Hetware
Posts:
148
Registered:
4/13/13


Re: Is (t^29)/(t3) defined at t=3?
Posted:
Oct 7, 2013 8:52 PM


On 10/6/2013 11:51 PM, Arturo Magidin wrote: > On Sunday, October 6, 2013 8:55:23 PM UTC5, Hetware wrote: >> On 9/29/2013 9:46 PM, Arturo Magidin wrote: >> >>> On Sunday, September 29, 2013 7:30:49 PM UTC5, 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^29)/(t3) 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 t3, 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. > > Then you won't get anywhere.
Too late, I already have. I now realize I was asserting my assumptions in the wrong order. I was assuming continuity after stating the definition of the function.
> What "first principles" did you apply to derive the notion of > "function"? Of "domain"? Of "ordered pair"?
Axiomatic set theory, though it's been a long time since I went through the entire exercise. Some concepts must be taken /a priori/, such as the dictates of logic.
> Again: get off the high horse; because you are being either ignorant, > or hypocritical.
In this context /argumentum ad hominem/ is clearly a fallacy.
> You aren't trying to "derive it in some sense from first principles." > Rather, you are trying to justify your initial mistake, instead of > simply accepting it and moving on.
No. I was attempting to understand why my mathematical intuition was telling me that I could treat (t^29)/(t3) as identical to t+3. The exercise was well worth the effort.
>> >> Thomas (1953) did not ascribe to that definition of a function. > > Thomas distinguishes between regular functions and "multvalued > functions". This, however, does not invalidate the definition of > function, because the other notion that you are now bringing to the > table extraneously is a different notion, that of "multivalued > function". The adjective signals that you are dealing with a > different notion.
No, Thomas distinguishes between "singlevalued" and "multiplevalued" functions.
"Suppose now that with each value of the variable x in its domain there is associated a value, or several values, of the variable y. We then say that y is a /function/ of x. If with each x there is associated a single value of y, then y is said to be a /singlevalued/ function of x. If several values of y correspond to each single value of x, then y is called a /multiplevalued/ function of x."
> Really. It's not that hard, once you stop trying to justify your > errors. >
Perhaps you were thinking of multivariable calculus, such as is treated in
http://www.amazon.com/AdvancedCalculusSeveralVariablesMathematics/dp/0486683362
or
http://www.amazon.com/IntroductionVectorTensorAnalysisMathematics/dp/B00A19P4EW/ref=sr_1_2?s=books&ie=UTF8&qid=1381192056&sr=12
or
http://www.amazon.com/SpaceMatterDoverBooksPhysicsebook/dp/B00C59C6JM/ref=dp_kinw_strp_1
or
http://www.amazon.com/TheoreticalPhysicsDoverBooksebook/dp/B00C8UR0B2/ref=pd_sim_kstore_2
or
http://www.amazon.com/FeynmanLecturesPhysicsboxedset/dp/0465023827/ref=sr_1_2?s=books&ie=UTF8&qid=1381192262&sr=12&keywords=feynman+lectures+on+physics
or
http://www.amazon.com/GravitationPhysicsCharlesWMisner/dp/0716703440/ref=sr_1_1?s=books&ie=UTF8&qid=1381192296&sr=11&keywords=gravitation+misner+thorne
or
http://www.amazon.com/TensorsDifferentialVariationalPrinciplesMathematics/dp/0486658406/ref=sr_1_1?s=books&ie=UTF8&qid=1381192367&sr=11&keywords=lovelock+and+rund
or for that matter, in the later chapters of Thomas.
No. That is not what Thomas was talking about.
> >> 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. > > And that question is meaningless in the abstract. > > A "contradiction" only arises **in context**. You must have some > axioms, some rules of inference, some background logic, and some > definitions. > > Just asking if something or other "leads to a contradiction" without > providing full context (or having that context understood) is > nonsense. The fact that you not only did it once but now insist on > repeating it, well, you can guess what that implies... >
That you have little of value to contribute. The context was reasonable clear. If you wanted clarification, you should have asked for it.
>> >> In this case, the domain is all real numbers were f(t) is defined. > > That was the point. But you are trying to extend this from "where it > is defined" to "wherever I can define it irrespective of the formula > given". > > I say right after: > >>> 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.
Ah, but there's the rub. It did make sense to me to treat the function as defined where t=3. Until I was able to sort out the axiomatic ordering of assumption, I was confused by applying an assumption that is generally applicable in the domain where I apply mathematics.
>> and the observation that >> >> (t^29)/(t3) is meaningful for all t!=3 dictates that f(3)=6. > > Nope. Because > > (i) "f(t) is continuous" does not mean the same thing as "f(t) is > continuous everywhere"; and
At this point, I already stated that it is continuous for all real values t. You simply chose to ignore that fact.
> (ii) Because by asserting that f(t) is continuous you are ASSUMING > that f is defined at 3.
You are finally catching on to what I was saying.
However, whether or not f(t) is defined at 3 > is *precisely* the crux of the matter here.
I defined it as continuous.
> > History teaches mathematicians what axioms and what definitions to > use. Refusing to accept them because you want to reject history makes > you a willful ignoramus, not the searcher for knowledge that you try > to tell yourself you are being.
You should study the discipline of axiomatics sometime.
> > Yes. The reasons you were given were the definitions.
Citations from previous posts, please.

