6 ms·

reading one of the PDFs referenced http://www.cs.cmu.edu/~aada/courses/15251f15/www/slides/lec05.pdf http://www.cs.cmu.edu/~aada/courses/15251f15/www/slides/lec

reading one of the PDFs referenced http://www.cs.cmu.edu/~aada/courses/15251f15/www/slides/lec05.pdf http://www.cs.cmu.edu/~aada/courses/15251f15/www/slides/lec0... Why other mathematicians at the time, were so nasty towards Cantor's ideas? The quotes noted in the PDF -- just sound nasty and dismissive (or may be this is not the whole story)

Nomentatus 9y ago
People don't self-select themselves to study and teach mathematics because they like surprises. This is true of academia in general, but very true re math.
- Ar-Curunir 9y ago
  What? Science and mathematics requires you to be open to new ideas, beause your ideas can be proven wrong easily.
  - Nomentatus 9y ago
    That's the ideal. But if you've spent much time in academia, you'd know that the personality types that choose science usually do so to wallow in certainty. High School doesn't tell you much about the real history and philosophy of science. I trust everyone who added a downvote has actually spent some time inside a math dept of a University (but I highly doubt that.)
voidhorse 9y ago
I think, as with many paradigm changing and major ideas, its novelty required a change in the prevailing thinking of the epoch, which always makes people uncomfortable--according to wikipedia most objections were to Cantor's notion of transfinite numbers: https://en.wikipedia.org/wiki/Georg_Cantor https://en.wikipedia.org/wiki/Georg_Cantor There's even a separate article on objections to his theory: https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_th... Seems like the notion of "completed infinity" which sort of gives infinity ontological status (? I'm no mathematician) did not sit well with some--I guess I could understand why--it definitely isn't an intuitive notion: "Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world; for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence.[18] "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already".[19] Carl Friedrich Gauss's views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics'.[20] In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets." I guess this shows that the overall pragmatic value of your work may eventually win out against formal refutations.
- chriswarbo 9y ago
  It may be that a more deep-seated change has taken place within mathematics: the rise of formalism, with Frege, Russell, etc. separated mathematics from intuitions about 'reality'; a formal theorem can't be refuted by saying it's 'unrealistic', since it's a watertight consequence of the chosen axioms and deduction rules. We can attack those axioms and deduction rules as being 'unrealistic', but since Goedel shot down the attempts to define "one true" formalism, it seems that mathematicians have come to accept axioms and logics based on their own merits (e.g. whether they lead to interesting, non-trivial consequences) rather than whether they're accurate models of 'reality'. In this relaxed view, Cantor's arguments make sense from his premises, and it's interesting to entertain the idea of his premises because they lead to such interesting results. Whether or not "completed infinities" are "real" is delegated to the philosophers, while mathematicians 'play the proof game', and change the rules if they make it more fun.
addicted 9y ago
You are highly underestimating the mental leaps required to understand infinity pre Cantor. It's not surprising at all that people should find it ridiculous and get especially angry about it because they couldn't really explain why it was wrong, despite it just seeming like complete BS to them. It's like telling a lay person (or even many mathematics students for that matter) that 0.999999.... = 1. They understand the proofs, but it's so against their intuition there is a visceral negative reaction to it.
- pacala 9y ago
  There is no such thing as 0.9... There is 0.9[k], for k as large as you'd like. However, no matter which k you pick, I can squirrel a number between 0.9[k] and 1, for example 0.9[k]5. Your alleged proof is wrong :p
  - alangpierce 9y ago
    Usually 0.9... is defined as the limit (from calculus) of the sequence 0.9, 0.99, 0.999, 0.9999, etc. So it's the unique number that the sequence gets arbitrarily close to (even if it never reaches it), which is exactly 1. With the proper definition in place, the statement becomes a lot less interesting, in my opinion, but regardless, I'm pretty sure the statement "0.9... = 1" is well-accepted in the math community. I agree that all of the proofs you normally see in high school are flawed in one way or another because they don't properly define what "..." means in the first place.
    - igravious 9y ago
      I may be out of my depth but I disagree. When I see … I read “and so on” so my brain when it sees 0.9… writes a function which first produces 0.9, then 0.99, then 0.999 and keeps going and never stops – it's return type is the bottom type to use type theoretic jargon. I say 'produces' but the function never returns so if there was some way to sample where the function was at or … (heh, that's an ordinary ellipsis). Or rather say it is a function where the first time it returns 0.9, the next time it returns 0.99 at a higher resolution, then 0.999, and so on. If I'm wrong then I'm wrong for this reason. Mathematicians are using the symbol … ambiguously. Because in some places in math it really does just mean “and so on“ and nothing more. Or more pedantically “and keep going in such a manner following that pattern“. If, as you say, in instances like this mathematicians take … to mean “the limit (from calculus) of the demonstrated sequence” this is news to me. Well, maybe it's not news to me but I'm not exactly buying it. What I'm saying is that the expression could be taken to be one of any number of things (not sure if I ought to qualify that with the phrase 'mutually exclusive'): (1) the (symbolic representation of the) sequence, (2) the most recently generated term, (3) the generator (4) the limit (from calculus) of the sequence. That you say the convention is that it most assuredly is 4 is not something I'm buying into. (I'd nearly go a step further and assert that it ought not be defined this way). Where we get into problems, I believe, is when people say nonsensical things like the sequence reaches 1. Or if they start talking about the actually completed sequence rather than a symbolic representation of it. And this, by the way, is the gripe that people have with Cantor's completed infinities. It really is an ontological claim, not a mathematical one that such things exist. It's a claim whose refutation I support. When this topic comes up on HN I invariably point[0] people towards the absolutely best book I've read on the subject. René Guénon's The Metaphysical Principles of the Infinitesimal Calculus. This book is lucid, compelling, precise. I urge anyone with even the slightest bit of interest in this topic to read it. The notion of the continuum is a very very slippery beast and I know of no other book that deals with the topic so well. Category theorists have a much better handle on the smoothness of the continuum than set theorists because they have far better machinery. [0] http://www.sophiaperennis.com/books/rene-guenon-series/the-metaphysical-principles-of-the-infinitesimal-calculus/ http://www.sophiaperennis.com/books/rene-guenon-series/the-m... Here's a link to a copy in various formats on Archive.org https://archive.org/stream/reneguenon/1946%20-%20The%20Metaphysical%20Principles%20of%20the%20Infinitesimal%20Calculus https://archive.org/stream/reneguenon/1946%20-%20The%20Metap...

