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Irrational numbers and Physical constants

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Ah, I got a little confused when you said artifact, earlier I read that as suggesting sets were invalid concepts with regards to math.

Virtually all of mathematics, ancient and modern rests on set theory,  either explicitly or implicitly.

 

It took a long time to recognize how basic and necessary the concept of set is.

 

ruveyn1

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I don't believe that LEM has much to do with the original question. Does anyone have any input on the concept of physical constants and how they should be treated or thought of?

properly measured physical quantities are how we get at the underlying facts of nature.  Anything in natured should be able to be measured, provided we are smart enough to do it and develop the technology to do it.  During the Victorian era,  a big leap was made in Britain in developing high resolution measuring instruments.  Mass producing them.

 

ruveyn1

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I don't believe that LEM has much to do with the original question. Does anyone have any input on the concept of physical constants and how they should be treated or thought of?

Here we have a problem, for without the Infinity Axiom we cannot construct the classical Natural Numbers, much less the Real Numbers. There exist non-standard maths that may do the trick, but what do objectivists accept?

Rejection of LEM is one way of avoiding the objections implicit in Ockham's Razor (and Rand's Razor) to distinguishable infinities. In classical mathematical theory, there are more transcendental numbers than algebraic numbers, though there are an infinity of each. There are so few algebraic numbers that their measure is zero. Now, all of this is absurd at some basic level. You can resolve these meaningless distinctions by rejection LEM but as Grand Minnow has pointed out that doesn't get you past the problems inherent in the Infinity Axiom. However without that axiom you cannot have the classical Natural Numbers. One might like to believe that Natural Numbers are observable properties of collection of observable objects. Therefore, a theory needs to account for them that doesn't assume the Infinity Axiom.

There are other reasons for rejecting LEM, but this is enough for now.

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Grand Minnow,

Your proof of CT is different than the one that I remember, and I thank you for giving me an Intuitionistically consistent proof. Of course, most mathematicians do not constrain themselves to the hypotheses you used and so may construct other proofs.

One must ask, what is the basis for acceptance of potential infinities? A potential infinity is an abstraction impossible to reduce to perception. They are dragons!

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Here we have a problem, for without the Infinity Axiom we cannot construct the classical Natural Numbers, much less the Real Numbers. There exist non-standard maths that may do the trick, but what do objectivists accept?

Rejection of LEM is one way of avoiding the objections implicit in Ockham's Razor (and Rand's Razor) to distinguishable infinities. In classical mathematical theory, there are more transcendental numbers than algebraic numbers, though there are an infinity of each. There are so few algebraic numbers that their measure is zero. Now, all of this is absurd at some basic level. You can resolve these meaningless distinctions by rejection LEM but as Grand Minnow has pointed out that doesn't get you past the problems inherent in the Infinity Axiom. However without that axiom you cannot have the classical Natural Numbers. One might like to believe that Natural Numbers are observable properties of collection of observable objects. Therefore, a theory needs to account for them that doesn't assume the Infinity Axiom.

There are other reasons for rejecting LEM, but this is enough for now.

 

What is the infinity axiom?

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What is the infinity axiom?

The Infinity Axiom says, There exists a set I such that the empty set is an element of I and for all elements x of I, the set (x union {x}) is also an element of I.

This set, in the lingo of set theory, is the set of Natural Numbers. There is no bound to the number of elements to this set.

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Objectivism rejects that there are sets.   Sets are epistemic artifacts, created by regarding some things as similar and others not similar. 

 

 

In Objectivist writings there are occasional uses of the word 'set' in the everyday sense of a collection. I don't know why you say Objectivism rejects that there are such things, even if those things are what you call 'epistemic artifacts'. Is there a passage in Objectivist writing (especially OPAR or ITOE) where I can see that Objectivism rejects that there are sets?

 

And, again, I understand that Objectivsm would reject that there are such things without consciousness to conceive them. That point is not at stake.

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what is the basis for acceptance of potential infinities? 

