For at least 2500-year people have been trying very hard to solve those defects disclosed by infinite related paradoxes.

But the newly discovered modern Harmonic Series Paradox as one of family members of ancient Zeno’s Paradox discloses relentlessly a fact that all our strives failed------the working train of thought is wrong!

How will be???:

On July 17, 2014, Seiichi Koshiba and Masami Yamane said at Gunma University:

The idea for the division of Hiroshi Michiwaki and Eko Michiwaki (6 years old daughter) is that division and product are different concepts and they were calculated independently for long old years, by repeated addition and subtraction, respectively.

Mathematicians made the serious mistake for very long years that the division by zero is impossible by considering that division is the inverse operation of product. The division by zero was, however, clear and trivial, as z/0=0, from the own nature of division.

There are two rigorous notions of infinitesimals available.  The more commonly cited and more generally known is that provided by Robinson's non-standard analysis.

The other, less well known, is provided by synthetic differential geometry, which escapes Bishop Berkeley's critique of infinitesimals (remembered for the poetic phrase "ghosts of departed quantities", but actually containing a rigorous critique in the midst of his polemics) by dropping classical logic in favor of intuitionistic logic.  In the context of SDG, infinitesmials are simply the bit of the line that satisfies x^2 = 0, which in models of SDG (for instance the Dubuc topos) for an object D large enough that the commutative ring R called "the line" satisfies the Koch-Lawvere axiom R^D \cong R x R, allowing the definition of derivatives for functions from R to R exactly as Newton and Leibniz gave it.

Dear Mr. S. Saitoh ，

I am sorry to say that in fact, infinitesimals can be nothing to do with “division by zero” or “z/0=0”.

“Division by zero” or “z/0=0” is only one of the ways people study and understand 0, it can be nothing to do with infinitesimals. Would you share with us what you think 0 is?

Sincerely yours, Geng

Dear Mr. David Yetter ，

Thank you very much for your summaries, but I am sorry to say that in fact, we human never have any rigorous definitions of infinitesimal at lease since Zeno’s time with his paradoxes. So, we have modern version of Zeno’s Paradox------ Harmonic Series Paradox.

Let’s see following divergent proof of Harmonic Series which 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）

Because of not knowing what infinitesimals are, the unavoidable practical problem has been troubling us ever since is how many items (including infinitesimals of cause) in infinite decreasing Harmonic Series can be added up by “brackets-placing rule" to produce infinite numbers each bigger than 1/2?

This kind of “infinite-infinitesimals paradox” tells us:

1, in Harmonic Series, we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite infinitesimals in Harmonic Series by “brackets-placing rule" to change an infinitely decreasing Harmonic 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;

2, the “brackets-placing rule" to get 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items in Harmonic Series corresponds to different runners with different speed in Zeno’s Paradox while the items in Harmonic Series corresponds to those steps of the tortoise in Zeno’s Paradox. So, not matter what kind of runner (even a runner with the speed of modern jet plane) held the race with the tortoise he will never catch up with it.

In this case of “how many number items bigger than 1/2 can we have by ‘brackets-placing rule’ in harmonic series”, either finite or infinite is “illogical” in present science.

Robinson's non-standard analysis can do nothing to solve any of those suspended “infinite-infinitesimals paradoxes”.

If no one is able to answer my infinite related questions here scientifically all over the world so far, it proves that this work is worthy continuing------these defects should be solved by some people sooner or later.

Sincerely yours, Geng

Dear Ms. Barbara Sawicka, you hit the point!

This is why we are here in this thread discussing for.

Sincerely yours, Geng

Of course we have rigorous definitions of infinitesimals.  Read

A. Robinson, "Non-standard analysis" , North-Holland (1966)

for one definition, and

A. Kock, "Synthetic differential geometry" , Cambridge Univ. Press (1981)

for the other.

In fact, your objections actually have nothing to do with infinitesimals.  Series, whether convergent or divergent, are composed of infinitely many finite summands:  there are no infinitesimals in sight in either your harmonic series objection or Zeno's paradox (which is quite handily resolved once one accepts the modern notion of limit of a geometric series and goes through the exercise of verifying that 1/2 + 1/4 + 1/8 +... = 1 in the sense that the limit of the series is 1, and is thus, quite uninteresting except as a historical curiosity).

Dear Mr. David Yetter ,

Thank you.

Yes, I agree with you Mr. David Yetter, we really have “some definitions” of infinitesimals in our history. But if they can do nothing to solve any of those suspended infinite-infinitesimals paradoxes, what is the use of those “rigorous definitions”, are the all “rigorous”?

