# Explain mathematical practice and axiomatization to non-mathematicians

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.

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What do you think about the idea to quote Banach Tarski's paradoxon to expound the problems of "matheatics (tries to) explain(s) reality" and the self-evidence of axiomas or how the measured against our intuition or experience in the physical world? – AminCophag Nov 29 '13 at 18:42
My favourite proof for the non-math specialist is the one about the hiker from Martin Gardiner's "Gotcha." On one day, a hiker follows a path from his home in the village to his cabin in mountains and stays overnight. The following day, he returns home along the same path. You can "prove" that, on his return journey, he will be at the same point at exactly the same time of day as the previous day. Just imagine both journeys happening simultaneously. The hiker would have to run into his double at some point, as long as he doesn't deviate from the path. See humorous diagrams in the book. – Dan Christensen Nov 30 '13 at 6:12
I like Gardiner's hiker because it demonstrates, in a cartoon sort of way, what a proof is. Most people don't have a clue. They will see that you are not just calculating where and when -- what they think mathematicians do -- but you are showing that such a point must exist. You could also make a simple distance-time graph to up the level of abstraction just a bit. – Dan Christensen Nov 30 '13 at 17:47
I don't know if you could prove it easily, but the logical extension of this example could be demonstrated with 2 identical sheets of paper. Crumple one and it lay it in top on top of the other, without the crumpled piece extending past past the edge of the flat sheet. At least one point on the crumpled sheet will be exactly above the corresponding point on the flat sheet. – Dan Christensen Nov 30 '13 at 17:51

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:

1. Add a U to anything ending in I
2. Replace Mx by Mxx
3. Replace 3 consecutive Is with a U
4. Delete 2 consecutive Us

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).

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Thanks for the pointer. It looks quite neat and shows the power of "mathematical reasoning". But e.g. Wikipedia's proof that MI can not be turne to MU is not a mathematical proof in the sense that it does not solve the problem by using the axioms but rather some "number theory". – AminCophag Nov 29 '13 at 18:22
Very true, but you could use the "axioms" inside the MU system to show how to prove a "theorem" such as "MUI" by using some of the axioms a few times - even ask the audience if they could see how to do it? I would show them one or two such theorems, and then tell them (but not go through a proof) that obtaining MU is not possible - i.e. there are some strings which are not theorems. – Old John Nov 29 '13 at 18:25
That's true. Maybe the Wikipedia proof could even serve as an example of what is not a mathematical proof despite it seems so appealing to intuitive logic. Afterwards I could present or ask the adience for a proof of another theorem. – AminCophag Nov 29 '13 at 18:29
Doesn't the wiki proof compare to Gentzen's proof of GCH using methods outside the theory under consideration? – AminCophag Nov 29 '13 at 18:55
In case you are interested, I already have a resolution the statement that MU cannot be changed into MI not only cannot be proven within the system, it cannot be even stated in the system. (because the system doesnt have notions like "change") – Adam Nov 30 '13 at 0:34