Your left sum is 0.111....11 with 1024 1 bits in binary. Your right sum is 1. Is 0.111...11 with 1024 1-bits greater than 1? — InPitzotl

The sum of terms 1/x is infinity if 1/x > 0.

...and you'll find the inequality always breaks down for some number of terms, and all terms after that. In fact, you can cheat... whatever integral x you specify, it will break down at the xth term. — InPitzotl

x is a power of 2.

1/2 + 1/4 + 1/8+....+1/1024 >

1/1024 + 1/1024 + 1/1024+....+1/1024 for the same number of terms.

Now go to infinity with x. You are still adding an infinity of positive terms.

This is false soon as the number of terms is greater than or equal to x, after which point the bottom sum is greater than 1 and the top sum is still less than 1. — Pfhorrest

x is not static. I'm saying if there are the same number of terms in each. Now increase x indefinitely with the same number of terms top and bottom.

1/2 + 1/4 + 1/8+....1/x >

1/x + 1/x + 1/x+...+1/x

Now let x go to infinity-

$\sum_{i=1}^{\infty} 1/x = \infty$

The point of a limit is that the sum never exceeds it. No matter how many terms you add to that convergent series, it will never exceed 1. Why then would you think it could ever add to to infinity? If it could, that would make it a divergent series, one with no limit, by definition. — Pfhorrest

I don't insist it doesn't sum to 1. You may well be right. I'm saying we don't know because we can never have an actual infinite sum. An infinity of positive quantities are being summed and any positive quantity summed infinitely, is infinity.

What is the 'last' term in the sequence 1/2, 1/4, 1/8...? No need to answer, it is a rhetorical question. But surely all terms - the whole infinity of them - are positive and > 0. Right? Now sum an infinity of positive quantities...

One of those series diverges; it does not have a limit. The other converges: it has a limit. The second one never gets anywhere close to infinity no matter how long you run it. It would only ever even get up to 1 if you ran it forever, with your “God-calculator”. — Pfhorrest

But that begs the question: you say it don't sum to infinity because it don't sum to infinity. That is the very thing that is being questioned. I know the limit is 1. But that limit is defined by finite arithmetic. I am asking what really happens at infinity. An infinity of positive quantities are being summed and any positive quantity summed infinitely, is infinity.

This becomes false as soon as k > 2. — Pfhorrest

I know, I made a mistake. Let me rethink how to formulate it...

rethinking...

you cannot talk about this reified "actual sum" unless you can talk about it, and I'm not sure you've convinced me there's a thing to talk about. — InPitzotl

What I'm saying is very simple. Suppose you had a kind of God calculator that would print out the actual addition of 9/10 + 9/100... to an infinity of terms, what would that be, 1 or infinity? That's what I mean by the actual sum.

To show how quirky infinite sums are consider the following (this is not meant to answer anything, it is just to illustrate how strange things become at infinity)

Theorem 1
Define 1/x such that 0 < 1/x < 1. If 1/x is summed to itself infinitely often, the sum is infinity. From this we conclude that any positive quantity added infinitely sums to infinity

Now sum 1/2 + 1/4 + 1/8...in view of the above theorem. No term in this series is zero, they are all positive quantities. So we are summing an infinity of positive quantities, some of the bigger than others...

Again, I'm not trying to answer anything here but it is worth contemplating.

Apparently not... see the underlined as evidence for your continued confusion of the same point. The sum is by definition the same as the limit. — InPitzotl

I missed this post. Yes, the definition of the sum is the same as the limit. But I am talking about an actual sum. An explicit infinite sum that you can write down. This of course is not possible because it requires an infinity of calculations. That's the point I'm making. I understand sums and limits but that is not really how the question can be answered. It can only be answered by an explicit infinite sum. What makes me suspicious is the paradoxes that exist at infinity.

1. Is there a theory of intelligence that explains these statistics?

No.

2. Are the statistics a reflection of systemic bias against women? — TheMadFool

No.

You have to account for social factors. Only in affluent societies do women have the means to peruse science etc. Women don't go into scientific areas as much as men, even when they have the means to. Men who are breadwinners will sometimes have jobs that could lead them a nobel prize. Women often choose types of work, like medical care, that won't lead to a nobel prize. There are dozens of reasons why.

