# The ADC space: a Taxonomy for Proofs?

It is useful in education, to use taxonomies. An example is the Bloom taxonomy. Taxonomies can help to organize the landscape of mathematics, classifying puzzles in the context of Connected Curriculum Subjects, Generating calculus problems Taxonomies can be useful even when teaching machines. Not only in mathematical problems, also mathematical proofs can be organized in a Abstraction-Difficulty-Complexity space. An example of a difficult proof is the Proof of Fermat's last theorem, an example of a complex proof is the proof of the Kepler conjecture which is so complex that it is still computer assisted, an example of a proof with high abstraction is Goedel's proof of the incompleteness theorem. There is a picture of the abstraction-difficulty-complexity parameter space. Of course, one could add more parameters, but with Taxonomies, it is similar than with proofs: simplicity rules! Also, having only three parameters allows to visualize the space.

# Abstraction

Complexity and difficulty can usually be evaluated pretty well. Things are complex if they need a lot of time to solve (but busy time, like grinding computations). Things are difficult, if one is just stuck because an idea is needed. The abstractness parameter however is hard to estimate. One reason is that it is a highly personal parameter. Some think that documents like what can be found on the Stacks project or the n-lab project are not that abstract, others disagree. That's why it is useful to ask.

Thanks for the questionnaire: here are the submitted data from October 9, 2018. It shows that we can make the lectures a bit lighter and reduce the level of abstraction slightly. About the homework: We want to keep one or two problems more challenging to learn to fight with harder questions and to encourage to make use of resources. But it is good to know that the homework level should not be harder:
too high Just right Too low
Level of Abstraction 13 28 2
Difficulty Homework 17 25 1
Difficulty Lecture 16 27 0
Difficulty Exam 3 39 1
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