if no one knows “what infinity is”, many topics relating to “infinity” in RG leads us nowhere.

Thank you for your question, Geng. I think that for teaching the idea of infinite one of the first acknowledgements is that infinite should be non only end-less but also bottom-less. The philosopher Lévinas used an amusing metaphor that could work with students: infinite is an-archical (literally “without beginning”). That opens the big question of a self-generating infinite and how this self-generation could happen…

Dear Mr. Francesco Spagna,

Thank you very much!

I don’t know now much we could do, but I am deeply convinced that this is something we should do.

My researches told me that Infinitude is really a very challenging and tough thing which many people would not touch or have to avoid. However, many facts have proved that the concept of objective rooted Infinitude exists in our science; it is right there and the infinitude things are there for us to face, cognize and treat. Because of the fundamental defects in present traditional Infinitude theory system, we human really meet many difficulties when working in the field relating to Infinitude.

To introduce the “idea of infinity” to students is really very easy as so many colleagues have done. But it is difficult to come to real life with the” idea of infinity” when facing and treating the “number forms relating to infinity", especially their relationship with zero; for example, why sometimes those X--->0 are allowed to be zero by limit theory in many actual calculations but sometimes not, such as in newly discovered Paradox of Harmonious Series as well as in ancient Zeno’s Paradox’s “Race of Achilles and Tortoise” .

@ Miguel,

Thank you for such a good question. I am sorry that I didn’t express clearly to avoid ambiguity.

In fact, I use “infinitude” as a general concept in the fields of mathematics and philosophy in most of my papers.

As we know, we now really have 4 confusing “infinitude” related things in our science:

(1) potential infinity as something unknown, as the pre-Aristotlean Greek believed,

(2) actual infinity as a property of sets, as Cantor thought on it,

(3) infinitesimal with the number form of X--->0,

(4) infinity as big number, as the pre-K12 child think on it.

Why I post such a topic, because according to my studies, we have a big trouble:

for the first 2, we have long-drawn-out and ceaseless “potential infinity--actual infinity” debates since antiquity; while for the last 2, people use them as “numbers relating to infinitude” in many kinds of practical calculations but they are “non-number number of variables（theoretically they should not be numbers but practically they should be numbers）”.

What can we do?

Hi Miguel，

Actually your answer touch the bottom of my topic: what is infinitude? Very good!

The suspended Zeno’s Paradox of “Achilles--turtle Race” (we can see from many papers that many people are still trying hard but fruitlessly) tells us that this question remain unsolved since Zeno’s time.

(1) Theoretically, in present traditional infinitude system, the “infinitude” concept can be divided into “potential infinitude” and “actual infinitude”. But no clear, self-justification definitions of “potential infinitude” and “actual infinitude” can be found in our philosophy and mathematics so far-------that is why we have long-drawn-out and ceaseless “potential infinity--actual infinity” debates since antiquity;

(2) Practically, when we meet infinite things, we have no ideas whether the infinite things we meet are “potential infinitude” or “actual infinitude”-------- that is why we have suspended “infinitude related paradoxes families”.

The students I mention in this line are undergraduate students, master students and doctoral students but not 10-years-old kids.

Hi Miguel，

It is good that you share with me your “infinitude” teaching experiences, thank you. We are talking freely and frankly, aren’t we?

1, Theoretically, when you explain that infinity is not 'something that has a meaning', but 'something that has a meaning for', but for what? I guess your students may emerge very interesting long-drawn-out and ceaseless discussions about the nature of the infinitude concept (just like our ancestors did), but in the end, they may be still unaware what infinitude is------ something that has a meaning for what? Is it just like someone said “a language game”?

2, Practically, is the infinite concept in Hilbert's Hotel “potential infinitude” or “actual infinitude”?

I will be very happy if you can give me your direct and frank opinion.

Regards

Hi Miguel，

It is good that our discussion going deeper. Thank you.

I am sorry; do you mean that if 100 persons who think on a mathematical concept then this concept can really have 100 definitions in mathematics? Frankly, I don’t agree that the definitions of potential or actual infinity (or any concepts) in mathematics just “depending on how does that person understand infinity (or any concepts)” but not depending on objectivity.

Now we come to the bottom: Is mathematics a science or a religion?

Regards!

Reconsidering René Guénon, the segments are three: Science; Philosophy & Religion; Traditional [Transcultural] Metaphysic. Notwithstanding all the controversies around this author, Guénon tried to bridge - through Metaphysic - Science to Philosophy and Religion. Not so different to what, later, tried to do Gregory Bateson. The "evil" aspect of infinity that affected Western thought from Aristotle to Borges, is the false infinity of indefinite. I believe interdisciplinary discourse about infinity could be enhanced considering infinity under a qualitative, not  quantitative, aspect.

Hi Francesco,

In fact, “infinite” is a very general concept existing in our science. People meet this concept in many different areas of our science. I agree with you that interdisciplinary discussions on the nature of infinite would be helpful, because the discussions many be free from some ill limitations (I call them “pollutions” in my paper uploaded).

