Math controversy should not exist

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Many things are "debatable." Certainly when it comes to art many things are up to interpretation. Even in science, there may be reasonable controversy. A certain set of available data may seem to suggest multiple different underlying facts. Math, however, is unique. By those who know it well, it is sometimes referred to as a sort of logical game. It has certain rules, and when applied in the right sequence, those rules can lead to much more complex things. Nothing is left to interpretation. Everything is known for certain. Even so, controversy manages to sneak in, especially among those who are less experienced. When people run into certain counterintuitive concepts, they often assume that mathematicians must surely be mistaken. In this post, I will cover a few of these areas of controversy, and then discuss how they come about, and what is actually true. I will try to keep this at a level that everyone should be able to understand, but for those who want the more technical details, I'll be covering them in a second post.

0.99999... = 1:

This is undoubtedly the most notorious area of controversy in math. There are numerous debates on this on DART alone. Many people see this counterintuitive fact for the first time and assume it must be wrong. In reality, due to a flaw in our decimal system, not every number has a unique representation. Nothing magical is happening, these are just two different ways to represent the same number, not unlike 1 + 1 and 2. One could reasonably ask what defines a decimal representation of a number, and I will cover that in my second post. For the less mathematically inclined, if you are willing to accept that 0.99999... is a well-defined quantity, and that 10(0.99999...) = 9.99999..., the following argument should be enough to convince you that 0.99999... = 1 is a logical necessity:

x = 0.99999...
10x = 9.99999... = 9 + 0.99999... = 9 + x
9x = 9
x = 1

1 + 2 + 3 + 4 + ... = -1/12:

This sum clearly diverges to infinity. What is really meant by this statement is that if we extend the notion of summation to divergent series like this one, we will get -1/12. (I'll discuss how this "extended notion" works in my second post.) Unfortunately, many people take the clickbait thumbnails a little two literally, and start arguing against this, even though there is nothing to argue about.

Infinity:

A significant number of people are now saying that infinity must be purged from mathematics, because it is not truly a valid concept. Here are the main three points of contention that I have seen:

1. Infinity does not exist in the real world.

Infinity is a tool for math which is useful for the real world. No one is claiming that infinity "exists" in a physical sense. This would be like pointing out that in the real world, there are no perfect geometric figures, so we should not use geometry.

2. Infinity is not a well-defined concept.

I'm not really sure where people get the idea that infinity is not well-defined. Perhaps it is because there are multiple different definitions for depending on the context, but in every context that it is used, it is very well-defined.

3. Infinity leads to paradoxes.

Every paradox out there, even seemingly "sharp" paradoxes like the liar's paradox, are simply things which humans find it difficult to understand. Just because something is difficult for humans to understand does not make it invalid.

The Sleeping Beauty Paradox:

This simple paradox is widely debating even among (in fact, especially among) professionals. The paradox goes as follows: Sleeping Beauty is put under anesthesia on Sunday. A coin is then flipped. Regardless of its outcome, she is awoken on Monday and asked to guess the outcome of the coin toss. Then, if the coin landed tails, she is put back to sleep and looses all memory of being awoken on Monday. She is then awoken on Tuesday and asked to guess the outcome of the coin toss. You're Sleeping Beauty and you have just woken up. What is the probability that the coin landed heads?

The "halfer" position:

Clearly the answer is 1/2. A coin flip is always 50-50.

The "thirder" position:

Monday and heads, Monday and tails, and Tuesday and tails are all equally likely. If Sleeping Beauty is told it is Monday, the coin could have landed heads or tails with equal probability, and if she is told that the coin landed tails, it could be Monday or Tuesday with equal probability. Because of this, the answer is 1/3. This comes down to the fact that she is woken up twice when the coin lands tails, and only once when it lands heads. Given information can affect a probability, and in this case we are given that Sleeping Beauty has just woken up.

In reality, the controversy here is not because of the math, but because of ambiguous wording. What is meant by "You're Sleeping Beauty and you have just woken up?" Is it irrelevant, (as the "halfer" sees it) or is it intended to convey the given information "Sleeping Beauty just woke up?" (as the "thirder" sees it)
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Well, infinity is logically impossible.

