Nonstandard analysis, as conceived by Abraham Robinson and his followers,  puts the intuitive notion of "infinitesimal number" on a firm foundational footing and frees the definition of continuity from the "tyranny" of epsilons and deltas.  From a formal point of view, everything in standard analysis that can be proved by nonstandard arguments can also be proved using standard ones.  Sometimes--and this is the holy grail of NA--an open problem in standard analysis is first proved using a NA approach.  A standard proof is usually soon discovered, but it tends to be not as slick.  I view NA as another area of mathematics, and not in competition with any other.  I don't think it will settle any "paradoxes," though.

Dear Miguel, please see the following example.

There are two reasonable infinitesimals treating operations in both “standard analysis” and “non-standard analysis”:

（1） During the whole process in calculations dealing with infinite substances (infinitesimals), no one dare to say “get the limit or get the stander number”. So, the infinitesimals in the calculating operations would never be too small to be out of the calculations and the calculations dealing with infinitesimals would be carried out forever. Take the infinitesimals calculation in Harmonious Series as an example------ those items of Un--->0 never be 0 or infinitesimals all the time, so we can use the “brackets-placing rule" to produce infinite number items bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite Un--->0 items in Harmonious Series and change an infinitely decreasing Harmonious Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity and Harmonious Series is divergent. Here we have one of the modern versions of ancient Zeno’s Paradox.

（2）During the whole process in calculations dealing with infinite substances (infinitesimals), someone suddenly cries “get the limit or get the stander number”. So, all in a sudden the infinitesimals in the calculations become too small to stay inside the calculations, they should disappear from (be out of) any infinitesimals calculation formulas immediately. Take the infinitesimals calculation in Harmonious Series as an example -------those items of Un--->0 must be 0 or infinitesimals from some time on, so we cannot use the “brackets-placing rule" to produce infinite number items bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or…from infinite Un--->0 items in Harmonious Series and Harmonious Series is not divergent. But if it is convergent, another paradox appears.

Sincerely yours,

Geng

Dear Paul, thank you very much for you frank opinion.

1, I agree with you that a very important feature acknowledged for “non-standard analysis” is its equivalence with “standard analysis” and we really see theoretically and operationally, they are really perfect equivalences in nature------just with different formal languages. But it is this deep structural equivalence that makes “non-standard analysis” inherits all the problems and defects disclosed by the suspended infinitesimal paradox family in “standard analysis”. A typical example is the Harmonious Series Paradox in above post.

2, if the NA related theory is not superior to the SA related theory in infinite relating number operations, what is it for as another area of mathematics?

Sincerely yours,

Geng

Dear Geng,

Nonstandard structures may be constructed within the confines of axiomatic set theory (ZFC, say); as such they have just as much reality as Euclidean space or any other mathematical object, and may be studied for their own sake.  A nonstandard model of the natural numbers, for example, has very interesting features as an ordered set. Also there are those who advocate the use of a nonstandard model of the real line to mathematically express the metaphysical notion of time, energy states of quanta, etc.

Dear Paul,

1, it is just because nonstandard structure is constructed within the confines of present classical infinite related theory system that make it have “deep structural equivalence” with “standard analysis”------ “standard analysis” can do anything “non-standard analysis” can. So, in infinitesimal related mathematical area, it has the same fate as any former theories and is also unable to solve any problems and defects disclosed by the suspended infinitesimal paradox family at all and it is impossible for “non-standard analysis” to become the future mathematical analysis.

2, although it is for sure that nonstandard structure can do nothing in solving any problems and defects disclosed by the members of suspended infinitesimal paradox family (such as Harmonious Series Paradox), we still have another trouble: how does nonstandard structures solve problems and defects of “infinite-big” in present infinite related theory system?

Is the nonstandard structure only for part of the infinite things and just purposely constructed for the infinitesimal related "tyranny of epsilons and deltas" formal language only?

Sincerely yours,

Geng

If we understand the foundation and meaning of a certain mathematical numbers, we can use them well and develop them into a system such as the invention of “0”; but if we don’t understand the foundation and meaning of a certain mathematical numbers, we may meet troubles-------people have been troubling by the “non-number infinitesimal (variables)” related "tyranny of epsilons and deltas formal language" in “standard analysis” ever since.

Now, “non-standard analysis” is on the way. A very important feature acknowledged for “non-standard analysis” is its deep structural equivalence with “standard analysis” and this is just because nonstandard structure is constructed within the confines of present classical infinite related theory system ------“standard analysis” can do anything “non-standard analysis” can, it means that we don’t understand the foundation and meaning of the newly invented “being-number infinitesimal” either. So, will the new “being-number infinitesimal” related "tyranny of infinitesimal formal language" in “non-standard analysis” trouble us from now on?

