"Mathematical realism holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. The term Platonism is used because such a view is seen to parallel Plato's belief in a "heaven of ideas", an unchanging ultimate reality that the everday world can only imperfectly approximate.
This is a perfect example of what I'm complaining about. The quote implies that the term "Platonism" can be used as just another term for "mathematical realism", but then it immediately provides a definition that goes beyond simple mathematical realism. The belief that mathematical entities exist independently of the human mind - that humans discover mathematics rather than invent it - does not automatically entail the belief that there's a "heaven of ideas" containing (say) the Essence of Horseness which everyday horses only imperfectly approximate. These two ideas are logically distinct, and it seems sensible to call them by two different names. I prefer "mathematical realism" and "essentialism", or maybe "Platonic essentialism". I'd prefer not to use the term "Platonism" all by itself, but if I had to use it, I'd use it to refer to "Platonic essentialism", not "mathematical realism".