I used Mathematica on a NeXT computer back in 1991. It was a beautiful machine to work on. I did a student project where I simulated the flow of the boundary of a plane region over time (like how the shape of a drop of oil in water changes over time) and it was very, very easy to write in Mathematica with cool graphics.

How could I forget this! He also recommended the name.

Wolfram had come up with some normal techie names "computron", "math-o-matic" or whatever. SJ said No those suck, use something simple like Mathematica.

(Theo Gray is in there too; he built the notebook interface, was introduced by SJ to show it at some event. Theo's parents were math profs in Urbana and I studied under his dad one summer, house-sat, and I too had NeXTstep access and Mathematica since it was ubiquitous there)

Probably the most general-purpose one is SageMath, which is open-source and basically Python with a ton of sophisticated math stuff built into it. Everything I used to do in Mathematica I now do in sage, and I don't think I'm the only one. Probably field-dependent though.

Of course there is a whole constellation of more specialized things in certain fields, that has come a long way in the last 15 years. So people needing things like that no longer kludge things together in Mathematica.

Maple would be the main proprietary alternative but yes, there's also SageMath.