Cantor and infinities

Question

I know we have accepted Cantor's ideas a long time ago and many mathematicians use sets and infinities without ever realizing that thinking about sets and infinities intuitively fails, because there are many paradoxes associated with naive set theory. However, why did mathematicians such as Kronecker regarded Cantor's ideas as absurdities (and as I remember accused Cantor of impiety and the corruption of youth)?

Also, I believe Poincare did not like Cantor's ideas, yet he did a lot of research in topology, which is based on the notion of an open set. Can any one explain why so many people apposed Cantor's ideas? Why is the traditional view of infinite so appealing although Cantor's proofs are valid?

Edit:

I posted this question back in 2015, and since then I have changed my views on the validity of Cantor's arguments. I no longer believe they are valid arguments. Many people nowadays just have "itching ears" so to speak and they want to turn away their ears from the truth, and turn to fables. Cantor's theorems are fables; they are hearsays because many mathematicians just repeat them without taking time to consider what they are teaching their students.

I have studied calculus and analysis and I know one can study it and prove theorems without even mentioning Cantor's ideas. You can completely avoid the modern definition of an infinite set. The modern definition of an infinite set is so "strange" that it creates paradoxes, and people have to constantly add more axioms to the Zermelo-Fraenkel theory to fit a square peg into a round hole. For example, instead of saying "x belongs to the set of real numbers" one can just say that "x is a number such that the square of x is greater or equal to zero" to avoid defining the set of real numbers. This is how people have thought of real numbers for centuries. Moreover, in calculus and analysis, we never actually use infinity: look up every definition in calculus and you will only find a symbol for infinity. For example, look up the definition of a limit "at infinity" and you will find that there is no mention of infinite sets or cardinals.

Answer 1

Kronecker studied philosophy in his youth, in particular the doctrines of Hegel. I think Hegel identified the absolute giest (mind/spirit), an actual infinite reached the final resolution of his dialectic, as God. Hegels doctrines were wildly popular. He had declared his faith in a 'concrete God'. I assume his phenomenology of spirit was to construct him, rather I should say conceptualise him. In the naive way we talk about the qualities of God, like goodness and mercy.

Kronecker converted from Jewish faith to Christianity. He was opposed to non-constructive proofs, he demanded a construction of the actual object. He was in fact against the unrestricted use of the infinite in arguments & that of the use of the excluded middle. In fact one could say he initiated a new school of mathematics - intuitionism.

Cantors theories of the infinite may have appeared to encroach on that territory as Dain points out in his answer; and Kronecker with a new converts zeal, may have attacked him on this very point, hence the call of impiety, an attack on his faith and, since he didn't believe in infinities or the reductio, his charge of the socratic corruption of youth away from truth, was him being simply consistent with his position.

But ironically, Cantor himself worried about that too, he had mystical tendencies towards the absolute infinite (perhaps he too had read Hegels doctrines). Aristotle, believed in a potential infinity, but not an actual one. Cantors theories of the infinite, far from disproving this claim, only affirmed them. It kind of leads one to suspect that Kronecker never spent the time to understand Cantors theories.

But they needn't have worried, current set theorists have considered large cardinal axioms that go far, far beyond the kind of infinities envisaged by Cantor. And still there is territory far, far beyond what they have currently surveyed. In a way, considering the territory yet to cover, it's as almost as though we've never even come out of the safe shelter of the finite.

I'm sure, but without proof, that Cantor was aware of this, hence his mystic leanings towards the absolute infinite, an idea, which had been possibly conditioned by Hegel, and perhaps platonism (I don't know if he was that way inclined). One would suspect so, since his emphasis on consistency being the sole judgement of mathematical character.

Poincare may have said that 'there is no absolute infinity', and that the Cantorians had forgotten this, but I very much doubt that Cantor held the same view. He shouldn't be judged on the polemics & antics of his band of followers.

(If one views Kronecker as the father of intuitionism arising in opposition to Cantors formalism; one can view todays multiverse of sets now only dimly seen, as through a mist, the synthesis of these two contradictory positions. A rather nice example of a Hegelian dialectical movement - if it works).

Answer 2

From what I can recall of a book I read over five years ago (please be sure to check my sources), a little of the opposition was due to the way Cantor couched his terms. He appears to have worried in turns that he was intruding on God's territory, a claim that would have offended the religious notions of Kroenecker, and it's anyone's guess why Poincare didn't like him. Cantor was a loony, and by some accounts just not pleasant to be around. He just happened to be brilliant, and obsessed with Infinity.

As for widespread dislike, much of it was the same objection Cantor felt himself- proving the Infinite had the potential to be a series of ever-larger sets of infinities, it threatened the most basic tenet of monotheistic faith, that God is single, infinite and unitary. Here was a demonstration from the most basic logic anyone knew at the time, which threatened a fundamental premise of monotheism itself: How can the infinite be unified if there's many kinds of infinity?

