Help me understand

To your apple point though, this is an issue many young students struggle with. It is also how many children learn mathematics. Let's talk about it a bit.

So, let us discuss the entirely similar situation: let us put two apples on a table - a sturdy table mind you that would not shake or move the apples or bring in other apples by accident and so on.  Now we then put another two apples on the table, and count those now there - we will likely get the result 4.  We can perform additional experiments like this, but we are fairly confident here that 'adding' works as we think it does. Yes, the results are consistent, and perhaps even corollary, but this is not the same thing as placing two numbers ones on a table and then two more number ones.

However, if we did the same procedure with sticks, fingers, line and most things this would show us that sums are completely silly. Look at this figure to see 2+2 = 4, which is similar to your argument:



If this is all we need, as you claim, to show that 2 + 2 = 4 and in your case 5 + 5 + 5 +5 = 20, then this is completely valid reasoning:


2 + 2 + 2 = 4.

Source: Remarks on the Foundations of Mathematics; Page 52 MIT Press 198

For similar reasons, your supplied example is not sufficient to show that 2 + 2 = 4 in reality, or that 5 * 4 = 20.

Edit: filling in the dots.

If you follow the agreed rules for addition then you get 2 + 2 = 4 from the first image, if you use a different rule you get 2 + 2 + 2 = 4.
This shows is that if you have two different definitions, then they might not define the same operation.
The second method is not what is normally called addition.

There a few points you could be making here, it isn't clear what sums being "completely silly" means.
I guess you either mean

1. The definition of sum is unclear, so it is reasonable to think it means something different to "normal". If there is no clear definition then sum can't be uniquely defined, therefore doesn't exist.

2. Different definitions can be given to "+" (which is a symbol we use in communication). "+" has different definitions, so sum has different definitions, so sum can't be uniquely defined, therefore doesn't exist.

or

3. It is silly to say that sum is reality based, as the other definition of sum is as reality based as the first, so you have no reason to chose the first over the second.

They seem like they are the same rule. I'm saying its a bad way to equate mathematics with a language of reality as its open to attack by points like this. Whether its 2 + 2 = 4 or 1 + Σ (9/(10^n)) = 1.99999... or 1.999...999 > 1.99999...923

Edit: oopsies!

They seem like they are the same rule. I'm saying its a bad way to equate mathematics with a language of reality as its open to attack by points like this. Whether its 2 + 2 = 4 or 1 + Σ (9/(10^n)) = 1.99999... or 1.999...999 > 1.99999...923

Edit: oopsies!

They are similar rules, but I don't see how they are the same, the normal rule only allows you to count each object once, the other does not have this restriction. I guess you could argue, how can you define once.

A programming language on a computer never gets confused between the two definitions. Open up python and type 2+2 and it will always give 4, 2+2+2 will always give 6. So they is a way to describe addition in reality.

"Same rule" must mean something different to normal in this context?

They seem like they are the same rule.

They are fundamentally different rules.
If you are referring to addition, in one case you combine two groups and count each member once in the product.
In the other, you count two of them twice.

To go back to the original debate, I would say that you are right to say 0.999...896 (where the ...  is an infinite number of digits, and will be for the rest of this post) isn't wrong because "it isn't real". If what the notation means is well defined, then they is nothing wrong with it, doesn't need to be in reality.

Real numbers is such an annoying name. They aren't any more real than other numbers. 0.999...896 isn't a member of the real numbers, which are what is normally represented by decimal digits.
It isn't a real because if you look at its sum, 0+0.9+0.99+... +8 * 10 ^ (-infinity?), wait, stop there, what is a power to minus infinity supposed to mean, either it is zero, so the extra digits are redundant (so it would actually be a real number, but the notation would be worthless), or it's an infinitesimal, which are not members of the reals.

You can define a new type of number which 0.999... 896 is a member of, but you can't use these new "longer reals" to prove stuff about the reals unless you first prove a relation between them.

Quote from: John Davis on December 11, 2017, 07:08:34 PM

They seem like they are the same rule. I'm saying its a bad way to equate mathematics with a language of reality as its open to attack by points like this. Whether its 2 + 2 = 4 or 1 + Σ (9/(10^n)) = 1.99999... or 1.999...999 > 1.99999...923

Edit: oopsies!

They are similar rules, but I don't see how they are the same, the normal rule only allows you to count each object once, the other does not have this restriction. I guess you could argue, how can you define once.

A programming language on a computer never gets confused between the two definitions. Open up python and type 2+2 and it will always give 4, 2+2+2 will always give 6. So they is a way to describe addition in reality.

"Same rule" must mean something different to normal in this context?

Unfortunately, this is not the case; you can make it say really whatever you want:
https://codegolf.stackexchange.com/questions/28786/write-a-program-that-makes-2-2-5

This is actually by design in many languages, especially mathematical ones!