Dear Geng,  I think the term "nonstandard" is misleading because its meaning has more of a stylistic significance than a mathematical one.  All the nonstandard number systems arising from the model-theoretic constructions of ultraproducts and elementary extensions are themselves borne of "standard" axiomatic set theory.  So this is not "nonstandard mathematics," as opposed to the mathematics we're used to: it's all just mathematics.  What's new, maybe are certain "reflection principles":  some formally describable  facts about the real line are true for any of its elementary extensions, and vice versa. 

Dear Paul,

We may consider form the angle of “number”:

In front of the newly discovered Harmonious Series Paradox, we can see the deep structural equivalence between “Non-number infinitesimal (variables)” related "tyranny of epsilons-deltas formal language" in “standard analysis” and “being-number infinitesimal” related "tyranny of infinitesimal formal language" in “non-standard analysis” produced in present classical infinite related theory system.

Dear Miguel，thank you.

I am sorry; I try very hard to present the answer with three formal languages in different period of time here but it seems some expressions are wrongly transmitted so I have to give up.

I have uploaded a paper《A New Way Out of Pending Problem in Mathematics》 on RG with all of these in it.  Would you please check them on my paper?

Dear Akira，

1, “So nobody try to investigate the connection between infinitesimals in physics and that in mathematics”.

According to my studies, because of the foundational defects in present infinite related mathematical theory system, people do what they need but ignore what the related theory says in many cases. The most typical example is many infinitesimals are treated exactly like numbers as all the numbers in the same calculating formulas but you have to cheat yourself as well as other people saying they are not numbers------ “non-number variables”(the same situations happen to those infinite-big numbers). I am very sure that at the time when we solve those defects in present infinite related mathematical theory system, it is a must for people to investigate the connection between infinitesimals in physics and that in mathematics; that is what mathematics for in our science.

2, “Why people say that everything nonstandard analysis can do can be done by the standard analysis.”

I have uploaded a paper 《A New Way Out of Pending Problem in Mathematics》 on RG answering this question.

Sincerely yours,

Geng

Dear Akira and Geng, 

One has to be careful when saying "everything nonstandard analysis can do can also be done by standard analysis."  That is patently false.  What I said is that every question about standard analysis that you can settle using nonstandard techniques can also be settled using standard ones.  (The nonstandard theory is called a conservative extension of the standard one.)  Also I would like to address the idea that "infinitesimal calculus  removes topology from analysis."  Let's take a concrete example (so to speak), that of a nonstandard extension R* of the real line R.  Both number systems are linearly ordered, with R properly contained in R*.  From the topological point of view, the topology one gets from the linear ordering on R* is T0; indeed it's hereditarily normal Hausdorff and zero-dimensional.  Moreover, the subspace topology on R is discrete!  It is for this reason that one may claim that analysis and topology on R* have parted the ways.  But that doesn't say the topology of R* is uninteresting.

Dear Paul,

The most defects in nonstandard analysis is its “deep structural equivalence” with “standard analysis” but not how many and what kinds of things it can do-----“Non-number infinitesimal (variables)” related "tyranny of epsilons-deltas formal language" in “standard analysis” and “being-number infinitesimal” related "tyranny of infinitesimal formal language" in “non-standard analysis” because they are both confined in present classical infinite related theory system.

It is why every question can be settled by standard analysis can also be settled by using nonstandard techniques and vise verse.

Sincerely yours,

Geng

Dear Geng, please have a look at my note under the title "Have I found a new way to the concept of DERIVATIVE?". Although only the first steps towards a new approach to calculus, it may be of interest for you. As I go on with the material, I will try to give further details.

Dear Akira,

Talking about Cantor's view of infinity, I am a little sorry to say that he cared little for infinitesimal. Does infinitesimal belong to infinite and is infinitesimal anything to do with cardinal numbers or ordinal numbers? So, that is why I call Cantor's view of infinity “half infinite”; I know many people nowadays will not agree with me, but what I have said is true.

Dear Akira,

1， It is no doubt that Cantor is one of the greatest mathematicians ever lived. But the full and correct understanding of infinity was blocked not because of the defects disclosed by Cantor’s paradox (Cantor’s paradox was solved easily soon afterwards) but by suspended Zeno’s Paradox 2500 years ago------we are still living in Zeno’s time. It is Cantor's “half infinite theory” that blocks something.

