Open thread, 14-20 July 2014

David_Gerard

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Previous thread

If it's worth saying, but not worth its own post (even in Discussion), then it goes here.

Notes for future OT posters:

1. Please add the 'open_thread' tag.

2. Check if there is an active Open Thread before posting a new one.

3. Open Threads should be posted in Discussion, and not Main.

4. Open Threads should start on Monday, and end on Sunday.

A couple of starting points to improve your investigation.

First, rigorous inductive reasoning, i.e. bayesian probability, includes as a special case deductive reasoning, at least in the case where deductive reasoning is conflated with "classical logic". AFAIK there have been only sparse and timid research into widening probability to fit other kinds of logic.

Second, the example you use to illustrate generalization is a case of what in logic is known as elementary embedding, and it's a pure application of deductive reasoning. Although the process that led the first mathematician to the invention of rational numbers might have very well been one of inductive reasoning.

Third, do not be tempted to think that abstraction is generally linear. It may very well be the case that the model C is generalized by model B, which is generalized by model A which is found to be a special example of C. This kind of "strange loops" (I credit Hofstadter for the invention of this term) happens all the time in category theory.
The very same four different foundations for mathematics (set theory, type theory, category theory, univalence theory) are all instances of one another (see for example the paper "From Sets to Types to Categories to Sets" by Awodey, which however was compiled before univalence was formalized).

Thanks for your enticing comment!

I understand your first point, but my math knowledge is not up to par to really understand point #2, and point #3 just makes me want to learn category theory. BTW, I also posted this question on the philosophy stackexchange: http://philosophy.stackexchange.com/questions/14689/how-does-abstraction-generalization-in-mathematics-fit-into-inductive-reasoning.

Do you have any recommendations of what to study to understand category theory and more about the foundations of math? (Logic, type theory, computability & logic, mode... (read more)