I thought you posed a dilemma of sorts?
This thread is only about base I0 Magnus and so talking about other bases or systems is not relevant here
Also all the relevant arguments have already been made which is why the thread stopped two years ago
Yes, but bases can be consolidated, no?
I don’t agree.
I don’t agree either. Other bases are fair game in mathematical debates. (Btw… Magnus is right)
When I was reading through this, I noticed the following that seems to bring it all to a salient conclusion.
It makes sense that “0.999…” is just an expression signifying that some ratio cant be expressed by a fixed number of digits (unlike 1). It isn’t actually a number. And neither are all of those other expressions that end with “…”. And apparently that is why the Wikipedia proofs are misleading.
To me, that seems like a “game over”.
Well, James was very firm that infinitesimals were useful, and usefully different than the “convergence”…
Hence, his InfA thing
This seems to explain why he would -
I’m no maths genius but isn’t calculus just the sum of the infinitesimals? Where would science be without calculus?
Obsrvr,
I’ve been in many threads with James …
So I’m certainly putting my own words in his mouth:
James would say that both infinitesimals and approximations are useful in their own right and for different reasons.
Calculus is awesome… it spans multi-disciplines… where would industry be without it.
I am not sure why you think it’s not a number i.e. a symbol representing some quantity. It appears to me that it clearly is. It has many properties that numbers have e.g. it’s greater than some numbers and less than others.
What it isn’t is a finite quantity, that’s for sure, and that’s why it can’t be 1.
Not really. I just expanded upon Gloominary’s post.
If it isn’t a “finite quantity” I imagine that it isn’t a number.
The whole question boils down to a confusion between the qualitative and the quantitative.
“1” is clearly a precise quantity, but as soon as you profess 0.(9) you add in the quality of “endlessness” to describe the repetition of the quantity of “9” for each decreasing power of 10 (or whatever base you’re using).
0.(9) is an attempt to restate the quantity “1” in a way that involves endlessness. As is 0.(3) to restate 1/3 when one divides 1 by 3. It’s an admission that one cannot denote 1/3 etc. entirely quantitatively without the use of the quality of endlessness. Multiplying 1/3 again by 3 is obviously 1 (3/3), yet multiplying 0.(3) by 3 is not so obviously 1 (0.(9)) precisely because of the injection of the qualitative into the otherwise entirely quantitative.
Subtracting 0.(9) from 9.(9) to get the exact quantity of 9 requires the same confusion.
As soon as you allow the notion of the qualitative into the quantitative you invite possibilities such as ε as an epsilon number and so on.
This is the same kind of mistake that every extended or “new” number set allows - much to the advancement of mathematics and other utilities… but not truths. Experientialism highlights the distinction.
So we see how useful it is to make particular types of mistakes that are not true, but are useful: such as the notion that 1 =/= 0.(9)
Is it really? No.
But that’s the wrong question.
The more useful question is whether any new knowledge can be gleaned from the possibility that 1 =/= 0.( 9)
That would be “1.000…”, not “0.999…”
No, I meant 0.(9)
The recurring 9 necessarily must recur endlessly without defined quantity (infinitely) for 1 to divide 3 times into 0.(3) and multiply back to 0.(9)
For 0.(9) the 9s must recur with the quality of endlessness for it to be subtracted from 9.(9) to equal 9 exactly. If they didn’t, you couldn’t get 9 exactly.
By contrast the quantity of 0s that you put after “1” doesn’t matter, whether it’s 1.0, 1.00 and so on - it doesn’t affect the quantity of 1 even if you try to impose the quality of endlessness with 1.(0)
The quantity of 9s that you put after “0” most definitely matters because 0.9 is different to 0.99 and so on with any non-endless (finite) quantity of 9s after the 0 different to 0.(9) with its quality of endlessness.
So 0.(9) is 1 only with the quality of endlessness, where 1.(0) is 1 with or without the quality of endlessness.
Endlessness is irrelevant to 1 when denoted as 1 so endlessness isn’t necessarily involved.
Endlessness is essential to 1 when denoted as 0.(9) so endlessness is necessarily involved.
There’s no issue dividing 9 by 3 to get 3, and then multiplying it back by 3 to get 9. The 9 is the same before and after the operations.
So why would dividing 10 by 3 to get 3.(3) and multiplying it back by 3 to get 9.(9) be any different? The 10 is the also same before and after the operations.
There’s no issue subtracting 1 from ten times that i.e. 10 to get 9.
