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.999999999999 repeating = 1

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WI_Rifleman

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Schmarksvillian: where people are taking issue is with the assumption that the concept of counting numbers is interchangeable with other mathematical concepts. They are not interchangeable. This can be seen by attempting to count imaginary/negative/decimal objects in the real world. One cannot say "I have 10i apples" or "I have negative six pears" or "I have exactly 17.21 hot dogs". The concept of a counting number is not interchangeable with these other mathematical concepts.

So what is happening here is that there is some equivocation regarding "1". You are referring to "1.000..." whereas others here are referring to the counting number "1". Mathematically, "1" and "1.000..." are equivalent, however they do refer to different concepts with different purposes. They are not simply "different in name" as you suggest - they are different concepts with different referents. The concept of a repeating decimal sources its referents in abstract mathematics. The concept of counting numbers sources its referents to physical objects that one has attempted to count.

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Here again is a rigorous proof. If you do not understand this proof then you don't understand the basics of this subject and you'd do better to STUDY the subject rather than ignorantly SPOUT off about it:

Definition: .999... = lim(k = 1 to inf) SUM(j = 1 to k) 9/(10^j).

It is also equally true that if you do not understand the objections to a claim that 1 is the same as .9 repeating then you do not understand the basics of philosophy, and Objectivism in particular, and you would do better to STUDY the subject rather than ignorantly SPOUT off about something that you have no clue about. It is irrelevant that you can construct a method of showing that 1 and .9... are "mathematically the same object", because "mathematical identity" is not the same as ontological identity.
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It is also equally true that if you do not understand the objections to a claim that 1 is the same as .9 repeating then you do not understand the basics of philosophy, and Objectivism in particular, and you would do better to STUDY the subject rather than ignorantly SPOUT off about something that you have no clue about. It is irrelevant that you can construct a method of showing that 1 and .9... are "mathematically the same object", because "mathematical identity" is not the same as ontological identity.

Often two things are the same in one respect, but different in another respect. Just as two identical twins can look the same, but in fact be different people, so can two signs have the same quantitative value, but look completely different. Four quarters are financially equivalent to one dollar, but they aren't the same thing.

I used to think about this all of the time as a kid. If you take any number of any # of decimal places, say, 4.789, you could rewrite it as 4.79-0.001. You can rewrite this as 4.8-0.011. You can rewrite this to be 5-0.211. Is 5-0.211 the same expression as 4.79-0.001? No! But they are equivalent. So consider 0.999999999...

One can rewrite this as 1.000... - limit as x goes to infinity of (1/(10^x)). The limit is 0 so 0.999...=1.000...

Caveat: Infinity is not an ordinary number, but rather an object that equals the sum of its parts. One could also rewrite 0.9... as 1.0-limit as x goes to infinity of (2/(10^x)). This DOES NOT MEAN that 1=2 here. Rather, it means that 1 and 2 are part of the same set of all real numbers.

Edited by Drregaleagle
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where people are taking issue is with the assumption that the concept of counting numbers is interchangeable with other mathematical concepts. They are not interchangeable. This can be seen by attempting to count imaginary/negative/decimal objects in the real world. One cannot say "I have 10i apples" or "I have negative six pears" or "I have exactly 17.21 hot dogs". The concept of a counting number is not interchangeable with these other mathematical concepts.
That 1 = .999... doesn't depend on such notions.

So what is happening here is that there is some equivocation regarding "1". You are referring to "1.000..." whereas others here are referring to the counting number "1".
This is dealt with rigorously in mathematics. There are various constructions of the real numbers. In some of them, the natural number 1 turns out to be the actual real number 1, while in other constructions, the natural number 1 is mapped to (per a certain embedding function) the real number 1. The proof I showed does not depend on which of these options is chosen; the proof works in any case.
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It is also equally true that if you do not understand the objections to a claim that 1 is the same as .9 repeating then you do not understand the basics of philosophy, and Objectivism in particular, and you would do better to STUDY the subject rather than ignorantly SPOUT off about something that you have no clue about.

And I have not done any such spouting.

It is irrelevant that you can construct a method of showing that 1 and .9... are "mathematically the same object", because "mathematical identity" is not the same as ontological identity.

