慕容青草的博客
哲學園地
戴榕菁
前文“哲學無窮大與數學無窮大”【[1]】中提到在academia.edu的GM Jackson的一篇試圖將康托關於無窮大的錯誤理論合理化的題為“Why Different Infinities Are Really Equal” 的文章的討論中，我指出在哲學上無窮大是一個標誌著大於任何一個數的概念，而在數學上無窮大則是基於哲學的無窮大概念的度量，而不同背景條件下的數學無窮大的度量意義是不同的，因此數學上的無窮是不會如Jackson他所要證明的那樣相同的。
雖然基本上沒有人提出什麽異議，我可以感受到那裏參加討論的人並沒能真正理解我上麵提到的論述。。。。相應地，他們之間的爭論繼續著，爭論的焦點變為無窮大到底是否為一個數值。從他們的討論可以看出那裏參加討論的基本上都是搞數論和集合論的專業數學界人士，對相關的定理及理論都很熟悉，但在與之相關的哲學上的認識上卻都相當欠缺。比如，由於康托的影響，使得該討論文章的作者Jackson對於n/0以及無窮大的意義產生了混亂和困惑。下麵是我針對無窮大是否為一個數值所表達的觀點整理後的中譯文（英文相關討論原文在附錄裏）：
（1）
從小我們都應該在學校裏學到n/0 是不存在的, 也就是說. n/0 不是一個數。類似地，n/0 = ∞是不ok的, 而當x → 0 時lim n/x = ∞ or y 是ok的。這是因為∞大於任何一個數，因此它本身就不是一個數。相應地，對於數值進行的運算就不宜用於∞。也就是說：
n × ∞ = ∞ 不ok 而 lim n × u = ∞ 當 u → ∞ 時 ok。
n + ∞ = ∞ 不ok 而lim n + u = ∞當u → ∞ 時 ok。
n / ∞ = 0 不ok 而lim n / u = 0, m, 或 ∞ 當u → ∞時 ok。
∞ + ∞ = ∞不ok 而lim n + u = ∞ 當n → ∞ 和u → ∞時 ok。
∞ × ∞ = ∞不ok 而lim n × u = ∞ 當n → ∞ 和u → ∞時 ok。
∞ / ∞ = M 不ok 而lim n / u = 0, m, 或 ∞ 當n → ∞ 和u → ∞時 ok。
（2）
人們可以任意定義他們自己想要的形式係統。比如，他們可以定義∞+ ∞= ∞, ∞ × ∞ = ∞, ∞ - ∞ = M, n × ∞ = ∞, n + ∞ = ∞, 等等。但是，當他們這麽做的時候，他們已經製造出了一個與具有“大於任意一個數”的哲學意義的∞相矛盾的特殊的∞。。。也就是說，他們製造了他們自己的一套遊戲體係而已。
人們是否可以將這樣的遊戲體係做實際的應用呢？
在特定的條件下，隻要不會引起邏輯上的衝突，人們也確實可以將他們自己隨心所欲定義的形式係統應用在現實世界的實踐中。。。。這就是好比每個人都可以有自己的速記的係統，但這並不意味著別人就能讀懂你用速記寫下的內容。至於你自己的速記是對是錯也隻有當你把你記的內容拿出來與現實對比時別人才知道。。。。如果你做速記隻是為了自嗨，那麽別人也完全沒必要在乎你記錄的是什麽。
所以，當人們將他們自己所定義的關於∞的形式係統用於現實生活時，他們應該記住這一點：他們的係統與現實之間是有著裂縫的，在某些情形下，他們的腳有可能被卡在那些縫隙中的。
比如，人們常把tan(π/2) = 1/0 寫成∞，但是實際上，tan(π/2)與tan(π/2-ε)在幾何意義上有著本質的區別，因為在一個直角三角形內，當一個銳角增大到π/2時，那個三角形就不複存在了。所以tan(π/2) = ∞ 就相當於人們用來記錄一個並不存在的內角的正切值的速記符號，一個用來表示當ε趨於0的時候，tan(π/2-ε) 之極限值的速記符號而已。
盡管上述內容非常地哲學但它們絲毫也不比數學來得不真實。
（3）
Jackson在回應我的上述論述時說【Real numbers require that there be an infinite number of numbers between, say, 0 and 1. It's tough to have a concept of real numbers without evoking infinity. 】
