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Irrational numbers and Physical constants

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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

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I consider this to be an important problem for Objectivism. There are non-standard maths that avoid the distinguishability of rationals and irrationals, but they often reject the Law of the Excluded Middle. Since LEM is one of the tenets of Objectivism, Oism would have to take a different route. Itvis unclear in my mind what that would be.

Edited by aleph_1
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I'm not bothered by numbers that can't be expressed as a ratio. We have a variety of ways to express quantities as it is because sometimes something which is useful for expressing many quantities doesn't work well with certain other quantities. The decimal system breaks basically if you try to write out how much you have when you cut one into three equal pieces and then take away two of them. Huge problem? Nope. Fractions handle that just fine. So the ratio system of expressing quantities has short comings sometimes? We just need to come up with another way to express certain quantities again. Obviously for something like pi you could draw two identical circles on a piece of paper, cut one out and then in half and then bend the flat edge of one of those halves along the outline of the other circle and lo and behold, you can see right there that there is a specific, finite amount involved in the diameter of a circle versus its circumference. Neither the world nor our minds broke and created something non-finite when we did that simple thing with paper circles even though if we tried to write out our findings in the decimal system we'd get an endless string of numbers and likewise writing it out like a ratio/fraction is no good either. (The wikipedia article on pi has a nifty little animation on finding the value of pi on a number line.)

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@bluecherry

Your little construction may seem adequate to you, but it has been proved that pi cannot be constructed with straightedge and compass. In fact, the number of non-constructible numbers is a higher order of infinity (aleph 1, to be exact) than the number of constructible numbers (aleph 0). Okay, so there are "roller" constructions outside of Euclidean straight-edge and compass geometry, but there is no clear construction for transcendental numbers generally. What is more, Oism does not address why Cantor's Theorem, which demonstrates the existence of an infinity of cardinal infinities, is wrong. The usual approach is to reject the LEM, which Oists do not do. What then?

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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

an infinitely long non-repeating decimal expansion is the name of an irrational real number.

Now consider this: 0.333.... (forever) what is wrong with that? nothing. It is just 1/3 in another guise.

ruveyn1

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This is silly. Whether any particular number is irrational or integer entirely depends on the base of the number system in which it is represented. In the base pi number system the value of pi is 1.

That some ratios are nonterminating is a quirk of whatever method is being used to manipulate numbers, method referring to and including number systems used and method of dividing (and possibly other more fundamental factors, caveat added because I am not a mathematician). Infinities that arise from the methods used are epistemological, not ontological. These methodological, epistemological infinities are entirely acceptable because they are NOT existential infinities.

Edited by Grames
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This is silly. Whether any particular number is irrational or integer entirely depends on the base of the number system in which it is represented. In the base pi number system the value of pi is 1.

That some ratios are nonterminating is a quirk of whatever method is being used to manipulate numbers, method referring to and including number systems used and method of dividing (and possibly other more fundamental factors, caveat added because I am not a mathematician). Infinities that arise from the methods used are epistemological, not ontological. These methodological, epistemological infinities are entirely acceptable because they are NOT existential infinities.

How does one represent rational numbers in a base pi system?

ruveyn1

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How does one represent rational numbers in a base pi system?

ruveyn1

I would suggest switching numbers systems as convenience dictates. My background is engineering not mathematics, so I don't get the attraction to try to use the same one tool for every problem just to prove it can be done. Choose the right tool for right problem.

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I would suggest switching numbers systems as convenience dictates. My background is engineering not mathematics, so I don't get the attraction to try to use the same one tool for every problem just to prove it can be done. Choose the right tool for right problem.

That is an interesting point, but it does not answer the question I put.

ruveyn1

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That is an interesting point, but it does not answer the question I put.

ruveyn1

There was an answer implied, but I shall spell it out. Every number system will have some quantities which are nonterminating ratios. It is unavoidable. There is no solving this problem, just accept it and work around it.

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This is silly. Whether any particular number is irrational or integer entirely depends on the base of the number system in which it is represented. In the base pi number system the value of pi is 1.

That some ratios are nonterminating is a quirk of whatever method is being used to manipulate numbers, method referring to and including number systems used and method of dividing (and possibly other more fundamental factors, caveat added because I am not a mathematician). Infinities that arise from the methods used are epistemological, not ontological. These methodological, epistemological infinities are entirely acceptable because they are NOT existential infinities.

Actually, what you claim about irrational numbers is absolutely false. By definition, a number is irrational if it cannot be expressed as the ratio of two integers. This is independent of what base you are in.

Your attempt to represent pi as 1 in the pi base system doesn' t work. Let [x]be the pi-base representation of a real number. Under this system, [pi]=(1). However, this number representation does not serve the same function as 1 inthe real number system. To see this, note that (1) times (1) is (10) in your system. The set of pi-based representations of the integers is not easily recognizable, but the collection of symbols (n) where n is an integer do not form a ring isomorphism with the integers. Since they don't behave like the integers, they are not the integers.

