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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

by 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.
  - dmfdmf 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. I agree, this is the crux of the issue. > It refers to the process of arbitrarily unbounded enumeration (having the potential to count to any arbitrary number). This is too narrow. The world is full of infinite processes including your life or the earth revolving around the sun, etc. These are the facts that give rise to the concept though enumeration is the archetypical example because its so easy to see. The danger is forgetting that that someone (or a TM) must be doing the enumeration and eventually he (it) will die, planets will be engulfed by the sun, etc. So no real process goes on literally for infinity. >Aristotle's argument is that the human mind (Turing Machine) is restricted to computing decidable problems and cannot compute undecidable ones. He went even farther than that -- he argued that what ever the mind (or Turing machine) is processing, becoming aware of or knowing has to be finite too. His basic principle of existence is that whatever exists must have identity, including the mind. > It does not have to physically exist, but we certainly can theoretically associate a finite characterization to it. Here is where we disagree or maybe misunderstand each other. You are equivocating on "it". "It" what? By definition, infinity leaves undefined the length, life or extent of the process. Whatever the process is, it is open-ended. Now you can't add back or sneak in some finite measure of the length or extent of the process without destroying its meaning. What you can do is say things about the process even while abstracting away its metaphysical and eventual termination. This is in fact the main value of infinity as a mathematical concept, such as with limits. > A similar argument would lead to the conclusion that pi does not exist... Well, what do you mean by "exist"? Clearly the relation it identifies exists but it does not denote a specific number. Pi denotes a specific open-ended process to calculate a number based on what you are trying to do with the math. So if you are buying tile for your circular patio you might use 3.14 to determine the area and how much tile you need but if you are a NASA engineer and want to land a rover on Mars you'll need to carry a few more decimal places, but not an infinite number of them, and the meaning is clear. That Pi is irrational means that curved paths are incommensurable with linear measures. This did not stop the mathematicians as they just defined a symbol to represent an infinite process (implicitly at first but later fully developed in calculus) that defines the ratio and treat it just like any other number in further equations and theory, and in that sense it is a specifically defined number, and math theory could proceed. The mistake that mathematicians made, primarily due to Plato, was thinking that Pi is a "completed" number in some super reality where ideal, abstract math exists. It was just a matter of time before Cantor came along and applied the same idea to infinity itself.

by mdxn 13y ago

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.
- dmfdmf 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. I agree, this is the crux of the issue. > It refers to the process of arbitrarily unbounded enumeration (having the potential to count to any arbitrary number). This is too narrow. The world is full of infinite processes including your life or the earth revolving around the sun, etc. These are the facts that give rise to the concept though enumeration is the archetypical example because its so easy to see. The danger is forgetting that that someone (or a TM) must be doing the enumeration and eventually he (it) will die, planets will be engulfed by the sun, etc. So no real process goes on literally for infinity. >Aristotle's argument is that the human mind (Turing Machine) is restricted to computing decidable problems and cannot compute undecidable ones. He went even farther than that -- he argued that what ever the mind (or Turing machine) is processing, becoming aware of or knowing has to be finite too. His basic principle of existence is that whatever exists must have identity, including the mind. > It does not have to physically exist, but we certainly can theoretically associate a finite characterization to it. Here is where we disagree or maybe misunderstand each other. You are equivocating on "it". "It" what? By definition, infinity leaves undefined the length, life or extent of the process. Whatever the process is, it is open-ended. Now you can't add back or sneak in some finite measure of the length or extent of the process without destroying its meaning. What you can do is say things about the process even while abstracting away its metaphysical and eventual termination. This is in fact the main value of infinity as a mathematical concept, such as with limits. > A similar argument would lead to the conclusion that pi does not exist... Well, what do you mean by "exist"? Clearly the relation it identifies exists but it does not denote a specific number. Pi denotes a specific open-ended process to calculate a number based on what you are trying to do with the math. So if you are buying tile for your circular patio you might use 3.14 to determine the area and how much tile you need but if you are a NASA engineer and want to land a rover on Mars you'll need to carry a few more decimal places, but not an infinite number of them, and the meaning is clear. That Pi is irrational means that curved paths are incommensurable with linear measures. This did not stop the mathematicians as they just defined a symbol to represent an infinite process (implicitly at first but later fully developed in calculus) that defines the ratio and treat it just like any other number in further equations and theory, and in that sense it is a specifically defined number, and math theory could proceed. The mistake that mathematicians made, primarily due to Plato, was thinking that Pi is a "completed" number in some super reality where ideal, abstract math exists. It was just a matter of time before Cantor came along and applied the same idea to infinity itself.