我對此回複說：當你說實數要求0和1之間有無窮多個數的時候，我猜在你的潛意識中那個無窮大就是“大於任何一個數”的哲學概念，這是沒有問題的。。。。但是，當你打算將∞ + ∞ = ∞應用於實數時，你的∞就不再是那個單純的古老的∞概念了，你步入了一個具體的度量係統，在那裏你將與常識或自伯努利和洛必達以來人們已經知道了數百年的知識發生衝突。比如，你會象康托一樣地聲稱一個正方形的一邊上的點與整個正方形上的點的數目完全一樣，那不但是錯的，而且是荒誕的。
（4）
關於無窮大的一個易讓人不小心犯錯的議題是所謂的空間中含有無限多個幾何點，那些點通過笛卡爾映射也與無限多個實數相對應。
空間中含有無限多個幾何點的概念是基於幾何點沒有體積這一概念構建的，而這一概念曾給芝諾和亞裏士多德造成麻煩，甚至在牛頓通過微積分解決了與所謂零體積幾何點和無限多個點相關的問題後，黑格爾和羅素（他同時也是一位專業的數學家）還在相關的問題上跌跤。 這裏的“零體積幾何點”是一個由人類想象力構造出來的抽象概念，這個抽象概念具有自身的矛盾性......如果一個點沒有體積，它不應該占據空間，但如果它不占據空間，無論沿著哪個方向堆疊多少個它都不會積累。
這時哲學便有了用武之地。。。。哲學的思維告訴我們：利用人類想象力的目的不是破壞人類對自然的反映，而是幫助人類更好地感知自然。顯然，將零體積幾何點的自相矛盾概念擴展到計數實踐中以使∞+∞=∞有效則會導致邏輯錯誤。因此，盡管沒有生物法則或心理法則來阻止你進行這樣的想象，你也不應該不合理地應用你的想象。
（5）
Jackson對我上麵的這個評論回應到：【Ron, I don't disagree. But I wonder how many zeros it would take to make 5? How about 5/0 zeros? Where 0 * 5/0 = 5.】
對此，我做了如下的回複：
不論有多少個0，它們都無法構成一個5。當黑格爾試圖證明無中生有時，他在這個問題上跌了跤。。。。他實際需要的應該是微積分，但微積分是一種無法用不具備無限逼近概念的自然語言來表達的特殊的數學語言。。。。黑格爾非常努力地做了論證（參見【[2]】），但我不認為他把問題說清楚了。
當羅素想用集合論來表達時間的瞬間時，他也在這個問題上跌了跟頭。。。。他遇到了你們現在麵臨的同樣的問題：因為集合論缺乏無限逼近的概念，所以它無法用來表達微積分。。。。與之相應的是人類的數字係統本身的局限（參見【[3]】）。
人類的數字係統來自數千年前的計數實踐。。。。但人類數字係統本身並不含有牛頓的微積分工具。。。。現在你和這裏其他的數論方麵的數學專業人員可能很不願意聽到我這麽說。。。。但我要說的是：你們最好還是接受人類的計數係統中所存在的不足之處吧！
很顯然，作為一個集合論的數學家，羅素沒有意識到人類計數係統中的上述缺陷，然後作為因其在哲學上的貢獻而獲得諾貝爾（文獻）獎的人，他最終也沒有認識到導致他無法用集合來表達時間瞬間的真正原因是什麽。。。。
結束語
本文的內容可以幫助讀者更好地理解“哲學無窮大與數學無窮大”【1】一文中提到的哲學無窮大與數學無窮大的區別。。。。
其實，作為這次討論之收獲的本文和上次的“哲學無窮大與數學無窮大”的意義有可能遠超出前兩次參加Jackson討論後所寫的兩篇英文文章【[4],[5]】及與之相關的中文文章【[6],[7],[8],[9],[10],[11],[12]】。那兩次的討論促使我推翻了著名的康托連續性假說（即希爾伯特第一問題）且提出了被數學界忽略但在物理實踐（尤其是在揭露物理學界實驗造假方麵）有著重要意義的一個定理，說起來都是非常有意義的具體收獲，但是這一次討論所促成的這兩篇文章為撼動被康托整歪了的現代數學奠定了堅實的哲學基礎！
這裏的要點是：即便在康托之前，雖然無窮大的概念是自古希臘時期就有了的，從未有人如本文和“哲學無窮大與數學無窮大”一文那樣對無窮大的哲學概念進行深入的剖析討論（否則的話那群數論專業的人也就不會那麽糊塗了，作為數論專家的羅素也不會在相應問題上跌跤了，希爾伯特也不會被康托忽悠了，而康托自己也不會忽悠他自己了）。。。。相信在不久的將來人們會在數學界看到對本文內容的回響。。。。
感謝上帝！榮耀歸於上帝！
附錄
本文內容在academia.edu的討論中的原文及上下文：
It is supposed to be a basic from grade school that n/0 does not exist, i.e. n/0 is NOT a number.
