Two kinds of generalization

I noticed that there are two different things in mathematics both referred as “generalization”.

The first is like replacing real numbers with complex numbers, that is replacing a set in consideration with its superset.

The second is like replacing a metric space with its topology, that is abstracting away some properties.

Why are both called with the same word “generalization”? What is common in these two? Please comment.

4 comments

  1. One path to generalization passes from a smaller space to a larger space.
    Another path to generalization passes from more axioms to fewer axioms.

    They sometimes converge, sometimes not.

    I tried making this point on MathOverFlow once but it was not well received.

    On a related note, C.S. Peirce recognized two types of abstraction —

    Prescisive Abstraction

    Hypostatic Abstraction

  2. I think mathematicians outside category theory often treat structures as if they were properties. For example, they treat metric spaces as just “metrizable topological spaces”. In that regard, then forgetting a structure is kind of like generalizing to the superset which may not have that selected property.
    It seems to me that category theorists have made this quite precise here: https://ncatlab.org/nlab/show/stuff%2C+structure%2C+property .

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