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.
For the first part (a),(b) of this 1 hour talk, I want to point out, what mathematical practice is, i.e., what mathematicians are doing, that they are not calculating anything aiming to explain the nature (that's the physicist's task). Raise the question why mathematics seems to reflect some natural aspects and allows application in nearly every field of science. Last but not least what it is about logic and axiomatization, which should be as complete and detailed enough to allow explaining Gödel on a more imprecise level.
I thought that due to the lack of time it would be helpful stating a toy example of an (very restricted) axiomatic system and to prove one simple statement. As I do not feel qualified to think of such an example myself I was looking for something similar around. Do you have any ideas?
Do you know illustrative examples or cartoons or analogies, which show what mathematicians are doing, that they actually just reconstruct tautologies or "unfold the axioms" since actually everything is already included in the axioms.
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.
One possible system that you might use for explaining a system of axioms without having to spend a huge amount of time might be Douglas Hofstadter's "MU" system.
The ideas are explained here on Wikipedia, where it is described as a puzzle, but it could be presented as a system of axioms.
You are initially given a string "MI" and a number of rules (axioms) that allow you to create other strings:
The aim is to produce MU given only the starting string MI. It is actually impossible to do so.
Incidentally, if you have not read Hofstadter's book "Gödel, Escher, Bach", it might be worth looking into, as it is written for people without massive mathematical experience and does talk about axiomatic systems and Gödel's theorem(s).