Sincerely yours, Geng

神田さま：　世に誤解があるのは残念です。 ゼロ除算は、除算の意味を 正確に定義せず、　除算を 積の逆と考えれば、 直ぐに、不可能、矛盾、定まらないとなりますが、 除算の意味を 正確に定義することによって、確かな世界が開かれます。 論文、解説記事などお送りしたいのですが、メールを頂けるでしょうか：　[email protected]

A cartilage growth mixture (CGM) model is linearized for infinitesimal elastic and growth strains. The inherent difficulties in using finite deformation theories, develop a CGM model for infinitesimal strains and seek analytical solutions to specific boundary

- value problems

http://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1011&context=meng_fac

Dear Mr. David Yetter ,

》IIn fact, your objections actually have nothing to do with infinitesimals. Series, whether convergent or divergent, are composed of infinitely many finite summands: there are no infinitesimals in sight in either your harmonic series objection or Zeno's paradox《

But, what are those items of Un--->0 in harmonic series called?

Can we call them “finite numbers”?

So, can we say we have infinite items of “finite numbers” in harmonic series?

Sincerely yours, Geng

Dear Mr. Krishnan Umachandran,

Can CGM theory you just mentioned above solve those defects disclosed by those suspended infinite-infinitesimals paradox families?

Sincerely yours, Geng

@ Geng

CGM theory may not solve numerical problems, but in reality solve infinite-infinitesimals in cartilage development

Dear Mr. Krishnan Umachandran,

Looking back into our science history, many infinitesimal cases in mathematics have very close relationship with numerical problems practically (although infinitesimals can not be accepted as “numbers” in present mathematics theoretically, that is why we start the discussion on this confusing stuff).

I am afraid any infinitesimal theories not solving infinite related numerical problems would not become real mathematical ones. What is your opinion?

Sincerely yours, Geng

Dear Ms. Ellis D. Cooper, thank you for your idea.

You are right; we really have many different theories about infinitesimals and many infinitesimal related cases. But the “deep structural equivalent relationship” among them is the biggest problem-----they are the same products out from the same “defected production line”. Let us take “nonstandard one” and “standard one” as an example.

We focus on the “deep structural equivalent relationship” between “nonstandard one” and “standard one”. Let’s exam following facts:

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 equivalence”, the “provable  equivalences” are guaranteed.

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 equivalence” .

 .

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.

Another serious problem-------if we have many definitions of infinitesimals and their related operations, should we choose one or busy with everyone?

Sincerely yours, Geng

There is an excellent book by John Lane Bell on the issue: A primer of infinitesimal analysis, Cambridge 2008, and other papers by him on specific aspects.

Dear Mr. Alberto Peruzzi , thank you for your helpful offer.

But it is very sorry that Mr. John Lane Bell’s works on infinitesimals mostly only deal with the employments of infinitesimals in present traditional infinite theory system with those fundamental defects disclosed by the suspended infinite related paradoxes.

Actually, the problems we meet are “confusions of what infinitesimals are” but not “how to employ infinitesimals in some situations” (it is believed that people could employ infinitesimals quite well in many situations before Newton’s time already).

So, Mr. John Lane Bell’s works can do nothing to help us to solve those fundamental defects disclosed by the suspended infinite related paradoxes either just because the natures of infinitesimals are still very confusing in present traditional infinite theory system.

Sincerely yours, Geng

Dear Ms. Ellis D. Cooper, thank you for your frank opinions. I sincerely hope you don’t mind if I present my frank opinions in our discussion.

I really agree with you Ms. Ellis D. Cooper that many people have done a lot in understanding “what infinitesimals are to those things accepted” so far in our science history. I think Mr. Solomon Feferman only introduced a few common ideas of infinitesimals. To say the truth, one can witness more if going into university classrooms.

You can see clearly in this thread, a very typical characteristic is we don’t join that kind of “understanding accepted things” but solve infinitesimals related mathematical paradox---------such as Harmonic Series Paradox.

Talking about something infinitesimal is easy but solving infinitesimals related mathematical paradox is not easy, that is why much work has been done during 2500 years but infinitesimals related mathematical paradoxes are still there challenging us in our science history, so we cannot just pick up easy ones.

The topic in this thread is not only mathematics but philosophy as well because infinitesimals related mathematical paradoxes are produced by the very mathematical fundamental defects.

Sincerely yours, Geng

Thank you Wes，I really like your frank and constructive opinions.

For your first post, I call it paradox in the case of “how many number items bigger than 1/2 can we have by ‘brackets-placing rule’ in harmonic series” because either finite or infinite is “illogical” in present science. But I can’t open the website you recommended, can you help in some way or just post me that paper?

For your second post, you are right, what I have presented in my paper is not a perfect ready fruit. On one hand, some of my papers have not being translated into English; one the other hand, I am still working hard to make things completed.

