6 ms·

Cantor is wrong, yes, but you won't prove that to contemporary mathematicians who believe his ideas because this is an epistemological issue and can't be resolv

by dmfdmf 13y ago

Cantor is wrong, yes, but you won't prove that to contemporary mathematicians who believe his ideas because this is an epistemological issue and can't be resolve with mathematical arguments. This is really an age-old philosophic dispute that the mystics won decades ago. As David Hubert described it "No one shall expel us from the Paradise that Cantor has created." It is the "paradise" of deuces wild for the mathematicians and it rests on an equivocation of the meaning of "infinity". Cantor's idea of a "completed infinity" is a self contradiction if you grasp what the concept of infinity actually means and keep it tied to reality. It is the error of treating infinity as a real thing and not an abstraction. A similar error, for similar reasons, is made in the history of the philosophy by the mystics of nihil who wanted to treat nothingness as on par with existence via the Reification of Zero.

mdxn 13y ago
Can you be more specific about the nature of the logic you are using when concluding that a "completed infinity" is a self-contradiction? Are you sure that you aren't naively equivocating abstract notions of infinity with cardinal numbers?
- dmfdmf 13y ago
  I am coming at this from a philosophic perspective so I don't know how much I can explain the mathematics. I am following Aristotle who held that infinity cannot actually exist; it is a potential, an abstraction. For something to actually exist it must be specific, it must be definite (which is a metaphysical principle) and for something to be "infinite" means a quantity, amount, characteristic, etc. that is not specified by definition. The genus of infinity is "process", some kind of identifiable change, such as adding one to a number to get the next number in a sequence, that is open-ended. Infinity identifies the open-endedness of the process but any real process (such as you counting "to infinity") must terminate as some point and must have a definite state at any given moment (such as being at 2,345,652 at such and such date and time). In sum, a completed infinity would mean an open-ended process that has stopped which contradicts the meaning of infinity, at least on my terms. Moreover, in my view Cantor makes exactly the same conceptual mistake as a child that uses the phrase "counting to infinity". NB: To be fair, infinity is a valid mathematical concept but it has to be used very carefully to avoid sliding into non-sense. Some of Cantor's work may actually be valid, I do not know. But given my views I suspect that his basic error is trying to "count the unspecified", i.e. the infinite. This whole approach appears to be invalid since the concept "counting" necessarily presupposes that what you are counting has a specific identity. Unfortunately, to separate the wheat from the chaff means some future mathematician will need to become an epistemologist to sort it all out as the problem is in philosophy not mathematics.
  - JohnHaugeland 13y ago
    "I am coming at this from a philosophic perspective" Philosophical* perspectives require clear, factually driven, referencable arguments. Philosophy is not a fancy word for opinion. . "which contradicts the meaning ... at least on my terms." When you have to change the meanings of words in order to feel like you've made a point; etc. . "Unfortunately, to separate the wheat from the chaff means some future mathematician will need to become an epistemologist" The person you're looking for is named Georg Cantor. That work was all done already. You are merely unfamiliar with it. . "the problem is in philosophy not mathematics." The philosophy of mathematics has considered this matter settled for several hundred years now. It turns out they're an existing community. You could try reaching out to them. Nuel Belnap is a very nice man, and would probably explain this to you if you asked, without presuming first that you knew someone to be wrong when you weren't familiar with their work.
    - dmfdmf 13y ago
      >... at least on my terms This was not a cop out as you portray but recognition that many people are not familiar with Aristole's position on infinity nor is it the popular view. I defined my terms and explained my position as to why Cantor's use of "completed infinity" is a contradiction. It is clear to anyone who wants to understand. I defy you to define, in your own words, your concept of infinity and what you actually mean by a completed one without degenerating into non-sense. > The philosophy of mathematics has considered this matter settled for several hundred years now. This was hardly a fair fight. As I conceded, the mystics won round one which is why Cantor's non-sense is accepted. The philosophy of mathematics is and was dominated by Platonists and other avowed irrationalists. So no surprise in this outcome.