Similarly, n/0 = ∞ is not ok while lim n/x = ∞ or y when x → 0 is okay.
This is because ∞ is a notion of “greater than any number” and thus is NOT a number itself.
Accordingly, any arithmetic operations cannot be applied to ∞, which means:
n × ∞ = ∞ is not okay while lim n × u = ∞ when u → ∞ is okay
n + ∞ = ∞ is not okay while lim n + u = ∞ when u → ∞ is okay
n / ∞ = 0 is not okay while lim n / u = 0, m, or ∞ when u → ∞ is okay
∞ + ∞ = ∞ is not okay while lim n + u = ∞ when n → ∞ and u → ∞ is okay
∞ × ∞ = ∞ is not okay while lim n × u = ∞ when n → ∞ and u → ∞ is okay
∞ / ∞ = anything is not okay while lim n / u = 0, m, or ∞ when n → ∞ and u → ∞ is okay
Summary:
1) n/0 is not a number;
2) ∞ is not a number….it is a notion of “greater than any number”….this notion cannot be counted or calculated like a number.
Cheers, Ron
Like1
We can compare infinity with cardinal and intercardinal points. We could, for instance, move towards the northeast indefinitely, but we would never arrive at a place called the northeast. And, if we are naive enough to believe in the existence of such a place, we just need to imagine ourselves being there to realize that, even from there, we could keep moving towards the northeast. And, if we keep moving in that direction forever, we would be going in circles, getting closer and closer to the north pole, but never getting there or at the northeast point.
Like
David this is true, but is the definition of the finite only rational in light of the irrationality of the actual infinite? The concepts are interdependent. Thus both are real in that sence. The finite expressions depend conceptually on the transcendent for intelligibility.
We cannot navigate 3d visual reality without the vanishing point. So the vanishing point although absurd in one sence is necessary and real.
Actual infinity is an irrational real object and we as fininte can never arrive but are dependent on the aim.
Cantor agreed with this and conceptualised mathematical ways, sets to bound the actual into intelligibility and mathematical expression. The symbol for Pi is a visual set that contains the actual infinite real numbers. We are also forced by this reality to say the verbal word "Pi", within that verbal set contains the infinity of digits. Imagine if I had to verbally express the digits of Pi instead of saying "Pi". I would stand muttering digits until I died. This is absurd but true.
Cantors impulse was an exploration into what we do with language and symbols implicitly. And in my estimation proves the Actual as a necessary irrational yet real thing.
What do you think about that?
Like
T Clark.
I'm sorry. I think all that is nonsense. We don't need actual infinity to understand infinity (potential infinity, as named by Aristotle). It seems that you an I understand irrational numbers in a different way.
Like2
Ron, thanks for your comment.
"It is supposed to be a basic from grade school that n/0 does not exist, i.e. n/0 is NOT a number."
Yes, I learned that too, but I now believe it is OK to divide by zero, since that is the only way you can reach absolute infinity.
"Similarly, n/0 = ∞ is not ok while lim n/x = ∞ or y when x → 0 is okay."
Yes, I agree, but if x has an absolute value greater than zero, then n/x doesn't equal infinity--just a finite number that tends to infinity.
"This is because ∞ is a notion of “greater than any number” and thus is NOT a number itself."
Actually infinity is a number. Here is the definition:
"a number greater than any assignable quantity or countable number (symbol ∞)."--Google search.
"Accordingly, any arithmetic operations cannot be applied to ∞, which means:
n × ∞ = ∞ is not okay while lim n × u = ∞ when u → ∞ is okay
n + ∞ = ∞ is not okay while lim n + u = ∞ when u → ∞ is okay
n / ∞ = 0 is not okay while lim n / u = 0, m, or ∞ when u → ∞ is okay
∞ + ∞ = ∞ is not okay while lim n + u = ∞ when n → ∞ and u → ∞ is okay
∞ × ∞ = ∞ is not okay while lim n × u = ∞ when n → ∞ and u → ∞ is okay
∞ / ∞ = anything is not okay while lim n / u = 0, m, or ∞ when n → ∞ and u → ∞ is okay"
True, but that doesn't stop mathematicians from trying to do operations with infinity. See Ramanujan, Cantor, etc.
