Very interesting problem!

see:http://www.journals.uchicago.edu/doi/pdfplus/10.1086/671348

Dear Dr. Joseph Dubrovkin, thank you for your offer.

Best regards,

 Geng

“Potential infinite” and “actual infinite” have been in our present classical infinite related science system since antiquity------dominating how all the “infinite things” exist in our science and how we human behave in our cognitions to all the “infinite things”. But, some say “all the infinite things in mathematics are potential infinite and they are treated by set theory and limit theory” while some others say “all the infinite things in mathematics are actual infinite and they are treated by set theory and limit theory”.

The confusing and debating on “potential infinite and actual infinite” are hard and endless at least since Zeno’s time 2500 years ago. In fact, the situation is: to many mathematicians, “the infinite related number forms are invisible and untouchable variables, they are not numbers” although they really see them, touch them and treat them as numbers very often in calculus while to many people who are working in practical science (such as physicists), they are brave enough to say “the infinite related number forms are visible and touchable numbers”.

The most trouble thing for me in such confusing and debating on “potential infinite and actual infinite” is: when facing and treating the infinite related number forms, we have to declare first that they are not numbers (such as infinitesimals and infinities) theoretically in mouth but tell in mind firmly for those very infinitesimals and infinities “forget what have been just said in mouth and they are in fact numbers other wise no practical numerical operations can happen”.

When I first come across such operations 40 years ago I feel very sorry for myself that I have been very often as so many people played the role of a minister in Andersen’s fairy tale The Emperor's New Clothes...

Certainly,there are actual infinitesimals in real fields. Such fields are called non-archimedean. In particular, so-called  "fields of hyper-reals".

I don't know what is meant by "potential" or "actual" infinitesimals so the following may not be helpful. Abraham Robinson developed "Non-Standard Analysis" in the 1960's based only on the axioms of ZFC set theory. In his book of the same name he shows how infinitesimals may be defined rigorously so as to present a complete basis for analysis, an alternative to the standard epsilon and delta basis. It becomes non-standard by including generalized functions. This approach has been criticized on a pedagogical basis because it seems to present a justification for physicists use of an intuitive form of infinitesimal reasoning without using the formal math. For example take the casual explanation of how Newton's fall of an apple is related to an object in orbit.

Nonstandard Analysis (N.A.) has also been commended just for the same reason -- that it has presented a strong basis for our vague intuitions of infinitesimally small magnitudes. There is a  good deal of  books and other texts on N.A., beginning from Keisler's textbook "Foundations of infinitesimal calculus" (1976) and including "Nonstandard analysis for the working mathematician" (2000) by Loeb & Wolff. Thus, though N.A. has not become very popular, we have “actual infinitesimals in our mathematics". Of course, there are also “potential infinitesimals” -- due to Cauchy.

Dear Dr. Artur Piękosz, Dr. James Langworthy and Dr. Jānis Cīrulis, thank you for your view points.

Yes, I agree with you that many “formal languages” have been created so far. But, looking back into our unsuccessful “paradoxes history”, in front of infinite related paradox families, these “formal languages” are equal and do nothing.

Such question was asked by Zeno 2500 years ago with his creative Achilles--Turtle Race Paradox and now by infinitesimal related Harmonic Series Paradox as well as infinite big related Russel’s Paradox…

Are there some more waiting ahead?

Thank you again for your ideas.

Best regards,

 Geng

Dear Geng

What your initial question was about? about infinitesimals in mathematics or infinitesimals in solving Zeno's and other paradoxes? In mathematics, they are there and work well. (By the way, I see nothing paradoxal in what you have called Harmonic Series Paradox in one of yours blogs: the lack of a constructive proof is not a paradox, -- but this is another story.)

You also asked about the "infinitesimal" dx --->0 in calculus. My opinion is that this notation does not present, in calculus, any infinitesimal at all. In Leibniz's time, dx itself was any (not definite) infinitesimal. The question, is dx  the same infinitesimal as Un--->0 in Harmonic Series looks like to the question, is n the same number as 5.

You declare that "in front of infinite related paradox families, these “formal languages” are equal and do nothing." Please, be more specific. There are several papers by various authors, where Achilles--Turtle Race and other Zeno's paradoxes have been analysed on the ground of Nonstandard analysis, for example. Do you consider all of them as unsatisfactory? What are yours contra-arguments, then, and where can one get acquainted with these?

Best,

 Janis

Dear Dr. Jānis Cīrulis, thank you for your insightful view points.

1, my initial question was about infinite related numerical cognizing in human science (including mathematics).

2, according to present science (mathematics) theories and practices, we can prove that infinite items each bigger than 1/2, or 100, or 1000000, or 1000000000000000,…can really be produced by the “brackets-placing rule" from Harmonic Series but in fact it is impossible-------- we can prove that Achilles will never catch up with the Turtle in Ancient Zeno’s Achilles--Turtle Race Paradox but in fact Achilles will easily catch up with the Turtle.

