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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 qu

by 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.
- mdxn 13y ago
  There are a lot of things here that I take issue with. Some things I find relatively insulting. I wrote up a list, but I'd rather you spend your time answering the following challenge: Describe to me any particular infinite process which does not derive or presume the existence of an actual infinity (e.g. an infinitely large set). You may also want to specify the details of your model for an infinite process to prove that your example isn't circular. --------------- I promise that I will not simply quantify over all the results of the process and call it an infinite set. You may use time as part of the definition of your process, but you may not prematurely assume that there is an infinite process for enumerating timesteps. If you did, the example would run the risk of being circular. If you can construct it, then you can use it. Likewise, you may include a finite number of atoms, but you can't sweep any actual infinite objects underneath them.

by mdxn 13y ago

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.

mdxn 13y ago
There are a lot of things here that I take issue with. Some things I find relatively insulting. I wrote up a list, but I'd rather you spend your time answering the following challenge: Describe to me any particular infinite process which does not derive or presume the existence of an actual infinity (e.g. an infinitely large set). You may also want to specify the details of your model for an infinite process to prove that your example isn't circular. --------------- I promise that I will not simply quantify over all the results of the process and call it an infinite set. You may use time as part of the definition of your process, but you may not prematurely assume that there is an infinite process for enumerating timesteps. If you did, the example would run the risk of being circular. If you can construct it, then you can use it. Likewise, you may include a finite number of atoms, but you can't sweep any actual infinite objects underneath them.