A ninth of that particular pie is a particular quantity. A ninth, or 1/9, is not a particular quantity. Are you capable of understanding this? — Metaphysician Undercover

1/9 is a proportional relationship. In geometry if the side of the square is $1$ the diagonal is $\sqrt{2}$ and the proportion is $1:\sqrt{2}$

Hmm, have you ever asked yourself a simple question "why"? Definitely we can, and we do, for centuries (since Leonard Euler's time) — Andrey Stolyarov

No, we know what a limit is. But as I keep saying, a limit is not a literal infinite sum. Weierstrass did not formulate calculus in terms of literal infinities. He formulated it in terms of finite sums converging to a limit. I'm not arguing that limits are not as they are defined. I'm saying a literal infinite sum is qualitatively different to a finite sum. The definition of a limit only makes sense in terms of finite sums converging to a limit. This is how the limit was rigorously defined by Weierstrass, Cauchy...That definition says that the finite sum can get ever closer to the limit. But you are jumping from the finite to the infinite and saying finite calculations still apply to infinite sums.

The limit of 1/2 + 1/4 + 1/8... is 1. But what is the infinite sum, as a literal infinity? We cannot assume it is 1 just because finite arithmetic points in that direction.

The infinite sum itself has been defined to be the limit... by mathematicians — InPitzotl

I'm aware of that. But it has not been explicitly defined. It has been defined in terms of limits which are limits of finite sums.

Mathematicians regularly compute what they mean by it. — InPitzotl

They compute limits which are not the same as sums. A limit is what a finite sum converges to.

Actual mathematicians do. — Pfhorrest

See next answer.

An infinite sum is defined because the mathematics community defined it; same as "twelve" is defined because English speakers defined it. — InPitzotl

That still begs the question what is an infinite sum if nobody has ever computed it? You can't jump from the finite to the infinite and expect finite rules to apply. And it is questionable that it has been defined. All that has been rigorously defined is a limit.

The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the serie — Wikipedia

Emphasis mine.

They're talking about infinite series, and saying that the limit is the sum of that series. I didn't quote the whole article, just the relevant part. Click the link and read for yourself. — Pfhorrest

It hardly matters. An infinite sum is undefined because nobody has ever computed it. The problem is that while it is ok to apply the logic to finite quantities, nobody knows what an infinite sum is.

When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. — Wikipedia

They don't say the infinite sum, ie the sum of all terms.

When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. — Wikipedia

You need to provide the link. As a working definition that may well be useful but there are still problems with infinite sums.

A limit is by definition something that will not be exceeded. — Pfhorrest

By a finite number of terms.

What do you think "the limit is 2" means, if not "this will never add up to more than 2, no matter how many terms you add"? — Pfhorrest

Yes, you are correct but what is an infinite sum of terms? What happens when you sum an infinity of terms? Calculus does not account for this.

A limit is by definition something that will not be exceeded. We can be absolutely sure that 1/1 + 1/2 + 1/4 + 1/8 ... will never add up to infinity, because the limit of the partial sums is 2, which means it will never ever ever add up to more than 2, and only "at infinity" will even add all the way up to 2. — Pfhorrest

No. Calculus is formulated in terms of finite sums and limits. You can't jump to infinity and expect the rules of finite arithmetic to apply. Jumping from the finite to the infinite is an infinite distance and we don't know what happens there.

It is not. For any decimal fraction, the limit of both the corresponding sequence and the corresponding series is always finite, which is obvious enough: it is always between 0 and 1, there can be no infinity here. — Andrey Stolyarov

You need to read back a few pages to see what I'm saying. It is like this-
It is being asserted that 1/10 + 1/100 + ...taken to an infinity of terms is 1.
If we take a finite number of terms they converge to a limit. But a limit is not a sum. It is what a finite sum converges to. We can't know what an infinite sum is. It may well be 1 but it may also be infinity. We don't know because you can't apply finite arithmetic to infinity. You can't jump to an infinite sum and assume it is 1.

For this particular case, you can either take the limit of the sequence 0.9, 0.99, 0.999, ..., or you also can work in terms of so called series (see https://en.wikipedia.org/wiki/Series_%28mathematics%29) and consider the series of 9/(10^n), that is, 9/10 + 9/100 + 9/1000 +..., the result will be exactly the same. — Andrey Stolyarov

Earlier on someone wrote a very convincing 'proof':
x = 0.999...
10x = 9.999...
10x = 9 + 0.999...
10x = 9 + x
9x = 9
x = 1

All well and good. But what does x = 0.999... mean? In terms of infinite sums let the sum be S.
The last two lines give:
9S = 9
S = 1.

But what if S is infinite? That is, what if an infinite sum of terms is infinite?
Then we have $9\infty = 9$
$\infty = 1$

That's the thing, we don't really know what S is because you can't apply finite arithmetic to infinite sums.

No, you're not getting the point. There's no need for any "infinite sum" — Andrey Stolyarov

But this is what is being asserted: 1/10 + 1/100 + ...taken to an infinite sum of terms. Let S be this literal infinite sum. What is S? Is it 1 or $\infty$? That is what the problem is, we don't know what a literal infinity is because finite algebra does not apply to literal infinities.