So why would subtracting 0.(9) from ten times that i.e. 9.(9) not get 9?
0.(9) not being 1 requires a double standard for (mod 0) versus some other modulus, which removes a fundamental necessity that’s essential to mathematics: that it’s consistent.
But that’s my whole point: introducing any notion of the quality of endlessness to quantities confuses everything. That’s why infinities are such a minefield.
Hence why “1 = 0.(9)?” is the wrong question - the more useful question is what happens if you go against the truth that they are equal and say they aren’t. It’s what we did for complex numbers - there is no square root of minus one in truth, but what if there was? What usefulness can “i” provide? Turns out it provides a lot of usefulness even though it’s not true that “i” exists any more or less than it’s true that epsilon numbers and hyperreals exist.
The whole debate behind this thread isn’t looking deep enough - and as always, Experientialism puts it all into perspective.
Ah…
I see that Magnus has come back to ILP much wiser than when you had departed, to KTS.
The concept of equality might be the problem.
In mathematics, “a = b” means that the two symbols “a” and “b” have one and the same meaning i.e. that everything that can be represented by the symbol “a” can also be represented by the symbol “b” and vice versa.
“2 + 2 = 4” is true because everything that can be represented by “2 + 2” can also be represented by “4” and vice versa.
However, when we say something like “2 + 2 = Finite number” we are working with a different concept of equality. In such a case, “a = b” means that everything that can be represented by one of the two symbols can also be represented by the other symbol but not necessarily the opposite. This is evident in the above expression. Namely, everything that can be represented by “2 + 2” can be represented by “Finite number” whereas the opposite does not hold true i.e. there are things that can be represented by “Finite number” that cannot be represented by “2 + 2” e.g. any finite number other than 4, such as -250, 0, 1/2, 5, 100, etc. This isn’t problematic per se unless it leads to fallacious arguments such as the following:
- 2 + 2 = Finite number
- 4 + 4 = Finite number
- Therefore, 2 + 2 = 4 + 4
The concept of equality we’re working with does not permit this kind of deduction.
Put another way:
A = The set of all things that can be represented by “2 + 2”
B = The set of all things that can be represented by “4 + 4”
C = The set of all things that can be represented by “Finite number”
- A is a subset of C
- B is a subset of C
- Therefore, A is a subset or a superset of B
It should be perfectly obvious now why the conclusion does not follow. Just because A and B are part of something it does not mean that A contains B or that B contains A or that they are the same part.
People can neglect this because they can forget (and in some cases not even notice) that they are working with a different concept of equality. You can’t abide by the rules that apply to one concept of equality if you’re working with another one (unless the same rules apply, which is not the case here.)
In the same exact way, one can make the following argument:
- (1+1+…+1) + 1 = Infinity
- (1+1+…+1) + 2 = Infinity
- Therefore, (1+1+…+1) + 1 = (1+1+…+1) + 2
The premises are correct but the conclusion does not follow – and it’s wrong.
You seem to be agreeing to James here-
And that was his purpose for creating the infA designation.
I do.
I also agree that, by definition, to add some quantity (finite or infinite) to some other quantity (finite or infinite) means to increase that other quantity. This means that, if you add 1 to infinity, you get a larger number. “Infinity + 1 = Infinity” is only true in the sense that both “Infinity + 1” and “Infinity” are infinite numbers (i.e. they both belong to the same class of numbers.) It’s not true in the sense that the two numbers are of the same size.
Given that it’s obvious that 0.999… =/= 1, the question is, why do Wikipedia proofs look so convincing? Because they do, at least to me, and figuring out where’s the flaw isn’t so trivial.
Take this proof, for example:
What’s wrong about it? Where’s the mistake?
After much pondering (and reading, of course) I’ve come to the conclusion that the mistake lies in the fact that the following two numbers aren’t the same.
They are equivalent in the sense that they belong to the same class of infinite numbers that can be represented using the symbol “0.999…” But they are not equivalent in the mathematical sense of the word. I am inclined to think that they are not equivalent because the number of digits after the dot in the second number is smaller than in the first one. And this is due to the fact that we got that number by taking 0.999…, multiplying it by 10 and then subtracting 9 from it. But when you take 0.999… and multiply it by 10, one of the digits from the right side moves to the left side. Sort of like how when you take 0.999 and multiply it by 10, all of the digits move one position to the left, with one leaving the right side and entering the left side. It becomes 9.99 which has two, rather than three, 9s after the decimal point.