[begin]
I disagree with both Gordon and Eric.
I shall give just two examples.
1. The delta-epsilon definition of continuity of a (real) function f at
a (real) point x is of the form Sigma^0_3. In other words, it has the
form
forall epsilon there exists delta such that forall x',
if |x' - x| < delta then ....
The equivalent NS characterization has the form Pi^0_1:
forall delta ~ 0: f(x+delta) ~ fx.
Instead of three quantifiers (of alternating kinds) there is just one.
This is a considerable logical and psychological simplification. Note
also another thing: the function f 'goes' from its domain to its range
(or codomain), and we instinctively tend to think of it as moving in that
direction. But the delta-epsilon definition starts at the 'end', at the
range of f, where the epsilon is given; then it moves to the domain,
where the delta is found, and then moves back to the range, where we
have |fx' - fx| < epsilon. By contrast, the NS characterization starts
at the beginning, in the domain, and goes to the range.
[end]
As is known, an optimal strategy is not always the shortest one. All is
defined by the optimization criterion.
I think the Non-Standard Analysis (NSA) definition " for all delta ~ 0:
f(x+delta) ~ fx " is a considerable simplification only in an
algorithmical complexity (a number of operations, steps) sense. All
logical, psychological and as a consequence - pedagogical aspects are
hidden into the symbol "~", and in order to explain a sense of the symbol
"~", we must explicate not only mathematical, but also logical,
ontological, philosophical, historical and even legal aspects of the
"infinitesimal" notion and its usage. As more that 100-years experience
of mathematical education shows, all students comprehend the
"delta-epsilon" definition of the continuity of a (real) function f at a
(real) point x without any problems. In particular (but not in the last
turn) because of the entire obviousness of its geometrical visual
interpretation. From this point of view, the geometrical sense of the
very notion of "infinitesimals" remains not quite clear even for
NSA-people themselves. For example, Vladimir A. Uspensky, in his known
remarkable book "What is the nonstandard analysis?" {"Chto takoje
nestandartny analiz?". - Moscow: "Nauka", 1987}, gives the well-known
Keisler geometrical "microscope-telescope" interpretation of
"infinitesimals" (see pp. 23-25). Using some other words, the same can
be said as follows: from the large height, we see a forest as a whole
{say, the segment [0,1]}. Making a descent, we begin to distinguish
individual trees {say, the single standard real numbers x (- [0,1]}. At
last, being touched down and using a microscope, we can distinguish new
wood objects (existing between the trees !) called nonstandard
e-infinitesimals of NSA (than we can see: e^2 between e, e^3 between
e^2, and so on). I think such the geometrical interpretation of the
nonstandard infinitesimals is not correct. Not only because it destroys
our absolute trust in the standard analysis statement according to which
the set of all standard real numbers of the segment [0,1] is complete
and has no gaps. But also because of the following reason. There is an
entire analogy between the algorithm for the algebraic extension of the
real numbers field R to the NSA-field *R by means of the symbol "e"
("epsilon" = "an infinite small positive hyperreal number") and the
algorithm for the algebraic extension of the real numbers field R to the
field of complex numbers by means of the symbol "i" ("square(-1)" =
"the imaginary unit"). Of course, "to within" the corresponding axioms
for the addition and the multiplication. In such the case, according to
the Keisler geometrical interpretation, we must search for complex
numbers not in the complex 2D-plane as usually, but between standard real
numbers-trees on the real axis. But we do that never.
However, there is another interpretation.
In the end of his book (see p.117), Vladimir Uspensky gives the
following quite different geometrical interpretation of the
"infinitesimals": "...what we see as a point on the real axis is, in
reality, an interval with the real point as its center, i.e. a monade."
I deeply believe that just the last interpretation of
"infinitesimals" is correct. But it leads to some very unexpected
conclusions. Some of such the conclusions are published:
in Russian:
2. A.A.Zenkin, Cognitive Visualization of some transfinite objects of
the Classical Cantor Set Theory. - In the Collection "Infinity in
Mathematics: Philosophical and Historical Aspects", Edr. Prof.
A.G.Barabashev. - Moscow: "Janus-K", 1997, pp. 77-91, 92-96, 184-189,
221-224.
9. A.A.Zenkin, On Logic of Some Quasi-Finite Reasonings of Set Theory
and Meta-Mathematics. New Paradox of Cantor's Set theory. - News of
Artificial Intelligence, 1997, no.1, 64-98, 156-160.
10 A.A.Zenkin, Whether the Lord exists in G.Cantor’s Transfinite
Paradise? – News of Artificial Intelligence, 1997, No. 1, pp. 156-160.
And short abstracts in English:
3. A.A.Zenkin, Cognitive Visualization Of The Continuum Problem And
Mirror Symmetric Proofs In Transfinite Numbers Mathematics. -
ISIS-Symmetry Congress and Exhibition. Abstracts. Haifa, Israel, 13-19
September, 1998.
14. A.A.Zenkin, Cognitive Visualization of the Continuum Problem and of
the Hyper-Real Numbers Theory. - International Conference "Analyse et
Logique", UMH, Mons, Belgia, 25-29 August, 1997. Abstracts, pp. 93-94
(1997).
I think that it would be more correct (especially according to the
last V.Uspensky interpretation of the geometrical nature of
"infinitesimals") to call the "infinitesimals" as the
TRANS-finitesimals. In particular, it permits to distinguish quite
naturally the "infinite small" value "lim v = 0 or v -> 0" in Cauchy
and Weierstrass'es sense from the value "delta ~ 0" in the NSA-sense.
Now some words about very soft and delicate doubts of the respected
colleagues Gordon Fisher and Eric Schechter concerning the usefulness of
NSA for computing mathematics. I think the following words of the same
Vladimir Uspensky {see his preface to his book} will help to make more
clear this question.
"The place {AZ: of NSA in modern mathematics} is quite modest.
Non-standard analysis does not intend to abolish the standard one. All
existing "standard" results remain valid. Moreover the non-standard
analysis does not pretend to obtain fundamentally new results: all
results obtaining by means of its methods can be proved by means of the
habitual methods."
A-Z