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Topic: Is (t^2-9)/(t-3) defined at t=3?
Replies: 166   Last Post: Oct 30, 2013 9:41 AM

 Messages: [ Previous | Next ]
 magidin@math.berkeley.edu Posts: 11,749 Registered: 12/4/04
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Oct 8, 2013 12:43 AM

On Monday, October 7, 2013 7:52:32 PM UTC-5, Hetware wrote:
> On 10/6/2013 11:51 PM, Arturo Magidin wrote:
>

> > On Sunday, October 6, 2013 8:55:23 PM UTC-5, Hetware wrote:
>
> >> 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
>
> >>
>
> >>>>
>
> >>
>
>
> >>
>
> >>>>
>
> >>
>
> >>>>>
>
> >>
>
> >>>>
>
> >>
>
> >>>>> 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.

It wasn't a problem of order. The problem was that you were asserting assumptions without warrant.

> I was assuming continuity after stating the
>
> definition of the function.

For which you had no warrant.

> > 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.

And where do you think that axiomatic set theory came from? From some immortal? You said you did not accept things handed down to you from mortals... but apparently you do; so it wasn't so much that you were against accepting things "from mortals", but rather something else. Which was... what?

>
>
>

> > Again: get off the high horse; because you are being either ignorant,
>
> > or hypocritical.
>
>
>
> In this context /argumentum ad hominem/ is clearly a fallacy.

Sigh...

First: saying that you are being ignorant or hypocritical is not, in and of itself, an "argumentum ad hominem". Contrary to common misconception, "argumentum ad hominem" is not merely the saying of bad things about others.

In order for what I said above to constitute an "argumentum ad hominem", I would have had to *draw a conclusion* from the assertion. I would have to say something like "what you said was pompous, therefore your argument is invalid." I did no such thing.

Instead, I drew the *conclusion* that your assertion demonstrates either ignorance or hypocrisy. That's not an argumentum ad hominem, because I am not using this alleged "attach at the man" as my premise. It's my conclusion, based on the evidence you present.

You wrote: "I don't answer to the authority of mortals."

Yet you admit that you take as authority axiomatic set theory, which comes from, surprise, mortals. So... either you did not realize that you were answering to *some* authority "of mortals" (ignorance), or else you knew perfectly well you were, but decided to make a grandeloquent statement to stake a position you did not in fact hold (hypocrisy).

Again, that's not an ad hominem, because I'm not using those conclusions as premises.

Do learn the meaning of the phrases you decide to employ in the future.

> > 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.

Yes.

> I was attempting to understand why my mathematical intuition was
>
> telling me that I could treat (t^2-9)/(t-3) as identical to t+3.

You were trying to justify your erroneous conclusion based on your (admitted) error of assuming something was defined at 3 in order to justify that it is defined at 3 (circular argument). When this was explained to you, you replied that you "don't answer to the authority of mortals", rejecting the explanation in an attempt to justify your mistake.

Ergo, yes. You were in fact trying to justify your original mistake instead of accepting it.

> The exercise was well worth the effort.

The exercise was a waste of time because you refused to listen in the first place, and instead spent time saying pompous things like "I don't answer to the authority of mortals". It may have been worth your time and effort, but you could have saved yourself quite a bit of both if you hadn't been stubborn and ignorant.

(And, no, that's not an argumentum ad hominem either, because again, I am not trying to challenge your conclusions or draw inferences from the facts that you were being both stubborn and ignorant; I am merely stating an observation about your behavior).
>
> > Thomas distinguishes between regular functions and "mult-valued
>
> > 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 "multi-valued
>
> > function". The adjective signals that you are dealing with a
>
> > different notion.
>
>
>
> No, Thomas distinguishes between "single-valued" and "multiple-valued"
>
> functions.

And "single-valued function" is what I called "function", and "multiple-valued function" is what I called "multi-valued function." So, yes. Thomas has TWO distinct notions that you claimed were the same. Ergo, you were wrong. Oh, and by the way, I was right. The fact that both notions have "function" in it does not mean both refer to the same thing, just like two people with last name "Smith" are not necessarily the same person.

>
> > 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.

To he that will not listen, it always seems like nobody else is talking.

> The context was reasonable
>
> clear.

To the person who assumes it; and, as it turns out, your assumptions were all over the place an incorrect in many places, as you admitted.

> If you wanted clarification, you should have asked for it.

I did not need clarification to point out your error; you needed clarification in order to understand that you were in fact in error. So who was it exactly who needed to ask for help, then?

> >> 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.

There's no "rub". There's a convention that you decided to flout, and then complained that the conclusions drawn from using the convention did not match the conclusions you drew by flouting those conventions.