Likewise, I could write my own math parser (many perform this task in low level programming courses at uni), and have it say whatever darned thing I choose.

I feel they are the same: group the apples into groups, then append the count of each group. I do see your point that the second set is more specific, if it were to reflect addition as we know it.

Quote from: Empirical on December 12, 2017, 12:23:26 AM

Quote from: John Davis on December 11, 2017, 07:08:34 PM

They seem like they are the same rule. I'm saying its a bad way to equate mathematics with a language of reality as its open to attack by points like this. Whether its 2 + 2 = 4 or 1 + Σ (9/(10^n)) = 1.99999... or 1.999...999 > 1.99999...923

Edit: oopsies!

They are similar rules, but I don't see how they are the same, the normal rule only allows you to count each object once, the other does not have this restriction. I guess you could argue, how can you define once.

A programming language on a computer never gets confused between the two definitions. Open up python and type 2+2 and it will always give 4, 2+2+2 will always give 6. So they is a way to describe addition in reality.

"Same rule" must mean something different to normal in this context?

Unfortunately, this is not the case; you can make it say really whatever you want:
https://codegolf.stackexchange.com/questions/28786/write-a-program-that-makes-2-2-5

This is actually by design in many languages, especially mathematical ones!

That java hack is hilarious  .
Anyway I would say that modifying addition means it is no longer the original addition, so you can get "additions" where 2+2=5, but in that original addition 2+2 will always equal 4.

I feel they are the same: group the apples into groups, then append the count of each group. I do see your point that the second set is more specific, if it were to reflect addition as we know it.

They both share the property that they are "grouping the apples into groups, then append the count of each group."
Normal addition extends on that property to "grouping the apples into non-intersecting groups, then append the count of each group."

Unfortunately, this is not the case; you can make it say really whatever you want:

Yes, by completely changing the meaning of the words.
For example, you can set up a different number system which would count something like:
1,2,3,5,...
in which case 2+2=5.
You can change the "+" function to be something other than addition, or change the meaning of "2" or "=".

Or you can just ignore all that and output the line "2+2=5" like what a lot of people there did.

That doesn't change what we have meant when we say 2+2=5.

And again, THIS IS NOT SPECIAL TO MATH!

You can do the exact same in English.

For example, I can arbitratily decide that your statement:
"Unfortunately, this is not the case;"
to instead mean something like:
"Yes, you are absolutely correct."

It doesn't actually mean you said that.

Or more to the topic of a flat Earth, you can simply switch the meaning of flat and round, so a flat Earth is one which approximates a globe, while a round Earth is one which has no curvature.

That doesn't mean Earth is actually flat, it just means you are pretending flat means something else.

Another victory for FE!

Another victory for FE!

Woo

1,2,3,5,...

Fibonacci 

Anyway, 0.9999=1

Quote from: John Davis on December 12, 2017, 11:04:12 AM

Unfortunately, this is not the case; you can make it say really whatever you want:

Yes, by completely changing the meaning of the words.
For example, you can set up a different number system which would count something like:
1,2,3,5,...
in which case 2+2=5.
You can change the "+" function to be something other than addition, or change the meaning of "2" or "=".

Or you can just ignore all that and output the line "2+2=5" like what a lot of people there did.

That doesn't change what we have meant when we say 2+2=5.

And again, THIS IS NOT SPECIAL TO MATH!

You can do the exact same in English.

For example, I can arbitratily decide that your statement:
"Unfortunately, this is not the case;"
to instead mean something like:
"Yes, you are absolutely correct."

It doesn't actually mean you said that.

Or more to the topic of a flat Earth, you can simply switch the meaning of flat and round, so a flat Earth is one which approximates a globe, while a round Earth is one which has no curvature.

That doesn't mean Earth is actually flat, it just means you are pretending flat means something else.

So you can't supply an examine where math works in the real world?

Can you at least show me a real live point or line?

Quote from: th3rm0m3t3r0 on December 11, 2017, 11:09:12 AM

Quote from: John Davis on December 10, 2017, 09:52:14 AM

Quote from: th3rm0m3t3r0 on December 08, 2017, 06:39:37 PM

Quote from: John Davis on December 08, 2017, 05:48:33 PM

I am not playing games. I am not describing a finite and undefined amount of digits by showing the series of 9s ends. I am describe an infinite amount of digits that ends. For example, I could use this notation to say something like 1.999...923.

You are playing games. That's an incredibly abstract way to use that and it doesn't really make much sense. It's like saying, "keep going this straight for an infinite amount of time, then take a right."

This doesn't work in reality.

Why would you have two notations if they do not differ in this way?

