Although I try to tell people about what we actually do in mathematics, sometimes I can't translate the activity very well. For example, when telling them that we study were do mathematical ideas come from, they usually understand it with an arbitrary self-explanation, for example: When someone learns the derivative, they are able to see that it comes from the concept of limit. But that wasn't really my point, for the concept of limits, there are some other concepts that ought to be learned in order to give the limits genesis and, who knows what other kinds of concepts could form a deeper construction that, perhaps, could be used to construct all mathematics?
Until the present date, I have only a handful of examples of mathematical phenomena and I guess that the most interesting question I've been able to produce is:
Are there simple objects which could be used in arbitrary combinations to build all known mathematics? If yes, what is the proof for that?
I have some questions now:
What are good questions that could be used to demonstrate the nature of mathematics study?
Is that question I've formulated suitable for my purpose? I believe it is but I'd like to hear what other people with more competence than me could think about it.