 

I'm just referring to the Objectivist acceptance of the notion as described in such quotes as in the Lexicon:

 

There is a use of [the concept] “infinity” which is valid, as Aristotle observed, and that is the mathematical use. It is valid only when used to indicate a potentiality, never an actuality. Take the number series as an example. You can say it is infinite in the sense that, no matter how many numbers you count, there is always another number. You can always keep on counting; there’s no end. In that sense it is infinite—as a potential. But notice that, actually, however many numbers you count, wherever you stop, you only reached that point, you only got so far. . . . That’s Aristotle’s point that the actual is always finite. Infinity exists only in the form of the ability of certain series to be extended indefinitely; but however much they are extended, in actual fact, wherever you stop it is finite.

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I'm just referring to the Objectivist acceptance of the notion as described in such quotes as in the Lexicon:

 

There is a use of [the concept] “infinity” which is valid, as Aristotle observed, and that is the mathematical use. It is valid only when used to indicate a potentiality, never an actuality. Take the number series as an example. You can say it is infinite in the sense that, no matter how many numbers you count, there is always another number. You can always keep on counting; there’s no end. In that sense it is infinite—as a potential. But notice that, actually, however many numbers you count, wherever you stop, you only reached that point, you only got so far. . . . That’s Aristotle’s point that the actual is always finite. Infinity exists only in the form of the ability of certain series to be extended indefinitely; but however much they are extended, in actual fact, wherever you stop it is finite.

How many points in the interior of a unit circle?

 

ruveyn1

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According to what I have read here: http://forum.objecti...showtopic=12514

... the extension of space and | or time cannot be infinite (and are also discrete) since infinity cannot be applied to physically existing things. If that is so can it be possible that the physical constants we use and are regarded to be irrational numbers can be interpreted in ways in which they would not need an infinitely long number to represent them? If we get the right point of reference shouldn't these constants be integers, or at least finitely long?

http://en.wikipedia....ysical_constant

http://en.wikipedia....rational_number

"Something needs an infinitely long number to represent it" = "something can't be represented with a number". 

 

That's all. That doesn't say anything about what that thing is. 

 

If we get the right point of reference shouldn't these constants be integers, or at least finitely long?

Yes, with the right point of reference, they could be integers or finitely long. If the point of reference were Pi-5 , then Pi would be 5, and the current integers would all be irrational numbers.

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"Objectivism rejects that there are sets. Sets are epistemic artifacts, created by regarding some things as similar and others not similar."

"I don't know why you say Objectivism rejects that there are such things, even if those things are what you call 'epistemic artifacts'."

 

The distinction that I think is meant to be drawn here is calling sets "epistemic artifacts" as opposed to entities. Sets, like all concepts and ideas in general, exist in our mind as opposed to existing "out there" in reality like the objects many of the concepts refer to. The actual objects "out there" in reality we'd generally refer to as entities. Sets are a concept people form in their mind based upon relationships among entities.

 

"The first concepts man forms are concepts of entities—since entities are the only primary existents. (Attributes cannot exist by themselves, they are merely the characteristics of entities; motions are motions of entities; relationships are relationships among entities.)"


“Concept-Formation,”Introduction to Objectivist Epistemology, 15
 

There's a little more on the lexicon page for "entity" that gets into how to distinguish what is and what is not an entity.

 

If that's not what you were getting at, Grames, sorry about that. Go ahead and correct me.

 

Also, if anybody knows of some more detailed official-like stuff on the fallacies spoken about earlier, feel free to post those too. While I'm pretty sure what I said earlier is based on reliable stuff, I can't recall exactly where that was. :(

 

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How many points in the interior of a unit circle?

 

ruveyn1

From what point of view? From the point of view of ordinary mathematics, the answer is that the cardinality of the set of points in the interior of a unit disc is the cardinality of the set of real numbers. I don't know what the answer would be for an Objectivist. 

Edited by GrandMinnow
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The distinction that I think is meant to be drawn here is calling sets "epistemic artifacts" as opposed to entities.

Sets, like all concepts and ideas in general, exist in our mind as opposed to existing "out there" in reality like the objects many of the concepts refer to. The actual objects "out there" in reality we'd generally refer to as entities.

