Suppose you have a society in which they have no concept of math at all - no written number system, nothing.
Now suppose we are to instruct them in the the world of math, but our time is very limited. Thus, we have to choose precisely which few rules will allow them to deduce the same mathematical techniques we have.
In other words, what are the basic axioms of math that would allow one to deduce our entire math knowledge?
BUT, perhaps more interestingly, I’d like to know if there exist any math principals which would not allow us to deduce the rest of the mathematical set?
sigh
I have been trying to find this out for a whil now.
Ya ain’t gonna get an answer. At least not a straight one, mainly because there are many disagreemnets on the matter. Do a search of the SDMB for axioms and math. You will find some of the threads on this exact question.
Hmm. A more round-about anwer. In his Physics Lectures, or some other book, Richard Feynman said that if he could impart one scientific principle to a primitive people, it would be that everything in the world is made up of tiny little thingies (not his exact words). Probably, the best path would be to find the most important kernel, such as, “things in the universe follow patterns, you see two fingers, two antelope, two rocks–this pattern is universal, and there are patterns like this for everything.” Okay, that sounds pretty boneheaded. But seriously, if you had a culture that knew nothing mathematical, then I thing you’d need something like that. No axioms. I think the last part of your question, the part I underlined, simply doesn’t apply to a culture with absolutely no math whatsoever. Only a mathematically adroit** listener could deal with the basics of set theory, or Euclids axioms, etc.
**Adroit isn’t the right word. Someone who has some familiarity with math is what I mean.
Perhaps I’m not quite understanding your response, but my question need not introduce anything involving set theory*
As you mentioned, if you have a tribe with no mathematical knowledge, there are certain “kernels” (as you say), which we could impart that would allow them to develop the math tools we have today.
What I’m asking is are there any kernels that would be completely useless?
Ie, if given A we can deduce all the alphabet A - Z.
But does there exist a “letter” wherein we could not deduce all of A - Z?
(*For you and me now, to actually solve this question, set theory may be necessary; but my underlined part is still valid - knowledge of set theory, or lackthereof, has nothing to do with actually carrying out this proposal.)
Fair enough, I probably wasn’t very clear in what I intended to say. It seemed like you were really asking three questions:
If you had a completely mathematically illiterate culture, what basic piece of mathematical info do I think would be most useful to them?
What are the most fundamental mathematical axioms?
What mathematical axioms are dead ends, i.e. from which mathematical axioms can nothing useful be derived?
I’m of the opinion that question 1 is unrelated to questions 2 & 3. It seems to me that alot of mathematical advancement came in the form of intuitive guesses and pragmatic experiences, for example showing why paradoxes aren’t paradoxes at all, or figuring out how much stone will be needed for my pyramid. The theoritical foundations of math came about really quite late, the 18th or 19th centuries, if I recall correctly, with the development of set theory.
With that in mind, it doesn’t seem to me that giving one or some axioms of set theory would be (most) useful to the development of math. Rather, I wanted to take a guess at a very primitive conceptual hurdle that needed to be crossed. I’ve read some on the history of mathematics, but I couldn’t recall any specific problems off hand–plus from the context of the question I inferred that you were interested in a society that would pre-date the “history” of math anyway. So that’s why I went with patterns. The universe follows patterns (some of which are next to impossible to discern), and that seemed to me to be a very big hurdle for a society which would probably be still in the magical/animistic stage of philosophical/religious development. The idea that the universe follows certain rules and that those rules can often be guessed at quite accurately seemed like a huge conceptual break from the idea that individual spirits animated the elements of the world and its events.
As I understand it, one can deduce more or less all of math from set theory and thus its axioms–whatever those axioms are. But one must be adept at math to do so. So, I guess that they are fundamental in the theoritical sense that they give a solid foundation to mathematics as a whole, but they aren’t fundamental to the development of mathematics because they really came around quite late.
As to axioms that are useless, I have no idea. I would imagine that those are lost because they were useless in the first place. Maybe a history of math would cover some, or maybe some early fallacies–but I really couldn’t hazard a guess.
Try checking out the book Number Sense. The author describes research into the inherent mathematical capabilities of the brain. That probably would be the first step–that natural ability. For example, we already have some number sensing ability built in (for small numbers), however going from merely detecting some numbers intuitively to explicitly using them I couldn’t say. Anyway, it’s a good book that may make a fine addition to your reading list.