The diagonal of a square, for example, measured in the units that the sides are measured in, is how long? Is that length not a number? Or did something magic happen? — tim wood

In geometry the length of the line - in this example $\sqrt{2}$ - is exact. But the decimal expansion representing it is not, unless we go to an infinite number of places.

that 0.999... is taken as another representation for 1 — Andrey Stolyarov

It may well be that the infinite sum is 1 but mathematicians were suspicious about such a concept because infinity is not a number. This is why calculus is formulated in terms of limits, not infinite sums.

I think the system of real numbers allows that "number" remain undefined, indefinite, and this is why "the real numbers" is not a fixed system. Rigorous defining of "number" has been withdrawn for the sake of producing the real numbers. — Metaphysician Undercover

Yes exactly, calculus works in practice. You can sum $10^{1000000000000000000000000}$ terms and that's fine because it is a finite sum.

But what is $10^{\infty}$?

Show me a definition of "number" which allows that .999... is a number. — Metaphysician Undercover

Kummer did not believe in real numbers; "God made the integers and all the rest is the work of man". This is a bit extreme as real numbers - whatever they are - are cool. Without them we would not have calculus.

It does though. It defines the sum of an infinite series as the limit that the partial sums approach. — Pfhorrest

I understand what you are saying but a literal infinite sum is not considered in calculus. If partial sums are added they approach the limit. If more terms are added it gets closer to the limit and so on. This is how people like Cauchy formulated calculus. They don't consider literal infinite sums. And this is the question I am raising: is the concept of an infinite sum coherent?
When infinitesimals were invented/discovered it was asserted that, they are so small, no matter how many of them are added together, you end up with another infinitesimal. This is why Berkeley balked at these 'Ghosts of departed quantities.' They were so small you couldn't do anything with them.

As I see it, the conclusion that was being reached was that an infinite sum of zeros add to 1. And this did not make sense. So infinitesimals were invented. This is why I have reservations about literal infinite sums.

$0.999...$ is the limit of the sequence of partial sums $s(n)$ — fdrake

Yes, 'partial' sums. That means the sums are finite. Calculus does not speak about literal infinite sums. It speaks about finite sums approaching a limit. As more terms are added indefinitely, the limit is approached more closely. That is what calculus is saying.

What I am saying is that a literal infinite sum is probably an incoherent concept.

0.999... is the limit of {0.9,0.99,0.999,...} — fdrake

https://en.wikipedia.org/wiki/Limit_(mathematics)#:~:text=In%20mathematics%2C%20a%20limit%20is,)%20%22approaches%22%20some%20value.&text=The%20concept%20of%20a%20limit,direct%20limit%20in%20category%20theory.

You just accepted that 0.999... is the limit of {0.9,0.99,0.999,...}, and equal to 1 — fdrake

Yes of course. I have not said it is not the limit. I said 0.999... is not equal to 1 if we are talking about a literal sum. If we are talking about a limit, yes, the limit is 1. I keep saying a sum and a limit are not the same thing.

Great. That means you accept
0.999... = 1 — fdrake

Only according to a strict interpretation of the ' = ' sign: 1 is not the sum. It is the limit of the sum. So 0.999... = 1 does not mean it is literally 1. It means 1 is the limit.

Just a comment about posting math material, symbols, equations, etc. I doubt if anyone here uses it, but MathType is very easy to use and is WYSIWYG rather than coding for each symbol. — jgill

The first post on this math forum explains how to us the math tags.

https://thephilosophyforum.com/discussion/5224/mathjax-tutorial-typeset-logic-neatly-so-that-people-read-your-posts

There isn't. What you're looking for is the infinity symbol above the Sigma. — Kenosha Kid

I just left it out for brevity. I'm sure you know what i mean.

That is exactly how it is meant. That is what 0.999... means. — fdrake

Ok, I'll accept that. But what we are talking about here is subtle and the " = " sign in calculus can be misleading:

$\sum 1/x$ is a literal sum.

lim $\sum 1/x$ is not a sum. It is the limit towards which the sum (over the range) converges.

That is the difference.

fdrake — EnPassant

What is the difference between
$\sum$ = x and
lim $\sum$ = x ?

What is the limit of the series {0.9,0.99,0.999,...}? Call this x.
What does the symbol "0.999..." represent? Call this y.

Is x=y ? — fdrake

It depends on how you read these expressions. I'll grant you that 0.999... can be identical to the limit of the series if that's how you interpret it. But if you do you interpret it as a limit not as equals.
You can say 0.999...= 1 if by that you mean the limit of 0.999...

fdrake — fdrake

Yes, but that is the limit which is different from equals. When you say 1 = you are saying 1 = the limit not simply 1 =

It should be written
lim $\sum$
not simply $\sum$