Are there unquestionable answers?
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So far, this seems like an unanswerable question.
I'm no genius, though.Wouldn't pi or the square root of 2 be the same quantity universally? It doesn't really matter what base is used, it still would be the same quantity. I suppose that that can be questioned as well. *Shrug*
Dorian Gray
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I think all answers are questionable. For any answer, you could ask "What do you mean by that?"
evildave
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xouper said:
And here's the problem with it: in a machine representation, there are a finite number of bits to store the value. You can't demostrate that the value doesn't stop at or after the mantissa after any arbitrary divide (or multiply by fraction). So the calculator (or FPU) has to do something to *approximate* the vinculum. It looks like it will be all nines, so fudge it. Add the 'epsilon' and call it a whole.
I agree with your explanation of the limitations of floating point numbers as used by computers. I just wanted to clarify (for the peanut gallery) that those limitations do not apply to the decimal expression of real numbers in mathematics. We can certainly discuss the problems of machine representations, but that has nothing to do with the previous discussion of 0.999... = 1.
evildave
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But the proofs for 0.999... = 1 are based on other numeric representations. You can't prove 0.999... is anything but LOTS of nines. Somewhere, somebody "fudges" it. Either by making the value between .999... and 1.0 "infinitely small" (i.e. sub-'epsilon'), or assuming it's zero. Any way you slice it, you get a fudge.
1/3 * 3/1 = 1, but 0.3333... * 3.0 isn't necessarily ever 1.0, except by "fudging" it.
Three times an imprecisely represented value is three times the error. The matter is complicated by math that results in lots of repeated numbers before some other value; enough that nobody bothers to check. 0.3333....345..., or 0.333....321... won't tally, but they can easily be assumed to when the error is remote enough. Just a matter of "close enough" again.
1/3 * 3/1 = 1, but 0.3333... * 3.0 isn't necessarily ever 1.0, except by "fudging" it.
Three times an imprecisely represented value is three times the error. The matter is complicated by math that results in lots of repeated numbers before some other value; enough that nobody bothers to check. 0.3333....345..., or 0.333....321... won't tally, but they can easily be assumed to when the error is remote enough. Just a matter of "close enough" again.
evildave said:
Please, please, please.
Go to read the above-linked thread before posting any more posts on this subject.
Please.
I really mean it.
In the other thread Suggestologist has been asked for quite a long time to provide a real number that lies between 0.99... and 1 but he has failed to do so. Can you give it? If they are different, then it should be an easy thing to do since there's an uncountable number of real numbers between any two real numbers.Though, I'd once more suggest you to read the other thread before answering to this, and to answer in the other thread.
0.33... is a precisely represented number.
Dave, you are wrong. Please take LW's advice and read that other thread before you make any more comments on this issue. And please take this conversation to that other thread instead of hijacking this one.
How many times does this need to be said???
DialecticMaterialist
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My answer to this would be two-fold.
First that there are no "unquestionable answers" in the sense that there are no "answers" we literally cannot question again. I can technically do this forever. For example I can ask "What color is three?", "Where is the universe?" and "What do you think of three sided squares?" Whether these questions are meaningful or have any epistemic merit is a different matter entirely though.
Thus my second point, if we mean by the original question however that all questions are meaningful or epistemologically worthwhile; the answer I will give is "no."
To illustrate why real fast I will have to invoke the regress model.
Basically it is that "conclusions" (answers) are based on "premises" (proof and evidence); which are themselves in their own way "conclusions", themselves based on "premises"; which are themselves based on premises. And so on, ad infinitum- or maybe not.
The latter comment refers to the fact that ultimately, we have to stop/begin somewhere, with some final premises or we go on for infinity.
This leads to two approaches- infinitism, and finitism/ foundationalism.
Infinitism is usually rejected as absurd, since we cannot ground knowledge in something infinite and it seems to lead to mere relativism. If we cannot ultimately justify any premise, how do we even justify infinitism over foundationalism?
This makes infitinism inconsistent with itself and thus a worse theory.
Foundationalism however is consistent, but it has to start somewhere. With some final epistemic premises or standards. These final standards are usually called axioms, and are considered the basis for all justification.
Now by final I don't mean the end road for all knowledge, so much as the beggining. They are only "final" in the sense that they are the last thing we can test extrinsically via by other standards.