Thank you again Wes!

Yours, Geng

Hi Wes，thank you for your suggestions.

In fact, it is the example of newly discovered modern Harmonic Series Paradox（new-ancient paradox）as one of family members of ancient Zeno’s Paradox that makes sense. This truth proves that we should do something different from what we have done during 2500 years in present traditional science system to solve those suspended fundamental defects disclosed by ancient infinite related paradoxes.

Wes，it is my turn to make a suggestion to you: start a topic in a question to discuss your new ideas with researchers from different angles, highlight the essences in your book. RG is a very good platform for us, I would like to read your book and discuss your new ideas with you interactively. Do remember to invite me to your discussion please.

You are right; many people are complaining that my papers are not mathematical enough. Wes，to say the truth, sometimes I doubt myself whether my project mathematical or philosophical, as it deals with very fundamental problems. I will accept your suggestion and try my best to use more math language.

Is Zeno’s Paradox of Runner-turtle’s Race mathematical or philosophical?

Yours, Geng

First, I agree with what Wes Raikowski already remarked above: the questions as they are raised here are too vague. It just seems that the history of both mathematical and philosophical understanding of the continuum is totally neglected.  The history of the Calculus was also due to an effort towards conceptual clarification, which is something of interest for philosophy, whereas it now seems that the whole set of issues has to be restarted from scratch with much less effort. The literature on infinitesimals is huge. Let me just remind Bertrand Russell's description, at the beginning of XXth century, of Zeno's paradoxes in terms of set theory: it is also a reference point in order to understand the links between mathematical and philosophical considerations about infinitesimals; and this answers your last question, even though it does not mean that Russell's analysis must be subscribed. A new start came with Abraham's Robinson's re-introduction of infinitesimals by means of logic (more specifically, through the compactness theorem as a tool in model theory to construct non-standard reals). Jerome Keisler wrote a clear textbook of Calculus based on infinitesimals. But the most radical step, and perhaps the philosophically most important one in recent times, was the purely algebraic notion which came with synthetic differential geometry, by means of category theory - this line of thought implies the rejection of the logical principle of the excluded middle. Such an algebraic notion makes no reference to transfinite sets. In parallel with the formal, axiomatic, presentation of infinitesimals, there is also a number of papers in contemporary philosophy of mathematics which already discuss the issue in precise terms. One remarkable aspect of John Bell's contribution is that such an algebraic notion can be presented without resorting to the language of category theory; at the same time that small book is plenty of philosophical remarks which do not ignore the history of both mathematics and philosophy. 

Dear Mr. Alberto Peruzzi , thank you very much for your summing up and helpful ideas.

1, the continuum is not totally neglected but will be more “scientific” than before.

2, some important reformations in the infinite related fundamental part of science (both mathematics and philosophy) will happen based on many fruits already gained by our ancestors but not the whole set of issues has to be restarted from scratch.

3，we human have been trying for at least 2500 years to solve those infinite related fundamental defects disclosed by the infinite paradoxes, the defects have to be solved sooner or later out of our will.

Thank you again Mr. Alberto Peruzzi!

Sincerely yours, Geng

Dear Wes,

Your above post is not negative but frankly sharing between friends, thank you Wes. 

After reading your post, I think a lot and suddenly understand many things. I agree your understanding and definition to RG------the science related seller’s market. I am sure many people, including me, have some of the same feeling with you. Visiting this kind of market is part of our researchers’ life.

It is great we have a very good managing, maintaining group of people working hard for this “market website” to ensure our “business”.

Wes, I came to RG at the beginning of Sep. 2014, but my situation is even poorer than yours. As you know, my work is “off stream”. So, many of my posts are regarded as “nonsense” and put downvoted; but it is lucky for me to receive some kind and friendly encourages too.

In this science related seller’s market, it is nature and understandable that people come and go busy buying or selling their products by their own will and don’t care who you are. I am very sure not matter buying, selling or window shopping, everyone benefit from it------especially different ideas from different fields. 

People can have their remarks to any products in the market, but the most important thing is we must be sure ourselves with strong scientific foundation.

I say thanks to the researchers who say “no” to my work because I got some ideas from them why “no”; I also say thanks to the researchers who care their own products more than mine because I may pick up some ideas from them.  Of cause I pay particular tributes to the researchers who offer me constructive and helpful suggestions and award me kind and friendly encouragements.

My best regards to you, Geng

I always related Zeno Achilles and the Tortoise paradox to the continuum of time rather than to infinitesimals.