In math, there are many paradoxes.

For example, me moving closer to an object with each step being twice shorter than the former. I will move closer to that object forever, but I will never reach it.
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@Math_Enthusiast
The Thirder position is correct, rather counterintuitively. Suppose the coin is flipped 100 times and SB is woken up each time accordingly. The coin will be tails 2/3 of the times she is woken up (on average). Now suppose she is offered $1 for every coin flip she answers correctly (once or both times, depending on the result). A strategy of always guessing "heads" and a strategy of always guessing "tails" both yield $50 on average. Now why is this the case?

A) The expected revenue from guessing heads is (1/3) * $1 (or 33 cents, approx)
B) The expected revenue from guessing tails once assuming the latter strategy is (2/3) * $0.50 (or 33 cents, approx). The lesser monetary value is because she gets $1 after 2 correct guesses, thus yielding an average of $0.50 for each correct guess.

Even with one flip, SB adopts the strategy before she is put to sleep, when the odds of either result are even. When she wakes up, the odds are 2/3 in favor of tails, but the expected revenue that SB will make in that instance of being woken up is the same regardless of the strategy chosen. This follows because SB will be woken up 150 times and earn $50 total, yielding an average of 33 cents approx. per instance regardless of the strategy chosen.

---

Suppose you are placed on a game show and told "We've randomly selected a value 'heads' or 'tails'. Through some obscure selection method, there is a 66% chance that we've selected the latter. If you guess heads correctly, we will give you $100. If you guess tails correctly, we will give you $50." Both selections have the same expected revenue.

You might think that SB is earning the same amount by getting the guess right in the original scenario, but she's not. If the value is tails, she's only getting half the money (on average) right now—she's not ensuring that she guesses tails later by doing so right now, she's already ensured that by choosing the strategy before she was put to sleep.

---

Now suppose that earlier decisions do influence later decisions. SB is told that her second guess won't count, even if she doesn't know it at the time. If her vote counts, she gains the same amount of revenue from heads and tails, and tails is more likely. But guessing heads correctly guarantees a payout, while guessing tails correctly could be inconsequential if this is her second guess. Hence, expected revenue is still equal:

Heads => $1.00 * (1/3) = 33 cents (approx)
Tails => $1.00 (2/3) * (1/2) = 33 cents (approx)
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Two maybe three kinds of infinite exist

1} macro-micro infinite truly non-occupied space,

2} Meta-space concept of infinite this or that....
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I should add that in the game show scenario, if the coin lands on tails (regardless of your guess), you will be brought on the show a second time. But this does not impact your expected revenue right now. In the original SB scenario, earlier guesses do not influence later guesses. Of course, assuming SB has adopted the strategy, she will guess the same thing both times, but this is correlation, not causation. (The strategy has already been adopted.)
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@Savant
While the coin will land heads in half of all trials of the experiment, it will have come up heads in only one third of the instances of Sleeping Beauty waking up. This is the discrepancy in how the two sides interpret the question. I must agree that considering "Sleeping Beauty has just woken up." to be given information is the more reasonable interpretation of the original question, but the "halfer" position cannot be discredited entirely, since the original wording is somewhat ambiguous. After all, maybe it is just a matter of whether or not sleeping beauty can get it right, rather than how many times she gets it right.
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I'm currently a bit low on time, but I will be sure to post the promised more technical details later today.
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@Best.Korea
Zeno's paradox is easily resolved: There may be infinitely many steps in the process, but each one takes a decreasing amount of time. If you must travel 1 meter and you are going 1 m/s, the first 1/2 meter will take you 1/2 second, the 1/4 meter after that will take you 1/4 second, etc. The result is that it takes you 1/2 + 1/4 + 1/8 + ... = 1 second to complete all infinitely many steps.
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@Math_Enthusiast
I believe most Halfers argue that SB should presume the coin to have an equal probability of heads or tails when she wakes up. That's why the issue is considered by many to be unresolved.


