3 ms·

Inheritance works great in mathematics because then it's all about using some laws to compute derived properties, but in programming we use objects for behavior

Inheritance works great in mathematics because then it's all about using some laws to compute derived properties, but in programming we use objects for behavior with side effects. So it's like apples and oranges.

dan-robertson 8y ago
I’m not really sure where you think mathematics is full of inheritance. Do you have an example? Off the top of my head: Subsets, sub-anything-else, not really inheritance Definitions of the form an X is a Y with the following extra properties, I think don’t really count as inheritance either. Example: A ring is a group with ..., but if you wanna to prove anything interesting about a ring, you can’t kust use the group properties or you would have proven a theorem about a group, and so the only useful thing about the group structure is that you know what it is. Any useful theorems will need other useful definitions (eg homomorphism) and these don’t get inherited and so the isomorphism theorem for groups doesn’t become the isomorphism theorem for rings “for free”. There’s also the fact that a field is a ring plus an extra property but most ring things become basically useless because much ring weirdness becomes trivial in a field and what is not trivial in a field is typically impossible in a ring, so knowing about rings doesn’t tell you much about fields. Example: a topological space is a set plus a subset of its power set such that ... but all the interesting things in topology typically have little to do with those sets and more to do with eg functions (but not just any set function) and the added properties. I suppose the theme is that one adds structure to (ie inherits from?) something because the new structure bans a lot of the weirdness of the bigger thing and allows for more interesting things to be done. The structure of the bigger thing is therefore not so useful because what is known about it is less relevant once more structure is imposed. In applied maths, the Fourier and Laplace transforms are part of a more general family of integral transforms yet this knowledge is basically useless in their application and it would not make much difference if they were not related. Fourier series fall into part of a bigger theory called Sturm-Liouville theory but in many of the places one uses a Fourier series this bigger picture is not relevant whereas the specific details about Fourier series are. This is not to say that Sturm-Liouville theory is not useful.

dan-robertson 8y ago

I’m not really sure where you think mathematics is full of inheritance. Do you have an example? Off the top of my head: Subsets, sub-anything-else, not really inheritance Definitions of the form an X is a Y with the following extra properties, I think don’t really count as inheritance either. Example: A ring is a group with ..., but if you wanna to prove anything interesting about a ring, you can’t kust use the group properties or you would have proven a theorem about a group, and so the only useful thing about the group structure is that you know what it is. Any useful theorems will need other useful definitions (eg homomorphism) and these don’t get inherited and so the isomorphism theorem for groups doesn’t become the isomorphism theorem for rings “for free”. There’s also the fact that a field is a ring plus an extra property but most ring things become basically useless because much ring weirdness becomes trivial in a field and what is not trivial in a field is typically impossible in a ring, so knowing about rings doesn’t tell you much about fields. Example: a topological space is a set plus a subset of its power set such that ... but all the interesting things in topology typically have little to do with those sets and more to do with eg functions (but not just any set function) and the added properties. I suppose the theme is that one adds structure to (ie inherits from?) something because the new structure bans a lot of the weirdness of the bigger thing and allows for more interesting things to be done. The structure of the bigger thing is therefore not so useful because what is known about it is less relevant once more structure is imposed. In applied maths, the Fourier and Laplace transforms are part of a more general family of integral transforms yet this knowledge is basically useless in their application and it would not make much difference if they were not related. Fourier series fall into part of a bigger theory called Sturm-Liouville theory but in many of the places one uses a Fourier series this bigger picture is not relevant whereas the specific details about Fourier series are. This is not to say that Sturm-Liouville theory is not useful.