4 ms·

OK, I'm blushing, because at first I thought it was going to be about Georg Cantor. I can still hope, I guess.

GFK_of_xmaspast 11y ago
Infinity's pretty big, there's lots of room for people to know it.
KKKKkkkk1 11y ago
At least based on Wikipedia, Cantor's life would make a lousy movie. The first half of his career was consumed by petty academic politics, and in the second half he lost interest in mathematics. Curiously, if we believe Wikipedia, Kronecker came up with the idea of constructive mathematics specifically in order to stymie Cantor. I always had the suspicion that constructivism was all about a distinction without a difference.
- fusiongyro 11y ago
  That's pretty far from what I get from the Wikipedia article. Kronecker sounds genuinely concerned about constructivism, not like he was disingenuously plotting the downfall of Cantor, which seems to be what you got from it. Constructivism has an important difference from non-constructivist math: it is computable. The math you can do with assistance from a theorem prover such as Coq is constructivist math. In fact, one of the labs in Pierce's _Software Foundations_, you get to show for yourself how various formulations of classical logic create problems. https://www.cis.upenn.edu/~bcpierce/sf/current/Logic.html#lab223 https://www.cis.upenn.edu/~bcpierce/sf/current/Logic.html#la... I think (in constrast to my friend teaching the class on theorem proving) that there is still room for non-constructivist math, but I can see why one would prefer constructivist math. We have computers.
  - fpoling 11y ago
    I see non-constructivist math not much different from theology. That is assume something that cannot be experimentally observable or constructable and then apply logic.
    - accountt 11y ago
      Suppose God doesn't exist. Then ... middle, middle, middle,...Contradiction. Thus by "excluded middle", God does exist. I have never seen theology do that. Theology proceeds by saying God exists with no proof, constructive or otherwise. Then it derives whatever it wants from that. Logic applies only when it's convenient and if you start from false premises, you can prove anything correct. Nonconstructive proofs usually tell you something exists, but don't give you an example of it. Theology asserts something exists(with no proof), gives you examples(which don't support the assertions). Quite a difference here.
      - fpoling 11y ago
        Consider medieval consideration about number of angels that can fit on a pin. Those were often pretty sofisticated deductions that are not much different in spirit from modern deductions from Continuum hypothesis.
      - jessriedel 11y ago
        You're not addressing these sorts of arguments, which is what fpoling is referring to: https://en.wikipedia.org/wiki/Existence_of_God#Deductive_arguments https://en.wikipedia.org/wiki/Existence_of_God#Deductive_arg...
        accountt 11y ago
        Can you, please, give me an example of a nonconstructive mathematical concept that's close to what you linked?
        jessriedel 11y ago
        You said > Suppose God doesn't exist. Then ... middle, middle, middle,...Contradiction. Thus by "excluded middle", God does exist. I have never seen theology do that. Here is a summary of Anselm's argument for the existence of God: > 1. It is a conceptual truth (or, so to speak, true by definition) that God is a being than which none greater can be imagined (that is, the greatest possible being that can be imagined). > 2. God exists as an idea in the mind. > 3. A being that exists as an idea in the mind and in reality is, other things being equal, greater than a being that exists only as an idea in the mind. > 4. Thus, if God exists only as an idea in the mind, then we can imagine something that is greater than God (that is, a greatest possible being that does exist). > 5. But we cannot imagine something that is greater than God (for it is a contradiction to suppose that we can imagine a being greater than the greatest possible being that can be imagined.) > 6. Therefore, God exists. (Emphasis mine.) https://en.wikipedia.org/wiki/Ontological_argument#Anselm https://en.wikipedia.org/wiki/Ontological_argument#Anselm But really, you don't even need something with such a close 1-to-1 mapping for fpoling's comment to be reasonable. Consider the Banach–Tarski paradox. In both cases, you're talking about "something that cannot be experimentally observable or constructable" -- a perfect being, or a certain disjoint decomposition of the 3-ball -- "and then apply logic".
  - deleted 11y ago
    [deleted]
justifier 11y ago
i think cantor would be more appropriately described as the man who knew finity
iainmerrick 11y ago
Maybe Cantor would be The Man Who Knew Infinities.

