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Both. Defining a mathematical object is creating it, but all of its properties are implicit in the definition, and so have to be discovered. That is, groups were created but the Sylow theorems were discovered.

Agreed. Axioms are created, everything else is a property of those axioms and 'discovered.'

Those basic axioms tend to be the most useful in our universe, but they were still 'created.' We often define or create other axioms, for example, taxicab geometries are imminently useful for modelling paths in cities. We created the rules for the taxicab geometry, as with Euclidian geometry, but we didn't create the properties that emerge from those rules.

I think maybe the confusion comes from a misunderstood analogy: people think 'if you build a castle of LEGOs, you created that castle; aren't maths likewise created?' The misunderstanding is that mathematical theorems and properties aren't like the castle, they're like the *potential to build a castle from those given LEGOs*. The child created the castle, but she only discovered the *possibility of building a castle*, which always existed as a property of those LEGOs.

I think mathematics in terms of physics, because that's where I did my heavy learning. So when I think 'operator' I think in terms of QM operators, which are defined actions, instead of assumed properties. I see the two as separate, although I suppose they probably seem less so when viewed through a purely mathematical lens.

I don't know that I totally agree. Defining something is not really creating it as whatever it is already exists.

I do somewhat agree in the sense that people create equations to explain things that happen in the real world. Nature doesn't give a fuck about equations. Also things like prediction of future problems and answers with equations people make with stats from present and past I feel has an aspect of creation.