NON STANDARD ANALYSIS AT PRE-UNIVERSITY LEVEL
– NAIVE MAGNITUDE ANALYSIS –
RICHARD O’DONOVAN AND JOHN KIMBER
Abstract. Non standard analysis can be introduced in a pre-university mathematics course. Examples are given of how an intuition of infinitesimals leads to a
mathematical concept. Using infinitesimals, students acquire the fundamental idea
of a differential which in turn introduces derivatives and integrals.
At this level, compared to epsilon-delta analysis, the non standard analysis approach allows for greater simplicity.
Introduction. In Collège André-Chavanne in Geneva, Switzerland, non
standard analysis is used at a pre-university level.1 For the mathematician,
it is interesting to observe how words encountered in everyday language,
such as “smaller than”, “infinitely close” or “infinitely large” lead to a
more refined and rigorous description of fundamental ideas. Here is a presentation of how infinitesimals are used to introduce the basic concepts of
analysis: differentials, derivatives and integrals. Comments are given about
the students’ response.
Workshop exercises are used throughout the course; these are the driving
force and motivation provided to illustrate and introduce new fields of
investigation. Theory is a result of discussions following the exercises which
introduce a concept. This type of pedagogical approach works well with
non standard analysis. Some of these exercises are given as examples.
At this pre-university level, ultrafilters are not mentioned but neither are
Dedekind cuts in an epsilon-delta analysis course. At this stage, “analysis”
whatever the approach, could be called “naive analysis”. Non standard
analysis at this level, emphasises simplicity.
At the AMS-UMI meeting in Pisa (June 2002) Richard O’Donovan received many
helpful remarks by Jerome Keisler, Peter Loeb, Nigel Cutland, David Ross, David Ballard, Mauro di Nasso, John Bell and Renling Jin. Many of their suggestions have been
included in this article.
1 For a short description of the mathematics curriculum in Geneva see appendix A
Meeting
Edited by Unknown
c 1000, Association for Symbolic Logic
°
1
2
RICHARD O’DONOVAN AND JOHN KIMBER
§1. Complexity. A difficulty systematically encountered by students
is the order in which quantifiers are used. It often requires some time for
them to realise that ∀x∃y is not the same as ∃y∀x.
An interesting link between logic and pedagogy is illustrated by the
prenex form of a formula. In [10] the authors quote the prenex rank of
a formula as the number of alternations of quantifiers, with rank 0 when no
alternation occurs. The higher the rank, the more difficult it is for students
to grasp the meaning and, the authors add, “A formula of prenex rank 4
would make any mathematician look twice.” Considering the prenex rank
as some measure of conceptual difficulty, analysis with infinitesimals rates
better than analysis with epsilons and deltas.
Example: the formula “∀² > 0 ∃δ > 0 ∀x ∈ A | x − x0 |< δ ⇒
| f (x) − f (x0 ) |< ²” which defines continuity at x0 ∈ A is of prenex rank 2,
whereas the formula “∀x ∈ A x ' x0 ⇒ f (x) ' f (x0 )” is of prenex rank
zero.
Other rankings are conceivable. The number of alternations of the variables, for example “which of δ or ² comes first?”, “which of | x − x0 | or
| f (x) − f (x0 ) | comes first?” These alternations cause many errors.
Another ranking compares the number of signs per theorem and per
proof. The more sophisticated the theorem, the greater the difference in
number of signs between standard and non standard proofs – again to the
advantage of non standard analysis.
This could be a good reason to use non standard analysis at pre-university
level.
§2. Making infinitesimals appear. The following exercises are introduced in class with no prior theory or definitions. The students, therefore,
do not necessarily know what kind of answer is expected of them. Finding
a correct answer is not the main goal of this type of exercise. Its purpose
is to start a discussion where students will first rely on their intuition, then
gradually – through a question and answer process – refine their ideas and
eventually feel the need for formalisation.
Exercise 1:
Hold up your pen. Now drop your pen. At first, the speed is zero
(there is no speed) then . . .
What is the first non-zero speed?
