What is the nature of mathematics? Becoming a mathematician in the 1960s, I swallowed hook, line, and sinker the Platonistic philosophy dominant at the time, that the objects of mathematics (the numbers, the geometric figures, the topological spaces, and so forth) had a form of existence in some abstract ("Platonic") realm. Their existence was independent of our existence as living, cognitive creatures, and searching for new mathematical knowledge was a process of explorative discovery not unlike geographic exploration or sending out probes to distant planets.
I now see mathematics as something entirely different, as the creation of the (collective) human mind. As such, mathematics says as much about we ourselves as it does about the external universe we inhabit. Mathematical facts are not eternal truths about the external universe, which held before we entered the picture and will endure long after we are gone. Rather, they are based on, and reflect, our interactions with that external environment.
This is not to say that mathematics is something we have freedom to invent. It's not like literature or music, where there are constraints on the form but writers and musicians exercise great creative freedom within those constraints. From the perspective of the individual human mathematician, mathematics is indeed a process of discovery. But what is being discovered is a product of the human (species)-environment interaction.
This view raises the fascinating possibility that other cognitive creatures in another part of the universe might have different mathematics. Of course, as a human, I cannot begin to imagine what that might mean. It would classify as "mathematics" only insofar as it amounted to that species analyzing the abstract structures that arose from their interactions with their environment.
This shift in philosophy has influenced the way I teach, in that I now stress social aspects of mathematics. But when I'm giving a specific lecture on, say, calculus or topology, my approach is entirely platonistic. We do our mathematics using a physical brain that evolved over hundreds of thousands of years by a process of natural selection to handle the physical and more recently the social environments in which our ancestors found themselves. As a result, the only way for the brain to actually do mathematics is to approach it "platonistically," treating mathematical abstractions as physical objects that exist.
A platonistic standpoint is essential to doing mathematics, just as Cartesian dualism is virtually impossible to dispense with in doing science or just plain communicating with one another ("one another"?). But ultimately, our mathematics is just that: our mathematics, not the universe's.