 

I already mentioned that it is not at issue that Objectivism does not allow mental objects to be taken as independent of consciousness.

 

But non-Objectivist mathematicians too would ordinarily allow that sets are mental objects or conceptual objects and not concrete objects, and even some mathematicians may deny that sets are platonic objects. So I don't see the point in saying that Objectivsts deny that there are sets when talking about the comparison between the ordinary mathematical notion of sets with an Objectivist notion of set (unless one take "ordinary mathemtatics" to include a platonic commitment, which is sticky question, since probably most mathematicians would be platonists or lean toward platonism if they were finally pressed to take a philosophic stance, though one might take the position that doing mathematics does not require a decision on that philosophical question let alone a decision in favor of platonism). 

Edited by GrandMinnow
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The distinction that I think is meant to be drawn here is calling sets "epistemic artifacts" as opposed to entities. Sets, like all concepts and ideas in general, exist in our mind as opposed to existing "out there" in reality like the objects many of the concepts refer to. The actual objects "out there" in reality we'd generally refer to as entities. Sets are a concept people form in their mind based upon relationships among entities.

 

According to Ayn Rand, concepts are mental entities. If, as you say, sets are concepts, then sets are mental entities. 

Edited by GrandMinnow
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"Epistemic artifact" is definitely not any kind of standard terminology used by Objectivism and/or Objectivists, It is something I've only seen used that one time in this thread. I thought the unusual terms were being used for trying to make clearer what distinction was being made - that the sets aren't existents the same way the tree in the front yard is - to somebody who may not be familiar with the same terminology we often use among ourselves. It wasn't about saying other people necessarily don't make the same distinction, just noting that we do. It's pretty common for us to use terms how we're used to using them with other people and then the other person or people misunderstands what we were trying to say, often to the point of even thinking we're disagreeing with them when we are agreeing. So, finding other terms to use with people is often a way to avoid a lot of confusion. Calling things entities versus mental entities I could easily see leading toward somebody concluding that since both cases use the term "entity" there must really be, according to our view point, no real significant difference between them. Then insert some accusations of contradicting ourselves and so on and so on.

 

I forget now though why this subject came up in the first place. There was the opening post about physical limitations and math, then some objections about LEM involving infinity and Cantor's Theorem, then some stuff about the definitions of certain fallacies, then I forget what was next. I think we were talking about something to do with what infinity is before getting into what sets are perhaps. I just saw that you seemed to have some objections with the terms Grames used at one point and he wasn't around, so I thought things could go faster if I just posted what I thought the explanation was right then and there, as long as later I wasn't told I was way off base from what he meant.

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In Objectivist writings there are occasional uses of the word 'set' in the everyday sense of a collection. I don't know why you say Objectivism rejects that there are such things, even if those things are what you call 'epistemic artifacts'. Is there a passage in Objectivist writing (especially OPAR or ITOE) where I can see that Objectivism rejects that there are sets?

 

And, again, I understand that Objectivsm would reject that there are such things without consciousness to conceive them. That point is not at stake.

Is that so?

 

I already mentioned that it is not at issue that Objectivism does not allow mental objects to be taken as independent of consciousness.

Fantastic.

 

According to Ayn Rand, concepts are mental entities. If, as you say, sets are concepts, then sets are mental entities.

What is this, an attempt at a "Gotcha"? A mental entity is not an entity independent of consciousness. For a mathematician you (surprisingly) seem to enjoy contradicting yourself. Edited by Grames
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The distinction that I think is meant to be drawn here is calling sets "epistemic artifacts" as opposed to entities. Sets, like all concepts and ideas in general, exist in our mind as opposed to existing "out there" in reality like the objects many of the concepts refer to. The actual objects "out there" in reality we'd generally refer to as entities. Sets are a concept people form in their mind based upon relationships among entities.

 

"The first concepts man forms are concepts of entities—since entities are the only primary existents. (Attributes cannot exist by themselves, they are merely the characteristics of entities; motions are motions of entities; relationships are relationships among entities.)"

