I am asked to give a talk about (a) mathematical practice, (b) axiomatization, (c) Gödel's theorems and (d) possible antimechanist arguments based on the incompleteness theorems (as mentioned in P Smith's Introduction to Gödel's theorems, 28.6., J Lucas or R Penrose) to non-mathematicians.
As all these points can not be explained in 1 hour, I need to cut it short while still being able to present a coherent talk. So I certainly will miss something out, the question is what. To transport the intuition/message I will probably even cite some things not completely correct.
The part about logic and Goedel's theorems (c), should be complete and detailed enough to allow explaining Gödel on a more imprecise level and finally to allow at least sketching possible antimechanist arguments based on the incompleteness theorems (d).
The question is: how to best present this part? As I can't replace a whole course in mathematical logic and another whole course about Goedel's theorems within half an hour in full detail, I need to find a way to cut it short while still making the most important points. Do you have ideas and recommendations how to manage this balancing act and allowing the audience to get the gist about what it actually really is Goedel is talking about and where his incompleteness theorem 'applies'? Maybe there are illustrative examples, toy examples or easy to get analogies..
As I feel not that comfortable with such a talk, I would appreciate any helpful comments on how to organize that talk. How to intuitively but still precisely present a coherent talk which transports the most important points. I would like to clarify some views of mathematics (mathematics is not being an expert in doing calculations) and what is wrong about a philosopher rather uncautious citing Gödel as "we can not prove everything" — it seems popular to unknowingly cite Gödel as it is to cite quantum mechanics.
Note: As I am new to this forum, I hope that I did it the correct way, to open a new question since it is another question despite being related to the same talk.
Is there someone who especially has some good ideas pointers for (d)? And any hints where intuitively logically reasoning leads to wrong conclusions? Like the harmonics series converging etc?