I don’t see how we can say that pi only contains a finite amount of information. I understand we can represent this infinite amount of information in a finite manner. But this is not the same as pi itself containing a finite amount of information. If pi consists of an infinity of digits, then that is an infinite amount of information because each number is a bit of information.
Where we can divide a circle into 3 wholly equal parts using pi, then pi is infinite. 1/3 is a finite part of something. Pi is more like a tool to do something with than a finite part of something.
I meant that it says that it consists of an infinity of digits.
See my above point on the representation of pi versus the value of pi.
I mean no offence by the following, just constructive feedback: I think you are too focused on representation and not focused enough on semantics.
I agree that you can get as close as you want. We were never in disagreement on this. And that was my point to you when you first said 1/3 = 0.333…
If you remember, I said that 1/3 cannot equal .333… because an infinity of 3s are impossible and an infinity of 3s are needed to fulfil the semantic of 1/3 as opposed to just get close to it. But then you made that point about circles and I reconsidered my position.
To emphasise, you say you do not need to talk about infinitely many 3s, but then semantically speaking you have not fulfilled the semantic of 1/3 have you? When the 3s are finite, you have only fulfilled close to 1/3. The only way you can say 1/3 = .333… is if you are saying that an infinity of 3s follow.
Certainly - 120° each slice = exactly 1/3 of a circle.
Still - you are confusing the digits involved with the value they represent. A very small number might be represented by an infinity of digits - that does NOTmake the number infinite. And it does not stop anyone from dividing something into 3 equal parts.
The decimal system of listing numbers has issues - problems. Some values (such as 1/3) cannot be entirely and precisely represented with decimal numbers. The issue is only in the language being used (base_10 digits) - not with the real values.
Pi is a value that cannot be exactly represented by decimal digits ever. The value of Pi doesn’t care whether you try to represent it in decimal form. The value of it doesn’t change to match your estimations. The value is a little over 3.14 and always will be even if you list an infinity more digits in the effort to be 100% exact. Pi will always just be a little over 3.14.
Similar with 1/3. It doesn’t matter how many digits your want to throw together to try to represent it. It will always remain merely 1/3 or a little over 0.33. Add a billion more 3’s at the end and you do not change what 1/3 is at all. You only change the accuracy of your string of decimals.
The whole thing is a language issue - the language of decimal numbers. It has nothing at all to do with reality or real values.
The language keeps saying that 1/3=0.333… and 1=0.999… through series convergence and the many other techniques that have been demonstrated in this thread.
Take the semantic of ‘21’ and the semantic of ‘divide’ and the semantic of ‘3’ and the semantic of ‘equals’. What does 21 divided by 3 equal? The answer is not 21/3 because that is the question that is being asked. It is not the answer. The answer is the semantic of 7. 7 is the answer in its actual purest form.
Now, what is 1 divide by 3 equal to? The answer is either absurd or .333… (where … stands for an infinity of 3s)
You cannot say the decimal system of listing numbers has issues. It is the semantic of ‘one’ and the semantic of ‘three’ and the semantic of ‘divided by’ such that when you ask what is one divide by 3? the answer is either you cannot divide 1 by 3 or it is zero point followed by an infinity of 3s.
Yes but semantics dictate that either 1/3 is absurd, or 1/3 is 0.333…
It is semantically inconsistent to say that 1 divide by 3 is 0.333… (follow by a vary large but finite number of 3s).
I see them as having the same identical answer. The semantic of 7. I don’t see them as being semantically identical. As in 21/3 is not semantically identical to 7/1. Their answers are semantically identical. This does not make them semantically identical.
It might help us (and you) if you would describe what certain words mean to you - define them in your own words - such as “semantics”, “infinite”, “…”, and “infinity”.
I don’t have any doubt that this whole thread is nothing but arguments over words and symbols.
Semantic = meaning
Infinite = that which has no beginning and no end
Infinity = that which no greater quantity can be conceived of
If I ask the question what is the answer to a third of 21 exactly, the answer I would get is 7. Not 21/3. Perhaps someone might say you will find the answer when you divide 70 by 10, but they are still saying you will find the answer after you do this division.
To convey what I’m trying to convey more clearly, if I ask the question what is a third of a 1 meter ruler, I would not get the answer, well that’s just a third of a one meter ruler. I would either get the answer you cannot have a third of a meter, or it’s .333…m or 33.333…cm
But that doesn’t mean that you can’t divide the meter. It only means that you cannot say it using decimal digits.
Bend that meter into a circle an cut it at exactly 120 degrees and you have your 1/3 meter. Ask someone how long those portions are in centimeters and they cannot give you an exact number.
I think I already did. (10^{\infty}) stands for “10 multiplied by itself infinitely”. Every term in that sentence (such as 10, multiplication and infinity) has an established definition. My claim is that all you have to do is use the established rules and deduce the meaning of the expression (in the same way one can deduce the meaning of “2 + 2” without having to look for an answer from mathematicians, books and online encyclopedias.) If you want a longer description, here is one: the statement tells you to calculate the result of (10 \times 10) and then take that result and multiply it by (10) to get a new result and then take that result and multiply it by (10) to get a new result and so on for an infinite number of times. What you get after you’re done multiplying all the (10)s is what the expression stands for. What exactly is unclear and/or unconventional about that? How can I help you if you are not willing to explain what’s unclear and/or unconventional? Merely repeating yourself by saying “It’s undefined”, “You haven’t defined it!” and “Define it!” will get you nowhere. A better approach would be to explain what it means for something to be undefined, and if it proves to be necessary, to present an argument that my expression is undefined.
Perhaps you should start by defining what the word “ambiguous” means.
When we say that what someone is saying is ambiguous, we are merely saying that we cannot determine exactly what that person is talking about (“He might be talking about X, but also, he might be talking about Y”.)
Wikipedia seems to be unambiguous about the meaning of the word “infinity”. It tells us right at the start that the word means “something that is larger than any real or natural number”. What exactly is ambiguous about that?
The fact that the word “infinity” can be used to represent more than one number does not mean that it is ambiguous. It merely means that it is not a specific number but a category of numbers (similar to the word “integer”.)
And even if “infinity” is an ambiguous term, what makes you think that logic can’t handle ambiguity? Of course, ambiguous terms have limited use but that’s a different subject.
The word “integer” cannot be used - as an integer - because it is mathematically ambiguous - it is a category of values, not a specific value. And “infinity” is equally a category of values - not a specific value.
(10^{\infty} = ?)
If you want to use maths and be specific just use (infA) instead -
(10^{infA} = 10^{Natural Number Count} = 10^1x10^2x10^3x…)
The only definition I could find for the ellipsis “…” is in Wikipedia -
I have never seen “…” being used to mean anything other than “continue indexed throughout the natural numbers”. So without a formal definition in our debate I would have stated - Ellipsis - “…” = “repeated throughout the natural numbers with index starting at either the first digit to the right of the decimal (when a decimal is to its left) or repeated throughout the natural numbers with index starting at the first left hand digit indicated at the left of the symbol.”
From there, most of our argumentation would never have occurred - although perhaps you want to debate a “better” definition - but that is a different debate.