"Summary:
"1) n/0 is not a number;"
Technically it is a number, but the number is undefined.
"2) ∞ is not a number….it is a notion of “greater than any number”….this notion cannot be counted or calculated like a number."
Infinity is a number, but it is not finite, so it does not behave like finite numbers when operations are performed with it.
Like
Well, people can define any formal system they want.....for example, they might define ∞ + ∞ = ∞, ∞ × ∞ = ∞, ∞ - ∞ = something, n × ∞ = ∞, n + ∞ = ∞, etc.....However, by doing this they have created an ∞ that is in conflict with the basic notion of ∞ for "greater than any number".....they just created a game system for themselves......
Can they put that system in practical use?
Sometimes they can as long as it does not cause logical trouble.....just like anyone can have his own shorthand system but not necessarily everyone else could recognize all the contents he put on paper for fast recording (Chinese people might appreciate this more than English people because Chinese shorthand writing of some people could really go wild)......
Therefore, when people do use their own formal system for ∞ in real life they need to bear in mind that there are cracks between their system and the reality and their feet might be stuck in the cracks sometimes.....
For example.....tan(π/2) = 1/0 is often written as ∞, but the geometrical meaning of tan(π/2) is fundamentally different from the geometrical meaning of tan(π/2-ε) because within a right triangle, when an acute angle increase to π/2, the triangle would no longer exist.....so the tan(π/2) = ∞ is just a shorthand for taking note of the tangent of a nonexistent internal angle of the right triangle, or a shorthand for noting the limit of tan(π/2-ε) when ε goes to zero.
All the above content is much more philosophical than mathematical.....nonetheless it is not less real because it is more philosophical than mathematical.....contrarily, it just exposes the shortcoming in mathematical system and it tells that mathematics cannot completely be independent of philosophy!
Cheers,
Ron
Like2
GM Jackson,
In mathematics we do not use Google definitions. We define objects by their properties, and those properties can be accepted only if they do not create contradictions. You can believe in whatever you want but in mathematics we are either right, if our statements do not produced contradictions, or wrong, if they do. Mathematics need to be consistent and coherent. This is not about believes and dictionaries but about rigor and proofs.
Thanks for inviting me to this discussion.
Like
Ron, thanks for your last comment. You wrote:
"Can they put that system in practical use?"
The only practical use I can think of is the idea of infinite precision and real numbers. Infinite precision is a worthy goal but probably can never be achieved. Real numbers require that there be an infinite number of numbers between, say, 0 and 1. It's tough to have a concept of real numbers without evoking infinity. Infinity does not behave like finite numbers when operations are performed on it, but I don't consider that a contradiction. It would only be a contradiction if infinity and finite are defined the same way.
Like1
David Peralta: Could you please tell me what book cites 400 people who question Cantor's arguments? I would like to see this book. Thank you.
Like
GM Jackson,
When you say 【Real numbers require that there be an infinite number of numbers between, say, 0 and 1. It's tough to have a concept of real numbers without evoking infinity. 】， subconsciously I guess you were referring to ∞ as a notion of "bigger than any number" which is consistent with the ancient knowledge since thousands of years ago......
However, if you intend to apply ∞ + ∞ = ∞ to real numbers then you are no longer in the realm of the ancient notion of ∞, but rather getting into a counting scheme of numbers, where you will get into logical contradiction with basic commonsense or the knowledge that people have been knowing since Bernoulli and L'Hôpital for centuries.....for example, you will conclude like Cantor did that the number of points on a side line of a square are the same as the number of points on the square itself, which is not only wrong but also absurd.
Cheers,
Ron
Like
David, you wrote ...
"In mathematics we do not use Google definitions."
After Googling the definition more than once I find there are two camps: those who define infinity as a number and those who don't. At this point in time I believe infinity is an absolute value greater than any number I can imagine. This belief is based on X = lim h --> 0 n/h, where h is taken all the way to the limit of zero. n/0 is an undefined number that must have an absolute value greater than any value imagined. It is undefined because no one can pin down that absolute value.