Nomentatus 9y ago

People don't self-select themselves to study and teach mathematics because they like surprises. This is true of academia in general, but very true re math.

Ar-Curunir 9y ago
What? Science and mathematics requires you to be open to new ideas, beause your ideas can be proven wrong easily.
- Nomentatus 9y ago
  That's the ideal. But if you've spent much time in academia, you'd know that the personality types that choose science usually do so to wallow in certainty. High School doesn't tell you much about the real history and philosophy of science. I trust everyone who added a downvote has actually spent some time inside a math dept of a University (but I highly doubt that.)

voidhorse 9y ago

I think, as with many paradigm changing and major ideas, its novelty required a change in the prevailing thinking of the epoch, which always makes people uncomfortable--according to wikipedia most objections were to Cantor's notion of transfinite numbers: https://en.wikipedia.org/wiki/Georg_Cantor https://en.wikipedia.org/wiki/Georg_Cantor There's even a separate article on objections to his theory: https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_th... Seems like the notion of "completed infinity" which sort of gives infinity ontological status (? I'm no mathematician) did not sit well with some--I guess I could understand why--it definitely isn't an intuitive notion: "Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world; for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence.[18] "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already".[19] Carl Friedrich Gauss's views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics'.[20] In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets." I guess this shows that the overall pragmatic value of your work may eventually win out against formal refutations.