2， Dear Akira, actually we are very happy to live in a modern world because our ancestors devoted us lots of heritages and we can stand on their shoulders, work and pioneer with more energies and experiences.

Dear Akira, why you are so sure that if present people maintain Cantor’s value system in today's science they will definitely fail?

History repeats!

Sincerely yours,

Geng

Dear Akira, Happy New Year!

 I think what you say is true and we have to accept the fact. But another thing is also true that there is still someone in present world like Cantor never receives million dollars research grant but wants to know the truth and maintain his value system to fulfill his dream------Man proposes and God disposes.

Sincerely yours,

Geng

Dear Geng and Akira,

Happy (western) New Year, and be of good cheer...

I personally know many mathematicians who definitely understand the distinction between countable and uncountable--although some don't see any point in distinguishing between aleph-one and the cardinality of the continuum (sigh)--and am here to even defend the physicists.  I've known one or two, and they're a smart lot.  But their focus is entirely different from a mathematician's:  for them the phenomenal world is there to be discovered, to be explicated using whatever descriptive language works.  For example, take time.  From our intuitions, borne of millennia of experience of the macroscopic world (definitely not the world of particle physics), time is a "linear continuum."  To a physicist, if it's convenient to talk of "infinitesimal displacements," then so be it.  It was the mathematics of the 19th and 20th centuries that gave us a precise construction of the "real line" R and locked us in to a clear model of the "continuum."  Of course all things infinitesimal had to be chucked out as a result if physics were to remain logically consistent, but the physicists didn't care; after all the phenomenal world is consistent in and of itself!   R has its mathematical properties--some decidable in ZFC, some not-- but has only a Platonic existence and may turn out not to be the best model for physics after all.   Something similar goes for modern quantum physics and its mathematical foundations using Hilbert space.  Maybe--as Akira suggested several posts ago--there is a "constructive infinitesimal functional analysis" that offers a better fit, both for the macro- and the microscopic worlds. 

Hi Akira, I think a firm theoretical bridge between discrete systems and continuous approximations to them may not be possible any time soon.  Again, if a physicist can find an excuse to treat a large discrete system (e.g., a container of gas) as a continuous entity, she will.  "Anything that gets answers."   As for supporting/refuting physical theories, only degrees of confidence are required to underpin belief.  I don't believe empirical data near as much as I believe a mathematical argument;  and the more complicated the argument, the more difficult it is for me to believe. 

I think we researchers are somewhat like a group of “working hard” bees, in front of people’s comments (it doesn’t matter how high or how low) we say “that’s ok, it’s our life and we really enjoy our life” (am I a very poor innocent boy?).

Dear Akira and Paul,

1, if we don’t have clear distinction between “potential infinite” and “actual infinite”, how can we be sure to definitely understand and operate well the distinction between continuity and uncontinuity, countable and uncountable?

2, is there something for us to worry about: it is a tragedy of mathematics if Physicist (in practical sciences) versions of “infinite numbers” are not the mathematical version of “infinite numbers”?

 Dear Miguel, Happy New Year!

I think if we treat mathematics just as a kind of handy tools, we need not care for any paradoxes in mathematics, may not trouble ourselves “understanding infinity to develop it's cognition about math, physics or science in general” as well as some other mathematical theories.

But, if we treat mathematics as a science, we must care for “understanding infinity to develop it's cognition about math, physics or science in general” as well as “continuity and uncontinuity, countable and uncountable” and Physicist (in practical sciences) versions of “infinite numbers” must be the mathematical version of “infinite numbers”.

Sincerely yours,

Geng

Dear Akira,

I think it is mathematicians who didn’t do well in the infinite field. We have failed to create a useful “number language” to tell what infinite is mathematically ever since.

Physicist (all kinds of practical sciences) versions of “infinite numbers” must be the mathematical version of “infinite numbers”-------“infinite numbers” have only one version in human science (physics is one of human science).

Sincerely yours,

Geng

Dear Akira,

So, it proves why our science really needs a mechanism of “metabolisms” to become a living thing synchronizing with us human.

Dear Akira, very philosophic, thanks.

It is true that there are so many things for us human to study and our research life is endless, still, we are able to try our best to avoid “narrowness and shallowness” perspective in our researches, avoid being “blind people arguing what an elephant is by touching part of elephant”.

Our science history proves that philosophic and logic ideas and techniques are needed, especially when we are doing some fundamental researches in some of our knowledge (science) areas, so that the researchers are able to have either deeper or wider insights to combine different ideas, methods and techniques and pay enough attention to the relationships between and among things in different branches of our knowledge (science), borrow some well (better) developed ideas, methods and techniques from each other.

