Actually, all that it shows is how poorly you understand maths.
jamest wrote: ... S = the sum of the series. But 1/2 + 1/4 + 1/8 + 1/16 + ...., is the series itself, not summed.
WRONG! The series itself is the sequence of discreet terms: 1/2, 1/4, 1/8, 1/16, etc.
1/2 + 1/4 + 1/8 + 1/16 + .... is all of these terms added together - that's what the '+' signs do!
So, what the author of this math has done, is that he's assumed that the series has a sum, so that he can say "But, if we discard the first term, this is just S.".
WRONG! I have simply given a name (S) to exactly what Zeno describes in his paradox. I have made no claims about that sum having any actual value at this stage. I could replace 'S' throughout with a phrase such as '
the result, whether that be a calculable number or not, of adding infinitely many parts of the whole, the first a half of the whole, the next a quarter, and each subsequent part exactly half the size of the previous' without changing anything fundamental about the proof.
I find 'S' simpler to handle. If you like, think of it as standing for 'something unknown' rather than 'sum'.
The whole point of the math is to try and prove that the series can be summed - not just to assume that it has a sum and then use that assumption to prove the sum of that series.
That is indeed the point of the maths but I never made such an assumption. All I did was to write Zeno's description in mathematical notation and give it a name.
In swapping 1/2 + 1/4 + 1/8 + 1/16 + ...., for S, the author of these math has made the very assumption that he was setting out to prove - that the series has a definite sum.
I have not 'swapped' anything for anything else. I have merely named something. I am sure that there are many interesting, ontological debates to be held upon the subject of a thing changing once it is named but that is irrelevant here. I will even rewrite the proof below
without ever using S, or any replacement name, in order to allay any suspicion that I am presuming anything.
Is that clear? Because if it is, you will see that the math instantly becomes null & void, at this very juncture.
If it was clear, it might well do that. All that is clear, however, is your complete lack of training in and/or understanding of formal mathematics.
I want to discuss tangible and conceptual infinities, at some stage of this conversation. But what I have said, above, will probably stir alot of dust, so I'll let that settle first.
I will look forward to that. In the meantime, here is the promised, alternative proof.
The paradox (paraphrased - I don't speak ancient Greek!): It is not possible to travel the whole distance between two points. This is because first you must travel half of the distance, then you must travel half of the remaining distance (1/4), then you must travel half of the remaining distance (1/8), then 1/16, then 1/32, etc
ad infinitum. Because an infinite number of actions need to be performed, no matter how many you have completed, there will always be an infinitesimal distance still to travel, hence the journey cannot be completed and all motion is illusory.
To solve this, I will show that the infinite sequence that Zeno describes sums to 1.
I have used no symbols whatsoever in this proof, simply the added sequences of numbers exactly as described by Zeno, lacking infinite time or space, I have adopted the standard shorthand and used ellipses to represent the continuation of any sequence of terms to infinity. I have described each step in words and provided the properties of
rational numbers that I am using in each step, since every term in Zeno's sequence is rational. (NB. Rational numbers (
Q) are numbers of the form p/q, where p and q are both integers (whole numbers) and q is not equal to 0. All integers are included in
Q as a subset where q is equal to 1.)
1. The result of adding together the infinite parts as described by Zeno, be it an actual number or not, is identical to itself.
(Equality is reflexive for all rational numbers)
[pre]1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 + ....[/pre]
2. If we double the terms on both sides of the equation, it is still true.
(Rational numbers are closed under multiplication & Equality is reflexive for all rational numbers.)
[pre]2(1/2 + 1/4 + 1/8 + 1/16 + .... ) = 2(1/2 + 1/4 + 1/8 + 1/16 + .... )[/pre]
3. We can then multiply out the contents of the bracket on the right hand side.
(Multiplication is distributive over addition for all rational numbers.)
[pre]2(1/2 + 1/4 + 1/8 + 1/16 + .... ) = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ....[/pre]
4. And re-bracket as follows.
(Addition is associative for all rational numbers.)
[pre]2(1/2 + 1/4 + 1/8 + 1/16 + .... ) = 1 + (1/2 + 1/4 + 1/8 + 1/16 + .... )[/pre]
5. We can subtract (1/2 + 1/4 + 1/8 + 1/16 + .... ) from both sides.
(Subtraction is closed for all rational numbers.)
[pre]2(1/2 + 1/4 + 1/8 + 1/16 + .... ) - (1/2 + 1/4 + 1/8 + 1/16 + .... )
= 1 + (1/2 + 1/4 + 1/8 + 1/16 + .... ) - (1/2 + 1/4 + 1/8 + 1/16 + .... )[/pre]
6. We can expand the brackets on both sides.
(Multiplication is distributive over addition for all rational numbers.)
[pre]2/2 + 2/4 + 2/8 + 2/16 + .... - 1/2 - 1/4 - 1/8 - 1/16 - ... = 1 + 1/2 + 1/4 + 1/8 + 1/16 + .... - 1/2 - 1/4 - 1/8 - 1/16 - ....[/pre]
7. We can reorder the terms.
(Addition is commutative for all rational numbers.)
[pre]2/2 - 1/2 + 2/4 - 1/4 + 2/8 - 1/8 + 2/16 - 1/16 + .... = 1 + 1/2 - 1/2 + 1/4 - 1/4 + 1/8 - 1/8 + 1/16 - 1/16 + ....[/pre]
8. We can rebracket.
(Addition is associative for all rational numbers.)
[pre](2/2 - 1/2) + (2/4 - 1/4) + (2/8 - 1/8) + (2/16 - 1/16) + .... = 1 + (1/2 - 1/2) + (1/4 - 1/4) + (1/8 - 1/8) + (1/16 - 1/16) + ....[/pre]
9. Finally, we can evaluate the terms inside the brackets.
[pre]1/2 + 1/4 + 1/8 + 1/16 + .... = 1 + 0 + 0 + 0 + 0 + ....[/pre]
We can now see that the left hand side of the equation is Zeno's original description and that this is equal to 1.
