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Re: A statement on what is wrong with standard calculus
Posted:
Feb 24, 1996 12:32 PM
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Jerry Uhl wrote:
>
> At 1:05 PM 1/31/22 +0000, nstahl@uwcmail.uwc.edu wrote:
> For instance, we don't know that someone who doesn't
> > know much algebra can learn much science.
> >
>
> Poincare and Einstein were very poor at algebra.
> And Archimedes didn't know any.
>
Geometry<->Algebra<->Arithmetic
[A lengthy contribution]
Tired to read over and over again the confused and confusing
debate about numerical <-> analytical and 'how much algebra
is needed' where mostly 'algebra' refers to anything but
algebra, I'll try to make more explicit my point of view
argumenting why not less but more algebra is feasable
using Computerdevices.
1) What I would define 'Algebra' (in most general terms)
Let A be some set (substitute :Naturals, Integers, Rationals
Reals, Complex) equipped with same binary and unary operations
(substitute addition, multiplication ....) which map this set
into itself. Asume that there are some distingushed elements
(substitute 0,1) and some properties of combining operations
[as those are mappings from the set into the set] (substitute
asociative, commutative, distributive ...)
then we may call the complex of set, operations, distingushed
elements and properties an algebraic structure.
Still we aren't doing algebra ! We just got the base structure !
Now asume another set -a set of names, constants and symbols- and
rules to form sentences out of these names and symbols (eg. 1+2, x+3,
x^n ....). Asume a base interpretation of this language, which maps
our constants onto the set of elements, the symbols onto the operations
of the above explained algebraic structure.
By means of this any sentence, which contains only symbols and constants
has a meaning (that is a value or corresponding element en A).
Now we may ask which of these sentences have identical meaning
and we will call them equivalent (say 1+2 equiv. 2+1 by means
of the properties of our algebraic structure).
Teaching how to 'prove' equivalence of this type of sentences,
extending the domain from natural numbers up to rationals (or reals)
is normaly covered by primary and the first years of secondary education.
Still, we aren't doing algebra but rather arithmetic, exploiting obviously
the algebraic properties of our structure.
In the next step we permit sentences, which contain not only constants and
symbols but also names. In first place it's not obvious to give those
sentences a meaning, in particular it's not obvious to extend the notion
of equivalence. So we define that we will consider two sentences with names
to be [universally] equivalent, iff for any substitution of names by
constants -uniform substition which means equal names by equal constants-
the resulting sentences are equivalent the base interpretation.
The definintion has far reaching consequences, as form now on we may
substitute any subsentence of a sentence by an universally equivalent
subsentence, 'knowing' that the original and the modified by substitution
will be equivalent. Inversely we 'know' that augmenting two equivalent
sentences both with the same sentence will preserve the original
equivalence. (Obviuos the inverse: if two sentences are not equivalent,
they can not be made equivalent, substituting ore augementing by equivalent
subsentences).
This definition of equivalence and its acompanying mecanics gives origin
to what we know as 'clever' symbol-shuffling, where on tries to reduce
two sentences by substitution and augmentation to sentences, where
in before we know or have proven that they are equivalent or not.
We might call this level 'primitive algebra'.
We now may restrict the concept of equivalence in the following way:
Asume two sentences and a subset of elements of our algebraic structure.
We define those sentences to be 'restricted' equivalent, iff for any substitution
of names using only elements of the restricted set, the resulting base-sentences
are equivalent. The pair of sentences we call equation and the restricted
subset the set of solution of that equation. (Some tecnical detail omitted,
as the definition of a more sofisticated substitution rule).
Asking inversely, whether there is a non-empty subset, which used as restriction
makes our sentences equivalent, introduces the concept of 'solving an equation'.
Now investigating/prooving which class of paired sentences has a solution and
deriving concepts of how to construct those clases or to prove that a given
class has always nonempty solution-sets brings us finally to true a l g e b r a.
2) Algebra and the computer
Obviously algebra relies havily on the mecanics of the primitive level. Obviously
again keeping track of substitutions and augmentations is by far easier
using the computer. Unfortunately f i n d i n g the apropriate substitution
is not always feasable by means of a computer ! Why ? Even for properly posed
problems (those which are decidable, Church, Turing and Post dedicated almost
there whole life to investigate the question which class of problems may
be properly posed), most 'substitution finding algorithms' are NP-hard or worse,
what means that the time to find a solution as function of the length of
the sentences increase more but polinomial. (Factorizacion for instances
is f a c t o r i a l, base of the security of all modern encryption algorithms).
This is a hard fact (Theorem) and not the result of insuficient research,
velocity of computers etc. etc. etc. (The barriers of complexity are as
insuperable as the velocity of light). Hence there is n o w a y the
either today or tomorrow the computer will render algebra (and knowlegde
of algebra of course) obsolete.
3) Teaching Calc with the aid of a computer
Some (like J.Uhl) will not get into arguments about 1+2 (they do have suficient
Math-Background), but will argue rather from a pedagogic point of few in favor
of reducing (I supose they never claimed eliminating) algebraic content
from calculus courses. With all respect & admiration I have for C&M -we will
use parts of their material in our courses-, I do have serious doubts, wether
this is both desirable and feasable to the extent Jerry is claiming.
Just to pose my questionmark simple: Consider the follwong problem: We're given
the initial conditions and the equation of the falling-body problem with
friction. Initial data is given en terms of seconds,inches and pounds as units of
measurement. The students use Euler to solve the problem numerically. Everything
works fine, til the teacher asks for conversion (Nicaragua is metric) to seconds,
meters and kilograms. What to do know ? [Not a hypothetic situation, as we adapted &
translated as student projects teaching material from the US & Great Britain].
T h e r e i s n o w a y to substitute algebraic/geometric reasoning by
pure number crunching to solve the conversion problem.
(I would like to enter here in a more filosofic/historic debate about
what meant the sucesive achievments of abstract geometry [Euclid]
and its algebraization, where metrics and units of mesurement are
carefully seperated from geometric invariants, which remain
intact applying the apropriate transformation groups [Riemann, Ricci,
Lie, Weyl, Dieudonne, Spencer ......] but time & space have run out already.
So just a cry: Don't reform away the content of 20 century science &
Mathematics )
Cornelio
_______________________________________________
Cornelio Hopmann Universidad Nacional de Ingenieria
cornelio@uni.rain.ni Facultad de Electrotecnia y Computacion
http://waltz.uni.rain.ni Avenida Universitaria, Managua Nicaragua
Apartado Postal 5595
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