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?

  • $\begingroup$ There is a thread about explaining the incompleteness theorem to high school kids. Isn't that enough? math.stackexchange.com/questions/453503/… $\endgroup$ – Asaf Karagila Nov 30 '13 at 15:36
  • $\begingroup$ Thanks for the pointer. Indeed very helpful. Though my question has the additional part (d) and also asks to explain it on basis of (a) and (b) within a talk. If it is assumed to be duplicate, I am sorry. $\endgroup$ – AminCophag Nov 30 '13 at 15:49

The audience will surely probably have a misleading expectation of just how powerful mechanical reasoning is: it is important to reset that expectation. For example, they might not realize that Go is a solved game -- in fact, it's a nearly trivial exercise to write a computer program that plays perfect Go. All of the challenge in playing Go "mechanically" is to find a program that only needs a few quadrillion calculations and a few trillion bytes of storage.

Working through a program that proves every theorem of Peano arithmetic might be an enlightening example.

But axiomatization is somewhat irrelevant to the topic. It's probably worth pointing out that another perfectly good axiomatization of Euclidean geometry is the one where every theorem is an axiom. All proofs of theorems are trivial one statements "It's an axiom. QED". The challenge with doing mechanical reasoning with this axiomatization is to figure out whether or not any given statement is an axiom or not.

If we had a model of the natural numbers, we could choose the set of all true statements about that model as an axiomatization of integer arithmetic. The problem here is that there is no computer program that can tell whether a statement is an axiom or not -- in fact, this was the first version of Gödel's incompleteness theorem that I had ever seen precisely stated.

It is also probably worth showing a familiar theory that Gödel's theorems don't apply to -- first-order Euclidean geometry is a lovely example. Tarski translates it into the language of real closed fields, and provides an algebraic tool that can solve any arithmetic problem, thus giving a computer program that can decide whether any particular statement of Euclidean geometry is a theorem or whether its negation is a theorem.

It's probably worth saying something about standard models. Many arguments don't simply invoke that Gödel's theorem proves some statement undecidable, but that it is actually true. It is worth pointing out that talking about "truth" makes some presuppositions about semantics. It may also be worth saying something about internal versus external reasoning -- many arguments I've seen involving Gödel's theorem make analogous mistakes to the reasoning that leads to Skolem's paradox.

But on these last points, I can't really offer much advice: it seems to be a difficult topic even for seasoned mathematicians.


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