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Zeno's paradox
I wasnt talking about Zeno's paradox.

I was talking about each step being twice shorter than the former.

Duration of a step is irrelevant, as long as each step is twice shorter than the former. You will literally never reach the destination.

Example:

Destination is 4 meters away.

First step is 1 meter.

Second is 0.5 meters.

Third is 0.25 meters.

Fourth is 0.125 meters.

Impossible to reach destination. You literally never reach the destination despite moving closer to it with each step.

There is no step at which you reach the destination.
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I wasnt talking about Zeno's paradox.

I was talking about each step being twice shorter than the former.
That's Zeno's paradox of the arrow. Here's a link if you doubt me: https://plato.stanford.edu/entries/paradox-zeno/#Arr

Perhaps you thought I meant Zeno's paradox of Achilles and the tortoise?

Duration of a step is irrelevant, as long as each step is twice shorter than the former. You will literally never reach the destination.
"Never" is a word relating to time. "Never" would suggest that it would require an infinite amount of time, but as I have already demonstrated at does not.

Example:

Destination is 4 meters away.

First step is 1 meter.

Second is 0.5 meters.

Third is 0.25 meters.

Fourth is 0.125 meters.
This is exactly what I thought you meant. This is Zeno's paradox of the arrow.

There is no step at which you reach the destination.
There is no individual step at which you reach the destination, but there is also no last step. Once all infinitely many steps have been completed, which as I have already demonstrated is possible in a finite amount of time, you will reach your destination.

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@Best.Korea
I forgot to add you as a mention on the above post, but it was in response to you.
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There is no individual step at which you reach the destination
Case closed.

Once all infinitely many steps have been completed, which as I have already demonstrated is possible 
They cannot be completed. It literally takes infinite amount of individual steps that you cannot possibly make. You may make one big step, but thats not what we are talking about here.

I am not saying that movement is impossible. I am saying that in the math, there are paradoxes that cannot be explained.
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As promised, I will now cover some of the more technical details of the topics I listed in the original post.

0.99999... = 1:

For those familiar with limits, infinite decimals like this are defined using an infinity sum, which is in turn defined by a limit. For example, saying that π = 3.141592... is the same as saying that π = 3 + 0.1 + 0.04 + 0.001 + 0.0005 + 0.00009 + 0.000002 + ..., where this infinity sum is defined as the limit of the sequence of partial sums 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, .... Similarly 0.99999... by definition represents the sum 0.9 + 0.09 + 0.009 + ..., or equivalently, the limit of the sequence 0.9, 0.99, 0.999, .... For any ε > 0 we can choose a number in this sequence which is less than ε away from 1, so by the definition of a limit, and in turn by the definition of 0.99999..., 0.99999... = 1. Many complain that this definition should not be accepted, as under any reasonable definition, 0.99999... is infinitesimally different from 1. This, however, would violate the Archimedean property, which would in turn force us to give up the defining property of the real numbers: That they are completely ordered. Here is a useful resource: https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind.pdf

1 + 2 + 3 + 4 + ... = -1/12:

The usual algebraic "proof" of this assumes that the sum is a real number, and then demonstrates that it must be -1/12. (Here is the video which includes this "proof" and made this sum popular if you are curious: https://www.youtube.com/watch?v=w-I6XTVZXww) While this is of course invalid since the sum diverges, it does demonstrate that if this sum were to have a real value, it would have to be -1/12. This becomes meaningful in the context of summation methods, where due to the algebraic properties they must satisfy, this particular sum can only ever be assigned the value -1/12. One such summation method is Ramanujan summation. (Technically this is not a "summation method" in the usual sense, because it depends not only on a discrete sequence, but on a function defined on all complex numbers, but it is still the same concept.) Ramanujan summation also applies to series such as 1 + 4 + 9 + 16 + ... and 1 + 8 + 27 + 64 + .... (Which are assigned 0 and 1/120 respectively.)