I mean Zeno used infinitesimals (continuos division) to state the paradox, but the solution to the paradox is that the time series running parallel to it is convergent, so you cannot say Achilles will never overtake the Tortoise, but only that the Tortoise will not be overtaken in a time t (limit of the convergent series)...

Dear Samuel,

How do you think of Harmonic Series Paradox presented in the first page of this thread, Mr. Samuel Arba Mosquera?

Sincerely yours, Geng

Thank you for your wonderful metaphor Ms. Ellis D. Cooper, it really helps!

My best regards to you, Geng

Thanks Geng,

which one do you mean?

The one of the rubber band?

If the rubber band is not stretched the worm reaches the other end for any v>0.

If the rubber band is stretched, I just say, the rubber band is stretched while the worm crawls over it, but the worm relative speed to the other end although decreasing remains always >0 so the worm always progresses along the rope, which in turn means that it will reach its end.

If I missed the paradox you meant just let me know. 

Dear Samuel,

I mean the Harmonic Series Paradox presented in the first page (chapter) of this thread.

Yours, Geng

Charles, people may have different names for ”the infinite related numerical things (number forms)”, but it doesn’t matter what they are called, they are there in our mathematics.

So dear Mr. Charles Francis, how do you think of the Harmonic Series Paradox presented in the first page (chapter) of this thread?

Yours, Geng

Dear Mr. Charles Francis, thank you for your frank ideas. I hope you don’t mind my frank ideas.

1，if "in our mathematics" is very different from "in our science", than the sentence in my post may be presented in the way of "in both our mathematics and our science".

2, if you “see no issue with infinite (infinities?) in our mathematics”, than there should be no infinite harmonic series and of cause no Harmonic Series Paradox.

My best regards to you, Geng

Charles, Infinities in mathematics are objects of thought, so, infinities and all their related “mathematical things” are all objects of thought (games of imaginations?) and very different from objects of reality. 

Very interesting idea about mathematics and science, although we have different understanding about what mathematics and science are, at least it should be without antinomies.

Is mathematics a science?

Dear Mr. Charles Francis, do you think antinomies and paradoxes arise in above mentioned cases of Harmonic Series and Zeno’s Paradox because it is incorrectly applied where there is no application.

Is mathematics a science?

My best regards to you, Geng

Dear Ramon，

Do you have any ideas of practical reconciling operations to them?

Sincerely yours, Geng

Mathematics as an art and as a tool for use in a range of fields including science is only one part of mathematics.

Charles,

1, if “art” is “tool” (“art”=“tool”) in your mathematics, than we just chose one: “mathematics is art” or “mathematics is tool”. So, we don’t have “neither or  nor” choice

2, if “art” is not “tool” (“art”≠“tool”) in your mathematics, than the part of “art” is not “tool” and the part of “tool” is not “art”.

My best regards to you, Geng

Dear Ms. Ellis D. Cooper, thank you.

I am sorry that I could not access the website you offered so I really don’t understand what kind of “Harmonic Series Paradox" you are talking about. But one thing is very sure that one in the website you offered is not the same one I presented in the first page (chapter) of this thread.  Mine is a newly discovered one, people can only find it in my papers or in my posts on RG.

Would you please go and see the one in the first page (chapter) of this thread and kindly let me what your ideas are about it?

My best regards to you, Geng

Dear Charles,

I really enjoy your frank ideas in our discussion as I got some ideas from you. It is true.

I am sorry that you think I changed your idea somewhere you mean in your post, would you please be so kind as to telling me my misunderstanding and your original meaning for that part?

I think if mathematics is described in your mind just “art” and “tool”, things will be in trouble because except “some mathematical things” are all objects of thought and very different from objects of reality (such as infinities and all their related mathematical things), one may say “everything is art” or “everything is tool”. In this way, mathematics is surely not human science.

Looking forward to your answer

My best, sincerely regards to you, Geng,

Thank you for your new idea about the relationship between mathematics and human science, Charles.

If possible, I would suggest you start a new question to build a thread on the relationship between mathematics and human science. I think many people would enjoy the new discussion.

Sincerely yours, Geng

Archimedes only asked an infinite series question: how to find the finite version for the area of a conic section written as a 1/4 geometric series:

4A/3 = A + A/4 + A/16 + A/64 + ...

Archimedes wrote down the answer

4A/3 = A + A/4 + A/12

discussed by http://planetmath.org/archimedescalculus

The infinite series question followed by a finite answer continued in the medieval era. For example, Fibonacci's famous Fibonacci Sequence was a question, how many pairs of breading rabbits can be obtained in one year that begins with one pair breading every 30 days. Fibonacci's answer 377.

Dear Mr. Milo Gardner ,

The infinite—finite problem is really a challenging one ever since. For example: in the case of “how many number items bigger than 1/2 can we have by ‘brackets-placing rule’ in harmonic series”, either finite or infinite is “illogical” in present science.