You're welcome to whatever distinction you wish to have between mathematical idenity and ontological identity. Meanwhile, in ordinary mathematics, 1 = .999..., where '=' stands for equality, which is taken in this mathematical context to mean 'the same object'. That you may have some other notion of ontological equality is not within the realm of my commentary in this particular context. I'm not talking about philosophy. I'm talking about ordinary mathematics.
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Often two things are the same in one respect, but different in another respect. Just as two identical twins can look the same, but in fact be different people, so can two signs have the same quantitative value, but look completely different. Four quarters are financially equivalent to one dollar, but they aren't the same thing.

No one is disputing that the "signs" '1' and '.999...' look completely different. But 1 IS .999... no matter that '1' is a different "sign" from ".999...".

'1' is a different sign from '2/2' but 1 = 2/2. '1' and '2/2' name the exact same natural number, viz. 1. That the SIGNS suggest different notions ('1' suggests the unity, or whatever, while '2/2' suggests division of the successor of unity by the successor of unity) is not in dispute; but still '1' and '2/2' NAME the same object (though, again, it is not in dispute that the NAMING is different, yet the object NAMED is the same). (By the way, the above about 1 and 2/2 modulo whatever construction of the rationals where the natural number 1 may be a rational or may only be mapped to a certain rational per a certain embedding; which does not vitiate the main point I'm making here, and I'l leave this qualification tacit for the remainder of any discussion in this context.)

That's just ordinary mathematics.

If you have some other philosophical qualifications, then I don't care to stop you about them, but they don't vitiate that in plain ordinary mathematics 1 IS .999...

Edited by Schmarksvillian
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if you do not understand the objections to a claim that 1 is the same as .9 repeating then you do not understand the basics of philosophy, and Objectivism in particular

Would you please provide an authoritative Objectivist article in which it is objected that 1 is not the same as .999... (Of course, I don't mean in the sense that the notation '1' is different from the notation '.999...'.) Edited by Schmarksvillian
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Schmarksvillian: where people are taking issue is with the assumption that the concept of counting numbers is interchangeable with other mathematical concepts. They are not interchangeable. This can be seen by attempting to count imaginary/negative/decimal objects in the real world. One cannot say "I have 10i apples" or "I have negative six pears" or "I have exactly 17.21 hot dogs". The concept of a counting number is not interchangeable with these other mathematical concepts.

If you can have 1 hot dog, you can have 17.21 hot dogs. Quantity is objective and precise. We estimate and state quantities as approximations, and therefore state them incorrectly unless we use significant figures, but the quantities are objective. One could have 10i apple, but it the quantity wouldn't be an apple so we'd call it something else. Try to conceive of an object whose existence would mitigate the existence of an apple. Now, try to conceive of an attribute that this anti-apple possesses in two ways. Say one could define the anti-apple as a pair of "somethings"(I have no damn clue what it would be). One of these somethings, by itself, could be considered i apple.

An easier way to picture it is to imagine a square with area of '-1'. The length of one of its sides is i.

Hope this helps.

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And I have not done any such spouting.

.....

I'm not talking about philosophy. I'm talking about ordinary mathematics.

There's the spouting. We're not talking mathematics, we're talking philosophy, Objectivism in specific. The context is established here. Thus your "proof" is simply irrelevant.
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To recapitulate, it would be good to know who disagrees that the following are results of ordinary mathematics:
It would be good to know what you think the question is at all relevant. Am I correct that you did not read this thread to determine what the issue is?
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To recapitulate, it would be good to know who disagrees that the following are results of ordinary mathematics:

1 = 1

1 = 2/2

2 = (5+3)/4

1 = .999...

You're absolutely right Schmarks. I think you and I are the only people on the forum right now defending mathematics. Do many Objectivists reject mathematical truth?

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I think you and I are the only people on the forum right now defending mathematics. Do many Objectivists reject mathematical truth?
I think you guys are the only Platonists who can't distinguish between a methodological abstraction and a claim about reality.
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To recapitulate, it would be good to know who disagrees that the following are results of ordinary mathematics:

1 = 1

1 = 2/2

2 = (5+3)/4

1 = .999...

