On Thu, Dec 23, 2010 at 8:41 AM, Clyde Greeno @.MALEI <email@example.com> wrote: > > You ask of a taxonomy of functions ... including a class of what you choose to call "really" functions. > > The latter seems to exclude the constant functions, such as g(x,y)=3 ... as needed for such intersections as 2x+5y=3 ... more precisely, 2x+5y=3sub-(x.y) > > Your class of "really functions" also seems to exclude what Karl Menger called the (First, Second, etc.) place-selector functions ... as with F(x,y)=x and S(x,y)=y ... on which all of function analysis depends. Apparently, your (postulational) criterion for your "really" functions is that for all (x,y,...) coordinate-points within the domain of a "really" (numeric-valued) function, the total range of numbers used in any one place always yields more than one value of the function. >
Hey cool, someone else who knows about / cites Karl Menger.
Karl contributed this essay to an anthology relating to Einstein's philosophy (that's what they called it back then), about how we might want to get off the Euclidean bandwagon some way *other than* jiggering with the fifth postulate: we could mess with the definitions. He called his invention a "geometry of lumps" and only started to derive the ripple effects, leaving it to others to perhaps build on this proposal.
Enter "claymation station" and other animation applications which focus on the distribution of light energy through a space. In order for target energy to register with an observer, it needs to occupy definite volume ("res extensa" in Descartes vocabulary). In this sense, all targets are the same in being volumetric, even if some are more point-like or line-like (or plane-like in the case of surfaces).
Of course this is an idealization in need of further elaboration, and instead of doing all that work from scratch, I hooked up a neighboring body of literature that was following just such a course. All the lumps were topologically tetrahedrons, if anchoring to the family of volumetric polyhedrons (because the tetrahedron is the simplex or "first volume" in the world of V,F,E). That brought edges (E), faces (F) and vertexes (V) into the picture (all at once, ab initio, V + F = E + 2) which resulted in further articulation and definition through two volumes of published work, endorsed on the dust jacket by such as Arthur C. Clarke, early speculator about satellite communications, and U Thant, a former UN general secretary.
The two tomes in question (by Dick Fuller and on the web for some years complete with illustrations, thx to Dr. Gray) were also in collaboration with a retired staff chief and deputy inspector general under another Dick: Helms (talking about Sonny). These many links to interesting historical personages opens a lot of windows (and/or Wikipedia links) for those wishing to explore Americana amidst a larger body of world literature (Arthur was a resident of Sri Lanka, while U was Burmese, Sonny lived in Georgetown).
Rather than channel all this through K-12 high school math courses directly, which would be hopeless, I've been promulgating the above in a commercial context. That's not to discourage teachers (see below), but simply to channel around an obviously rickety infrastructure that doesn't respond to external stimuli very quickly or with much coordination. The private sector has some faster switches, as do elements of the public sector. LCDs (flatscreens) bring geometry directly into the home, as well as geography (the two main subjects we seek to overlap i.e. res extensa and res cogitans, to get philosophical about it).
High schoolers visiting one of our coffee shops after school or on weekends see allusions to these ideas on the LCDs and/or surrounding poster art and decor (we design in lots of Phi 'n stuff, per Dr. Livio's influence etc.). They click around on the web and find out about Karl Menger et al, and start wowing their teachers and friends with all this esoteric knowledge that actually sounds employable (as in applicable) to real world problem solving.
> > Your question seems to be of whether or not there exists a very simple, classical name for your "really" functions. Probably not. Therefore, you are free to assign to such functions whatever title you choose. As a word of caution, the word, "real" commonly refers to the "golden" numbers as a continuum. So, -- as opposed to "real functions", you might wish to consider using an alternative title, such as "fully variable functions" ... meaning variations in each place yields variations in the function values. > > Good query. Good luck on your efforts to clarify the cognitive structure of the underlying mathematics. > > Cordially, > Clyde >
I'd remind the questioner that you don't need to know the rule for a function to have a function.
As we all learned in New Math, a function is no more or less than a set of tuples relating input states to output states. These pairs should have the property that the same input is never seen with two different outputs (domain members are monogamous with range members), however it's fine for any number of input elements to "date" the same output.
In terms of "correlation", you really don't always know.
That's the game of statistics, to figure out if "x" (e.g. aspartic acid) is really an influence (e.g. on the "attention span" however measured). You get a lot of data and search for some rule, perhaps a linear relationship, though given this is biochemisty, you know there's a point at which too much of anything is gonna be bad for ya. Someone might be wanting to test Nutrasweet against Ritalin for example, perhaps in a bid to get those Pepsi vending machines back in the schools, dispensing "smart drinks" this time (not my bailiwick, but I have been following research on Nutrasweet and MSG).
The people working with me on the coffee shops network (I've got chief officers displayed in the biz logs or chronofile) share readings. I'll likely point some of them back to this thread.
I'm on the marketing side (CMO) which is what's most active right now, as coffee shops (some are full bars, but not necessarily those frequented by high schoolers) won't want to subscribe to the LCD feed or games if they don't understand what's in it for them (e.g. lots of high school aged customers geocaching for geometric esoterica, some kind of "outdoor math" exercise perhaps organized by the math circles in participating schools, with faculty oversight and approval).
> - -------------------------------------------------- > From: "June Lester" <firstname.lastname@example.org> > Sent: Wednesday, December 22, 2010 5:30 PM > To: <email@example.com> > Subject: proper functions? > >> Just a quick question for the knowledgeable people of this forum: is there an official term for functions which are "really" functions of their arguments, i.e. their values actually vary with their arguments. So f(x, y) = x + y is one such, while g(x, y) = x is not. I tend to say f is a "proper" function of y, but g is not, but I would rather use the official terminology, if there is indeed one. >> >> Thanks