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Can you really eliminate the infinite?

8 posts in this topic

A circle is much bigger than a point. If a circle is a set of points then a circle is a set of infinitely many points. If this is unacceptable, then what is acceptable?

Should we say that conventional points don't actually exist? Should we say that a circle actually consists of finitely many "big points"?

Should we refuse to describe a circle as being a set of points? Then how would we describe a circle?

On the topic of the infinite, ewv wrote the following:

There is no such thing as a number with an infinite number of digits because there is no actual infinity. There is only .9, .99, .999, etc., each with a finite number of digits. The repeating expression, usually written as ".999..." (or with a dot over the last 9) refers to this sequence [...]

Source:

http://forums.4aynrandfans.com/index.php?s...572entry21572

Each term in the sequence .9, .99, .999, etc. has a finite number of digits, but are there infinitely many terms? If there are infinitely many terms in the sequence, then have you actually succeeded in eliminating the infinite from the discussion?

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A circle is much bigger than a point. If a circle is a set of points then a circle is a set of infinitely many points. If this is unacceptable, then what is acceptable?

Should we say that conventional points don't actually exist?

Of course "conventional points" do not actually exist, at least not in physical reality. A "conventional point" is a mathematical abstraction, a concept of method, not a physical existent.

Should we refuse to describe a circle as being a set of points? Then how would we describe a circle?

There are many ways to describe a circle, depending on context. For instance, a traditional definition, in the context of Euclidean geometry, is the locus of all points in a plane equidistant from a given point within.

But, regardless, I am not sure what you are after. The most important point here to grasp is the notion of concept of method, for without such a concept it is easy to fall into the trap of attributing to physical reality that which is really a product of consciousness. I would recommend reading Ayn Rand's Introduction to Objectivist Epistemology for a good grounding in this particular area, and epistemology in general.

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The concept of "infinity" is perfectly valid, but is limited to the realm of mathematics. It merely describes a range or set without limit.

You can divide any mathematical unit into an infinite number of parts. See also Xeno's Paradox.

But there is no such thing as an infinite thing existentially. Meaning, there is no infinity and there is no thing in existence that has no physical limits. Even "space" is not infinite because it is always defined by the distance between this and that.

I believe Ayn Rand discusses infinity briefly in the Introduction to Objectivist Epistemology as a special concept. Perhaps when I get home I will remember to go look it up for you.

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I believe Ayn Rand discusses infinity briefly in the Introduction to Objectivist Epistemology as a special concept.

That "special concept" is what I identified for the poster as being a product of consciousness, namely that sub-category of concepts of consciousness that Miss Rand identifies as concepts of method. This is all discussed in detail in Chapter 4 of ITOE, titled "Concepts of Consciousness."

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Here's the quote that made the issue more clear for me:

The concept of "infinity" has a very definite purpose in mathematical calculation, and there it is a concept of method. But that isn't what is meanth by the term "infinity" as such. "Infinity" in the metaphysical sense, as something existing in reality, is another invalid concept. The concept "infinity," in that sense, means something without identity, something not limited by anything, not definable. Therefore, the measurements ommitted here are all measurements and all reality.

That is from the Q&A section at the end of the book under the heading "Three 'Hard Cases.' "

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Here's a simple example of something that is infinite.

First, assign names to some sentences as follows:

Sentence #1 is (1x1x1)=1x1

Sentence #2 is :(1x1x1)+(2x2x2)=(1+2)x(1+2)

Sentence #3 is: (1x1x1)+(2x2x2)+(3x3x3)=(1+2+3)x(1+2+3)

Sentence #4 is: (1x1x1)+(2x2x2)+(3x3x3)+(4x4x4)=(1+2+3+4)x(1+2+3+4)

and so on

Now consider the set of sentences that consists of Sentence#n for every positive integer n. That's an infinite set.

It seems unlikely that anyone will discover an infinitely long piece of paper that has all of those sentences written on it. However, it also seems unlikely that anyone will ever discover or create a piece of paper that has Sentence#n written on it for all integer values of n from 1 to 10^1000. Is the set of sentences that consists of Sentence#n for all integer values of n from 1 to 10^1000 in some sense not legitimate?

Now, you can claim that the infinite set of sentences described above is not definable, but wouldn't you be relying on some particular concept of definability? Is there a unique correct concept of the "definable"?

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Here's a simple example of something that is infinite.

First, assign names to some sentences as follows:

Sentence #1 is (1x1x1)=1x1

Sentence #2 is :(1x1x1)+(2x2x2)=(1+2)x(1+2)

Sentence #3 is: (1x1x1)+(2x2x2)+(3x3x3)=(1+2+3)x(1+2+3)

Sentence #4 is: (1x1x1)+(2x2x2)+(3x3x3)+(4x4x4)=(1+2+3+4)x(1+2+3+4)

and so on ...

At best the process is infinite as a potential, not as an actual. Whatever particular number of steps have been taken at any given time, that number of steps is finite.

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At best the process is infinite as a potential, not as an actual. Whatever particular number of steps have been taken at any given time, that number of steps is finite.

So no matter how far you go, there you are? B)

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