Sincerely yours, Geng

If we don’t really understand what infinite and ”the infinite related very small numerical things (such as infinitesimals)” are, our related mathematical operations (such as differentiation and integration) become the operations on “operation lines”-------fingers and hands moving up and down, pressing buttons, …; just do, it is impossible and no need to have any analysis knowing what the operators are treating in front of them and why infinite related numbers in and out of the formulas------whatever they are called: very small numbers, infinitesimals, variables, monads, …; products are out and paradoxes are still there, …. It is one of the reasons that some people working in the physics field even say “no infinite things in mathematics and science”.

Hi Wes，

Thank you for the website, I can access it this time.

I really question something more down to earth, after 40 years “down to earth” work I discovered the reasons for the defects disclosed 2500 years ago by ancient Zeno. I try to solve those defects disclosed by the suspended infinite related paradoxes and I am on the way.

I hope to see more of your books published for your academic work in new ideas.

Regards, Geng

As most people agree that human science is a kind of human’s property (some other creatures in universe may also “enjoy” this property) composed by human’s mental work (subjective world) and universe (objective world) ------- no human science without human. So, the unbalanced things of subjective and objective compounds of our knowledge are not scientific. Sometimes the subjective factor or objective factor is overstated, each causes confusion.

Human science is human’s, so we must present all the things in our science with our human’s ways. Our science history proves that this is not easy even very hard, the infinite related things in our science are typical examples-------- people have been trying very hard for at least 2500 years so far but the arguments about “finite-finite”, “potential infinite-actual infinite”, “infinitesimal with numerical value meaning-infinitesimal without numerical value meaning”,… aroused by the suspended infinite related paradoxes are still on and on endless. Something should be done in this field to improve our mathematics and science.

The main intellectual reason for Zeno's paradoxes was that infinite series questions were attempted to be answered by finite math. Many classes of infinite series questions were asked:

1. estimate an irrational number by finding its square root in three steps?.

http://planetmath.org/squarerootof3567and29

2. find the formula for the geometric  shapes, such as the  area of a conic section

a. 4A/3 = A + A/4 + A/16 + A/64 + ... . by:

b. 4A/3 = A + A/4 + A/12

3. pi = 4(1 - 1/3 + 1/5 -1/7 + 1/9 - ...)  to find the surface area of a sphere and cylinder

a1. A = 4(PI)r^2

a2  A = (PI) D

b. set D = 1 such that

A = 4(PI)(1/2^2 = PI

and so forth.

The infinite series question model that searched for finite answers was applied by Eudoxus, Archimedes, Fibonacci and as late as Galileo. The modern reason for the model's non-use since 1600 AD has been that infinite series arithmetic has tended to solved for finite and infinite series answers as if the domains of each are equal, which  the domains were not equal for the previous 2,000 years.

.

Dear Mr. Milo Gardner, how do you think of the Harmonic Series Paradox presented in the first page (chapter) of this thread?

Yours, Geng

Well, the fascinating domain of infinitely big (and conversely infinity small) numbers is nicely addressed by a family of numbers called surreal numbers which can be see as an extension of cardinal numbers as explored by G. Cantor through an outstanding construction process using infinite transduction, discovered by John Conway in the early 70's.

What is fascinating with surreal numbers is that the arithmetic is "natural" (this is not the case for Cantor's ordinals, up to my opinion), letting the conceptual oddities of infinite numbers standing right in front of us. Formally speaking, surreal numbers form an algebraic structure which has been proved to be the largest ordered field.

I recommend reading [1] for a lively and didactic introduction by Donald Knuth, and also more in-depht presentation on wikipedia:

[...] the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals.[...]

[1] 

How two ex-students turned on to pure mathematics and found total happiness

by Donald E. Knuth (Reading, Massachusetts: Addison-Wesley, 1974), vi+119pp.

ISBN 0-201-03812-9

Illustrated by Jill C. Knuth

see: attached link

http://www-cs-faculty.stanford.edu/~uno/sn.html

http://en.wikipedia.org/wiki/Surreal_number

Dear Ms. J.-Y. Vion-Dury, thank you for your recommendation.

From at least Zeon’s time, the problem of “what are the infinite related small number forms?” have been troubling us human at least 2500 years. “Many infinite related small number forms” really have been designated with different titles in different periods of time through out our science history, but the problem is: these “delicate infinite related number forms” did nothing to solve the suspended infinite related paradoxes ever since and such games may be played for some more times.

Dear Ms. J.-Y. Vion-Dury, would you please be so kind as to telling me how you think of the Harmonic Series Paradox presented in the first page (chapter) of this thread?