"We define objects by their properties, and those properties can be accepted only if they do not create contradictions."
But humans can be walking contradictions and are also considered objects.
" You can believe in whatever you want but in mathematics we are either right, if our statements do not produced contradictions, or wrong, if they do."
The word "contradiction" needs to be defined as well. If a statement is consistent with the nature of infinity, then there is no contradiction. For example, if I claim that infinity behaves like a finite number and then write infinity + 1 = infinity, that is a contradiction. But if I claim infinity does not behave like a finite number, then there is no contradiction.
"Mathematics need to be consistent and coherent."
But your math won't be if you don't keep your definitions straight and true. Your error is you assume the infinite and the finite should all be the same and behave the same way. That's the contradiction.
"This is not about believes and dictionaries but about rigor and proofs."
Rigor and proof have no value without dictionaries that contain the agreed upon terminology used in proofs. Further, you can't prove that infinity is or is not a number. It depends on how it is defined, and definitions are subjective agreements within communities, so members of a community can communicate effectively. There is no golden tablet I'm aware of that lays out definitions of words.
"Thanks for inviting me to this discussion."
Thanks for participating.
Like
One tricky issue concerning infinity is the so-called infinite number of points in space which is also corresponding to the infinitely many real numbers through Descartes mapping.
The notion of infinite number of points in space is constructed on the notion of point without volume which is a notion that has stumbled Zeno, Aristotle, and even Hegel and Russell (who happened to be also a professional mathematician) after Newton resolved the issues related to zero volume points and the infinitely many points with calculus that is constructed on top of the notion of infinitely approaching the limit.
Here the zero volume point is an abstract notion that is constructed by human imagination and causes logical contradiction........if a point has no volume, it should not occupy space, but if it does not occupy space, it should not accumulate no matter how many you pile them up along one direction.....
This is where philosophy comes to play in a realistic way: the purpose of using human imagination is not to ruin human reflection of nature but to aid human better perceive nature.
Obviously, the self-contradictory notion of zero volume point would cause logical blunder if you extend it into the counting scheme to make ∞ + ∞ = ∞ valid, therefore you should not do it even though there is no biological or psychological law to prevent you from doing so when you make the imagination.....
Happy Thanksgiving,
Like1
Ron, I don't disagree. But I wonder how many zeros it would take to make 5? How about 5/0 zeros? Where 0 * 5/0 = 5.
Like
Well, no matter how many zeros, they won't make 5.....That is the point.....Hegel stumbled when he tried to prove how nothing could become something.....he needs calculus....but calculus is a special math language that cannot be expressed using natural language without the notion of infinitely approaching......Hegel tried very hard, but I don't think he made it very clear with natural language......
Russell stumbled on the issue again when he tried to use set theory to express the instant moment of time......there he encountered what you are encountering here: set theory cannot be used to express calculus because itself lacks the notion of infinitely approaching.....
Numbers came from counting thousands years ago.....but number system does not contain the tool of Newton's calculus.......Now, people like you or others here who are professional mathematicians specialized in number theory might not like this philosophical insight.....
However, philosophically I have to say this: you have to accept the defect in human counting system!
Obviously, as a mathematician specialized in set theory, Russell did not realize this defect of human counting system,....then as a Nobel laureate for his contribution in philosophy he FAILED to know what the real cause was for his failure of expressing the instant moment issue with set theory....
Cheers,
Ron
[[4]]Dai, R. (2022). Solution to Hilbert First Problem against the Illusion of Cantorian Cardinal System. Retrieved from: https://wp.me/pkz9Y-8A
[[5]]Dai, R. (2024). Theorem of the Impossibility to Precisely Match Nonlinearity with Averages of Scattered Data. Retrieved from: https://murongqingcao.wordpress.com/2024/06/12/664/
【[6]】戴榕菁 （2022）一不小心破解連續性假說（CH）？
【[7]】戴榕菁 （2022） 此事非同小可。。。
【[8]】戴榕菁 （2022） 康托集合論之哲學誤區
【[9]】戴榕菁 （2022） 那麽為什麽會有希爾伯特第一問題呢？
【[10]】戴榕菁 （2022） 這回他們真急了。。。
【[11]】戴榕菁 （2023） 康托碰不得？
【[12]】戴榕菁 （2024） 離散數據擬合非線性之不可能定理