Students never mention the idea that there is no first non zero speed.2
They do, however, attempt to write down the concept of a first non zero
2 The non existence of a first non zero speed – even in ∗ R – is discussed at a later
stage when a working knowledge of infinitesimals has been acquired.
NON STANDARD ANALYSIS AT PRE-UNIVERSITY LEVEL
3
speed symbolically. One common claim is 0.01. To the students, it “feels”
smaller than any real positive number.
But what then would 0.012 be? Sometimes the answer 0.0 01 is suggested
by students. It is quite
interesting to notice that it is a purely symbolic
√
answer. Of course 0.01 will prove to be an insoluble problem to them;
thus 0.01 cannot be kept as a decent notation for an infinitesimal – though
certainly not a stupid suggestion.
Another common suggestion is 10−∞ . The use of the infinity symbol is
not surprising: it does appear as part of everyday culture – which does not
mean that the mathematical meaning of the symbol is understood. That 1,
2, 3, etc. goes on indefinitely seems an obvious fact to most students, probably since primary school when children are challenged with the question:
“how far can you count?” For some of them, indefinitely means infinity,
whereas for others the limit is when you die... The concept of infinitesimals,
on the other hand, does not appear with the same ease and requires more
thinking.
The following exercise helps students make the link between infinitely
small and infinitely large.
Exercise 2:
Take 11 ; 12 ; 31 ; 14 ; 51 . . . and go on forever. Where does it go to?
– because it must get somewhere . . . (just an informal restatement of
the compactness theorem.)
No student suggests that it would reach zero.
Even though the students’ first intuition is that counting can go on to
infinity, infinity is not thought of as a number but rather as an unfinished
process. On the contrary, the infinitely small, situated between numbers
(which – for them – are already there), once accepted, are considered quite
naturally to be numbers. This in turn allows the infinitely large to be
accepted as numbers.
The following exercise gives an immediate perception of what can be
done with these new number-like objects. At this stage, students have no
idea of what a derivative is.
Exercise 3:
Physicists have found that a falling object satisfies d(t) = 12 gt2 where
d(t) is the vertical distance after t seconds and g ≈ 10[ sm2 ]
How far has the object fallen in 3 seconds?
What is its speed after 3 seconds?
Of course the average speed 15m · s−1 is suggested as a first answer;
however, on second thought, students realise that after 1 second the object
has only fallen 5m thus the average speed will not do. As a next step,
some students think of calculating the average speed over the last instant
4
RICHARD O’DONOVAN AND JOHN KIMBER
– which they note as x because, as they say “that is the name we give to
unknown values.” The computation follows the rule that the average speed
0)
between t0 and t1 is d(t1t1)−d(t
where t1 = 3 and t0 = 3 − x. The result is
−t0
(30 − 5x)m · s−1 .
A discussion takes place in class about the following statement: If an
instant x is smaller than any measurable duration, then a speed (30 −
5x)m · s−1 is not measurably different from 30m · s−1 .
Everything is here: a derivative, an infinitesimal, infinite closeness and
even the concept of the standard part.
A colleague, attending a class, to become acquainted with teaching non
standard analysis, observed that nothing the students do is contrary to
their intuition. This reinforces the idea that mathematics need not stand
out as that part of philosophy where intuition is systematically proven to
be wrong. It is not an absolute necessity that students should “hate the
subject”.
§3. Definitions and proofs. The question of considering zero as an
infinitesimal or not was discussed more than once by the authors. In spite
of the fact that zero is considered in many articles as an infinitesimal, it
seems that this approach is less appealing from a pedagogical point of view.
The following definition is now used throughout the course.
Definition: An infinitesimal is a number which is less in modulus than any
positive real number, yet not zero.
This approach appears to be safer and closer to intuition. Zero is a
real number, infinitesimals are “new” numbers, zero is not new, therefore
zero is not an infinitesimal! In other words: 0 ∈
/ IS (where IS is the set
of infinitesimals, IS+ the set of positive infinitesimals and IS− the set of
negative infinitesimals) and whenever δ ∈ IS it is possible to divide by
δ. There is a distinct advantage in allowing division by all infinitesimals,
baring in mind that division by zero is precisely one of the main pitfalls of
analysis.