Show me any math that works in reality, and I'll give you 100$. I forewarn you that I am a nominalist, so it would not be an easy belief to shake, as it is supported quite well.

Math attempts to describe reality and works quite well in doing so.

Again, what you're saying there is akin to saying, "swim across this infinite body of water, my house is right on the other side."

Which you have yet to show is invalid.

Would you be willing to go to my house if those were the directions to get there?

So you can't supply an examine where math works in the real world?

I did. You were unable to refute it.
All you did was try to lie about addition to try and pretend that 2+2, somehow magically allows you to count 2 of the entities twice to end up with 6.

Do you need a more pictographic representation to understand it?

Quote from: John Davis on December 11, 2017, 11:11:27 AM

Quote from: th3rm0m3t3r0 on December 11, 2017, 11:09:12 AM

Quote from: John Davis on December 10, 2017, 09:52:14 AM

Quote from: th3rm0m3t3r0 on December 08, 2017, 06:39:37 PM

Quote from: John Davis on December 08, 2017, 05:48:33 PM

I am not playing games. I am not describing a finite and undefined amount of digits by showing the series of 9s ends. I am describe an infinite amount of digits that ends. For example, I could use this notation to say something like 1.999...923.

You are playing games. That's an incredibly abstract way to use that and it doesn't really make much sense. It's like saying, "keep going this straight for an infinite amount of time, then take a right."

This doesn't work in reality.

Why would you have two notations if they do not differ in this way?

Show me any math that works in reality, and I'll give you 100$. I forewarn you that I am a nominalist, so it would not be an easy belief to shake, as it is supported quite well.

Math attempts to describe reality and works quite well in doing so.

Again, what you're saying there is akin to saying, "swim across this infinite body of water, my house is right on the other side."

Which you have yet to show is invalid.

Would you be willing to go to my house if those were the directions to get there?

Reality does not relate to math.

Quote from: John Davis on December 13, 2017, 11:15:10 AM

So you can't supply an examine where math works in the real world?

I did. You were unable to refute it.
All you did was try to lie about addition to try and pretend that 2+2, somehow magically allows you to count 2 of the entities twice to end up with 6.

Do you need a more pictographic representation to understand it?

Yeah you are right.  My cited example by an expert that almost directly mirrored your example was "pretending that 2+2 magically allows you to count ... to end up with 6." Sounds legit to me.

Gosh those book covers must be heavy!

Yeah you are right.  My cited example by an expert that almost directly mirrored your example was "pretending that 2+2 magically allows you to count ... to end up with 6." Sounds legit to me.

So you just have an appeal to authority which you haven't substantiated?

I have explained the fundamental flaw in that example, and you have been completely unable to defend it.
It is only by changing what addition means that you arrived at 2+2=6, or alternatively, that 2+2+2=4. This was due to you counting 2 of the "X"s twice.

To go back to the original debate, I would say that you are right to say 0.999...896 (where the ...  is an infinite number of digits, and will be for the rest of this post) isn't wrong because "it isn't real". If what the notation means is well defined, then they is nothing wrong with it, doesn't need to be in reality.

Real numbers is such an annoying name. They aren't any more real than other numbers. 0.999...896 isn't a member of the real numbers, which are what is normally represented by decimal digits.
It isn't a real because if you look at its sum, 0+0.9+0.99+... +8 * 10 ^ (-infinity?), wait, stop there, what is a power to minus infinity supposed to mean, either it is zero, so the extra digits are redundant (so it would actually be a real number, but the notation would be worthless), or it's an infinitesimal, which are not members of the reals.

You can define a new type of number which 0.999... 896 is a member of, but you can't use these new "longer reals" to prove stuff about the reals unless you first prove a relation between them.

This got a bit forgotten, and it's quite relevant to the debate.

Now substitute in 1 = 0.999...9999. This leaves you with -1/12 - ( 0.0000...01)

Trying to interpret 0.0000...01 as a real isn't clear, I assume it means 10^(-infinite), but that's either going to be 0, so there is no problem, or an infinitesimal, but if your subtraction gives you an infinitesimal then you didn't have real numbers in the first place.

Quote from: John Davis on December 13, 2017, 02:26:54 PM

Yeah you are right.  My cited example by an expert that almost directly mirrored your example was "pretending that 2+2 magically allows you to count ... to end up with 6." Sounds legit to me.

So you just have an appeal to authority which you haven't substantiated?

I have explained the fundamental flaw in that example, and you have been completely unable to defend it.
It is only by changing what addition means that you arrived at 2+2=6, or alternatively, that 2+2+2=4. This was due to you counting 2 of the "X"s twice.

Nothing about what you described, or reality, prohibits this action. You didn't defend - you moved the goalpost.