Yours, Geng

Dear Claus, thank you.

But if infinity (of the denominator) can be "reached" or “stopped” in this case, there would be no relating theories of divergent proof of Harmonic Series. It would cause much, much more troubles in present science system.

Yours, Geng

Dear Claus, thank you.

But the fact is:

1, in our actual operations, no one knows “when infinity is considered as reached (as actual)”; by personal wishes? Actually in many cases, people are even unable to distinguish in present science system what kind of infinite we are treating in front of us------- actual one or potential one?

2, in Zeno’s Paradox, not matter what kind of runner (even a runner with the speed of modern jet plane) held the race with the tortoise he will never catch up with it-------- infinite is never considered as reached so far in our modern world.

3, in the Harmonic Series Paradox presented in the first page (chapter) of this thread, can we produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite infinitesimals in Harmonic Series by “brackets-placing rule" to change an infinitely decreasing Harmonic 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?

Yours, Geng

Dear Claus, you hit the point and many people ask the same question to me, thank you.

I know very well that my work is against “the right track” in present science system. But if the defects in “the right track” disclosed by the suspended infinite related paradoxes since Zeno’s time are still there producing its family members (such as Harmonic Series Paradox), there surely be something wrong in present science system of “the right track” and it is our liability to remove such wrongness in “the right track”-------we should do something.

So undoubtedly, on the shoulders of saints I will continue their ideas and work; not only try my best to ask the infinite related questions here but also dig deeper and find a new way out in the rest of my life, it doesn’t matter how much I can do.

If no one is able to answer my infinite related questions here scientifically all over the world so far, it proves that my work is worthy continuing------these defects should be solved by some people sooner or later.

Yours, Geng

Dear Claus, thank you very much for your understanding and encourages.

I sometimes wondered: why we scientists have failed to solve such “ready defects ------they are so clear and definite, so dazzling” in our science disclosed by the suspended infinite related paradoxes since Zeno’s time?

The infinite related paradoxes is really very challenging-----a huge working field; but not matter how difficult the work is, it is impossible for us human to skipped over such defects in our own mathematics and science.

My best regards to you, Geng

2500 years ago.  Many problems were stated as an infinite series...areas of shapes. Conic section 4A/3 = A + A/4 + A/16 + A/64 + ... and solved by finite series. 4A/3 = A + A/4+A/12.  And irrational numbers estimated by the square root of a Prime number by a three step inverse proportion...and so forth

Dear Mr. Milo Gardner, thank you.

I agree with you that in many “applied area” even now (not only 2500 years ago); many infinitesimals are just taken and really operated as very, very small numbers, it is unnecessary for those people working in that areas to ask what they really are-----they don’t care, too. So, many infinite related problems are “estimated” and solved by finite ways, this never trouble us because we know very well that it is quite enough in some “applied areas”.

But in mathematics, especially in theoretical mathematics, infinitesimals can not just be “estimated” as some applied scientists do. On one hand, it is not allowed (scientific) for mathematicians just “estimated”; on the other hand, they really don’t know how to “estimate” practically in present mathematics and science. For example, no one so far in the world are able to “estimate” how many infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or…can be produced from infinite  Harmonic Series by “brackets-placing rule".

Yours, Geng

Geng

WhatI read in your wordsi

 Is that it is okay to answer an infinite series question by linking it to another infinite series. That is fine only if at some point a theoretical finite series is found like the formula for the area of a conic section and the differential and integral calculus

Best regards

Milo

ps. read Kevin Brown's 1995 paper

http://www.mathpages.com/home/kmath340/kmath340.htm

http://www.mathpages.com/home/kmath340/kmath340.htm

Dear Mr. Milo Gardner, thank you.

That is what we are discussing here about, a newly discovered family member of Zeno’s Paradox---------Harmonic Series Paradox.

Finite-infinite, finitesimal-infinitesimal, … confusions in the infinite related fundamental area of our mathematics (science) have been troubling us human ever since and this decides the unsolvable infinite paradoxes since Zeno’s time.

Yours, Geng

Geng,

Kevin Brown in 1995 discussed all arithmetic, geometric and harmonic means as methods used by Greeks and Egyptians to generally convert rational numbers to concise unit fraction series. Read Kevin's paper closely. At that point a deeper discussion can commence of Greek paradoxes of solving infinite series  by finding a finite series, a method of scientific thought that lasted until 1600 AD, and the rise of base 10 decimals that erased the method from our immediate memories.

http://www.mathpages.com/home/kmath340/kmath340.htm

Dear Mr. Milo Gardner, thank you.