> It did make sense to me to treat the function
>
> as defined where t=3.

Because you made unwarranted and incorrect assumptions and ignored the standard conventions.

> Until I was able to sort out the axiomatic
>
> ordering of assumption,

Nonsense. There is not "axiomatic ordering of assumption."

You know, just writing down fancy words in semi-random order does not make you wise.

What you needed to realize is what the conventions were, and that you were making an UNWARRANTED assumption because you were ignoring those conventions and because you were making a circular argument (the function must be defined at 3 because it is continuous, and it is continuous because it must be defined at 3). Nothing to do with "axiomatic ordering of assumptions", everything to do with you making a mistake.

> I was confused by applying an assumption that is
>
> generally applicable in the domain where I apply mathematics.

And which is not an axiom and not an ordering of assumption. So no such thing as "axiomatic ordering of assumptions" (what immortal authority taught you about such a thing, anyway?)

>
>
>

> >> and the observation that
>
> >>
>
> >> (t^2-9)/(t-3) 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.

I'm once again pointing out that you are being sloppy and hence incorrect because you are misusing the terminology as it is defined. The fact that you fail to understand the difference between "f is continuous" and "f is continuous everywhere" is just one of the many places were you **CONTINUE** to make errors, despite your belief that you have discovered the One True Source Of All Your Mistakes.

> > (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.

I "caught on" well before you even realized you were wrong; but I realize that it's hard to get past your ego to get you to realize it.

> However, whether or not f(t) is defined at 3
>

> > is *precisely* the crux of the matter here.
>
>
>
> I defined it as continuous.

No, you ASSERTED, WITHOUT WARRANT, that it was continuous everywhere.

Again: the function f(t) = (t^2-9)/(t-3) **IS** "continuous", AND it is not defined at t=3. Because saying that the function is "continuous" only asserts that it is continuous at every non-isolated point in its domain, and that is the case with f.

What you actually assumed (not "defined") was that f was continuous EVERYWHERE. For a function f whose domain is not all real numbers,

"f is continuous"

DOES NOT mean the same thing as

"f is continuous everywhere"

Again: you are being sloppy, and that' just one of the reasons you continue to make mistakes.

>
> > 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.

You should teach your grandmother to suck eggs sometime.

> > Yes. The reasons you were given were the definitions.
>
>
>
> Citations from previous posts, please.

You mean, things like:

-- Begin Quote --

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. 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.

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.

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.

-- end quote --

Or, like my definition in the post you are replying to which you deleted, where I stated:

-- Begin Quote --

And again you run into trouble, because you insist on being careless.

A real valued function of real variable f(t) is "continuous at x=a" if and only if:

(i) It is defined at a;
(ii) The limit of f(t) as t approaches a exists; and
(iii) the value of the function at a equals the value of the limit of f(t) as t approaches a.

A real valued function of real variable f(t) is "continuous at all real numbers" or "continuous everywhere" if and only if it is continuous at x=a for all real numbers a.

A real valued function of real variable f(t) is "continuous" if and only if it is continuous at all points in its domain.

-- End Quote --

and which you deleted, and then continued to conflate "continuous" and "continuous everywhere" as if they meant the same thing?

What's the point of me giving you citations and quotes, if all you want to do is stroke your ego saying that *you* discovered your error (though you do not correctly identify what the error is), or that you don't answer to "authority of mortals"?

Simple fact: you screwed up; you screwed up because you ignored the conventions at play. Then you tried to justify your screw up through some rather impressive, but ultimately doomed, mental acrobatics. When this was explained to you clearly and simply, you got off on a high horse and made grand-elloquent but ultimately pompous and ignorant statements such as "I don't answer to the authority of mortals". Eventually, you conceded error, but were not willing or able to admit the explanations of others, and came up with an "explanation" of your error which is itself in error and inaccurate. Given that you have failed to adequately understand what your multiple errors are, it is unlikely that you will be able to proceed without committing even more in the future.

But thanks for playing.