But can we have only one definition for the concept of “infinite” in our science (including mathematics) or many, such as （1） “potential infinitude”, “actual infinitude” with different natures; （2） different degree of “infinite”------more “infinite”, more and more “infinite”,…; less “infinite”, less and less “infinite”, … ;…?

Thank you!

When we discuss our infinite related knowledge since antiquity, intuition was always mentioned. But one thing is important that our science needs ontology and form, we need exact theory and practical operations to deal with real infinite things.

Yes, we don’t reject the role “intuition” plays but it can only offer us inspirations and ideas.

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.

It is possible that peoples of different linguistic backgrounds conceptualise infinity differently due to peculiarities of their languages. I came across an interesting article about this problem on RG . The paper was written by Dong-Joong Kim, Joan Ferrini-Mundy and Anna Sfard; it is titled: "How does language impact the learning of mathematics? Comparison of English and Korean speaking university students’ discourses on infinity".

The article claims that "It was found that in spite of the comparable levels of mathematical performance, there was, indeed, a visible dissimilarity between mathematical discourses on infinity of Korean- and English speaking students. In general, whereas no group could pride itself on a well-developed mathematical discourse on infinity, the mathematical discourse of the English speakers, just like their colloquial discourse, was predominantly processual, whereas the Korean speaking students’ talk on infinity was more structural and, in an admittedly superficial way, closer to the formal mathematical discourse." 

This could explain the inconclusive results of many interesting discussions initiated by Geng. You can find the article on Anna Sfard ResearchGate site. 

Dear Mr., Untangling Math, thank your for your frank view points.

But I think there may be 2 other possible situations too:

1, mistaken understanding or mistaken translation,

2, some new ideas on infinite,

thank you again!

Yours,

Geng.

Hi, 

RE: "But I think there may be 2 other possible situations too: 1, mistaken understanding or mistaken translation,"

True, wrong translation is always a possibility. As far I know, the authors of the article include native speakers of Korean. 

RE: "2, some new ideas on infinite," 

There is nothing wrong with wanting to redefine mathematical ideas. The problem is that it is not clear whether you want to redefine the concept of infinity within the context of mathematics because you seem to jump between common understanding and mathematical understanding of it. The way I see it, you criticise mathematical definitions for not always relating well to our everyday experiences and notions. 

Regards, UM

Dear Mr. Untangling Math，thank you for your frank view points.

1， The suspended infinite related paradoxes drive us logically to solve the defects in the foundation of our present science and mathematics, including redefining some mistaken concepts and develop some new theories within the context of mathematics.

2， Would you please tell me what you mean by “common understanding and mathematical understanding of infinite” and “our everyday experiences and notions for infinite”?

Best regards,

Geng

Dear Mr. George Stoica, thank you.

I think infinite is a “scientific content”-------a kind of our cognized fruits. So it should exist theoretically and practically, it would be with quality and quantity characteristics. The simple definition of “Infinite is a symbol that designates the process of increasing without bound” has produced many suspended infinite related paradoxes and lots of troubles in the debates of “potential infinite—actual infinite”, “more infinite—less infinite”...

Best regards,

Geng

You are right dear Mr. George Stoica, philosophy plays very important roles in the foundations of many branches of our science. But, because of the serious defects in the “standard mathematical definition of the concept of infinite” theoretically and practically in present classical mathematical theory system, many suspended infinite related paradoxes have been produced-------not just “the subjective nature of the narrative” but the “2500-year-old huge black cloud over mathematics sky”.

Sincerely yours,

Geng

Infinite calls us for the existence of God. God is Omni potent,Omni Present,Omni scent . If we believe it God is within every mind & heart of the human beings.God does not require any introduction as it is within of the heart of every human beings.

For any justification of God it is up to every human being to decide to understand him either with the help of theoretically mode & with their their understanding ,he may certainly express understanding of God in a practically mode .

This is my personal opinion 

In present infinite related science system, people have been admitting the being of “potential infinite, actual infinite” concepts, unable to deny their qualitative differences and the important roles they play in the foundation of present classical infinite theory system and, unable to deny that the present classical infinite related mathematics is basing on present classical infinite theory system.

The fact is: on the one hand, present classical infinite related science system and mathematics can not avoid the constraining of “potential infinite--actual infinite” concepts and their relating “potential infinite things (such as potential infinite numbers)--actual infinite things (such as actual infinite numbers)”; on the other hand, no clear definitions for these two concepts of “potential infinite--actual infinite” and their relating “potential infinite things --actual infinite things” have been constructed since antiquity, thus naturally lead to following two unavoidable fatal defects in present classical mathematicas:

（1）it is impossible (unable) 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 (unable) 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)”.

This is the very reason for the thousands-year old suspended infinite related paradox families in present classical infinite related science and mathematics.



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