Mystical infinities and measurements with the unknown as the standard are in fact meaningless, not a dictate. When you embrace mystical infinities, contradictions, and meaningless measurement then stopping is all that is left!

On that basis you will never understand the mathematical infinity and infinite series, whose explanation has been well established since the 19th century in what was called the "arithmetization of analysis" that showed how to understand infinite limits in terms of finite numbers, which is all they can be.

The "..." means an infinite sum, which means the limit of the sequence of partial sums, not simple addition. There is no infinity of addition. The series diverges, oscillating forever. It has no unique number as a limit. Terms in divergent series cannot be arbitrarily combined. When you (invalidly) group them like this you get a different, inconsistent result:
S =-1+(1-1)+(1-1)+(1-1)... "=" 1

I find this useful in considering a a radical proposition, cannot recall the source-
But what does it un-explain.
This an heuristic not a rule.

An example is how to view the global temperature Hockey Stick which contradicted a vast set of knowledge accumulated from many sources and different cultures over centuries. Thus it was viewed with caution, this caution no doubt led to the contrivances and biased statistical analysis becoming exposed with the whole edifice collapsing.

As you dictate what is meaningless or not, we can't discuss anything. By the way, I don't consider meaningless nothing of what you said. I may have doubts about your premises.
Thank you for the discussion, anyway. I will leave it here.

The Sum S = (-1+1)+(-1+1)+(-1+1)... = 0 which has nothing to do with limits or series. It is simple addition.

Einstein did not "rewrite the rules of physics". He added new knowledge required by physics in some realms. New discoveries in physics do not negate what was known, they build on it. Science is not a sequence of "rewritten" exploded fallacies.

The cowardly hit and run jerks who systematically 'downvote' rejection of mysticism do not belong on this forum. Claimed mystical insights and sacred texts are not knowledge of reality. Knowledge requires method because it is not infallible. There is no exemption from rational concepts and validation of knowledge. Those declare their ideas immune to validation exclude themselves from the realm of knowledge.

You didn't provide any alleged proof to discuss. No one can tell you where a flaw is in a purported proof that you don't give.

Nor is it required when it results in a contradiction. Whatever details you have in mind don't matter because your assertion that the sum of positive integers is -1/12 is demonstrably false.

That it is false isn't "according to me", it is false in fact as established by mathematics, which you were shown in three proofs showing what is already obvious but formulated in terms of the mathematical meaning of an infinite sum.

Of course whatever "proof" you have in mind is flawed. It is flawed because it results in a falsehood. Valid proofs don't result in proving the false to be true. It is a bald contradiction, with an appeal to a meaningless mysticism of infinity to try to save it.

Your assertion that "Maybe the flawed assumption is that we understand infinity" is true, but it applies only to you: The mathematical infinite is a process not a mystical place or thing or some undefinable incomprehensibly mystic realm, and that process is understood by those who learn the mathematics.

You cannot understand how to apply mathematical processes of infinite limits without understanding some basic concepts of mathematical analysis which are well established and understood. But you would have to learn them, not arbitrarily manipulate some symbols and then ask 'what went wrong', while grasping at a mystical notion of the infinite.

The principle of "reductio ad absurdum" rejecting a premise that leads to a false conclusion is also a matter of the logic of mathematics, not of I "don't believe". There are no logical exemptions form "jumping so easily to conclude the contradiction". The contradiction is staring you in the face. Abandoning the principle of non-contradiction in logic for the sake of a mystical infinite makes rational thought and discussion impossible.

Your own lack of understanding of the mathematical infinite does not justify the non-sequitur: "Therefore, if you can't say where the proof is flawed, but you discard it because you don't 'believe' can be true puts you in the category you are accusing me of." Mysticism is a rejection of logic, not an alternative form of knowledge relegating logic to no better than the "same category" of "belief".

This isn't a matter of competing authorities vying for your attention in the name of science. You either understand the science yourself or you don't.

If you don't want to learn it then you don't have to, but that doesn't justify infinity mysticism and denouncing those who understand as engaging in personal subjective "beliefs" for following logic claimed to be no better.

You did not say what you mean by "deeply believe (no one can prove it, though) that knowledge is unlimited." Whose knowledge? Obtained how? Do you mean omniscience?

There is no infinite knowledge. The infinite, as Aristotle identified, is a process not a thing (called "potential" versus "actual"). Knowledge is unlimited in that nothing limits us from learning more, and there is always more to learn, but it is always finite knowledge. Is that what you meant?

Unlimited knowledge and limited human capacity do not "lead to ... [N]o matter how much knowledge human kind has accumulated, it is nothing compared to the unknown". You don't know anything about the unknown. If you did it would not be unknown. We measure things by what we know, not by what is "unknown". Knowledge as "nothing compared to the unknown" is meaningless.

It is true that we expect to learn a lot more, possibly much more than is known now, as has happened in the exponential knowledge explosion of the last several centuries. That is not measurement with respect to "the unknown", which is impossible.