Another pedagogical difficulty arises when considering variables and constants. δ ∈ IS is usually an undetermined infinitesimal, intrinsically a
variable, this collides with fact that zero is a uniquely defined number. By
stressing “infinitesimal or zero” whenever necessary, the teacher emphasises
this dual aspect.
Two fundamental ideas related to the above discussion are those of ”getting infinitely close to” and ”reaching” which are confusing for students.
This can be seen in most examples of convergence. A function represented
by a straight line is not easily considered as converging to the line, precisely
because it reaches it.
The idea of closeness is formalised using the following concept:
NON STANDARD ANALYSIS AT PRE-UNIVERSITY LEVEL
5
Definition: a ' b ⇔ a − b ∈ IS or a − b = 0
The ' sign is read “equivalent to”, merging into a new concept infinite
closeness, and equality – or in a dynamic view, the ideas of approaching
and reaching – which appear in their own right as distinct concepts.
As the students have already used “infinitely smaller” and “infinitely
larger”, it is necessary to formalise the definitions.
Definition: If ab ∈ IS then a is infinitely smaller than b – which leads to
the converse: b is infinitely larger than a.
a
a
b ∈ IS ⇒ b 6= 0 ⇒ a 6= 0 (by definition of infinitesimals) so it cannot be
said that 0 is infinitely smaller than 1 – or that 1 is infinitely larger than
0, implying that 1 could be zero multiplied by an infinitely large number.
Once again, everything is set up so that students are not even remotely
tempted into wanting to divide by zero. If a is infinitely smaller than b
then a is said to be “invisible” at the scale of b. “The visible part of a” is
used as an informal statement for the standard part of a. “The visible part
of a at the δ-scale” is what is left after discarding all the additive terms
infinitely smaller than δ: i.e. it is what would be the standard part after a
division by δ.
Cancellation is traditionally done by crossing out. Invisibility is indicated
by square brackets.
2 + 4δ +
2+
[δ 2 ]
|{z}
invisible at the δ-scale
2
[4δ + δ ]
| {z }
invisible at the standard scale
The definition of a standard part is introduced later. The words “visible” and “invisible” appear to be fully operational metaphors enabling the
students to start working more formally on the following exercises:
Exercise 4:
Prove:
∀x ∈ R∗ ∀δ ∈ IS x · δ < 1
Hint: Otherwise, assume x · δ = b where b is in R and show that δ
would be in R.
Exercise 5:
Prove that if ω is infinitely larger than some real number then it is
infinitely larger than any real number.
The infinitely large (IL ) are defined to be all numbers infinitely larger
than some real number.
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RICHARD O’DONOVAN AND JOHN KIMBER
§4. Standard part. The ∗ R symbol is used for hyperreals but aside
from that, the left star notation is never used. Theorems are about real
functions, thus inputs are either real numbers or hyperreals explicitly written in the form x + δ (with x ∈ R and δ ' 0). The syllabus does not require
the study of the difference between continuity and uniform continuity, so
x, y ∈ ∗ A (as in [2]) is not a required statement.
The definition of the standard part relies on the existence and uniqueness
of a real number equivalent to a given hyperreal. At this pre-university
level, finite hyperreals are defined as numbers of the form a + δ for a ∈ R
and δ ' 0 thus the existence of a real number equivalent to a given finite
hyperreal is a consequence of this definition; its uniqueness is probably
the first acquaintance of students with uniqueness theorems. The notation
st(a + dx) = a is used.
§5. Magnitudes and scales.
Exercise 6:
Draw a straight line scaled from -4 to 4 with 2 squares per unit. On
this line, plot δ , −2 + δ , 1 + 105 · δ [for δ ∈ IS+ ]
1
Redraw this line after magnification by a factor of centred on 0 [for
δ
the same δ ∈ IS+ ]
On the second drawing, what is 2 squares to the left of 0? What is 4
squares to the right of 0? Plot 3δ on this line. Plot√ δ 2 on this line.
How many squares are there between 0 and 1? Plot δ on the line.
