On 9/29/2013 9:46 PM, Arturo Magidin wrote: > On Sunday, September 29, 2013 7:30:49 PM UTC-5, Hetware wrote: >> On 9/29/2013 8:06 PM, quasi wrote: >> >>> Hetware wrote: >> >>>> >> >>>> What I am saying is that if I encountered an expression such >> >>>> as (t^2-9)/(t-3) in the course of solving a problem in >> >>>> applied math, I would not hesitate to treat it as t+3 and not >> >>>> haggle over the case where t = 3. >> >>> >> >>> And you would be wrong unless either >> >>> >> >>> (1) You know by the context of the application that the value >> >>> t = 3 is impossible. >> >>> >> >>> (2) You know by the context that the underlying function must >> >>> be continuous, thus providing justification for canceling the >> >>> common factor of t-3, effectively removing the discontinuity. >> >>> >> >>> I challenged you to find a book -- _any_ book, which agrees >> >>> with your naive preconception. >> >>> >> >>> Math book, applied math book, physics book, chemistry book, >> >>> economics book -- whatever. >> >>> >> >>> If all the books and all the teachers say you're wrong, >> >>> don't you think that maybe it's time to admit that you >> >>> had a flawed conception about this issue and move on? >> >>> >> >>> quasi >> >>> >> >> >> >> I don't answer to the authority of mortals. I answer to the >> dictates of >> >> reason. > > Would that be the reason that you have learned through the authority > of mortals, or through direct contact with an immortal of some sort? > Come on, now. Get off that high horse...
It does me no good to simply accept what someone tells me without being able to derive it in some sense from first principles.
> >> I say that it is logically consistent to view >> >> >> >> (t^2-9)/(t-3) = t+3 >> >> >> >> as valid when t = 3. If a contradiction can be demonstrated, then >> the >> >> proposition is clearly wrong. Note clearly that I am defining >> >> (t-3)/(t-3)=1. I am not appealing to a more fundamental meaning >> for the >> >> algebraic form. >> > > Here's the thing: what is your definition of a function? How do you > determine whether two functions are equal or not? Are you viewing the > expression as purely algebraic expression or as a function? > > In the context of calculus, which is after all the context you find > yourself in, a function is a rule that assigns to every valid input > one and only one output.
Thomas (1953) did not ascribe to that definition of a function. Though yours is the one that I prefer. I didn't ask if what I did violated tradition. I asked if it leads to a contradiction.
> Two functions are considered to be the same > function if **and only if** they have the same domain (the same set > of "valid inputs") and the same value at each element of their > domain. > > Strictly speaking, then, in order to discuss a function, we must > agree on two things: (i) what is the domain of the function; and (ii) > what is the rule that assigns to each element of the domain a value. > That means that each and every time we mention a function, we must > say what the domain is.
In this case, the domain is all real numbers were f(t) is defined.
> Because this becomes both onerous and complicated, there is a > standard convention that is, I am positive, mentioned in your book. > This convention is: > > If a function is described by giving a formula, and no domain is > explicitly specified, then it is agreed that the domain of the > function is the natural domain: that is, the domain is the set of all > numbers for which the expression, *as given*, makes sense. > > Now, the function > > f(t) = (t^2-9)/(t-3) > > with no domain specified, is therefore assumed to have as domain the > real numbers, **and only the real numbers** for which the expression > *as given* makes sense. And this collection is exactly the real > numbers different from 3.
If I say that f(t) = (t^2-9)/(t-3) is continuous for all reals, then you can protest that the expression (t^2-9)/(t-3) is meaningless when t=3. The assumption that f(t) is continuous, and the observation that (t^2-9)/(t-3) is meaningful for all t!=3 dictates that f(3)=6.
> On the other hand, the function g(t) = t+3 with no domain specified > is assumed to have as domain the real numbers, **and only the real > numbers** for which the expression, *as given* makes sense. And this > collection is exactly the set of all real numbers. > > That means that the function f(t) and g(t) have different domains, > and therefore are different functions. > > This is simply a matter of definitions and conventions. Those > definitions and conventions exist because they are *important* and > *necessary*. The fact that you don't see this yet is not a slight on > you: it is the consequence of centuries of work by many > mathematicians who have actually thought about this issue. You have > the benefit of being the inheritor of these many years of thought, > given to you distilled as precise, clear definitions that you are > expected to abide by.
History is not reason. Mathematics is founded on axioms and constructed of definitions conjoined with logic.
> The "contradiction" in your assertion arises simply because you > assert that f(t) is equal to g(t), when it can be demonstrated that > they are not equal: as functions, they have different domains and > therefore are different, not equal. The fact that you refuse to apply > the definition and say "I don't answer to mortals" does not excuse > this fact, nor does your claim that your assertions follows from > "reason". The plain definitions contradict your assertion, and that's > the end of it.
No. The reasons I was given for not accepting Thomas's development were not presented as sound and valid. They were predominately appeals to authority.
> You are free of course to make your own definitions; but then you > aren't doing the same calculus as the rest of us: you are doing > "hetware-calculus", perhaps.
All I was proposing by (t-3)/(t-3)===1 was an intellectual exercise to illuminate the nature of the reasoning involved in interpreting such statements as: let f(t) be a function defined for all real numbers where it is meaningful, and let f(t)=(t^2-9)/(t-3).
Compare that with: let f(t) be a continuous function defined for all real numbers, and let f(t)=(t^2-9)/(t-3) defined f(t) everywhere it is meaningful.