“Concept-Formation,”Introduction to Objectivist Epistemology, 15

 

There's a little more on the lexicon page for "entity" that gets into how to distinguish what is and what is not an entity.

 

If that's not what you were getting at, Grames, sorry about that. Go ahead and correct me.

Completely correct. Plasmatic also understood and stated so in post #40. It is good to know I am not being unclear in my writing.
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Also, if anybody knows of some more detailed official-like stuff on the fallacies spoken about earlier, feel free to post those too. While I'm pretty sure what I said earlier is based on reliable stuff, I can't recall exactly where that was.


http://books.google.com/books?id=VttF6CuC-cQC&pg=PT144&source=gbs_toc_r&cad=4#v=onepage&q=floating&f=false

This is the best I found for "floating abstraction". Really there isn't a lot to say about it anyway, except that the abstraction is not connected to anything on the perceptual level. The link earlier for "stolen concept" is fine. Stolen concepts are all floating abstractions I'd say, because a stolen concept presumes that someone is using a particular concept in a totally untenable away.

The bit about "rejecting sets" was part of GrandMinnow's response to me regarding greater and greater infinities. There are no actual infinities because that would imply an existent that is literally boundless and without physical constraint. Infinities only exist as mental entities, and not as anything that is perceivable. In that sense, infinity is an "artifact" because it is created for an epistemic purpose, even though there are no infinities "out there" metaphysically. (Some of this post is probably stated elsewhere, but this also partly answers Daniel's post #54.)

It is good to know I am not being unclear in my writing.


Well, you were very unclear in your word choice for me, hence my clarification post. I think we're all on the same page now.
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In addition to understanding "infinity" as Aristotle used it , is grasping "infinity" as a "concept of method". As a concept of method, the counting numbers, in a base 10 method, allows us to use "number" to count number with. This relies on our ability to take our grasp of 10 as a group of 10 units and consider it as a unit. Thus, 10, 20, 30, etc, are 1 group of 10 units, 2 groups of 10 units, 3 groups of 10 units,  etc. 100, 200 300, etc are 1 group of 100 units or 10 groups of 10 units, 2 groups of 100 units or 20 groups of 10 units or 3 groups of 100 units or 3 groups of 10 units, etc.

 

By using number to count number, this gives us the sense that we can extend the (number) sequence without limit (i.e. infinitely), because we know exactly what to do (just add another symbol (number) to the left, representing 10 of the group unitized to its right), in order to accomplish it. (Pat Corvini, "Two, Three, Four and All That")

 

As to the physical constants, this requires the knowledge of mathematics as a human construct. The decimal equivalence of fractions do not always result in a number that can be expressed in a finite decimal equivalent. The resulting decimal representation can easily exceed our ability to discern man-made methods of resolution. The decimal equivalence is a man-made result of a man-made description of metaphysical phenomenon.

Edited by dream_weaver
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How far down the rabbit hole are Oists willing to go? Are you willing to accept the Continuum Hypothesis, for example? The Continuum Hypothesis may be phrased that every uncountable set of real numbers is equinumerous to the set of real numbers. Surely this is meaningless gibberish to someone who demands reduction!

What about the property of Completeness of the real numbers? An ordered field is complete if and only if it has the least upper bound property. This means that every bounded set has a least upper bound within that set. The real numbers can be shown, up to isomorphism, to be the unique complete ordered field. Does this depend on the Union Axiom as it applies to infinite sets that you are willing to accept due to the Infinity Axiom?

Or, are you willing to accept the real numbers as a stolen concept?

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Not a -stolen- concept but an idealization.  There can only be a countable number of real numbers that can be computed to any desired degree of accuracy.   However,  that leaves the real number field full of holes unless we assume some property that guarantees local compactness.   Most mathematicians prefer a number field that  has no holes  rather than one in which a Cauchy Sequence does not converge to a member of the field.   Being able to take limits makes calculation a lot easier.  

 

That is the good news.

 

The bad news might be that physical reality is not locally compact and may be full of holes at the lowest end of resolution.

 

ruveyn1

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