For the scientists working with the style of "empiricism" in the area of applied science, Biomimetic is a very typical example to avoid “narrowness and shallowness” perspective in their researches, avoid being “blind people arguing what an elephant is by touching part of elephant”.

Sincerely yours,

Geng

Dear Paul, I loved your remark: 'I don't believe empirical data near as much as I believe a mathematical argument; and the more complicated the argument, the more difficult it is for me to believe.' I add that for me the most complicated concept in mathematics and elsewhere to tell apart 'simple' from 'complicated'. This is the very undertaking of any system of axioms, be it about straight lines or infinity.

Nonstandard analysis is much more a way of thinking about analysis, as a different analysis. Of course, the objects of non-standard analysis have been described algebraically. R* is a valued field, and R is its quotient field, considering as valuation ring the set of non-infinite numbers (the monad of infinitesimals being its unique maximal ideal). The other approach, by Edward Nelson, adds some axiom schemes to ZFC (schemes called Idealisation, Standardisation and Transfer - in order to build the Theory IST of Nonstandard Mathematics. Nelson proved that IST is consistent if ZFC also is consistent, and that the expansion is CONSERVATIVE: all theorems that do not contain the predicate "standard" and are provable in IST are already provable in ZFC. So: we have a new way of thinking about analysis, and is better to have more methods as less methods. This is almost all to say about nonstandard analysis: another way of thinking. (the methods are difficult because there are some canonical mistakes one can do. One can manage not to do them, just because there are so canonical!)

Dear Akira，

>Empiricism is just like a bunch of blind people arguing how the elephant looks like by touching part of an elephant.

Dear Mihai，the following three points in your post is really true:

1, as “monad of infinitesimals” has much to do with analysis; nonstandard analysis is much more a way of thinking about analysis, as a different analysis------simpler than standard one.

2, CONSERVATIVE is the nature and a must for Nonstandard Analysis or Nonstandard Mathematics, it is called a conservative extension of the standard one.

3, because of the “deep structural CONSERVATIVE”, the “provable” equivalence are guaranteed.

We focus on the “deep structural relationship” between “nonstandard one” and “standard one”. If there are “no defects” in the “standard one”, the “CONSERVATIVE guaranteed nonstandard” work would be really meaningful.

Now the problem is “nonstandard one” inherits all the fundamental defects disclosed by “infinite related paradoxes” from “standard one” since Zeno’s time 2500years ago------guaranteed by the “deep structural CONSERVATIVE” (some of the discussions shown in some previous posts in this thread).

Theoretically and operationally, “nonstandard one” is exactly the same as those of “standard one” with suspended infinite related defects in nature. Simpler or not weights nothing here.

Sincerely yours,

Geng

Just two thoughts:  First, suppose you notice a numerical pattern, say 1=1, 1+2=3, 1+2+3=6, and maybe you go on like this for quite a way before making the hypothesis that the sum of the first n integers is quadratically related to n, determine the coefficients, and come up with a  formula.  That's akin to the blind person coming up with a decent picture of the elephant, based on some well-chosen experimental data obtained by "feeling around."  A lot of mathematical discovery is obtained empirically;  I dare say that's most likely how Fermat was convinced of his "last theorem." Fortunately there are mathematical tools to prove/refute the hypotheses generated empirically, and mathematicians all understand the difference.  (Scientific induction and mathematical induction are worlds apart.)  Second,  Mihai  made the important point (made only incompletely by me in an earlier post) that Edward Nelson's Internal Set Theory (IST, there's a nice explanation on Wikipedia) is both consistent with and a conservative extension of  classical ZFC.  These are fairly basic metamathematical notions having nothing to do with nonstandard analysis per se, but which buttress many of the well-known assertions concerning the capabilities and limits of nonstandard analysis.   I know there is interest in a more "constructive" approach to infinitesimals; perhaps some version of IST overlaying, say, Bishop's constructivism might be a way to go. 

Dear Paul,

I think it is true that “experiments” is only one of the means for well developed researchers (but not Empiricists) and the defects disclosed in “blind person and the elephant story” can be avoided.

Akira, you bring up an interesting point:  suppose you have an algorithm for spitting out 0s and 1s, and you want to decide whether the resulting "real" .a_1a_2a_3... is rational.  I'll bet that's recursively undecidable, and somebody's looked into it, but I don't know.

Ok, the answer to my recent post is definitely no; see H.G. Rice, Recursive real numbers, P.A.M.S. 5 (1954), 784--791 (MR reviewed by G. Kreisel). 