chriswarbo 9y ago
It may be that a more deep-seated change has taken place within mathematics: the rise of formalism, with Frege, Russell, etc. separated mathematics from intuitions about 'reality'; a formal theorem can't be refuted by saying it's 'unrealistic', since it's a watertight consequence of the chosen axioms and deduction rules. We can attack those axioms and deduction rules as being 'unrealistic', but since Goedel shot down the attempts to define "one true" formalism, it seems that mathematicians have come to accept axioms and logics based on their own merits (e.g. whether they lead to interesting, non-trivial consequences) rather than whether they're accurate models of 'reality'. In this relaxed view, Cantor's arguments make sense from his premises, and it's interesting to entertain the idea of his premises because they lead to such interesting results. Whether or not "completed infinities" are "real" is delegated to the philosophers, while mathematicians 'play the proof game', and change the rules if they make it more fun.

addicted 9y ago

You are highly underestimating the mental leaps required to understand infinity pre Cantor. It's not surprising at all that people should find it ridiculous and get especially angry about it because they couldn't really explain why it was wrong, despite it just seeming like complete BS to them. It's like telling a lay person (or even many mathematics students for that matter) that 0.999999.... = 1. They understand the proofs, but it's so against their intuition there is a visceral negative reaction to it.

pacala 9y ago
There is no such thing as 0.9... There is 0.9[k], for k as large as you'd like. However, no matter which k you pick, I can squirrel a number between 0.9[k] and 1, for example 0.9[k]5. Your alleged proof is wrong :p
- alangpierce 9y ago
  Usually 0.9... is defined as the limit (from calculus) of the sequence 0.9, 0.99, 0.999, 0.9999, etc. So it's the unique number that the sequence gets arbitrarily close to (even if it never reaches it), which is exactly 1. With the proper definition in place, the statement becomes a lot less interesting, in my opinion, but regardless, I'm pretty sure the statement "0.9... = 1" is well-accepted in the math community. I agree that all of the proofs you normally see in high school are flawed in one way or another because they don't properly define what "..." means in the first place.
  - igravious 9y ago
    I may be out of my depth but I disagree. When I see … I read “and so on” so my brain when it sees 0.9… writes a function which first produces 0.9, then 0.99, then 0.999 and keeps going and never stops – it's return type is the bottom type to use type theoretic jargon. I say 'produces' but the function never returns so if there was some way to sample where the function was at or … (heh, that's an ordinary ellipsis). Or rather say it is a function where the first time it returns 0.9, the next time it returns 0.99 at a higher resolution, then 0.999, and so on. If I'm wrong then I'm wrong for this reason. Mathematicians are using the symbol … ambiguously. Because in some places in math it really does just mean “and so on“ and nothing more. Or more pedantically “and keep going in such a manner following that pattern“. If, as you say, in instances like this mathematicians take … to mean “the limit (from calculus) of the demonstrated sequence” this is news to me. Well, maybe it's not news to me but I'm not exactly buying it. What I'm saying is that the expression could be taken to be one of any number of things (not sure if I ought to qualify that with the phrase 'mutually exclusive'): (1) the (symbolic representation of the) sequence, (2) the most recently generated term, (3) the generator (4) the limit (from calculus) of the sequence. That you say the convention is that it most assuredly is 4 is not something I'm buying into. (I'd nearly go a step further and assert that it ought not be defined this way). Where we get into problems, I believe, is when people say nonsensical things like the sequence reaches 1. Or if they start talking about the actually completed sequence rather than a symbolic representation of it. And this, by the way, is the gripe that people have with Cantor's completed infinities. It really is an ontological claim, not a mathematical one that such things exist. It's a claim whose refutation I support. When this topic comes up on HN I invariably point[0] people towards the absolutely best book I've read on the subject. René Guénon's The Metaphysical Principles of the Infinitesimal Calculus. This book is lucid, compelling, precise. I urge anyone with even the slightest bit of interest in this topic to read it. The notion of the continuum is a very very slippery beast and I know of no other book that deals with the topic so well. Category theorists have a much better handle on the smoothness of the continuum than set theorists because they have far better machinery. [0] http://www.sophiaperennis.com/books/rene-guenon-series/the-metaphysical-principles-of-the-infinitesimal-calculus/ http://www.sophiaperennis.com/books/rene-guenon-series/the-m... Here's a link to a copy in various formats on Archive.org https://archive.org/stream/reneguenon/1946%20-%20The%20Metaphysical%20Principles%20of%20the%20Infinitesimal%20Calculus https://archive.org/stream/reneguenon/1946%20-%20The%20Metap...