DIFFERENTIABILITY AND CONTINUITY

[Nick Thompson]

Thank you, John,  It DOES help.  

Just as a check on my understanding: what are the "objects" of the calculus.

N


----- Original Message -----
From: John F. Kennison
To: nickthompson at earthlink.net;friam at redfish.com
Cc: Lee N. Rudolph; David Joyce
Sent: 7/26/2007 12:55:38 PM
Subject: RE: DIFFERENTIABILITY AND CONTINUITY





Nick,

Yes math does seem to have a remarkable unity. we agree on what a proof is and we agree on when a proof has been achieved. Sometimes, a new proof is annouced and for a while there are differences of opinions as to whether the proof is valid. But after suffucient study, a consensus forms and the math world is virtually unanimous.

It wasn't always like this. In the late nineteenth to early twentieth century (Dave Joyce can correct these dates) different schools (incuding a formalist and an intuitionist school) had different standards for what a proof was. (This may be the opposite of what you would expect, but the intuitionist school had the more difficulty standard to meet.) Now we have agreed on standards for a "proof"and we use the term "Intuitionistic proof"  for a proof that meets some difficult standards, which most of us regard as unreasonably difficult.

Many of your questions seem to really be asking, "What are the objects of Mathematical study?".   The intuitionistic svhool would say they are mental constructs (which makes it hard to say what is really true about them). The formalist school would say we study nothing but systems of formal rules. Present day Mathematicians would say we study nothing but sets  For example the real numbers is a set R, which comes equipped with operations such as addition. But even addition is a set A of triples (r,s,t) and when we sasy r+s = t, we are really saying that (r,s,t) is in A. so the commutative rule, that x + y = y + x says that if (x,y,z) is in a, them so is (y,x,z). There are two other
sets  that we need to give the structure of R, the set M (for multiplication) and the set P (of positive numbers) These three sewts plus about 20 axioms on their properties define the reals, All of calculus can be constructed from these sets. The nature of the axioms are intrinsically algebraic.

Finally, mathematricians think of their subject matter as being about abstract sets. A concrete, real-life situation may have similar properies as an abstracrt set in which casde we have a simulation, or model. It is the job of a scientist to find good models.

Hoped thisd helps, John











-----Original Message-----
From: Nicholas Thompson [mailto:[hidden email]]
Sent: Thu 7/26/2007 12:14 PM
To: John F. Kennison; friam at redfish.com
Cc: Lee N. Rudolph; David Joyce
Subject: RE: DIFFERENTIABILITY AND CONTINUITY

Thanks, John.  this is REALLY interesting.  I will also use this opportunity to respond to Lee's suggestion that I have retreated from the field of battle beaten and bloodied.

Well, Lee is to some extent correct.  I believe deeply in the usefulness of occasionally tieing bright people up with dumb questions, but there is a limit, and I try not to broach it.  And, I have learned a lot.   I have learned, for one thing, that algebra is more fundamental to you folks than I ever took it to be.   Lee is correct in his suspicion that I dont quite "get" why a simple logical account does not produce the mean valule theorem in three easy steps, but I think it is time for me to go away and think about it by myself or with the help of private tutors.  One glaring weakness in my suppositioin  was that the mean is itself an algebraic concept.

One thing I did not find, which i expected to, was a philosophic split amongst mathematicians, along some sort of dimentions such as inuitionist and formalist (I am making these words up).   You guys did close ranks to a remarkable degree, and as a psychologist, where there are 20 ideologies for every ten scientists, I found that quite remarkable.

there are still questions kicking aroiund in my haid   In what way is algebra NOT a simulation?   But unless somebody is really keen to go on educating me, perhaps we should save it for another day.

nick

----- Original Message -----
From: John F. Kennison
To: nickthompson at earthlink.net;friam at redfish.com
Cc: Lee N. Rudolph; David Joyce
Sent: 7/25/2007 6:39:10 PM
Subject: RE: DIFFERENTIABILITY AND CONTINUITY





Nick,

The gem that you found is well put, and it shows Lee's style. I have said that "algebra" is needed, but I find I would like to clarify that statement. Continuity and differentiability are defined in terms of the real number system, so any proof of any statement would have to go back to axioms we use for the real numbers. The simplest axioms are essentially "algebraic" in that they involve manipulating variables and operations, such as addition and multiplication. One could try to imagine a purely geometric set of axioms, but it is not clear (to me at least) what they would be like. My guess is that the proofs would be much harder. One attempt at a more intuitive foundation for calculus uses infinitesimals and some algebra --and some subtle logical maneuvres.

You raise the issue of pedagogy. We use different pedagogical approaches to meet different goals. We do want our students to leave calculus with an intuitive understanding of how it works --and an intuitive appreciation of its beauty.  For this purpose, appeals to intuition and geometry are emphasized. Since rigorous proofs usually fail to develop intuition, we do not spend all our time doing proofs. But another goal is that students understand that the results of calculus do have logical proofs and that intuition sometimes is misleading, so we do spend some times on proofs. And we are careful to distinguish intuitive plausibility arguments from logical proofs.  It probably seems to the students that most of our time is spent on proofs but to us it seems that relatively little time is spent on rigor.

--John


----Original Message-----
From: Nicholas Thompson [mailto:[hidden email]]
Sent: Wed 7/25/2007 2:09 PM
To: friam at redfish.com
Cc: John F. Kennison; Lee N. Rudolph; David Joyce
Subject: DIFFERENTIABILITY AND CONTINUITY



Deep down in the tangle of >>>>>'s I just found this gem.  The record is
two confused for me to know who to thank so I will thank you ALL.

> What you have given is the "handwaving" version of the proof. The
> trouble is that human imagination can easily get us into trouble when
> dealing with infinities, which is necessarily involved in dealing with
> the concept of continuity. In the above example, you mention that
> continuity is important, but say nothing about differentiability. Are
> you aware that continuous curves that are nowhere differentiable
> exist? I fact most continuous curves are not differentiable. By most,
> I mean infinitely more continuous curves are not differentiable than
> those that are, a concept handled by "sets of measure zero".

OK.  I AM BEING CALLED TO A MEAL AND YOU ALL KNOW WHAT HAPPENS WHEN ONE
DOESNT ANSWER THAT CALL.  BAD KARMA

AM I WRONG THAT BOTH CONTINUITY AND DIFFERENTIABILTY OF AT LEAST THE
primary FUNCTION ARE A PREMISE OF THE MEAN VALUE THEOREM.

MORE TO THE POINT,  ARE YOU ALL CONVERGING AROUND THE ASSERTION THAT THE
MEAN VALUE THEOREM CANNOT BE DONE WITH OUT ALGEBRA?  AS OPPOSED THE THE
VIEW I WAS ENTERTAINING THAT THE MEAN VALUE THEORY IS A LOGICAL PROOF THAT
IS REPRESENTED ALGEBRAICALLY FOR PEDIGOGICAL PURPOSES.

SORRY TO TWIST EVERYBODY'S KNICKERS ABOUT THIS.  BUT IRRITATING AS IT MAY
BE TO YOU ALL, THIS CONVERSATION HAS BEEN VERY HELPFUL TO ME.

NICK

nick
>
>
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