I'd hazard to guess that it was this conflict which was responsible for most of the opposition you'd have found against Cantor at the time.

Answer 3

The criticism of Cantor's ideas by Kronecker and Poincare is hardly comparable for several reasons.

• Poincare voiced his criticism in 1905 when Cantor's ideas were already accepted. He could see how people drew questionable conclusions from it and engaged in unproductive investigations.
• Kronecker was even opposed to the use of irrational numbers, and tried to prevent publication of papers using such ideas as early as 1870. He is famously quoted as having said: "God made natural numbers; all else is the work of man".

As Joseph W. Dauben explains, the opposition of Kronecker was part of the reason that Cantor gave philosophical arguments allowing him to defend set theory:

The philosophical arguments, however, were essential to Cantor, if not to Mittag- Leffler. They were essential because they were part of the elaborate defense he had begun to construct to subvert opposition from any quarter, but especially from Kronecker. The ploy was to advance a justification based upon the freedom of mathematics to admit any self-consistent theory. Applications might eventually determine which mathematical theories were useful, but for mathematicians, Cantor insisted that the only real question was consistency. This of course was just the interpretation he needed to challenge an established mathematician like Kronecker.

Cantor did more than just insisting that only consistency was important for mathematics. He even founded the "Deutsche Mathematiker Vereinigung" to be able to better prevent similar abuse of power by old established mathematicians against new ideas.

One way to better understand why some mathematicians are unhappy with consistency as the only important criterion for mathematics is to look at their mathematical hopes and dreams. Many hope to learn about nice and aesthetically pleasing theorems and proofs with important applications to problems outside of mathematics itself. Consistency is probably a prerequisite for both, but it is far from enough. The increasing separation between physics and mathematics (similar to todays separation between informatics and mathematics) further aggravated by increasing specialization in all sciences lead to frustration on the part of people like Henri Poincaré. In a series of essays in 1905-1906 he raised a number of valid concerns, but according to Zermelo was more hostile towards set theory than justified by his concerns. He was strongly opposed against transfinite numbers and the actual infinite:

Poincaré was dismayed by Georg Cantor's theory of transfinite numbers, and referred to it as a "disease" from which mathematics would eventually be cured. Poincaré said, "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction."

More valid are his philosophical concerns regarding impredicative definitions (even so Zermelo reported that he misclassified some valid predicative definitions as impredicative):

Poincaré believed that arithmetic is a synthetic science. He argued that Peano's axioms cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is a priori synthetic and not analytic. Poincaré then went on to say that mathematics cannot be deduced from logic since it is not analytic. His views were similar to those of Immanuel Kant (Kolak, 2001, Folina 1992). He strongly opposed Cantorian set theory, objecting to its use of impredicative definitions.

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This answer has become long and excessively cites external material. I initially wanted to answer this question, because I had an own opinion about set theory. Then I was afraid that my own opinion might not actually reflect the historical reality, so I tried to read at least a bit about the actual historical development. Then I had so much material that I struggled to turn it into a coherent answer, even if I accepted the fact that my initial opinion regarding set theory was completely irrelevant.

So let me conclude by giving two quotes for V.I. Arnold, which at least partly reflect my own opinions about axiomatic theories when they lead to a separation between mathematics and the real world:

In the last thirty years the prestige of mathematics has declined in all countries. I think that mathematicians are partially to be blamed as well (foremost, Hilbert and Bourbaki), particularly the ones who proclaimed that the goal of their science was investigation of all corollaries of arbitrary systems of axioms.

In On teaching mathematics, Arnold becomes even more explicit that mathematics has to solve real problems instead of just being consistent:

The ugly building, built by undereducated mathematicians who were exhausted by their inferiority complex and who were unable to make themselves familiar with physics, reminds one of the rigorous axiomatic theory of odd numbers. Obviously, it is possible to create such a theory and make pupils admire the perfection and internal consistency of the resulting structure (in which, for example, the sum of an odd number of terms and the product of any number of factors are defined). From this sectarian point of view, even numbers could either be declared a heresy or, with passage of time, be introduced into the theory supplemented with a few "ideal" objects (in order to comply with the needs of physics and the real world).

There is a huge difference between believing that the (potential) infinite exists as a useful abstraction of properties present in the real world (and being able to give concrete examples for this), and believing that the (actual) infinite is consistent but dismissing the importance of its relation to the real world.

Answer 4

Some short answers:

"I know we have accepted Cantor's ideas a long time ago"

Who is we? Look here into Mückenheim's source book on transfinity for over 250 sceptical voices: https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, chapter V.

"and many mathematicians use sets"

No mathematician has ever used an actually infinite set with any reasonable mathematical result. The notion of countability is already self-contradictory as has been shown formally on p. 258 of Mückenheim's source book.

"Can any one explain why so many people apposed Cantor's ideas? Why is the traditional view of infinite so appealing although Cantor's proofs are valid?"

Look through chapter VI to see that Cantor's ideas are invalid.