Now this being the case, and axioms be the bottom line or final premises, questioning them cannot really be too constructive. This is because any "questioning" would already presuppose the truth of these axioms, so all answers would be in some sense circular.
Now I am not saying questioning axioms in general is not good however. This is because since axioms are not extrinsically proven, (proven by other standards) which is what a "question" ultimately demands, false axioms can be extrinsically disproven. Thus asking questions may help us find false axioms, perhaps by showing that they can be disproven in theory, or that they contradict others axioms, or that they are not evident in the sense of being similiar to other axioms and necessary for logic and reasoning.
Axioms for example do have certain traits: they cannot be disproven even in theory, they cannot contradict other axioms, they are not sufficiently upheld by earlier premises and they are evident (provide a basis for evidence.)
An example of this is the statement " I am having sensations right now."
That is almost certainly true, it cannot be disproven in theory, it is not sufficiently proven by other premises, and it is evident in that it is necessary for future reasoning.
Another example is the claim "I exist." I cannot disprove that in theory, as it leads to contradictions.
Some others are the laws of identity and noncontradiction, objectivism, the basics of geometry. etc.
Lastly, since axioms are considered "evident" in that they can prove latter claims, which would not work if they were not "evident" as we cannot support something evident from something not evident-- and since axioms are not proven by any outside standard, i.e. not made evident by external premises- they are considered self-evident. For what makes them evident can only come from the self.
First that there are no "unquestionable answers" in the sense that there are no "answers" we literally cannot question again. I can technically do this forever. For example I can ask "What color is three?", "Where is the universe?" and "What do you think of three sided squares?" Whether these questions are meaningful or have any epistemic merit is a different matter entirely though.
Thus my second point, if we mean by the original question however that all questions are meaningful or epistemologically worthwhile; the answer I will give is "no."
To illustrate why real fast I will have to invoke the regress model.
Basically it is that "conclusions" (answers) are based on "premises" (proof and evidence); which are themselves in their own way "conclusions", themselves based on "premises"; which are themselves based on premises. And so on, ad infinitum- or maybe not.
The latter comment refers to the fact that ultimately, we have to stop/begin somewhere, with some final premises or we go on for infinity.
This leads to two approaches- infinitism, and finitism/ foundationalism.
Infinitism is usually rejected as absurd, since we cannot ground knowledge in something infinite and it seems to lead to mere relativism. If we cannot ultimately justify any premise, how do we even justify infinitism over foundationalism?
This makes infitinism inconsistent with itself and thus a worse theory.
Foundationalism however is consistent, but it has to start somewhere. With some final epistemic premises or standards. These final standards are usually called axioms, and are considered the basis for all justification.
Now by final I don't mean the end road for all knowledge, so much as the beggining. They are only "final" in the sense that they are the last thing we can test extrinsically via by other standards.
Now this being the case, and axioms be the bottom line or final premises, questioning them cannot really be too constructive. This is because any "questioning" would already presuppose the truth of these axioms, so all answers would be in some sense circular.
Now I am not saying questioning axioms in general is not good however. This is because since axioms are not extrinsically proven, (proven by other standards) which is what a "question" ultimately demands, false axioms can be extrinsically disproven. Thus asking questions may help us find false axioms, perhaps by showing that they can be disproven in theory, or that they contradict others axioms, or that they are not evident in the sense of being similiar to other axioms and necessary for logic and reasoning.
Axioms for example do have certain traits: they cannot be disproven even in theory, they cannot contradict other axioms, they are not sufficiently upheld by earlier premises and they are evident (provide a basis for evidence.)
An example of this is the statement " I am having sensations right now."
That is almost certainly true, it cannot be disproven in theory, it is not sufficiently proven by other premises, and it is evident in that it is necessary for future reasoning.
Another example is the claim "I exist." I cannot disprove that in theory, as it leads to contradictions.
Some others are the laws of identity and noncontradiction, objectivism, the basics of geometry. etc.
Lastly, since axioms are considered "evident" in that they can prove latter claims, which would not work if they were not "evident" as we cannot support something evident from something not evident-- and since axioms are not proven by any outside standard, i.e. not made evident by external premises- they are considered self-evident. For what makes them evident can only come from the self.
I assume you mean that within any given consistent system axioms cannot contradict other axioms within that same system, since that would render the system inconsistent. There is no requirement, however, that axioms from different systems be non-contradictory. Euclid's Fifth Postulate and its contradictions are an example.