At this point most students are hesitant because infinitesimals are considered to be invisible and the objective here is to make these invisibles
visible! The students are asked to look at the Alps 80 kilometres away and
visualise adding a snowball to the top of the highest mountain.3 Would
the difference be visible from where they stand? Would the difference be
visible through a sufficiently powerful telescope? They easily accept that
what can be seen depends on the scale√of observation.
When attempting to draw δ 2 and δ, students realise that there are
different magnitudes of infinitesimals, formalised by the following:
Definition: If ab is finite but not infinitesimal nor zero, then a and b are of
the same magnitude.
This definition, together with those of infinitely smaller and infinitely
larger, leads to δ-equivalence:
Definition: For δ ∈ IS, two numbers a and b are δ-equivalent if their
difference is infinitely smaller than δ or if they are equal. Written
a 'δ b
3 Mont
Blanc, 4807m
NON STANDARD ANALYSIS AT PRE-UNIVERSITY LEVEL
7
where the subscript is mentioned explicitly for a given scale. Thus the
scale of dx would be characterised by dx-equivalence: 2+dx 'dx 2+dx+dx2
The symbol “'” without a subscript refers to equivalence at the standard
scale.
Now the students are ready for two dimensions.
§6. The differential and the derivative. The next step for the students is to understand the differential which in this approach is a key concept.
Exercise 7:
Let f : x 7→ x2 , g : x 7→ x3 , and h : x 7→ |x| be real functions and
δ ∈ IS+
1
Zoom in at a magnification factor of , centred on < 1; 1 > of f , g
δ
and h.
1
Zoom in at a magnification factor of , centred on < 0; 0 > of f , g
δ
and h.
Before this lesson, the only observation that the students could have
made was that these functions are equal at some points in the sense that
f (1) = g(1) = h(1) (same for 0). They now realise that something more
can be observed.4 If an interval of size dx can be drawn, then an interval
of size dx2 can neither be seen nor drawn at the dx-scale. The functions
f and g appear to be almost straight lines in the interval. The modulus
function doesn’t share this newly observed characteristic.
The next step in class is formalisation. As the differential is considered
fundamental it is defined before the derivative.
∆f is defined as f (x + dx) − f (x) as usual in non standard analysis.
If, through a dx-zoom on < x0 ; y0 > , the plot is infinitely close to a
straight line then the function is said to be locally (no steps, no “angular”
singularities). If the function is locally then dy is defined as the visible part
of ∆y at the dx-scale (otherwise, dy is not defined at x0 .) The derivative
y 0 = dy/dx can be described as the slope of the line which is visible at the
dx-scale. It is the average slope of the curve on an infinitesimal interval.
The fact that dx and dy can be drawn is the fundamental reason to define
dx and dy as basic objects.
Then y 0 = st (∆y/dx) is shown to be an equivalent definition of the
derivative.
4 “Zoom in” now appears to be used both for microscope close-ups and for telescope
views probably due to the widespread use of video cameras, special effects and computer
programs.
8
RICHARD O’DONOVAN AND JOHN KIMBER
f (x + dx)
¶H
YH
difference between ∆f (x) and df (x)
¶
invisible at this scale
¶
¶
¶
¶
∆f (x)
¶
¶
¶
f (x)
¶
¶
dx
x
x + dx
Figure 1. A locally linear function
These concepts are introduced with polynomials and rational functions,
thus the expansion of f (x + dx) is simple. For y = x3 students compute
∆y = 3x2 ·dx+3x·dx2 +dx2 hence dy = ¡3x2 ·dx and dy/dx = 3x2 . With¢the
equivalent
definition, st
= st (3x2 · dx + 3x · dx2 + dx2 )/dx =
¡ 2
¢ (∆y/dx)
2
2
st 3x + 3x · dx + dx = 3x (for x ∈ R).
Figure 1 is fundamental and students must be able to draw it for any
function placing the various numbers relative to a given dx-scale. Though
simple, this sketch contains all the fundamental ideas of analysis with infinitesimals. It is also fundamentally different from what is rigorously acceptable in ²-δ theory.