Infinity:

I mentioned in my first post that infinity is well defined, but its definition may vary by context. Here is the definition in set theory. In other areas things are a bit more complicated. Calculus never actually uses infinity directly. Rather, it is a stand-in for that something increases without bound. For example "as x -> ∞" really means "as x increases past any given finite number." Algebraic structures such as the Riemann sphere or the surreal numbers often include their own distinct notion of an "infinite element."

The Sleeping Beauty Paradox:

Everything was covered in my first post.
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@Best.Korea
You seem to think that because infinitely many steps are required, those infinitely many steps cannot be completed. It may defy human intuition, but there is no real reason that this should be impossible. Every time you move from one location to another, you complete these infinitely many steps in a finite amount of time. I have already shown an exact calculation which demonstrates this:

If you must travel 1 meter and you are going 1 m/s, the first 1/2 meter will take you 1/2 second, the 1/4 meter after that will take you 1/4 second, etc. The result is that it takes you 1/2 + 1/4 + 1/8 + ... = 1 second to complete all infinitely many steps.

If you have an argument as to why this is invalid, or why infinitely many steps cannot be completed in a finite amount of time, please share it. I have fulfilled my BoP by demonstrating this is possible, but you have failed to give any reason why it shouldn't be. Instead, you continue to say things like "It literally takes infinite amount of individual steps that you cannot possibly make." without backing up your claim as to why "you cannot possibly perform" these infinitely many steps. 
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Because whenever something is fundamentally glitchy in mathematics, we just redefine the system itself. Check Russell's paradox.
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We also had disputes on what "infinity" is, resulting eventually in the ideas such as "aleph null" and "aleph one".
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@Intelligence_06
Because whenever something is fundamentally glitchy in mathematics, we just redefine the system itself. Check Russell's paradox.
Yep, pretty much. It isn't even known if our current set of axioms (ZFC) is consistent or not! Maybe someone will discover a contradiction and we'll have to start from scratch again. (For anyone who is going to call me out on this, no, I'm not contradicting myself. Regardless of consistency, everything we have proven thus far within ZFC is known to be a theorem in ZFC.)

We also had disputes on what "infinity" is, resulting eventually in the ideas such as "aleph null" and "aleph one".
I don't think this is entirely correct.  There were certainly disputes, and the discovery of the existence of multiple different "sizes of infinity" was part of it, but I wouldn't call it a result of it.
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@Savant
Thank you for informing me. I had assumed otherwise given that this can be disproved empirically with a simple simulation, even if one doubts the mathematical logic. It would appear that this is just another instance of people rejecting a concept because they find it counterintuitive without any consideration that they might be wrong.
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Yeah, disputes on maths are pretty much on definitions. If "infinity" is confusing and contradictory, just revise the definition or have several sub-concepts referring to different cases of infinity.
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Does 0.9 = 1?

No because the difference is significant.

Does 0.99 = 1?

No because the difference is still significant.

So how many 9's are required to make the difference insignificant?
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@zedvictor4
Does 0.9 = 1?

No because the difference is significant.

Does 0.99 = 1?

No because the difference is still significant.

So how many 9's are required to make the difference insignificant?
You seem to be holding the following common misconception: That 0.99999... = 1 because the difference is so small that it is insignificant. 0.99999... is exactly equal to 1. They are the same number. No rounding is happening, nor is anything being approximated. We generally would intuit a decimal like 0.99999... to be a number ever so slightly less than one, but such a difference would have to be infinitesimal, and infinitesimals do not exist in the real numbers thanks to the Archimedean property.* I suppose in a sense, 0.99999... = 1 because the difference is so small that it can't even exist!

This proof isn't exactly my favorite, since it doesn't provide any intuition on what 0.99999... and other such decimals are defined to be, but it does demonstrate something else valuable that other proofs cannot: Any reasonable definition of 0.99999... (where 0.99999... is at least 0.999...9 for any finite string of nines, and is no more than 1) must have 0.99999... = 1. This is important because it shows that this is not a quirk in the way we define infinite decimals like 0.99999..., but a quirk in the real numbers themselves.