Fish: "You say that if you don't understand, you don't know. I'm afraid that almost all knowledge you and I have is under that umbrella: we don't understand it. One simple example: do you understand gravitational force?"

We certainly do understand gravitational force. We do not know why masses attract or the full mechanism by which gravitational force is transmitted from one mass to another. We do understand the nature of the force and how it depends on mass and distance. We know in detail how to measure, use and predict it. Understanding of gravity has been fundamental in physics and engineering knowledge for centuries. That is understanding gravitation. It does not mean that there isn't more that could be discovered.

Understanding does not mean omniscience. That we don't know everything that does not mean we don't know anything. We know what we know based on sense perception of the world and our conceptual organization of it, not by mystically wrapping our consciousness around things with some kind of infinite insight. The standard of measurement of knowledge and its accumulation is what we do know.

If you don't understand what something means and why it is true then you do not know it. Knowledge is mental grasp of reality. If you don't understand how your knowledge is validated and what it means then you don't have that mental grasp of reality; you can't claim that the idea is true of reality -- you don't know. Someone else may understand and you may understand the meaning of the individual words he uses, but you don't know it yourself without adequate understanding. Floating abstractions are not knowledge.

The formula sum k=1->n {k} = 1+2+...+n = n(n+1)/2 means that the nth partial sums 1+2+3+..+n of the infinite sum have the value n(n+1)/2. The limit of the partial sums is therefore the limit of n(n+1)/2, showing that the process of the infinite sum is also unbounded. They both increase as n^2.

It does not mean that n can ever become infinite. Infinity refers to the open-ended process; it is not a numerical value for anything. n(n+1)/2 does not have a value "at" infinity and neither does the infinite sum have a value. It means that the farther out you go in the sum process the larger the result is without bound.

It does not permit a meaning that the n(n+1)/2 formula is somehow different in value from the sums or their limits, giving a different result for the two limits.

There are no loopholes to make 1+2+3+... = -1/12 and no reason to "trust" otherwise in the name of science. The assertion is flat out false.

That someone may think he "needs" "S=-1/12" as a mathematical rationalization for his physics speculations is not an argument that either must be true.

The value -1/12 "associated" with the series as used in string speculation has an entirely different meaning than an infinite sum = -1/12. It arises from the extension of something called the "zeta function" into the complex plane with the value -1/12 at x=-1. It does not mean that the infinite sum of positive integers is -1/12. It also still doesn't justify the string speculation.

Mathematical axioms are not arbitrary rules. Mathematics is a science of method and its basic principles are formulated for a purpose in accordance with standards.

The infinite sum -1+1-1+1... is not zero. The finite partial sums oscillate and the series does not converge to anything.

The limit S=1/2 cannot be obtained by arbitrary arithmetic, but can under a different meaning of the limit process that encapsulate an averaging process and is useful for some purposes, but that cannot be understood by someone who doesn't know the basic principle of limits and is still trying manipulate symbols without regard to the meaning of infinite sums as limits.

The mathematical demonstration is not correct. You cannot arbitrarily manipulate infinities as if they were numbers.

I didn't change the definition of S. The article did and I explained how it did it, which is what my first paragraph above is all about so you could see where it went wrong. Too many words? Apparently not enough as you missed the point. Sorry about that as I'm not trying to argue here, but impart information.

Yes, S(1) is an infinite sum without a finite solution, which is why it is written as it is in an equation. "(1-1+1-1+1-1+....)" is encapsulated in parenthesis because it is an unresolved expression and you can't remove the parenthesis until it is resolved as was falsely done to create the third expression above. The assumption that move could be made correctly is a false assumption.

You cannot arbitrarily manipulate terms in a divergent series. The meaning of an infinite sum is the limit of the sequence of partial, finite sums, in this case S = lim as n->inf of S_n where S_n is the partial sum of the first n terms.

The limit for your "S(1)" does not exist because the sum is unbounded: lim n->inf of S_n = infinity. Infinity is not a number and cannot be treated arithmetically as a number. You cannot add and subtract infinities and numbers to manipulate infinity as if it were a number.

The false assumptions in the symbolic manipulations you quoted are that S(1) converges to a number, that infinity is a number, and that you can arbitrarily use the rules of arithmetic the way you did with infinities.

There is a whole theory of how to mathematically deal with limits, including infinite sums. If you want to "understand infinity" in that context that is what you must learn. You cannot just arbitrarily manipulate symbols and then ask what went wrong.

In life- false positives, false negatives have their own risks and dangers. As for string theory- a hypothesis, it is not theory, it is not reality. Afaik string theory sometimes explains but does not predict.
2. In the fallacious proof, the component series may, or do not, stop. But they do not combine into the series being studied, to define thus is an error.
Come back when you can demolish the rebuttal in-
https://skullsinthestars.com/2014/01/...

A thing may be defined as something which it is not. But it still is what it is.