The pedagogically important point is that the slope of a function is the
slope of a line through two points (infinitely close). The tangent line is
otherwise usually defined as the limit of secants. Students often have a
problem with this definition since, when the two points meet, the tangent
can twirl around the point and it no longer qualifies as a “tangent”. The
metaphor is that of a line resting on two supports. As the tangent is about
to be defined, it disappears. This anomaly is avoided with non standard
analysis: with two points to rest on, the tangent remains firmly anchored.
These ideas are summarised in two basic equations:
(1)
dy = y 0 · dx
and the microscope equation given by Stroyan [7].
(2)
f (x + dx) = f (x) + df (x) + ² · dx
with ² ' 0
provided that the function is differentiable, of course.
Theorems about derivatives are generally simple and their proofs a direct
consequence of the concept of differential. For example, the proof concerning the derivative of the composition of functions can be obtained by the
NON STANDARD ANALYSIS AT PRE-UNIVERSITY LEVEL
9
systematic use of (1). There is no need to add and subtract convenient
terms (or other white-rabbit-out-of-a-hat performance). Students can then
discover the proof on their own. It is sufficient to assume that u and v are
differentiable, that u depends on v and v depends on x, therefore
d(u ◦ v) = u0 · dv = u0 · v 0 · dx
Dividing by dx, which is not zero since the independent variable is chosen
in IS yields the well known result. This works even if dv = 0, unlike the
du
dv
frequently used du
dx = dv · dx . This elegant proof often encourages other
teachers to consider non standard analysis as a worthwhile approach.
Proofs also often use Thompson’s simple approach [9]. Once the theorems
about derivatives have been proven, many exercises follow the same path
as in any other analysis course. There is, however, a major change in the
syllabus. Integration can be introduced shortly after the derivative.
§7. Integrals. The concept of integrals is introduced in class by considering areas under curves. The students are made aware of the fact that
the function describing an area is assumed to exist. The differential of this
function yields the concept of antiderivatives. The area is then shown to
be a sum of rectangles, where areas of rectangles are assumed to be well
defined. This confirms the existence of an area function, which in turn
proves the fundamental theorem of calculus.
Exercise 8:
Let f : x 7→ x2 Assume that A(x) is the function which describes the
area under the curve of f between 0 and x.
Sketch the curve of f and indicate where ∆A(x) would be. Zoom in
at < x; f (x) >.
Calculate ∆A(x) and dA(x) and show where these are on the drawing.
Students are asked to generalise to any continuous function, which leads
to figure 2.
It follows that
∆f (x) · dx
∆A(x) 'dx f (x) · dx +
2
Since f is assumed to be continuous, ∆f (x)·dx
is invisible at the dx scale,
2
thus
dA(x) = f (x) · dx
and therefore
A0 (x) = f (x)
This is probably one of the easiest ways to show the students that the
area under a curve is given by the inverse operation of finding the derivative:
10
RICHARD O’DONOVAN AND JOHN KIMBER
f
A(x)
a
x
f(x+dx)
b
∆ f(x)
f(x)
dx
∆ A(x)
Figure 2. Area under the curve defined by f
the antiderivative. In many ²-δ courses figure 2 is used as a metaphor to
illustrate the meaning of the fundamental theorem.
Exercise 9:
Partition the interval [a; b] into pieces dx long (dx ∈ IS). Put all these
pieces together again (add all the dx). What is the result?
Apply figure 2 to each of these pieces. Add all these pieces together
again. What is the result?
P
The “ ” sign indicates sums with an integer increment. It is re-interpreted here by indicating the step in the subscript.
b
X
x=a
by steps of dx
The transfer theorem is not used to show that sums can be extended to
a hyperfinite number of terms. Here also, the extension of the concept of
sum follows what intuition expects it to be.
NON STANDARD ANALYSIS AT PRE-UNIVERSITY LEVEL
11
The following discussion ensues: On an interval [a; b] on which the funcPb
tion is increasing, the area is greater than x=a f (x) · dx and less than
Pb
x=a f (x + dx) · dx
A close study of the endpoints through a dx-zoom shows that the end
Pb−dx
slice goes from b − dx to b and these sums are in fact
x=a f (x) · dx
Pb−dx
f
(x
+
dx)
·
dx.
