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Invisible wrote:
> I'm failing to see how equivalence and definition are different.
I'm thinking you have the choice of
x == y
x.isEqual(y)
and
(let x := y in (....)) or {const int x = y}
for equality, equivalence, and definition?
Two abstract structures can be equivalent without being equal. Definition is
an arbitrary assignment of a meaning to a formal name.
--
Darren New, San Diego CA, USA (PST)
I ordered stamps from Zazzle that read "Place Stamp Here".
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Darren New wrote:
> Invisible wrote:
>> I'm failing to see how equivalence and definition are different.
> Two abstract structures can be equivalent without being equal.
Oh, I see.
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On 11/17/09 09:50, scott wrote:
>> Sure, but not that much. I know people who've correctly used computers
>> to do this while they thought that the integral of a product is the
>> product of the integrals.
>
> Let's hope they're not involved in any safety critical projects then :-)
You're in the UK, and that may make the difference. I know that in the
distant past, it was significantly challenging to get admission into a
UK university to do engineering. So it would make sense if over there
it's common for engineers to use calculus. For lesser jobs, you have
people who don't have engineering degrees (technical degrees, associates
degrees, etc).
In the US, people with engineering degrees do both those kinds of jobs
- it spans a wider spectrum. Which is why perhaps most jobs here that
require an engineering degree often don't require calculus. Hence the
analogy with programming and computer science.
Sadly, here in the US it's common to have people graduating with a
bachelor's degree in engineering who've forgotten most of calculus (they
weren't bad at it - that person I mentioned never got less than an A in
any math course). So perhaps the industry in the US has somewhat shifted
not to expect them to know it, and thus most don't hire them for jobs
that require it.
Of course, some jobs simply *require* it (e.g. electromagnetics related
jobs). And they do hire people with BSc degrees. But of course, they
make sure they know calculus in the interview.
My experience is limited to electrical engineering. It may be different
in other engineerings.
--
Be independent! No, not that way! This way!
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scott wrote:
>> As far as I can tell, the majority of people in the world have jobs
>> like... Telemarketing. Driving trucks. Working in an office doing filing.
>> Fitting central heating systems. Hotel receptionists.
>
> Yes, the majority of people do a wide variety of jobs :-)
And most natural numbers are very very big.
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On 17-11-2009 10:20, Invisible wrote:
>> * equivalence, equality, definition, EXNOR, assignment and perhaps one
>> or more that don't have names.
>
> I'm failing to see how equivalence and definition are different.
If you say that f(x)=x^2+4x+1 then that is an equivalence, because that
will allow you to conclude say that x^2+3x+3=f(x)-x+2. If, however, this
is the first time you mention f(x) it is probably meant as it's
definition. Difference is subtle and has mathematically no significance,
but it feels different. Just as x=5 seen as a statement that the
solution to the problem is when you take x to be 5, feels different from
viewing = as a boolean function of two variables, x and 5.
>(I know some people use the triple-line symbol for this though.)
>
> The main confusion is between assignment and equality, generally.
That is why most languages have separate symbols for both meanings (and
why I am advocating imperative languages without assignment).
> Or, in mathematics, between a test for equality and a statement of equality.
Can you give an example of that?
>> ** See also the concept of '=' in OO languages. Are two objects the
>> same if all fields are the same?
>
> You would first have to define the concept of fields being "the same". ;-)
spot on
> Fortunately, in a pure functional language, the question becomes a lot
> simpler.
Not so sure about that. Anyway even if that would be the case it doesn't
solve the 'problem' that in real life you have multiple views to look at
an equation.
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> Sadly, here in the US it's common to have people graduating with a
> bachelor's degree in engineering who've forgotten most of calculus
Yeh see I don't think that could happen here. My course was 4 years (as are
most Engineering degrees in the UK now), in the 1st year they *really*
quickly skipped over all A level maths (which was the foundations of
calculus) and then jumped on to fourier transforms (and friends), vector
calculus and stuff. The reason for this was clear, all the other lecturers
were obviously waiting for the maths course to be done so they could start
using the "real" maths in their courses.
> any math course). So perhaps the industry in the US has somewhat shifted
> not to expect them to know it, and thus most don't hire them for jobs that
> require it.
I don't think many jobs here explicitly require it, but it is kind of
expected that if you have an Engineering degree you will at least know how
to perform basic calculus by hand, and of course know the concepts.