Dear Akira，you are right, monads really appear in many places not only in mathematics but philosophy as well; I am confusing, too. In nonstandard analysis, this concept of monad is used to describe infinitesimals related mathematical structure; but I don’t think it works, because the theory of nonstandard analysis can do nothing to solve those defects disclosed by the old and new suspended infinite related paradoxes.

Sincerely yours, Geng

How does the “infinite related small number forms” be called is not really important but how does it work is important. Long before Nonstandard Analysis and Standard Analysis (since Zeno’s time), people can do most of the infinite related calculations happily-------people have been treating exactly the same “scientific carrier” with different names (titles).

Dear Akira，in my opinion, what kind of name (title) would be given to a scientific concept is not really important but how does it work proves; different names and different languages (English, Japanese, Chinese, …) can be exactly the same “scientific carrier”.

We are still in Zeno’s time in front of his paradoxes!?

Sincerely yours, Geng

The fact is: on the one hand, present classical mathematical analysis can not avoid the constraining of “potential infinite--actual infinite” concepts and their relating “potential infinite number forms--actual infinite number forms”; on the other hand, no clear definitions for these two concepts of “potential infinite--actual infinite” and their relating “potential infinite number forms--actual infinite number forms” have been constructed since antiquity, thus naturally lead to following two unavoidable fatal defects in present classical mathematical analysis:

（1）it is impossible to understand theoretically what the important basic concepts of “potential infinite, actual infinite” and their relating “potential infinite number forms--actual infinite number forms” are. So, in many “qualitative cognizing activities on infinite relating mathematical things (infinite relating number forms)” in present classical mathematical analysis, many people actually don’t know or even deny the being of “potential infinite, actual infinite” concepts and their relating “potential infinite number forms--actual infinite number forms”--------the “qualitative cognizing defects on infinite relating mathematical things (infinite relating number forms)”.

（2）it is impossible to understand operationally what kind of relationship among the important basic concepts of “potential infinite, actual infinite”, their relating “potential infinite number forms--actual infinite number forms” and all the“infinite number forms as well as their quantitative cognizing operations” are. So, in many “quantitative cognizing activities on infinite things (infinite number forms)” in present classical mathematical analysis, many people have been unable to know whether the infinite relating number forms being treated are “potential infinite number forms” or “actual infinite number forms”, no one has been able to avoid the confusing of “potential infinite number forms” and “actual infinite number forms”, no one has been able to know whether or not treating “potential infinite number forms” or “actual infinite number forms” with the same way or different ways. What is more, many people actually don’t know or even deny the being of “potential infinite number forms” and “actual infinite number forms”--------the “quantitative cognizing defects on infinite relating mathematical things (infinite relating number forms)”.

The above two fatal defects have decided since antiquity the absence of “infinite carrier theory” and the confusion of “abstract infinite concept and its concrete infinite mathematical carrier” which have been making us impossible to construct the scientific and systematic theory of “infinite related concrete number form (the mathematical carriers of abstract infinite concept)”, impossible to construct the “potential infinite--actual infinite” relating number forms as well as the “potential infinite--actual infinite” relating number system and its relating treating theory, thus unavoidable forming an insurmountable obstacle in the “quantitative cognizing process on infinite things (infinite number forms)” in present classical mathematical analysis. And, it is impossible to know what those “’both potential infinite number forms and actual infinite number forms’, ‘neither potential infinite number forms nor actual infinite number forms’ mathematical things” are, and can only treat all of them with the “flow line (pipelining)” unified approach （such as the three formal languages in three generations of mathematical analysis）. So, many members of different infinite related paradox families in present “potential infinite--actual infinite” based classical mathematical analysis have been produced one by one naturally (the newly discovered “strict mathematical proven” Harmonic Series Paradox is a typical example), forming a thousands-year old suspended huge black cloud of paradoxes over present classical infinite related science and mathematical analysis. It is pitty that, our science history has proved that it is impossible at all to resolve the above two fundamental defects in present classical “potential infinite--actual infinite” related science and mathematical analysis.

Five of my published papers have been up loaded onto RG to answer such questions:

1，On the Quantitative Cognitions to “Infinite Things” (I)

https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I

2，On the Quantitative Cognitions to “Infinite Things” (II)

https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II

3，On the Quantitative Cognitions to “Infinite Things” (III)

https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III

4 On the Quantitative Cognitions to “Infinite Things” (IV)

https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV

5 On the Quantitative Cognitions to “Infinite Things” (V)

https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V