*If you aren't sure why this implies that infinitesimals can't exist, if a positive infinitesimal is defined as a number greater than zero but less than 1/n for every natural number n, (which 1 - 0.99999... would be if it were positive) than if x is a positive infinitesimal, x < 1/n for every natural number n, and so 1/x > n for every natural number n, a violation of the Archimedean property.

A bonus more technical note if you're interested:

In other number systems like the surreal numbers, there are infinitesimals. It is not true, however, that 1 - 0.99999... is an infinitesimal in this context. Rather, all decimals retain their original values when extending like this, we just also add new numbers which cannot be represented as decimals. How we denote numbers is of course a matter of convention, but as the algebraic argument (see my original post) demonstrates, if we want decimals to satisfy certain nice algebraic properties, we must have 0.99999... = 1. For that matter, number systems like the surreals aren't used as often because by getting rid of the Archimedean property, we also have to give up completeness, which is really important for calculus to work the way it does.
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So how many 9's are required to make the difference insignificant?
10 million.
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@Intelligence_06
There exists only two primary kinds of infinite:

1} Meta-space mind/intellect/concepts of infinite this or that, --ex an infinite set of numbers---,

2} the eternally existent, macro-infinite non-occupied space, that, embraces/surrounds our eternally existent, finite, occupied space, of which, would allow for expansion or contraction of finite, occupied space Universe.

Some here repeatedly make the error of associating infinite with time.

.."Eternity is to time, as,

...Infinite is to space."...Bucky Fuller

[email protected].................   wherein the @ is representative of  finite, occupied space Universe. Simple, even for Forest Gump

Math can do many thinks, some of which are irrelevant to Universe.

Math also has limits. Ex there is no math that can create a 6th, regular/symmetrical and convex polyhedron. 5 is the limit.  There are limits to how many identical right-triangles can exist on surface of a sphere.


901.03..." The utmost number of geometrically similar subdivisions is 120 triangles, because further spherical-
triangular subdivisions are no longer similar. The largest number of similar triangles in a sphere that spheric unity will accommodate is 120: 60 positive and


453.01..."we become satisfied that theicosahedron's set of 480 is indeed the cosmic maximum-limitcase of system-self-spunsubdivisioning of its self into tetrahedra, which 480consist of four sets of 120 similartetrahedra each."...http://www.rwgrayprojects.com/synergetics/s09/p86430.html#986.470
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How we come to learn about something that matters. How I learned about the mathematics decimal issue is over studying to be a machinist while in high school. A training process I never completed for personal issues distracted me. Is (0.99999 = 1) no it is not the reason why though is not due to size but rather expansion and contraction in proportion to hot and cold temperatures. As a machinist this was instructed to me to be the important issue as we are to understand how parts interact with friction to create mechanical failure. In the world of mechanical machining by hand something that is turned down or cut to 1.0 today may a few months from not be measurable less than 1.0 in a change of season. It may also change in a much faster sense if the part is operating under workload conditions and is exposed to heat as friction.
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@FLRW
Are you sure that's enough?

And is that the point at which matter ceases to exist?
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@Math_Enthusiast
0.9 = nine tenths.

Therefore ten tenths = 1.


0.99 = 99  hundredths

Therefore a hundred hundredths = 1


0.999 = 999 thousandths.

Therefore a thousand thousandths = 1


And so on.


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@zedvictor4
Are you sure that's enough?

And is that the point at which matter ceases to exist?
He was making a joke.

0.9 = nine tenths.

Therefore ten tenths = 1.


0.99 = 99  hundredths

Therefore a hundred hundredths = 1


0.999 = 999 thousandths.

Therefore a thousand thousandths = 1


And so on.
What's your point?
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0.9999....Never equals 1.


And joke or not. .....It's actually a reasonable consideration, relative to reductionism.
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0.9999....Never equals 1.
You have only demonstrated this for finitely many nines. "0.99999..." is intended to denote infinitely many nines after the zero. I demonstrated rigorously 0.99999... = 1 in posts #14 and #22 and demonstrated it nonrigorously in post #1. I await a counterargument which applies to the case of infinitely many nines.