In daily life and applied mathematics of present science system, we really can use finite ways to solve infinite problems, because that is enough and ok. But for theoretical mathematics we can not operate in the same way, because it will produce paradoxes such as ancient Zeno’s Paradox and modern -Harmonic Series Paradox.

Yours, Geng

Geng,

Theoretical math  before 1600 CE , 3,600 years to be exact, was based on finding finite solutions to  infinite series. You are discussing 300 BCE, a time period that falls within the finite solutions to infinite series problems ... harmonic series includes. Your analysis oddly limits yourself and your one algorithm size fits all during the last 500 years. You are correct that  algorithms have been applied to solve other algorithmic problems. Modern computers struggled to solve rough off issues ... IEEE sets standards that every good mathematician solved before 1600 BCE with exact or near exact answers

Dear Mr. Milo Gardner, thank you.

In fact, people used infinite ways to solve infinite Harmonic Series problem, so it is divergent and produce Harmonic Series Paradox of modern Zeno’s Paradox version.

Yours, Geng

nothing divergent about Kevin Brown's harmonic mean solution to

2/91 = 1/70 + 1/130

Occams Razors solved the problem by scaled 2/91 by 70/70 = (91 + 49)/(91*70) ... a point that seems odd in your world it seems

Thanks for the chat

Milo

Dear Mr. Milo Gardner, thank you for the case example. Please see another case:

In the divergent proof of Harmonic Series which 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） 

We have met 2 problems:

1, how many items each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… can be produced from infinite Harmonic Series by “brackets-placing rule"?

2, can we really produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or…as in the above proof to change an infinitely decreasing Harmonic 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?

Yours, Geng

Dear Geng,

As you know $\lim_{n\to\infty \frac{1}{n}=0$ and also $\lim_{n\to\infty \frac{1}{2n}=0$. We could describe infinitesimals as sequences like $\{a_n\}$ such that $\lim_{n\to\infty}a_n=0$. However we have an order! $\{a_n\}\le\{n_n\}$ iff for ever n $a_n\le b_n$. So, for example we have infinitesimals like $\{\frac{1}{n}\}$ and $\{\frac{1}{2n}\}$ where $\lim_{n\to\infty \frac{1}{n}=\lim_{n\to\infty \frac{1}{2n}=0$ while $\{\frac{1}{n}\}\le \{\frac{1}{2n}\}$. But this order is not linear. By the concept of ultrafilter we could define the order such that the order between  infinitesimals is linear. Furthermore we could develop the concept of infinitesimals on the real  line ($\mathbb{R}^*$).

Ref: Non-standard Analysis. By Abraham Robinson. Princeton University Press, 1974

Dear Mr. Seyed Mohammad Amin Khatami

Thank you for your point of view.

It is true that there have been many “understandings, interpretations, language…(of cause including Non-standard Analysis. By Abraham Robinson)” for the infinitesimals now in our mathematics since Zeno’s time more than 2500 years ago. The fact is no matter what kind of “understandings or languages” people chose, Zeno’s Paradox is still there challenging us human’s intelligent and the family members are getting more and more---------Harmonic Series Paradox as a “strict mathematical proven” suspended Zeno’s Achilles--Turtle Race Paradox is right there in our modern mathematics.

According to my studies, in front of the “2500 years old huge black cloud of infinite related paradoxes over mathematics sky”, we see clearly that those “understandings, interpretations, language…” for the infinitesimals are just a kind of “self-consolation game” doing nothing in solving practical problem.

Is it the time for us to stand on giants’ shoulders to work on a revolution in the infinite related philosophy of mathematics and the infinite related foundation of mathematics?

Thank you again for your frankness and kindness.

Best regards,

Geng

“Potential infinite” and “actual infinite” have been two important concepts in our present classical infinite related foundation of science (mathematics) theory system at lest since Zeno’s time more than 2500 years ago. The defects in this field have produced “the 2500 years old black cloud of infinite related paradoxes over mathematics sky.”

1, When paying equal attention on “potential infinite” and “actual infinite”, we meet the uncompromised logical contradiction and the typical case is the “strict mathematical proven” ancient Zeno’s Paradoxes and modern Paradox of Harmonic Series.

2, When paying more attention on “potential infinite” and neglect “actual infinite”, we meet all the ideas and theories of “all infinite things are the same endless without any inborn and numerical natures” and the typical cases are all the ideas and theories of the countable infinite sets.