Infinity's pretty big, there's lots of room for people to know it.

At least based on Wikipedia, Cantor's life would make a lousy movie. The first half of his career was consumed by petty academic politics, and in the second half he lost interest in mathematics. Curiously, if we believe Wikipedia, Kronecker came up with the idea of constructive mathematics specifically in order to stymie Cantor. I always had the suspicion that constructivism was all about a distinction without a difference.

fusiongyro 11y ago
That's pretty far from what I get from the Wikipedia article. Kronecker sounds genuinely concerned about constructivism, not like he was disingenuously plotting the downfall of Cantor, which seems to be what you got from it. Constructivism has an important difference from non-constructivist math: it is computable. The math you can do with assistance from a theorem prover such as Coq is constructivist math. In fact, one of the labs in Pierce's _Software Foundations_, you get to show for yourself how various formulations of classical logic create problems. https://www.cis.upenn.edu/~bcpierce/sf/current/Logic.html#lab223 https://www.cis.upenn.edu/~bcpierce/sf/current/Logic.html#la... I think (in constrast to my friend teaching the class on theorem proving) that there is still room for non-constructivist math, but I can see why one would prefer constructivist math. We have computers.
- fpoling 11y ago
  I see non-constructivist math not much different from theology. That is assume something that cannot be experimentally observable or constructable and then apply logic.
  - accountt 11y ago
    Suppose God doesn't exist. Then ... middle, middle, middle,...Contradiction. Thus by "excluded middle", God does exist. I have never seen theology do that. Theology proceeds by saying God exists with no proof, constructive or otherwise. Then it derives whatever it wants from that. Logic applies only when it's convenient and if you start from false premises, you can prove anything correct. Nonconstructive proofs usually tell you something exists, but don't give you an example of it. Theology asserts something exists(with no proof), gives you examples(which don't support the assertions). Quite a difference here.
    - fpoling 11y ago
      Consider medieval consideration about number of angels that can fit on a pin. Those were often pretty sofisticated deductions that are not much different in spirit from modern deductions from Continuum hypothesis.
    - jessriedel 11y ago
      You're not addressing these sorts of arguments, which is what fpoling is referring to: https://en.wikipedia.org/wiki/Existence_of_God#Deductive_arguments https://en.wikipedia.org/wiki/Existence_of_God#Deductive_arg...
      - accountt 11y ago
        Can you, please, give me an example of a nonconstructive mathematical concept that's close to what you linked?
        jessriedel 11y ago
        You said > Suppose God doesn't exist. Then ... middle, middle, middle,...Contradiction. Thus by "excluded middle", God does exist. I have never seen theology do that. Here is a summary of Anselm's argument for the existence of God: > 1. It is a conceptual truth (or, so to speak, true by definition) that God is a being than which none greater can be imagined (that is, the greatest possible being that can be imagined). > 2. God exists as an idea in the mind. > 3. A being that exists as an idea in the mind and in reality is, other things being equal, greater than a being that exists only as an idea in the mind. > 4. Thus, if God exists only as an idea in the mind, then we can imagine something that is greater than God (that is, a greatest possible being that does exist). > 5. But we cannot imagine something that is greater than God (for it is a contradiction to suppose that we can imagine a being greater than the greatest possible being that can be imagined.) > 6. Therefore, God exists. (Emphasis mine.) https://en.wikipedia.org/wiki/Ontological_argument#Anselm https://en.wikipedia.org/wiki/Ontological_argument#Anselm But really, you don't even need something with such a close 1-to-1 mapping for fpoling's comment to be reasonable. Consider the Banach–Tarski paradox. In both cases, you're talking about "something that cannot be experimentally observable or constructable" -- a perfect being, or a certain disjoint decomposition of the 3-ball -- "and then apply logic".
- deleted 11y ago
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justifier 11y ago

i think cantor would be more appropriately described as the man who knew finity

iainmerrick 11y ago

Maybe Cantor would be The Man Who Knew Infinities.