I'm curious:

1 = 1

1 - 1 = 0

What does 1 - .999... equal?

0 or .111...?

Is .111... = 0?

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Infinity is not an ordinary number, but rather an object that equals the sum of its parts.
Just to be clear, in the proof I gave, 'infinity' is not used in a sense such as just mentioned. Whatever one's notions of mathematical infinity, the notation '-> inf' and the notions of a converging sequence and of a series are well understood in mathematics as basic as freshman calculus. Edited by Schmarksvillian
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It would be good to know what you think the question is at all relevant. Am I correct that you did not read this thread to determine what the issue is?

It's relevant as you can see the analogy from one equation to another.

'1' and '2/2' name the same number, though '1' and '2/2' are different names for that number.

'1' and '.999...' name the same number, though '1' and '.999...' are different names for that number.

As to the thread, I've read it quite carefully. Now enough about ME and what I've read, I hope. Meanwhile, if one objects to the claim

1 = .999...

on grounds that '1' and '.999...' are different names with different conceptualization involved, then does one also object to the claim

1 = 2/2

on the grounds that '1' and '2/2' are different names with different conceptualizations involved?

Edited by Schmarksvillian
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I don't get it.

If 1 - .999... is not equal to .111..., but is equal to 0,

What then is 1 - .888... equal to?

Edit to change "is not = to" to "is not equal to" for clarity.

Edited by Trebor
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There's the spouting. We're not talking mathematics, we're talking philosophy, Objectivism in specific. The context is established here. Thus your "proof" is simply irrelevant.

(1) You don't need quote marks around 'proof' as you use. The proof is indeed a quite ordinary mathematical proof. (2) Yes, indeed, I'm talking about mathematics and not claiming anything in the matter with Objectivism, since I've not read an Objectivist article that claims 1 is not .999... Just as, by analogy, when I mention some ordinary aspect of computing, or carpentry, or stamp collecting, I'm not opining about Objectivism. (3) Would you please mention such an Objectivist article, as I am eager to be apprised of some authoritative Objectivist decision that 1 is not the same as .999... Edited by Schmarksvillian
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Meanwhile, if one objects to the claim

1 = .999...

on grounds that '1' and '.999...' are different names with different conceptualization involved, then does one also object to the claim

1 = 2/2

on the grounds that '1' and '2/2' are different names with different conceptualizations involved?

One objects to the claim that "one is the same thing as .9999999999999999999 blah blah", because "one" has a different ontological status. "One" is a basic (first-order) measurement concept, and it is trivial to point to instances of it. You can point to "one cat", and you can distinguish "one cat" from "two cats"; likewise you can point to "one dog" and "one car", and abstract "one" away from that. On the other hand, you cannot point to ".99999... cat", or ".99999... dog".

In fact, it also the case that "one cat" is not the same as "two cats distributed to two people". So your conjecture turns out to be correct. 2/2 is an intermediate level concept, standing between integers and repeating fractions.

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What then is 1 - .888... equal to?

Edit to change "is not = to" to "is not equal to" for clarity.

1-.8888888888...=0.111... or one-ninth

By the way, does anyone know how to use mathematical notation on this forum? I tried copy-paste from Microsoft Word with the equation editor, but it didn't paste.

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One objects to the claim that "one is the same thing as .9999999999999999999 blah blah", because "one" has a different ontological status. "One" is a basic (first-order) measurement concept, and it is trivial to point to instances of it. You can point to "one cat", and you can distinguish "one cat" from "two cats"; likewise you can point to "one dog" and "one car", and abstract "one" away from that. On the other hand, you cannot point to ".99999... cat", or ".99999... dog".

In fact, it also the case that "one cat" is not the same as "two cats distributed to two people". So your conjecture turns out to be correct. 2/2 is an intermediate level concept, standing between integers and repeating fractions.

(1) I have no comment about any of that (except I made no CONJECTURE in my post), since the plain ordinary mathematics I'm referring to does not depend on such a context. (2) Would you refer me to an authoritative Objectivist text in which it is claimed that 1 is not .999...? Edited by Schmarksvillian
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