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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

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Hetware

Posts: 148
Registered: 4/13/13
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Oct 7, 2013 8:52 PM
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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
>>
>>>>
>>
>>>>> 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^2-9)/(t-3) 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 "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.

"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 /single-valued/ function of x.
If several values of y correspond to each single value of x, then y is
called a /multiple-valued/ function of x."

> Really. It's not that hard, once you stop trying to justify your
> errors.
>


Perhaps you were thinking of multi-variable calculus, such as is treated in

http://www.amazon.com/Advanced-Calculus-Several-Variables-Mathematics/dp/0486683362


or

http://www.amazon.com/Introduction-Vector-Tensor-Analysis-Mathematics/dp/B00A19P4EW/ref=sr_1_2?s=books&ie=UTF8&qid=1381192056&sr=1-2


or

http://www.amazon.com/Space-Matter-Dover-Books-Physics-ebook/dp/B00C59C6JM/ref=dp_kinw_strp_1

or

http://www.amazon.com/Theoretical-Physics-Dover-Books-ebook/dp/B00C8UR0B2/ref=pd_sim_kstore_2

or

http://www.amazon.com/Feynman-Lectures-Physics-boxed-set/dp/0465023827/ref=sr_1_2?s=books&ie=UTF8&qid=1381192262&sr=1-2&keywords=feynman+lectures+on+physics

or

http://www.amazon.com/Gravitation-Physics-Charles-W-Misner/dp/0716703440/ref=sr_1_1?s=books&ie=UTF8&qid=1381192296&sr=1-1&keywords=gravitation+misner+thorne

or

http://www.amazon.com/Tensors-Differential-Variational-Principles-Mathematics/dp/0486658406/ref=sr_1_1?s=books&ie=UTF8&qid=1381192367&sr=1-1&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^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.

> (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.



Date Subject Author
9/28/13
Read Is (t^2-9)/(t-3) defined at t=3?
Hetware
9/28/13
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Michael F. Stemper
9/28/13
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Hetware
9/28/13
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quasi
9/28/13
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Hetware
9/28/13
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quasi
9/28/13
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Peter Percival
9/29/13
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quasi
9/28/13
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Richard Tobin
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Hetware
9/28/13
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9/29/13
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Hetware
10/6/13
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Hetware
10/6/13
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10/6/13
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Hetware
10/6/13
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quasi
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magidin@math.berkeley.edu
10/6/13
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Hetware
10/6/13
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magidin@math.berkeley.edu
10/7/13
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Hetware
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LudovicoVan
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Peter Percival
10/19/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
fom
10/19/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
Peter Percival
10/19/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
Hetware
10/19/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
Peter Percival
10/19/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
Hetware
10/19/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
fom
10/19/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
magidin@math.berkeley.edu
10/19/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
Hetware
10/19/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
magidin@math.berkeley.edu
10/20/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
Hetware
10/20/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
quasi
10/20/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
quasi
10/20/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
Hetware
10/20/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
Peter Percival
10/20/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
magidin@math.berkeley.edu
10/20/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
Hetware
10/20/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
Arturo Magidin
10/20/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
Hetware
10/20/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
magidin@math.berkeley.edu
10/19/13
Read Re: Is (t^2-9)/(t-3) defined at t=3?
fom

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