3, When paying more attention on “actual infinite” and neglect “potential infinite”, we meet all the ideas and theories of “all infinite things are not the same limitless and endless, they have inborn and numerical natures” and the typical cases are all the ideas and theories of power set and uncountable infinite sets (infinite Set R has more elements than infinite Set N): more infinite, more more infinite, more more more infinite,…; higher infinite, higher higher infinite, higher higher higher infinite,…; bigger infinite, bigger bigger infinite, bigger bigger bigger infinite,…; supper infinite, supper supper infinite, supper supper supper infinite,…; Continuum Hypothesis,…

I would commend to your attention Conway's surreal numbers, which include both Robinson-style infinitesimals and their infinite reciprocals. Wikipedia as a very good article on their construction, but as you are fascinated by things involving infinite series as well as infinitesimals, you might look at the article by Rubinstein-Salzedo and Swaminathan at arXiv:1307.7392v3 that gives, among other things, a formula for the limit of a sequence of surreal numbers.

How will our papers?

Saburou Saitoh, Mysterious Properties of the Point at Infinity、 arXiv:1712.09467 [math.GM]

H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,

Journal of Technology and Social Science (JTSS), 1(2017), 70-77.

I like the old-style approach to infinitesimals as in Granville's Calculus. In this view an infinitesimal * is a variable (real) quantity that is always decreasing in absolute value and * is never zero. Lets say *>0 for this discussion. Then * has the temporal property that if you specify any constant real number r, then 0

Please look the related papers, because 1/0=0 in the natural sense.

How will be the relation between \infty and 0?

\bibitem{os}

H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).

ｘ\bibitem{oku18}

H. Okumura, Is It Really Impossible To Divide By Zero? Biostat Biometrics Open Acc J. 2018; 7(1): 555703.

DOI: 10.19080/BBOJ.2018.07.555703.

ｘ\bibitem{o}

H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947. International Journal of Geometry, {\bf 7}(2018), No. 1, 17-20.

\bibitem{os18a}

H. Okumura and S. Saitoh,

Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum, {\bf 18}(2018), 97-100.

\bibitem{os18b}

H. Okumura and S. Saitoh,

Applications of the division by zero calculus to Wasan geometry.

GLOBAL JOURNAL OF ADVANCED RESEARCH ON CLASSICAL AND MODERN GEOMETRIES” (GJARCMG) {\bf 7}(2018), 2, 44--49.

Saburou Saitoh, Mysterious Properties of the Point at Infinity、 arXiv:1712.09467 [math.GM]

L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.

M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,

New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$, Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.

T. Matsuura and S. Saitoh,

Matrices and division by zero z/0=0,

Advances in Linear Algebra \& Matrix Theory, 2016, 6, 51-58

Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt

\\ http://dx.doi.org/10.4236/alamt.2016.62007.

ｘT. Matsuura and S. Saitoh,

Division by zero calculus and singular integrals. (Submitted for publication).

T. Matsuura, H. Michiwaki and S. Saitoh,

$\log 0= \log \infty =0$ and applications. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.)

H. Michiwaki, S. Saitoh and M.Yamada,

Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1.html

H. Michiwaki, H. Okumura and S. Saitoh,

Division by Zero $z/0 = 0$ in Euclidean Spaces,

International Journal of Mathematics and Computation, 28(2017); Issue 1, 2017), 1-16.

H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,

Journal of Technology and Social Science (JTSS), 1(2017), 70-77.

S. Pinelas and S. Saitoh,

Division by zero calculus and differential equations. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics).

S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/

S. Saitoh, A reproducing kernel theory with some general applications,

Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .

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

Geng Ouyang points out that, in the proof of the divergence of the harmonic series, the number of terms that have to be added together to make each summed increment >1/2 grows exponentially. Thus a "person" calculating the series would, after a certain time, run out of time and/or computational space to make the increments needed. This does not change the proof that the series diverges, but it does point out that in practice, with some specific limitations on our computational resources, we would not be able to model the increase beyond all bounds. Most theorems about integers have a similar property. For example.

There are infinitely many primes.

Proof: Consider n! + 1. This number has no prime divisors n for every n. Q.E.D. The proof does give a method for finding primes >n for any n: Factorize n! + 1. This problem "Factorize (n!+1)" will, for sufficiently large n, break any computer on the planet and will break any computer that we build in the future, including quantum computers. The difference between theory and practice is large. Some have believed that there is an inconsistency in number theory related to this (Edward Nelson suggested that for example.) but none has yet appeared.

Dear Mr. Louis Kauffman , thank you very much for your idea!

You admit there is really “The large difference between theory and practice” in our infinite related mathematics.

The problem is, I believe (in fact I am on the way doing something) that just because mathematics is human’s, we can try some way to fill up “The large difference between theory and practice” in our infinite related mathematics to solve many suspended “infinite related paradoxes” (paradox syndrome complex) in present mathematical analysis and set theory---------the "thousands--year old huge black clouds of infinite related paradox families over the sky of present classical mathematical analysis".

Regards!