mathematics is done. Here are some examples.

*

*The law of the excluded middle. Perhaps so unanimous, that we take it for granted, having it being at the heart of nearly every proof. Or more generally, the use of classical argument forms like modus-ponens, arising from the widespread use of binary logic.

*Generalization. We seem to love to make every proof a "special case" of even more powerful theorems.

*Occam's razor. The formulae and notation of mathematics wouldn't be quite the same if we didn't attempt to simply everything we can. Would $e^{i\pi}=-1$ really be as beautiful if we had written it out as
$$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n\sqrt{-1}\left(4\sum_{k=0}^n\frac{(-1)^k}{2k+1}\right)}=-1?$$
But these are just my thoughts, at least. However I do hope they provide some insight. Also, I recommend OP add the soft-question tag to warrant against closure of this question.
