“important” math concepts to pass on to next generation of creatures at some cataclysm [closed]

This may be somewhat silly to ask, but I couldn't resist the temptation. The idiosyncratic physicist Richard Feynman was once asked

If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words?

And his answer was that the world is made of atoms and the qualitative description of attractive electromagnetic interaction at long distances and repulsive nuclear interaction at short distances that governs the atoms:

I believe it is the atomic hypothesis that all things are made of atoms — little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.

If any mathematician was to be asked such as question, but restricted to mathematics, and perhaps, for it to be more interesting and more answerable, not necessarily restricted to a single sentence, but perhaps several ones, or several key concepts or ideas, what would those be?

And to make it possible for your answer to be appreciated and be interpreted not solely as a personal preference, provide an explanatory argument especially from math history point of view like @Anupam did, or a logical one. Feynman's answer is more or less easy to appreciate, as without knowing about atoms, people will be limited to many phenomenological interpretations of the nature, and not start to look at inner-workings of the world from that level of microscopic angle.

I would tell them:

With a set of assumptions (Axioms) we can prove a lot of things, but by the incompleteness theorem not everything.

This may stop alot of these creatures from going mad, and it will always give them something to do.

I am not a profound Mathematician but I am very much interested in Number theory. Reading the elementary maths especially the Numeral system leads me to believe that the most important, yet simple, but far more mysterious to be discovered is, the Zero. The importance of Zero does not lie in representing it as number which is an additive identity. The importance lies in its use as the tenth digit of our base-10 numeral system. Without $0$, that is the place value system, I cannot imagine the existence of computers. A small calculation would be a difficult task without 0. Without having calculation made easy modern technology would not have been developed as it is now.
Any new civilization born after the cataclysm would learn elementary mathematics like addition,subtraction and multiplication but it would take a long time for them to recognize the zero.
So if I were to pass on an idea to the next generation I would tell them the Numeral system.

I myself is very interested to see how mathematics would evolve if it is stared all over again. So I prefer to leave nothing and let people's imagination guide them.

If we have to leave something, personally I would tell them:

You can use algebra to do geometry.

And I believe they will find ways to achieve this that surprise all of us.

"This sentence is false."

(This sentence gets me past the SE minimum character limit.)

• This is more like "how to troll the generation of the future", no? – Bruno Stonek Apr 6 '14 at 8:36
• @BrunoStonek: Indeed not! The point is to prod post-apocalyptic peoples into taking a good hard look at the nature of truth and untruth, of implication, ... of logic. They're going to need that to rebuild mathematics from its foundations. I'm surprised and disappointed by the down-votes. – Blue Apr 6 '14 at 16:09
• In a post-apocalyptic world rebuilding mathematics axiomatic foundations wouldn't be among my top priorities. – Mark Fantini Apr 6 '14 at 16:39
• @Fantini: I suspect that pondering the fundamentals of atomic theory wouldn't rate very highly, either. :) I'll admit that mine isn't the most-serious (or most-potentially-effective) answer possible; of course, this isn't the most-serious question available. Nevertheless, this is a not an illegitimate response; heck, it's essentially the crux of the argument behind Gödel's Incompleteness Theorem, which says profound things about the nature of axiomatic systems. I don't understand the negativity. – Blue Apr 6 '14 at 17:18
• If you add this explanation in the original answer, it probably would not get down votes. – qazwsx Apr 6 '14 at 18:46