Students
can
then
transform
the
latter into
and
x=a
Pb
x=a+dx f (x) · dx. The difference between the two sums is
b
X
x=a+dx
f (x) · dx −
b−dx
X
f (x) · dx = f (b) · dx − f (a) · dx = (f (b) − f (a)) · dx
x=a
Since f (b)−f (a) is a standard number, the difference between the two sums
is a standard number multiplied by an infinitesimal, hence an infinitesimal
or zero.
The same holds for closed intervals on which the function is decreasing.
So the area is squeezed between two infinitely close numbers, therefore
Pb
A ' a f (x) · dx (Adding an extra slice, so that the endpoints are a and
b, only adds an infinitesimal quantity.) This number is not necessarily
standard.
Therefore the area is redefined to be the standard number:
à b
!
Z b
X
by definition
A = st
f (x) · dx
=
f (x) · dx
a
a
Z
Z
=
Z
tandard part of the
um
As f (x) · dx = dA(x) then
Z
A=
b
dA(x)
a
In Greek Antiquity: “the sum of the parts is equal to the whole”. It was
not: the whole is the limit of the sum of the limits of its parts.
As a student put it: “seeing the area as a sum of slices does not enable
us to compute the value. It shows what is happening. To compute the
value, we need to study the variation of the area.”
The next exercise is difficult to solve in ²-δ analysis.
12
RICHARD O’DONOVAN AND JOHN KIMBER
Exercise 10:
2
Given that the gravitational force between two masses is F = k m1d·m
2
(where d is the distance between the two masses and k a constant),
what is the force between objects A and B in the following situation?
A is a sphere considered as a point (for simplicity), B is a bar.
18kg
6m B
¾
6kg
t
3m - A
Students answer this by finding the weight of an infinitesimal piece of
the bar B of length dx. Finding how many pieces of length dx there are
is not easy for the students. They tend to answer something like ω ∈ IL.
They usually need to be guided, so that they give an answer with respect
to dx. Hence the resulting corresponding infinitesimal force depends on
the distance x from A: ∆F (x) = 18k
x2 dx. The students can check that it
satisfies the conditionsR to be written it as dF (x) They then calculate the
sum of all the forces: dF = F .
For these students, the dx is not “a mere symbol to point out which letter
Rb
is the independent variable.” f (x) · dx really is a product and a f (x) · dx
really is a sum – the standard part of a sum.
Exercise 11:
Prove that if ∀x ∈ [a; b] f 0 (x) = 0 then ∀x ∈ [a; b] df (x) = 0.
Calculate the variation of f between a and u ∈ [a; b].
Hence, deduce that if ∀ξ ∈ [a; b] f 0 (ξ) = 0 then f is constant on
[a; b].
This leads to:
Z
f (x) − f (a) =
Z
x
df (t) =
a
x
0=0
a
thus f (x) = f (a). An immediate consequence is the uniqueness of the
antiderivative up to an additive constant.
§8. More complicated functions. After polynomials, rational and
root functions, the class studies trigonometric, exponential and logarithmic
functions. It can remain simple. For instance, the hard way to compute
the derivative of the sine function is to expand sin(a + dx). Alternately, in
figure 3, d cos(θ) is negative because the horizontal variation is negative.
In each quadrant there is always one and only one of the two variations
adjacent
which is negative. The trigonometric definition yields: cosine = hypotenuse
,
= sin0 (θ).
therefore cos(θ) = d sin(θ)
dθ
The higher level students also study differential equations. Though solving the equations may remain difficult, the underlying concept is, in this
NON STANDARD ANALYSIS AT PRE-UNIVERSITY LEVEL
13
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approach, simple and several previous exercises already used it. No abstract
“differential form” is needed.
These exercises give a flavour of the content of the course and how students respond. Some more general aspects remain to be discussed.
§9. Terminology. The problem with the title “non standard analysis”
seems to imply that the course will not be “normal”. And then follows
the obvious question: “What is standard analysis?” a title nowhere to be
found.
