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Think axiomatically even about concrete objects

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There are many mathematical definitions that are slightly artificial: consider for example the definition \{x,\{x,y\}\} of the ordered pair (x,y), or the definition \{\emptyset,\{\emptyset\}\} of the number 2. The point of these definitions is not the definitions themselves but the fact that they establish the consistency of certain properties. In proofs, it is almost always better to use the important properties of a definition than to argue directly from the definition. Even when a concrete definition arises naturally and specifies a unique object, it can still be better to focus on characteristic properties and use those.


Algorithms and Invariants

In computer science:
Algorithms are like definitions.
Invariants (used to prove algorithms correct) are properties of the algorithms.
The invariants describe the higher-level ideas in the lower-level algorithm.
Invariants are closer to the "understanding" in the algorithm than the algorithm itself.
As a set of instructions for computing something, the algorithm is superior (a computer can use it to do the computation). But for "understanding the algorithm", the invariant is key.


I am sorry. I know that the definition is not *wrong* with a correct definition of set theory (which is usually not given in the books that use the above definition of ordered pair), so I reverted it. I just think that it is crazy to unnecessarily use that as definition, but it seems that many people love it.

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