Here it is, as promised ;p
I started a little on this earlier, but I'll elaborate on a couple things to preface this post. You have to understand that the whole concept of infinity actually existing is definitely a subject that is shared between mathematics, philosophy, and religion. Personally, as a fairly religious man, I absolutely think that there is a relationship between God and mathematics. I understand that not everyone agrees on perhaps the form which this takes or have varying beliefs on this (for example, that idiotic constructionist Leopold Kronecker -who, by the way, constantly rejected my reasonings and continually attempted to block my appointment to faculty at the University of Berlin- only believed that God created the integers, but then man did all the rest of the work), but I definitely know there is at least some intersection between these disciplines (as I wrote in Grundlagen einer allgemeinen Mannigfaltigkeitslehre ("Foundations of a General Theory of Aggregates")), and my personal belief suggests an extremely strong relationship.
However, my theorem about how power sets have greater cardinality than that of the original set (which is fairly obvious because a set can never be put into a 1-1 correspondence with its power set) was quite controversial among my fellow mathematicians and philosophers (mostly concerning my idea that there are infinite sizes of infinity, as is easily seen when you do consider power sets). Again, I'll call out Kronecker who hilariously claimed "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure there is no mathematics there", but there were others like Solomon Feferman who claimed that my theories were "simply not relevant to everyday mathematics". I even attempted to appease such criticism by later by improving my argument so it relied on less axioms of set theory, yet it was still hotly debated against by some.
I believe most of this criticism is due to changing times and how my brilliance has blessed the field of mathematics. This is because before me, this whole notion of infinity was really only taken as a useful abstract idea which was then used to help us reason things about this finite world. A lot of this early knowledge came from Carl Gauss since "infinity is nothing more than a figure of speech which helps us talk about limits", so it's as if a kind of completed infinity (as many reason would be necessary in order to reach greater infinities) doesn't actually exist in mathematics. Furthering this idea, some mathematicians thought some of my ideas were absurd because they think that you cannot treat infinite sets (and like, their existence) similarly to finite sets. So some would argue that yes, while it makes sense to say that 2>1, you can't say that one infinity is larger than another because it doesn't make sense, they think that you shouldn't be able to do that since we aren't talking about finite things.
That is the reasoning behind the objections to my theory, and the axiom of infinity, some (like Hermann Weyl, who, by the way, to a certain degree, agrees to my concept of divine absolute infinity) say that "classical logic was abstracted from mathematics of finite sets and their subsets... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets... [which is] the fall and original sin of [Cantor's] set theory".
The good thing, however, is that even with a ton of controversy, my ideas were mostly accepted eventually, and even the rejection of some of my ideas have helped influence various philosophies of mathematics (most notably constructivism and intuitionism since the former has roots in finite theory and the latter deals a lot with proof and also finite and infinite theory). However, when it came to the theologians, I don't think I'll ever really get a break. Most saw my work as a challenge to their beliefs, moreover that God's exclusive claim to "supreme infinity" (as opposed to the various other infinities that exist and are smaller) was a challenge to that power and absolute infinity of God, but I think they're just misinterpreting me, but even set theory couldn't correct this mistake so I guess I'll just leave it alone and let 'em have that one hahaha
However, my theorem about how power sets have greater cardinality than that of the original set (which is fairly obvious because a set can never be put into a 1-1 correspondence with its power set) was quite controversial among my fellow mathematicians and philosophers (mostly concerning my idea that there are infinite sizes of infinity, as is easily seen when you do consider power sets). Again, I'll call out Kronecker who hilariously claimed "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure there is no mathematics there", but there were others like Solomon Feferman who claimed that my theories were "simply not relevant to everyday mathematics". I even attempted to appease such criticism by later by improving my argument so it relied on less axioms of set theory, yet it was still hotly debated against by some.
I believe most of this criticism is due to changing times and how my brilliance has blessed the field of mathematics. This is because before me, this whole notion of infinity was really only taken as a useful abstract idea which was then used to help us reason things about this finite world. A lot of this early knowledge came from Carl Gauss since "infinity is nothing more than a figure of speech which helps us talk about limits", so it's as if a kind of completed infinity (as many reason would be necessary in order to reach greater infinities) doesn't actually exist in mathematics. Furthering this idea, some mathematicians thought some of my ideas were absurd because they think that you cannot treat infinite sets (and like, their existence) similarly to finite sets. So some would argue that yes, while it makes sense to say that 2>1, you can't say that one infinity is larger than another because it doesn't make sense, they think that you shouldn't be able to do that since we aren't talking about finite things.
That is the reasoning behind the objections to my theory, and the axiom of infinity, some (like Hermann Weyl, who, by the way, to a certain degree, agrees to my concept of divine absolute infinity) say that "classical logic was abstracted from mathematics of finite sets and their subsets... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets... [which is] the fall and original sin of [Cantor's] set theory".
The good thing, however, is that even with a ton of controversy, my ideas were mostly accepted eventually, and even the rejection of some of my ideas have helped influence various philosophies of mathematics (most notably constructivism and intuitionism since the former has roots in finite theory and the latter deals a lot with proof and also finite and infinite theory). However, when it came to the theologians, I don't think I'll ever really get a break. Most saw my work as a challenge to their beliefs, moreover that God's exclusive claim to "supreme infinity" (as opposed to the various other infinities that exist and are smaller) was a challenge to that power and absolute infinity of God, but I think they're just misinterpreting me, but even set theory couldn't correct this mistake so I guess I'll just leave it alone and let 'em have that one hahaha


Links to sites and other citations for this one:
ReplyDeletehttps://www.britannica.com/biography/Georg-Ferdinand-Ludwig-Philipp-Cantor#ref199439
https://www.scientificamerican.com/article/infinity-logic-law/
https://en.wikipedia.org/wiki/Intuitionism
https://en.wikipedia.org/wiki/Constructivism_(mathematics)
https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory
"Mathematics and logic: A brief survey serving as a preface to a review of The Philosophy of Bertrand Russell", American Mathematical Monthly
Dauben, Joseph W. "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite” Journal of the History of Ideas
Dauben, Joseph W. (1979), Georg Cantor: his mathematics and philosophy of the infinite, Boston: Harvard University Press
Dunham, W. (1990). Journey through Genius: The Great Theorems of Mathematics. New York: Wiley.
A few suggestions:
ReplyDelete" there", (commas and all punctuation goes inside of quotes. There are many places you do this in this post.)
"by later by" (get rid of one "by")
"less axioms of set theory" change less to "fewer"
"agrees to my concept" change to agrees "with"
"let 'em have that one hahaha " Needs SOME kind of punctuation at end.
This post reads a little rougher for me and I think it's because you use parentheses so often that the ideas don't flow easily for the reader.