Concerning infinities your point is well taken. Language requires additional validation. Otherwise you permit nonsense, such as dragons, into your system of concepts. However, what if a philosophical principle leads you to be able to prove the existence of something that it is not possible to physically demonstrate or refute? Is not that philosophical principle the foundation of meaninglessness? If so, shouldn't you root it out and find a replacement? LEM is such a principle.

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I should also mention that Grames is abusing the language in a way that should make an objectivist blush. In any base system, if b is the base, then single digit representations are numbers less than b and are equal to n* b^0. In all base systems, 1=1*b^0. Grams is asserting the existence of a base system where [pi]=1. This would imply that 1and pi have the same representation. Therefore, Grames system is not a base system.

Bluecherry, the problem of cardinal infinities naturally arises in logical systems possessing LEM. This is a consequence of the easily proved Cantor's Theorem. Grams is arguing that these are floating abstractions, but I am arguing that they are floating abstraction consequent to LEM.

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Actually, what you claim about irrational numbers is absolutely false. By definition, a number is irrational if it cannot be expressed as the ratio of two integers. This is independent of what base you are in.

Your attempt to represent pi as 1 in the pi base system doesn' t work. Let [x]be the pi-base representation of a real number. Under this system, [pi]=(1). However, this number representation does not serve the same function as 1 inthe real number system. To see this, note that (1) times (1) is (10) in your system. The set of pi-based representations of the integers is not easily recognizable, but the collection of symbols (n) where n is an integer do not form a ring isomorphism with the integers. Since they don't behave like the integers, they are not the integers.

Feature, not a bug. If there were full equivalence between two number systems there would be no point to preferring one over another. It would be awkward and impractical to count units in base pi, so I wouldn't suggest using it for that. Frankly I don't know what it could be good for (if anything) beyond drawing attention to the issue of representation.

Concerning infinities your point is well taken. Language requires additional validation. Otherwise you permit nonsense, such as dragons, into your system of concepts. However, what if a philosophical principle leads you to be able to prove the existence of something that it is not possible to physically demonstrate or refute? Is not that philosophical principle the foundation of meaninglessness? If so, shouldn't you root it out and find a replacement? LEM is such a principle.

I invoke Wikipedia: Law of the excluded middle and Wikipedia: Quine's Paradox as the reference for what follows.

LEM is not such a principle. Only by accepting the practice of assigning truth-values to arbitrary propositions does it become possible to create logical paradoxes. One supposed counterexample to LEM is Quine's Paradox. Quine's Paradox is:

"Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.

Given Rand's teaching that words are symbols for concepts, and further that the meaning of a concept is what it refers to, the concept of falsehood has no meaning within Quine's paradoxical proposition. Quine's falsehood is an attribute of a proposition rather than a relationship between what exists and what is stated. Not knowing what is your (aleph_1) exact objection to the LEM, I will say the problem probably falls into this pattern as well.

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There are non-standard maths that avoid the distinguishability of rationals and irrationals, but they often reject the Law of the Excluded Middle. Since LEM is one of the tenets of Objectivism, Oism would have to take a different route. Itvis unclear in my mind what that would be.

LEM isn't a tenet to Objectivism so much as it is crucial to logic. All LEM really says is that there is no third option for certain concepts. For Objectivism and its epistemology, the only implication really is that there is no third option for existing or not existing. What would the third option plausibly be? Beyond that, it's possible to use LEM wrong, and false dichotomies are examples of LEM being used wrong.

Secondly, how does Cantor's theorem have any implication on the way I described LEM? I really don't know, so I'm asking you to spell it out.

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"the problem of cardinal infinities"

What problem are you referring to exactly?

"Cantor's argument"

I looked up the wikipedia article on this one. I *might* know what this is from a previous math class, but I'm not sure. I may be recalling the wrong thing since as I'm trying to read this article it is like the words are going in one eye and out the other. It just doesn't seem to want to stick in my head and add up to any kind of meaning right now. Do you have a short summary of it?

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In Cantor's theory, a line of two inches of length contains an infinite number of points, a square of two inches contains an infinite number of points, and a cube of two inches contain an infinite number of points. To this, he wrote to Dedekind, "I see it, but I don't believe it."

If you look at a one inch line and a two inch line, both posited to contain an infinite number of points. This is done by accepting that a point, denotationally, has no breadth, width or depth.

Aristotle held that if you can continue to subdivide a line, but when you stop subdividing it, you have a finite number of segments. So, if we subdivide the 1" line 10 times we end up with 1024 1/1024" segments, and if we subdivide the 2" line 10 times we end up with 1024 1/512" segments. In each case we end up with the same number of unequal length line segments. If we began with a 1" line and a 2" line where the 2" line were already subdivided once, we would have 10 1/1024" long line segments from the 1" line and 20 1/1024" line segments from the 2" line.

