Wolfram would like to say that he invented Mathematica all by himself, but in its core, it is basically a lisp: everything is an expression, and mathematics are just transformations of the expressions. Afaik this makes it the best tool (conceptually and practically) for generic symbolic manipulations. For example, `1 + 1` in Mathematica is just syntactic sugar for `Plus[1, 1]`, and `a = 1` is `Set[a, 1]`.
I am a PhD student in theoretical physics and almost everyone in our field has no choice but to use it (I do know one or two people that use maple, but the overwhelming majority chooses mathematica).
Despite its elegant design, many people hate it with a passion, as it has grown to be a huge bloated mess that takes forever to run. Also, due to the closed source nature, it is very hard to debug when something goes wrong. For example, it is quite often for the basic functions like `Simplify` and `Integrate` to get stuck running forever, but there is no way to keep track of the internal transformations that mathematica is doing, since everything is sealed up.
I am a PhD student in theoretical physics and almost everyone in our field has no choice but to use it (I do know one or two people that use maple, but the overwhelming majority chooses mathematica).
Despite its elegant design, many people hate it with a passion, as it has grown to be a huge bloated mess that takes forever to run. Also, due to the closed source nature, it is very hard to debug when something goes wrong. For example, it is quite often for the basic functions like `Simplify` and `Integrate` to get stuck running forever, but there is no way to keep track of the internal transformations that mathematica is doing, since everything is sealed up.