I guess it's similar to whether you know how to write Excel macros or not -
hardly any job is going to ask for it, yet if you know how to do it it's
going to make life quite a lot easier in many situations that arise (and
make your boss think you are clever!).
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Nicolas Alvarez wrote:
> And most natural numbers are very very big.
Unnatural numbers FTW!
7
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>> (I know some people use the triple-line symbol for this though.)
>>
>> The main confusion is between assignment and equality, generally.
>
> That is why most languages have separate symbols for both meanings (and
> why I am advocating imperative languages without assignment).
Heh. How many C programs fail because people use "=" instead of "=="?
(Most strongly-typed languages manage to catch this mistake. C is
deliberately designed to make this mistake a valid program construct.
Friggin' weirdos...)
>> Or, in mathematics, between a test for equality and a statement of
>> equality.
>
> Can you give an example of that?
If I say "x = sqrt(y/z)", do I mean that x *is* equal to this RHS? Or is
it an equation that must be solved by *making* x equal to the RHS?
Similarly, this fragment can appear inside a statement, such as "f(x) =
3 if x = sqrt(y/z), otherwise 9". In that case, the "x = sqrt(y/z)" part
is clearly a conditional test.
So "x = sqrt(y/z)" can mean:
- x *is* equal to this.
- you must *make* x equal to this.
- please *test* whether x is equal to this.
As you point out, there's also the distinction between "x just happens
to equal this thing" and "x is defined as being this thing".
>>> ** See also the concept of '=' in OO languages. Are two objects the
>>> same if all fields are the same?
>>
>> You would first have to define the concept of fields being "the same".
>> ;-)
>
> spot on
Hence deep-equal and shallow equal, value-equal and reference-equal.
>> Fortunately, in a pure functional language, the question becomes a lot
>> simpler.
>
> Not so sure about that.
If data cannot be altered, it is not possible to distinguish between two
objects which are reference-equal and two objects which are merely
value-equal.
The question of deep/shallow still arises, and you still have to decide
whether you mean "identical" or just "equivilent" (which will generally
depend on what the data *means*). But you don't have to worry about
reference-equal any more, which is a significant simplification.
(Until you start dealing with mutable data. In that case, since equality
is a pure operation, it is natural that you should consider only
reference-equality. Indeed, Haskell's type system *demands* this.)
> Anyway even if that would be the case it doesn't
> solve the 'problem' that in real life you have multiple views to look at
> an equation.
Indeed.
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On 11/18/09 02:10, scott wrote:
>> Sadly, here in the US it's common to have people graduating with a
>> bachelor's degree in engineering who've forgotten most of calculus
>
> Yeh see I don't think that could happen here. My course was 4 years (as
> are most Engineering degrees in the UK now), in the 1st year they
> *really* quickly skipped over all A level maths (which was the
> foundations of calculus) and then jumped on to fourier transforms (and
> friends), vector calculus and stuff. The reason for this was clear, all
Over here, while many do it in high school, almost all engineering
programs have up to 3 semesters of calculus in undergrad (sometimes one
can "test out" of those if their high school education was good enough).
I've been to both a low ranked university (undergrad) and a top
engineering university (grad). In both places, I complained that the
upper level (undergrad) engineering courses rarely required the students
to be able to solve differential equations or perform integrals. The
exception was electromagnetics, where you just can't do without them.
And some courses would require it for calculating Fourier
coefficients/transforms, but those weren't challenging integrals. In all
other courses where, say, a diff eq would crop up, the professor would
almost always say - "Let's not get distracted by the math - this is an
engineering course. Here's the solution: You can verify it by plugging
it back into the diff eq."
It probably wasn't always this way.
--
I didn't know my husband drank until one day he came home sober.
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"Fredrik Eriksson" <fe79}--at--{yahoo}--dot--{com> wrote:
> On Mon, 16 Nov 2009 13:39:50 +0100, Invisible <voi### [at] dev null> wrote:
> > Suppose that the first derivative of f(x) is g(x).
> >
> > What the hell is the derivative of f(f(x))?
>
> http://www.google.com/search?q=What+is+the+derivative+of+a+composite+function%3F
>
> http://en.wikipedia.org/wiki/Chain_rule
>
> Basic high-school level math. I am slightly surprised that you did not
> know it, though I am not in the least surprised that you did not bother to
> google for it.
>
Already knowing the answer could mean already being familiar with the terms
"composite function" and "chain rule". It wasn't necessarily a stupid question.
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