  - mdxn 13y ago
    What you are talking about is actually the difference between two different notions of infinity: actual and potential. I think you identify these, but do not quite separate them as much as I believe you should. Potential infinity aligns with the explanation you provide in the second paragraph. It refers to the process of arbitrarily unbounded enumeration (having the potential to count to any arbitrary number). This is equivalent to the capabilities of a Turing Machine. Aristotle's argument is that the human mind (Turing Machine) is restricted to computing decidable problems and cannot compute undecidable ones. The whole point of the application of Cantor's diagonalization argument is to demonstrate this limitation. Cantor rather argues that assuming the Axiom of Infinity does not necessarily lead to a contradiction (unless you assume the opposite of course). The assumption simply states that some infinitely large set exists, in particular, the natural numbers. It does not have to physically exist, but we certainly can theoretically associate a finite characterization to it. You should look into Kolmogorov complexity for this. Note that this is NOT at all the same thing as "counting to infinity". A lot of people will counterargue that infinity is just a concept, but I feel as if they miss part of the point. A similar argument would lead to the conclusion that pi does not exist and neither does the number, 2. The only difference is that we apply the concept of two-ness to discrete objects we can compute with. We do have recursive descriptions (programs) that can describe a countable infinity (aleph null) or even pi. We might as well use these descriptions as placeholders for the actual thing. While we cannot contain the base 10 encoding of pi, we have another encoding of it of finite length (the program). Who is to say that a base 10 encoding of numbers is a better proof of existence than one written in C++? You might have more success with arguing against the existence of undefinable numbers. These do not have a description of finite length and are definitely numbers that we cannot conceptualize with our current assumed limitations.
jfarmer 13y ago
You out yourself by using the phrase "Reification of Zero," which is something I've only ever heard from Objectivists. It's funny to hear you say this, since Objectivists are usually so apoplectic about axioms and premises. You can object to ZF(C) as the basis for set theory, but I don't see how one can call it "wrong." If you want to build up a theory without the Axiom of Infinity you're free to do so — many mathematicians have. Either Cantor's arguments follow from ZF(C) or they don't (hint: they do). It's not a philosophical problem at all. One can certainly explore what it would mean to do mathematics without access to "infinity", as many mathematicians in the first half of the 20th century did. IMO this conversation is all air until you propose your own precise set of axioms.
- dmfdmf 13y ago
  > It's not a philosophical problem at all.... Well, that was my main point. You are free to disagree. > IMO this conversation is all air until you propose your own precise set of axioms. Apoplectically, the three basic axioms are Existence, Identity and Consciousness. Whatever the proper mathematical axioms (and methods) are they have to be consistent with these three. Cantor's method treats infinity as a real thing, which violates the axiom of identity. You could even call it the Reification of Infinity but the error is easier to see in Rand's identification of Reification of Zero by the mystics who attempt to treat nothingness as real.
  - jfarmer 13y ago
    The three basic (metaphysical) axioms according to Objectivists are Existence, Identity and Consciousness. Anyhow, as far as I can tell, all you're saying is that either (1) Cantor's argument is invalid within ZF(C) or (2) you find ZF(C) objectionable on the grounds that it conflicts with the basis of your metaphysics. That's fine — I don't care whether it does or not — but mathematics has no opinion on the matter. You're free to choose a set of mathematical axioms which you don't find objectionable and go about doing your math in that universe. If you want to claim Cantor's argument is invalid then you need to tell me what axioms you're taking as the foundation for your mathematics. It is valid in ZF(C) and I'm 99% sure it's valid in plain ZF. Which of the axioms of ZF do you find objectionable and what would you replace them with? Assuming you're trying to engage sincerely, I'd ask you to be precise. This means listing whatever foundations of mathematics you don't find objectionable and talking about how they differ from ZF(C). If you're unfamiliar with it, I'd read up on proof theory: http://en.wikipedia.org/wiki/Proof_theory http://en.wikipedia.org/wiki/Proof_theory