--
Arturo Magidin

Date Subject Author
9/28/13 Hetware
9/28/13 Michael F. Stemper
9/28/13 scattered
9/28/13 Hetware
9/28/13 quasi
9/28/13 Hetware
9/28/13 quasi
9/28/13 Peter Percival
9/29/13 quasi
9/28/13 Hetware
9/28/13 Richard Tobin
9/28/13 Hetware
9/28/13 tommyrjensen@gmail.com
9/29/13 Hetware
10/6/13 Hetware
10/6/13 Peter Percival
10/6/13 Hetware
10/6/13 quasi
10/8/13 quasi
10/7/13 Peter Percival
9/29/13 Michael F. Stemper
9/29/13 Hetware
9/29/13 quasi
9/29/13 Hetware
9/29/13 magidin@math.berkeley.edu
10/6/13 Hetware
10/6/13 magidin@math.berkeley.edu
10/7/13 Hetware
10/7/13 LudovicoVan
10/7/13 Peter Percival
10/8/13 magidin@math.berkeley.edu
10/12/13 Hetware
10/12/13 fom
10/13/13 magidin@math.berkeley.edu
10/13/13 Richard Tobin
10/13/13 Hetware
10/13/13 Peter Percival
10/13/13 fom
10/13/13 magidin@math.berkeley.edu
10/13/13 magidin@math.berkeley.edu
10/8/13 quasi
10/8/13 magidin@math.berkeley.edu
10/8/13 quasi
10/8/13 quasi
10/12/13 Hetware
10/13/13 quasi
10/13/13 Peter Percival
10/9/13 magidin@math.berkeley.edu
10/9/13 fom
10/10/13 magidin@math.berkeley.edu
10/10/13 fom
10/7/13 Peter Percival
10/7/13 Hetware
10/7/13 fom
10/7/13 Peter Percival
9/29/13 quasi
9/30/13 Peter Percival
9/30/13 Peter Percival
9/30/13 Peter Percival
9/30/13 RGVickson@shaw.ca
9/30/13 Roland Franzius
9/30/13 Richard Tobin
9/30/13 RGVickson@shaw.ca
9/28/13 Peter Percival
9/28/13 Hetware
9/29/13 Peter Percival
9/28/13 Virgil
9/29/13 quasi
9/29/13 Virgil
9/29/13 Hetware
9/29/13 quasi
9/29/13 Hetware
9/29/13 LudovicoVan
9/29/13 quasi
9/29/13 Virgil
9/29/13 magidin@math.berkeley.edu
9/29/13 Peter Percival
9/29/13 FredJeffries@gmail.com
9/30/13 Hetware
9/30/13 magidin@math.berkeley.edu
10/6/13 Hetware
10/6/13 Peter Percival
10/6/13 Peter Percival
10/6/13 magidin@math.berkeley.edu
10/6/13 Peter Percival
10/6/13 magidin@math.berkeley.edu
10/6/13 David Bernier
9/29/13 Peter Percival
9/28/13 Hetware
9/29/13 Richard Tobin
9/30/13 Ciekaw
9/30/13 Robin Chapman
9/30/13 Virgil
9/30/13 LudovicoVan
9/30/13 LudovicoVan
10/6/13 Hetware
10/7/13 Robin Chapman
10/7/13 David Bernier
10/7/13 Hetware
10/7/13 LudovicoVan
10/8/13 Hetware
10/9/13 Peter Percival
10/9/13 Richard Tobin
10/7/13 Peter Percival
10/8/13 Hetware
10/8/13 Virgil
10/8/13 Hetware
10/9/13 magidin@math.berkeley.edu
10/9/13 Peter Percival
10/10/13 Ciekaw
10/9/13 Peter Percival
10/10/13 Tim Golden BandTech.com
10/13/13 Hetware
10/13/13 Peter Percival
10/13/13 Hetware
10/14/13 Peter Percival
10/13/13 Hetware
10/13/13 fom
10/13/13 Hetware
10/13/13 fom
10/14/13 fom
10/14/13 Hetware
10/14/13 magidin@math.berkeley.edu
10/14/13 magidin@math.berkeley.edu
10/14/13 Peter Percival
10/14/13 Hetware
10/14/13 quasi
10/16/13 @less@ndro
10/16/13 quasi
10/19/13 Hetware
10/19/13 quasi
10/19/13 Hetware
10/20/13 fom
10/20/13 quasi
10/20/13 Hetware
10/20/13 fom
10/20/13 Hetware
10/20/13 Peter Percival
10/20/13 Richard Tobin
10/20/13 Hetware
10/30/13 @less@ndro
10/19/13 Hetware
10/10/13 Ronald Benedik
10/10/13 Peter Percival
10/10/13 Virgil
10/18/13 Hetware
10/19/13 Peter Percival
10/19/13 fom
10/19/13 Peter Percival
10/19/13 Hetware
10/19/13 Peter Percival
10/19/13 Hetware
10/19/13 fom
10/19/13 magidin@math.berkeley.edu
10/19/13 Hetware
10/19/13 magidin@math.berkeley.edu
10/20/13 Hetware
10/20/13 quasi
10/20/13 quasi
10/20/13 Hetware
10/20/13 Peter Percival
10/20/13 magidin@math.berkeley.edu
10/20/13 Hetware
10/20/13 Arturo Magidin
10/20/13 Hetware
10/20/13 magidin@math.berkeley.edu
10/19/13 fom