All mathematics proceeds from a set of arbitrary rules called axioms. Once these rules are agreed upon, then the mathematical truth of any assertion depends on the strict logical adherence to these axioms. Consider the following quote:

"From the axioms of addition, two other properties can be derived. One is the additive
identity property which says that for any real number a + 0 = a. The other is the additive
inverse property which suggests that for every number a, there is a number −a such that −a + a = 0."

It follows that any string S = -1+1-1+1... no matter how long must equal zero. There is
no stopping axiom. You are free to create one if you so inclined. But in no case can you derive a solution S(1) = 1/2 from the currently accepted axioms of addition.

https://science.jrank.org/pages/505/A...

Problem- If you accept the viewpoint about systems and infinity, then, all decisions are useless.
Decisions are made on a set of data and objectives. Data sets are limited, but the universe, being unlimited, influences outcomes not contained in the data used. Proper decisions must take a wide view, a systems approach. But any system is part of a bigger system which is part of an unlimited universe. Decisions made on less than infinite data cannot therefore be correct.

Solution- Re-examine the premises.

Cost accounting as an evil, as an opponent of Operations Research (!) -
speaking from qualifications and experience in both-
Cost accounting, is now up-marketed to Management Accounting.
It makes use of data, arithmetic operations, and a host of assumptions, with the aim of guiding management decisions. This it does as limited by the sense and skills of the people involved. Operations Research, with less data and more mathematics, same same.

The view of cost accounting not having a sufficiently systems view should not be given to the public service, those guys already are expert at analyzing problems to death, they need no more excuses.

I do not feel any danger or fear to face reality, do you?
"The fallacy here is in the implicit assumption of when the series stops" - exactly, I agree. As long as it never stops, the mathematical demonstration is correct.

Too many words, no explanation at all.
S is defined as an infinite sum, which is exactly what happened there in all the steps.
You, on the other hand, did not use the original definition of S, so it is easy to demonstrate anything if you change the definition.

For a bit of Ramanujan history.
1. https://plus.maths.org/content/disapp...
I am a great fan of Ramanujan. From memory- he was a semi-literate clerk who came across a school book, a summary of geometry theorems. He read, memorized, and developed wide ranging views in mathematics. He sent his treatise to renowned mathematician GE Hardy in Cambridge UK. Hardy, who got much junk mail from cranks, accidentally read some of it and was 'knocked off his feet'.

.. for a good explanation of the s=-1/12 see-
2. https://skullsinthestars.com/2014/01/...
There is a nice example of the type of error in the s=1/12 'proof'.

y j_IR1776wg- thanks for the ref which I followed up on.

Mathematical operations using zero are fraught with danger.
- .. Infinity, same.
- Calculations using an average as a representation of a numerical amount can be completely misleading. (eg- global average temperature).
Often the zero, the infinity and the averaging are implicit and will lead to conundrums or worse.

- The fallacy here is in the implicit assumption of when the series stops.
If on an odd number the answer is inadequate, if an even number likewise, so you take an average assuming it is one or the other. Wrong, an infinite series does not stop half way between an odd and an even number .. It keeps on going (like Ol' Man River) - until like the ooslum-bird going in ever decreasing circles it disappears up its own orifice, or the opposite, at the same time.

The pootaa is the assumption the infinite expression S(1) can be expressed or solved by the finite expression (0+1/2) = 1/2. Supposing, as in the article, you can randomly stop an iteration of S(1) at some point and get 0 and then randomly stop it again to get 1 is a contrived scenario that converts the infinite S(1) to two separate finite expressions where neither are actually S(1). Assuming you can now average the results of the two separate finite expressions to solve S(1) is a pootaa.

Your first expression "S(1) = 1-1+1-1+1-1+...." defines S(1) as an infinite expression. It's like asking "How high is up?". If S(1) were "up", then your second expression "1-(S(1)) = 1 - (1-1+1-1+1-1+....)" could be shown as 1-((up)) = 1-(up). Solve parenthesis first is a rule of math so you can clean up the left side of the expression by removing one set of parenthesis thus: 1-(up) = 1-(up), but your third expression "1-S(1) = 1-1+1-1+1-1+......" makes the false assumption that simply removing the parenthesis on the right side of the equation is valid to make the fourth expression 1-up = up. You can't do that because those parenthesis encapsulate an unsolved expression. The third expression is the pootaa and everything after it is as well.

S(1) = 1-1+1-1+1-1+....
1-(S(1)) = 1 - (1-1+1-1+1-1+....)
1-S(1) = 1-1+1-1+1-1+......
1-S(1))=S(1)
1=2S(1)
S(1)=1/2
Where is the pulled-out-of-the-ass assumption? I just see a logical derivation with no assumptions besides that you need S(1) gong infinitely for this to be true. I just say, this shows me that I don't understand infinity.

Agreed.
More precisely, a theory that worked well under certain circumstances, can be proved wrong in other circumstances, and we have found the boundaries of application of that theory (example of Newton and Einstein). And then, knowledge was expanded. Of course you can destroy a full theory as well (I've seen a couple).