Infinite, in this regard, allows subdivision as a concept of method. Repeating this methodology 112 times starting with an inch before ultimately arriving at a Planck length. Is a line segment measuring 4.10515E-34" considered a point? Mathematics is considered "the science of measurement" by ITOE. While epistemologically (or conceptually) we can look at the following three numbers and determine which is greater or lessor:

0.00000000000000000000000000000000041051499999999999999999 . . . 999999999999999999"

0.000000000000000000000000000000000410515"

0.00000000000000000000000000000000041051500000000000000000 . . . 000000000000000001"

is there a means to perceptually (or just conceptually) differentiate between line segments of these three different lengths?

Perceptually, we derive the 2d planar face or surface from the 3d cube, and the "theoretical" corner (or point) of a square from the 2d planar face or surface. Mathematically, we can extend the number system (or subdivide it) by simply applying the methodology of adding another "1" in front of (or subtracting another "0.1" just after the last position following) the decimal place. This method is what permits us to apply the concept of "infinity" as a concept of method..

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Ah, I was indeed recalling the correct thing from my math class. Yay for me. :smartass:

I'm . . . still not sure what this has to do with the law of excluded middle though. An explicit statement of how there's a problem with the LEM because of this would be appreciated. I have some vague guesses at what somebody may find objectionable here, but I'd like to make sure that I'm addressing what is actually up for discussion here already.

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Cantor's Theorem states that any set has a cardinality that is strictly less than the cardinality of it's power set. This applies to infinite sets, and so it implies that there are an infinite number of infinite cardinalities of different sizes. There are infinities that are strictly larger than others. The proof of Cantor's Theorem is very simple and is a reductio ad absurdum proof. As such it is founded on LEM. That is one reason Brouwer rejected LEM and started his non-standard math. Strict constructionists often reject LEM.

I understand that a rich philosophy must be capable of creating floating abstractions. Oism should say something about this. The strength of Oism is that it suggests that more than deductive proof is required for concept validation. Reduction to perception is required as well. However, it is disconcerting that LEM implies the existence of different cardinal infinities.

(I meant 'integral domain' isomorphism above.)

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"any set has a cardinality that is strictly less than the cardinality of it's power set"

1

One is what you need to press generally to get answered in English, yes? I'm pressing it. ;)

The cardinality of a set is the number of elements it has. For example, {1,2,7} has cardinality 3 since it has three elements. The power set of a set is the set of all subsets of a set. For example, the above set has power set { {}, {1}, {2}, {7}, {1,2}, {1,7}, {2,7}, {1,2,7}}, which has cardinality 8. The technique Cantor used to prove his theorem does not require the set to be finite. He discovered distinct infinite cardinalities which retain the names he assigned: Aleph_0, Aleph_1, etc. While infinite, he showed Aleph_0<Aleph_1.

The existence of distinct infinite cardinalities is a logical consequence of LEM.

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However, it is disconcerting that LEM implies the existence of different cardinal infinities.

There is nothing that implies the actual existential existence of existing infinities (redundancy deliberate). It is non sequitor to leap from a mathematical proof about dimensionless points (literally 'nothings') to assert infinities exist. How does an infinite amount of nothings (points) add up to something which exists and participates in causal relationships according to its identity?

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Alright, I think we're getting close to explained now.

"He discovered distinct infinite cardinalities"

What does "distinct" mean in this context?

"The existence of distinct infinite cardinalities is a logical consequence of LEM."

Still a bit fuzzy how you got from premise A to conclusion B here, but that may be due to the one term I'm still unclear on.

If you mean to say the question comes down to "Is this infinity bigger than that infinity?" then the question is invalid because neither of them exist any more than ghosts do. Infinity is a concept which results from omitting the measurements of an ending and/or starting point. Actual stuff out there includes those starting and/or ending points though.

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"He discovered distinct infinite cardinalities"

What does "distinct" mean in this context?

"The existence of distinct infinite cardinalities is a logical consequence of LEM."

Still a bit fuzzy how you got from premise A to conclusion B here, but that may be due to the one term I'm still unclear on.

If you mean to say the question comes down to "Is this infinity bigger than that infinity?" then the question is invalid because neither of them exist any more than ghosts do. Infinity is a concept which results from omitting the measurements of an ending and/or starting point. Actual stuff out there includes those starting and/or ending points though.

Distinct means "not equal". There are infinities with different sizes. To understand how the LEM implies the existence of different cardinal infinities requires that you learn the proof of Cantor's Theorem.

To you and Grames I would like to point out that while it is apparent that the existence of different cardinal infinities is playing with floating abstractions, the logical deduction of the existence of these infinities descends from LEM. Therefore, LEM comes prebuilt with floating abstractions as part of it. Your objection is also with LEM. When you reject the infinities as invalid, you reject LEM.

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Distinct means "not equal". There are infinities with different sizes. To understand how the LEM implies the existence of different cardinal infinities requires that you learn the proof of Cantor's Theorem.

Presumably if LEM implies the existence of different cardinal infinities is problematic; you were saying rejection of LEM is proper because LEM leads to that conclusion. I get that much of your position. But... why is having different cardinality infinities a problem anyway? Maybe a formal definition of LEM would help here.

Edited by Eiuol
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