by dmfdmf 13y ago

mdxn 13y ago

Can you be more specific about the nature of the logic you are using when concluding that a "completed infinity" is a self-contradiction? Are you sure that you aren't naively equivocating abstract notions of infinity with cardinal numbers?

dmfdmf 13y ago
I am coming at this from a philosophic perspective so I don't know how much I can explain the mathematics. I am following Aristotle who held that infinity cannot actually exist; it is a potential, an abstraction. For something to actually exist it must be specific, it must be definite (which is a metaphysical principle) and for something to be "infinite" means a quantity, amount, characteristic, etc. that is not specified by definition. The genus of infinity is "process", some kind of identifiable change, such as adding one to a number to get the next number in a sequence, that is open-ended. Infinity identifies the open-endedness of the process but any real process (such as you counting "to infinity") must terminate as some point and must have a definite state at any given moment (such as being at 2,345,652 at such and such date and time). In sum, a completed infinity would mean an open-ended process that has stopped which contradicts the meaning of infinity, at least on my terms. Moreover, in my view Cantor makes exactly the same conceptual mistake as a child that uses the phrase "counting to infinity". NB: To be fair, infinity is a valid mathematical concept but it has to be used very carefully to avoid sliding into non-sense. Some of Cantor's work may actually be valid, I do not know. But given my views I suspect that his basic error is trying to "count the unspecified", i.e. the infinite. This whole approach appears to be invalid since the concept "counting" necessarily presupposes that what you are counting has a specific identity. Unfortunately, to separate the wheat from the chaff means some future mathematician will need to become an epistemologist to sort it all out as the problem is in philosophy not mathematics.
- JohnHaugeland 13y ago
  "I am coming at this from a philosophic perspective" Philosophical* perspectives require clear, factually driven, referencable arguments. Philosophy is not a fancy word for opinion. . "which contradicts the meaning ... at least on my terms." When you have to change the meanings of words in order to feel like you've made a point; etc. . "Unfortunately, to separate the wheat from the chaff means some future mathematician will need to become an epistemologist" The person you're looking for is named Georg Cantor. That work was all done already. You are merely unfamiliar with it. . "the problem is in philosophy not mathematics." The philosophy of mathematics has considered this matter settled for several hundred years now. It turns out they're an existing community. You could try reaching out to them. Nuel Belnap is a very nice man, and would probably explain this to you if you asked, without presuming first that you knew someone to be wrong when you weren't familiar with their work.
  - dmfdmf 13y ago
    >... at least on my terms This was not a cop out as you portray but recognition that many people are not familiar with Aristole's position on infinity nor is it the popular view. I defined my terms and explained my position as to why Cantor's use of "completed infinity" is a contradiction. It is clear to anyone who wants to understand. I defy you to define, in your own words, your concept of infinity and what you actually mean by a completed one without degenerating into non-sense. > The philosophy of mathematics has considered this matter settled for several hundred years now. This was hardly a fair fight. As I conceded, the mystics won round one which is why Cantor's non-sense is accepted. The philosophy of mathematics is and was dominated by Platonists and other avowed irrationalists. So no surprise in this outcome.
- mdxn 13y ago
  What you are talking about is actually the difference between two different notions of infinity: actual and potential. I think you identify these, but do not quite separate them as much as I believe you should. Potential infinity aligns with the explanation you provide in the second paragraph. It refers to the process of arbitrarily unbounded enumeration (having the potential to count to any arbitrary number). This is equivalent to the capabilities of a Turing Machine. Aristotle's argument is that the human mind (Turing Machine) is restricted to computing decidable problems and cannot compute undecidable ones. The whole point of the application of Cantor's diagonalization argument is to demonstrate this limitation. Cantor rather argues that assuming the Axiom of Infinity does not necessarily lead to a contradiction (unless you assume the opposite of course). The assumption simply states that some infinitely large set exists, in particular, the natural numbers. It does not have to physically exist, but we certainly can theoretically associate a finite characterization to it. You should look into Kolmogorov complexity for this. Note that this is NOT at all the same thing as "counting to infinity". A lot of people will counterargue that infinity is just a concept, but I feel as if they miss part of the point. A similar argument would lead to the conclusion that pi does not exist and neither does the number, 2. The only difference is that we apply the concept of two-ness to discrete objects we can compute with. We do have recursive descriptions (programs) that can describe a countable infinity (aleph null) or even pi. We might as well use these descriptions as placeholders for the actual thing. While we cannot contain the base 10 encoding of pi, we have another encoding of it of finite length (the program). Who is to say that a base 10 encoding of numbers is a better proof of existence than one written in C++? You might have more success with arguing against the existence of undefinable numbers. These do not have a description of finite length and are definitely numbers that we cannot conceptualize with our current assumed limitations.

jfarmer 13y ago

You out yourself by using the phrase "Reification of Zero," which is something I've only ever heard from Objectivists. It's funny to hear you say this, since Objectivists are usually so apoplectic about axioms and premises. You can object to ZF(C) as the basis for set theory, but I don't see how one can call it "wrong." If you want to build up a theory without the Axiom of Infinity you're free to do so — many mathematicians have. Either Cantor's arguments follow from ZF(C) or they don't (hint: they do). It's not a philosophical problem at all. One can certainly explore what it would mean to do mathematics without access to "infinity", as many mathematicians in the first half of the 20th century did. IMO this conversation is all air until you propose your own precise set of axioms.

dmfdmf 13y ago
> It's not a philosophical problem at all.... Well, that was my main point. You are free to disagree. > IMO this conversation is all air until you propose your own precise set of axioms. Apoplectically, the three basic axioms are Existence, Identity and Consciousness. Whatever the proper mathematical axioms (and methods) are they have to be consistent with these three. Cantor's method treats infinity as a real thing, which violates the axiom of identity. You could even call it the Reification of Infinity but the error is easier to see in Rand's identification of Reification of Zero by the mystics who attempt to treat nothingness as real.
- jfarmer 13y ago
  The three basic (metaphysical) axioms according to Objectivists are Existence, Identity and Consciousness. Anyhow, as far as I can tell, all you're saying is that either (1) Cantor's argument is invalid within ZF(C) or (2) you find ZF(C) objectionable on the grounds that it conflicts with the basis of your metaphysics. That's fine — I don't care whether it does or not — but mathematics has no opinion on the matter. You're free to choose a set of mathematical axioms which you don't find objectionable and go about doing your math in that universe. If you want to claim Cantor's argument is invalid then you need to tell me what axioms you're taking as the foundation for your mathematics. It is valid in ZF(C) and I'm 99% sure it's valid in plain ZF. Which of the axioms of ZF do you find objectionable and what would you replace them with? Assuming you're trying to engage sincerely, I'd ask you to be precise. This means listing whatever foundations of mathematics you don't find objectionable and talking about how they differ from ZF(C). If you're unfamiliar with it, I'd read up on proof theory: http://en.wikipedia.org/wiki/Proof_theory http://en.wikipedia.org/wiki/Proof_theory