“Infinitesimal Calculus” or “Differential Calculus” sound old fashioned
and “Leibnizian Analysis” – though a deserved homage – roots the subject
back a few centuries and would be unfair to Newton. Non standard analysis
is one of the rare chapters where 20th century mathematics can be taught
at this level. In this context, “Magnitude Analysis” conveys the idea that a
function is analysed by changing the scale of observation and allowing the
use of infinitesimals as well as infinitely large. It also follows the French
“Analyse en ordres de grandeurs” by F. Diener and R. Lutz. We merely
suggest this title as a synonym.
Another suggestion arises from the intuitive feeling that there are different “layers” rather than magnitudes. Thus “Strata analysis” could be
used.
§10. What next? Students will choose to study law, medicine, languages, literature, some may read physics or mathematics. The majority
will never really study mathematics, thus the question of how their knowledge fits in with mainstream ²-δ analysis is irrelevant. These students will
hopefully remember that they did, once upon a time, study concepts related
to the idea of infinity and recall that it was not counter-intuitive.
14
RICHARD O’DONOVAN AND JOHN KIMBER
Those who choose to study Science will take some introductory course
in analysis starting from scratch, covering all necessary concepts to grasp
the subject. For these students, will the ²-δ approach be seen as:
• an enrichment?
• an unnecessary complication?
If the analysis course at University turned out to be given as non standard
analysis, we would see mathematicians who would not have studied with
epsilons and deltas or at least: not as the basic approach. How would these
mathematicians tackle problems? What would their insight of problems be?
Would they come up with novel solutions and new problems?
There is a heated debate whether non standard analysis should be introduced at pre-university level. It has been demonstrated herein that it is
possible, emphasising that in mathematics, simplicity rhymes with beauty.
REFERENCES
[1] John L. Bell, A primer of infinitesimal analysis, Cambridge University Press,
1998.
[2] Robert Goldblatt, Lectures on the hyperreals, Springer, 1998.
[3] Jerome Keisler, Foundations of infinitesimal calculus, Prindle, Weber &
Schmidt, 1976.
[4]
,
Elementary
calculus,
University
of
Wisconsin,
2000,
www.math.wisc.edu/˜keisler/calc.html.
[5] Alain Robert, Analyse non standard, Presses polytechniques romandes, 1985.
[6] Abraham Robinson, Non standard analysis, Princeton University Press, 1966.
[7] K. D. Stroyan, Mathematical background: Foundations of infinitesimal calculus, Academic Press, 1997, www.math.uiowa.edu/%7Estroyan/backgndctlc.html.
[8] K.D. Stroyan and W.A.J. Luxemburg, Introduction to the theory of infinitesimals, Academic Press, 1976.
[9] Silvanus P. Thompson and Martin Gardner, Calculus made easy, MacMillan,
1998.
[10] William Weiss and Cherie D’Mello, Fundamentals of model theory, Dep. of
Mathematics, University of Toronto, 1997.
Appendix A. The school. The school system in Geneva encompasses
13 years of basic academic education before University. During six years
of primary school children learn to count and do arithmetic along with
some basic geometry. The three year middle school includes the study of
simple algebraic manipulations, linear equations, polynomials and rational
functions. By the end of middle school students are 14 or 15 years old.
Compulsory education comes to an end.
Students interested in further academic education can go to “Collège”,
grades permitting. In Collège, mathematics is a compulsory subject. During the first two years, courses include trigonometric relations, solving
NON STANDARD ANALYSIS AT PRE-UNIVERSITY LEVEL
15
quadratics and linear systems with more than one unknown. In third and
fourth year emphasis is on analysis. Linear algebra, probability and statistics amount to no more than half the teaching time. Mathematics is one of
ten to twelve subjects, all of which are tested in final exams.
COLLÈGE ANDRÉ-CHAVANNE
14 AV. TREMBLEY, 1209 GENEVA, SWITZERLAND
Correspondence address: Richard O’Donovan, 3 place des Eaux-Vives, 1207 Geneva,
Switzerland
E-mail: richard.o-donovan@edu.ge.ch
E-mail: john.kimber@edu.ge.ch
URL: http://hypo.ge.ch:8080/chavanne/Enseignement/Disciplines/Math