3,in both Nonstandard analysis and Standard analysis, we all prove that infinite items each bigger than 1/2, or 100, or 1000000, or 1000000000000000,…can really be produced from Harmonic Series-------- Nonstandard analysis and Standard analysis are equal and do nothing in solving any members of Zeno’s Paradox family.

4, “is n the same number as 5 in Un--->0 in Harmonic Series”? No, such as “d” and “x” is not the same number as 5 or 6 in dx --->0 in calculus.

Thank you again for your ideas.

Best regards,

 Geng

There are both types of infinitesimals, plus Geometric infinitesimals! Actual infinitesimals are the Robinson's Nonstandard Analysis. On the other hand infinitesimals based on potential infinity is the Vopenka's Alternative Set Theory, which is based in turn on Nelson's Internal Set Theory (see e.g. Alain Robert Nonstandard Analysis)

Finally Geometric infinitesimals are the topos-theoretic ones, see e.g. Bell, J. L., A Primer of Infinitesimal Analysis. Cambridge 1998. Finaly there are Boolean-valued non-standard mathematics, (see, e.g. C. A. DROSSOS, G. MARKAKIS and P. L. THEODOROPOULOS,  B-FUZZY STOCHASTICS.)

Dear Dr. Costas Drossos，thank you.

I agree with you that there may be as many different theories of infinitesimals as we like, but because of the confusing “actual infinite and potential infinite”, none can offer a non-paradoxical answer to the very practical mathematical question: how many items each bigger than 1/2, or 100, or 1000000, or 1000000000000000,…can really be produced by the ‘brackets-placing rule’ from Un--->0 Harmonic Series in present mathematics.

So, something must be wrong in present classical “actual infinite” and “potential infinite” related philosophy and mathematics!

Thank you again dear Dr. Costas Drossos, your view points are really meaningful and appreciated.

Best regards,

 Geng

“Potential infinite” or “actual infinite” decides our infinite related behaviors in mathematics. The newly discovered Harmonic Series Paradox is a most typical example.

Such divergent proof of Harmonic Series can still be found in many current higher mathematical books written in all kinds of languages:

 1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．      （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．           （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                         （4）

Each operation in this proof is really unassailable within present classical “actual infinite--potential infinite” related science theory system. But, the unavoidable practical trouble is how many items in （1） can be added up by “brackets-placing rule" to produce the infinite items in （3） each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or…?

“Potential infinite” or “actual infinite” decides whether or not we can produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from Un--->0 infinite Harmonic Series by “brackets-placing rule".

（a）, the confusing and mixing of “potential infinite” and “actual infinite” in present classical infinite related philosophy and mathematics decides we can produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from above operation, but the fact is not------whether Achilles can catch up with the Turtle or not in Zeno's Achilles--Turtle Race Paradox.

（b）, most people in the world now are deeply convinced that we can produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from Un--->0 infinite Harmonic Series by “brackets-placing rule"------not matter how fast Achilles runs, he will never catch up with the Turtle in Zeno's Achilles--Turtle Race Paradox.

（c）, the unavoidable fatal confusing and mixing up errors of “potential infinite” and “actual infinite” in the foundation of present classical infinite related philosophy and mathematics create all the suspended “potential infinite--actual infinite” related paradoxes families------ the 2500 years old huge black cloud of infinite related paradoxes over mathematics sky.

Looking back into our hard and troubling “infinite things cognizing history”, we can see clearly that following five unavoidable questions closely relating to the definition of “infinite” in present classical science and mathematics have been harassing, troubling and challenging us human since antiquity.

(1), in human science, can we have only one definition with only one nature for “infinite” or we can have many definitions with many different natures for “infinite” (such as what we can see in present classical science and mathematics: actual infinite, potential infinite, infinite, less infinite, more infinite, big infinite, bigger infinite, small infinite, smaller infinite, super infinite, super super infinite, …)?

(2), what are the scientific-mathematical definitions (without contradictions and confusions of ontology--form and theory--operation) for “finite, infinite, actual infinite, potential infinite, less infinite, more infinite, big infinite, bigger infinite, small infinite, smaller infinite, super infinite, super super infinite, …”?

(3), can “infinite, actual infinite, potential infinite, less infinite, more infinite, big infinite, bigger infinite, small infinite, smaller infinite, super infinite, super super infinite, …” be mutual transformed and/or mutual contained among each other?

(4), among “infinite, actual infinite, potential infinite, less infinite, more infinite, big infinite, bigger infinite, small infinite, smaller infinite, super infinite, super super infinite, …”, which is most infinite, smallest infinite, biggest infinite, most super infinite?

(5)Do we have “actual infinite number, actual infinite mathematical analysis, potential infinite number,, potential infinite mathematical analysis”; “actual infinite set”, “actual infinite set theory”, “potential infinite set” , “potential infinite set theory “ in our mathematics?