"Ostap S. B. M. Bender Jr." <email@example.com> wrote: > On Jan 28, 11:05 am, Bill Dubuque <w...@nestle.csail.mit.edu> wrote: >> "Ostap S. B. M. Bender Jr." <ostap_bender_1...@hotmail.com> wrote: >>> On Jan 26, 4:48 pm, Bill Dubuque <w...@nestle.csail.mit.edu> wrote: >>>> "porky_pig...@my-deja.com" <porky_pig...@my-deja.com> wrote: >>>>> On Jan 25, 8:52 pm, eratosthenes <rehamkcir...@gmail.com> wrote: >> >>>>>> I am currently tutoring at a community college and one of my students >>>>>> is a sixth grade teacher going back to school because she needs a >>>>>> calculus credit to complete her certification or something, but that >>>>>> is beside the point. >> >>>>>> She asked me if I could give her a way to explain what a mathematical >>>>>> function is to a sixth grader other than the standard explanation >>>>>> about it being a machine that you put one number into and receive >>>>>> another out of. She said that this does not work. >> >>>>>> I tried explaining how I learned long ago: As a map where the >>>>>> equation is the directions or something like that. She was also >>>>>> dissatisfied with that. >> >>>> How does one make any sense of "map where equation is the directions"? >> >>>>>> Any thoughts? >> >>>>> If f: R -> R, where R is a real line, then thinking of f as a machine >>>>> that you put one number into and receive another out is adequate. >> >>> Sort of like changing hands in draw poker? You discard a card and >>> get a new one from the deck? >> >>> To me, if we are talking about R -> R maps, this "black box" model is >>> highly anti-intuitive, as it hides the continuous nature of R. To me, >>> real functions are usually associated with graphs (as in "plots"). >> >> But we're talking about abritrary functions, not just continuous ones. > > Very few children in 6th grade are familiar with many R -> R functions > that aren't continuous almost everywhere.
But we're talking about arbitrary functions, not only (continuous) real.
>>>>> In fact, I can't think of anything better than that. There may be some >>>>> simple formula associated with matching the input with the output, or >>>>> not. I don't know if they also need to know what is injection, >>>>> surjection and bijection, but that comes after the basic definition. >> >>>> The problem with "thinking of f as a machine" is that it is far too >>>> intensional to convey the modern set-theoretic extensional concept >>>> of function, i.e. as a single-valued total relation between sets. >> >>> You are right. No mathematician, other than a logician or a set >>> theorist, would think of real functions as black boxes. >> >> You're confused. The set-theoretical reduction of the notion of function >> is as I said. Whether or not that counts as a "black-box" definition >> I can't say since you haven't defined what you mean by that vague term. > > How does the statement "No mathematician, other than a logician or a > set theorist, would think of real functions as black boxes" contradict > your statement: "The set-theoretical reduction of the notion of > function is as I said"? I didn't say anything about set-theoretical > reduction, did I?
I didn't say it did. "You're confused" refers to you apparently believing that I agree with what you wrote after "you are right".
>>>> Indeed, said "definition" doesn't even specify what it means for two >>>> functions to be equal, so it does not make it clear that the concept >>>> of function is independent of any particular representation (e.g. >>>> analytic, rule-based, computable, etc). The concept of a relation >>>> and its associated properties of being total, single-valued, etc >>>> are certainly elementary enough that they could easily be taught >>>> at an early age, and motivated with many concrete examples. >> >>> The only definition of a function f: A -> B that I am aware of, is >>> that of a set of tuples (relations) in AxB, where each element of A >> >> You mean "a relation" not "relations". > > Yes. > >>> occurs exactly once. Of course, this is not the most intuitive >>> definition of the real functions either. >> >> Again, we're talking about general functions, But, out of curiosity, >> what rigorous definition of a real function do you think is more >> "intuitive than the standard set-theoretical definition? > > Please remind me what you call "the standard set-theoretical > definition". I am not a set theorist.
I said that above: "a single-valued total relation between sets"