> 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.
I disagree. As I explain in my posting to Eric Schechter, which I copy
to you, the main gain of NSA is in the reduction of the (apparent) type
of the objects that one needs to consider and visualize.
> 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 to explicate not only mathematical, but also logical,
> ontological, philosophical, historical and even legal aspects of the
> "infinitesimal" notion and its usage.
Yes; but once this is done, you can (almost) forget about these
explanations, and work intuitively with ~ .
> 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.
*All* students? Without *any* problems? You must have very unusual
students. In fact, these concepts are quite difficult to assimilate. Of
course, once they are assimilated, they may seem easy. `All difficulties
are but easy, once they are known' (W Shakespeare, Measure for Measure).
I think that a muzhik who sees a motor-car for the first time may think
that a simple horse and cart is much simpler to use--as has been proved for
several centuries. The problems of horses and carts are much simpler; or at
least they are much more familiar to our muzhik. Personally, I prefer a
motor-car.
> 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.
I can only speak for myself. To me it is clear.
> 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": "Öwhat 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."
This is a mistake. A monad is a whole cluster of *points. The monad of 0 on
the *real line is the collection of all *real infinitesimals.
> I deeply believe that just the last interpretation of
> "infinitesimals" is correct.
I believe this is a mistake.
> 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.Cantorís 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 shall be grateful if you send me reprints/copies of these papers. (I can
read Russian, with some difficulty.)
> I think that it would be more correct (especially in 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."
Yes; this is quite true. Technically speaking, NSA is conservative. But so
are many mathematical methods. For example, any result in functional
analysis that does not mention explicitly Banach or Hilbert spaces can be
proved without using these concepts. But in many cases it is much easier to
prove the result using these spaces.
Best wishes,
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