[plt-scheme] The Lambda Calculus behind functional programming

On 8/31/07, Jos Koot <jos.koot at telefonica.net> wrote:
> Would it make sense to present a formal mathematical definition of a real number
> on primary school as a starting point for elementary arithmetics?

Starting with real numbers would probably be a bad idea.  But I do
think the peano axioms would be a good place to start.

Now, when I say that, I don't necicarly mean that kids have to learn
what the "upside down A" means (for all), and what the "backwards E"
means (there exists), etc.  You can teach anything isomorphic to the
symbols.  (for a LC example, see "the aligator eggs" link posted
earlier http://worrydream.com/AlligatorEggs/ ).

So the problems in the course would be essentially the same, just "tweaked".

I'm not the first person to think this.  I heard that they tried
teaching the axioms at the beginning years ago, and gave it up.  I
briefly looked into it a while back, and the reason appeared to be
that most grade school teachers didn't understand what they were
teaching.

As for math, "what is this good for?":  By knowing the axioms, the
student can check their own work.

Before I move on, I'll give a example isomorphism for some of the peano axioms:

Put up your right hand.  We will call your palm is "zero".
Put up some fingers.  Each finger will be called "successor".

Peano axiom:  You are allowed to "move" a finger from one hand to another.
     i.e. (Sx + y) = (x + Sy)

Peano axiom:  You are allowed to raise, or put down your left hand at
any time iff you have no fingers up on it.
     i.e. (0 + x) = x

You probably get the idea from here....



Similarly, I think we might be differing on how we imagine a
introductory functional programming course in LC.  The LC syntax can
be adjusted to look almost like the scheme syntax, and the problems in
the book could be the same problems.

so the "answer" to a problem in LC might be:
 (map (+ 2)  (list 2 3 4))

and the "answer" to the same problem in scheme:
(map (lambda (x) (+ 2 x))  (list 2 3 4))

So the way I see it, the course syllabus would mostly be unaffected by
the move to LC.  However with LC we now have the ability to reference
+ as something other then just a primative when the student asks.


Does that make sence to you?

(P.S. say yes, cause I've already implemented the scheme-like LC syntax. heh)

Corey


> I think that for most students LC is too abstract as long as they have not yet
> done some fp.
> I would have been such a student, I think.
> I try always to be prepared for the question "what is this good for?"
> When working with future programmers, an answer might be: "to better understand
> the essentials of computation" but this would not make sense to someone who does
> not yet have any idea of computation at all.
> Jos koot
>
> ((((lambda(x)((((((x x)x)x)x)x)x))
>    (lambda(x)(lambda(y)(x(x y)))))
>   (lambda(x)(write x)x))
>  'greeting)
> ----- Original Message -----
> From: "Corey Sweeney" <corey.sweeney at gmail.com>
> To: "Jos Koot" <jos.koot at telefonica.net>
> Cc: "Grant Rettke" <grettke at acm.org>; "PLT Scheme"
> <plt-scheme at list.cs.brown.edu>
> Sent: Thursday, August 30, 2007 4:39 PM
> Subject: Re: [plt-scheme] The Lambda Calculus behind functional programming
>
>
> > On 8/29/07, Jos Koot <jos.koot at telefonica.net> wrote:
> >> For me discovering Lambda Calculus (and combinatory logic) was fabulous in
> >> itself. It is beauty.
> >> Whether or not it makes you a better programmer. I am inclined to think so,
> >> but
> >> I don't have any kind of evidence,
> >> Anyway, when my friends ask me what I am doing these days, I have much
> >> trouble
> >> making an understandable answer.
> >> My conclusion:
> >> 1: Lambda calculus is not the starting point of learning programming, I
> >> think.
> >
> > I personally would tend to disagree with the idea of lambda calucus
> > not being a good starting point.  (well, technically typing would be
> > the starting point :) .  Why do you think it's not a good starting
> > point?  Is it the lack of a good IDE?  (which by the way, i have a
> > solution for ;) .   I would agree that paper and pencil is not a good
> > way to start.  But for the motivated learner, I think that the lambda
> > calculus with a few moddification could be a near optimal environment.
> >
> > What are other peoples thoughts?
> >
> > Corey
> >
> >> 2: At some stage you wonder: what are the essential principles of
> >> programming.
> >> 3: At that point lambda Calculus may be an answer.
> >> 4: Wait for your students to ask the question before answering it
> >> 5: Not all of us can claim to be as clever as Socrates, but asking questions
> >> is
> >> a good method to make your students ask the right questions.
> >> 6: I know I have failed on this many times
> >> Jos Koot
> >>
> >>
> >> ((((lambda(x)((((((x x)x)x)x)x)x))
> >>    (lambda(x)(lambda(y)(x(x y)))))
> >>   (lambda(x)(write x)x))
> >>  'greeting)
> >> ----- Original Message -----
> >> From: "Grant Rettke" <grettke at acm.org>
> >> To: "PLT Scheme" <plt-scheme at list.cs.brown.edu>
> >> Sent: Wednesday, August 29, 2007 8:27 PM
> >> Subject: [plt-scheme] The Lambda Calculus behind functional programming
> >>
> >>
> >> > When I tell people that I'm learning Scheme the first thing they ask
> >> > me is "So have you learned lambda calculus?". I've learned enough to
> >> > say the words "lambda calculus" and that everyone says that lambda
> >> > calculus is the theoretical backbone of functional programming; but
> >> > that is it.
> >> >
> >> > The following question is a simple one, it makes no attempt to
> >> > generalize whether anything is worth learning, I think you get the
> >> > idea.
> >> >
> >> > So to revisit this again, what do you need to learn of the lambda
> >> > calculus relative to FP?
> >> >
> >> > What would you tell folks about who have never looked at FP when they
> >> > ask you about lambda calculus?
> >> > _________________________________________________
> >> >  For list-related administrative tasks:
> >> >  http://list.cs.brown.edu/mailman/listinfo/plt-scheme
> >> >
> >>
> >> _________________________________________________
> >>   For list-related administrative tasks:
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> >>
> >
> >
> > --
> > ((lambda (y) (y y)) (lambda (y) (y y)))
> >
>
>


-- 
((lambda (y) (y y)) (lambda (y) (y y)))