Quote from: Empirical on December 12, 2017, 01:06:02 AM

To go back to the original debate, I would say that you are right to say 0.999...896 (where the ...  is an infinite number of digits, and will be for the rest of this post) isn't wrong because "it isn't real". If what the notation means is well defined, then they is nothing wrong with it, doesn't need to be in reality.

Real numbers is such an annoying name. They aren't any more real than other numbers. 0.999...896 isn't a member of the real numbers, which are what is normally represented by decimal digits.
It isn't a real because if you look at its sum, 0+0.9+0.99+... +8 * 10 ^ (-infinity?), wait, stop there, what is a power to minus infinity supposed to mean, either it is zero, so the extra digits are redundant (so it would actually be a real number, but the notation would be worthless), or it's an infinitesimal, which are not members of the reals.

You can define a new type of number which 0.999... 896 is a member of, but you can't use these new "longer reals" to prove stuff about the reals unless you first prove a relation between them.

This got a bit forgotten, and it's quite relevant to the debate.

Quote from: John Davis on December 07, 2017, 07:18:42 PM

Now substitute in 1 = 0.999...9999. This leaves you with -1/12 - ( 0.0000...01)

Trying to interpret 0.0000...01 as a real isn't clear, I assume it means 10^(-infinite), but that's either going to be 0, so there is no problem, or an infinitesimal, but if your subtraction gives you an infinitesimal then you didn't have real numbers in the first place.

Either way its a negative number. So the sum of all positive integers is negative?! Round earth mathematics folks! That one is straight from a string theory textbook.

Quote from: JackBlack on December 13, 2017, 02:41:20 PM

Quote from: John Davis on December 13, 2017, 02:26:54 PM

Yeah you are right.  My cited example by an expert that almost directly mirrored your example was "pretending that 2+2 magically allows you to count ... to end up with 6." Sounds legit to me.

So you just have an appeal to authority which you haven't substantiated?

I have explained the fundamental flaw in that example, and you have been completely unable to defend it.
It is only by changing what addition means that you arrived at 2+2=6, or alternatively, that 2+2+2=4. This was due to you counting 2 of the "X"s twice.

Nothing about what you described, or reality, prohibits this action. You didn't defend - you moved the goalpost.

I did show that they are different rules though,

Quote from: John Davis on December 12, 2017, 11:09:02 AM

I feel they are the same: group the apples into groups, then append the count of each group. I do see your point that the second set is more specific, if it were to reflect addition as we know it.

They both share the property that they are "grouping the apples into groups, then append the count of each group."
Normal addition extends on that property to "grouping the apples into non-intersecting groups, then append the count of each group."

So if we are talking about what most people mean by addition, that is "grouping the apples into non-intersecting groups, then append the count of each group.", then your grouping doesn't show that 2+2+2=6, as the groups intersect.

Quote from: Empirical on December 13, 2017, 02:42:48 PM

Quote from: Empirical on December 12, 2017, 01:06:02 AM

To go back to the original debate, I would say that you are right to say 0.999...896 (where the ...  is an infinite number of digits, and will be for the rest of this post) isn't wrong because "it isn't real". If what the notation means is well defined, then they is nothing wrong with it, doesn't need to be in reality.

Real numbers is such an annoying name. They aren't any more real than other numbers. 0.999...896 isn't a member of the real numbers, which are what is normally represented by decimal digits.
It isn't a real because if you look at its sum, 0+0.9+0.99+... +8 * 10 ^ (-infinity?), wait, stop there, what is a power to minus infinity supposed to mean, either it is zero, so the extra digits are redundant (so it would actually be a real number, but the notation would be worthless), or it's an infinitesimal, which are not members of the reals.

You can define a new type of number which 0.999... 896 is a member of, but you can't use these new "longer reals" to prove stuff about the reals unless you first prove a relation between them.

This got a bit forgotten, and it's quite relevant to the debate.

Quote from: John Davis on December 07, 2017, 07:18:42 PM

Now substitute in 1 = 0.999...9999. This leaves you with -1/12 - ( 0.0000...01)

Trying to interpret 0.0000...01 as a real isn't clear, I assume it means 10^(-infinite), but that's either going to be 0, so there is no problem, or an infinitesimal, but if your subtraction gives you an infinitesimal then you didn't have real numbers in the first place.

Either way its a negative number. So the sum of all positive integers is negative?! Round earth mathematics folks! That one is straight from a string theory textbook.

The debate was about 0.999..., so if the value of 0.999... doesn't effect the result either way, why was this brought up and then argued for the last 5 pages?
I'm unsure what the debate is actually about? You do know that -1/12 isn't the sum of 1+2+3+... right? It is the Ramanujan sum of 1+2+3+..., but not the sum. I can prove that it isn't the sum if you doubt me. Assume for contradiction that the limit of the partial sums is -1/12. x_n is the partial sum of the first n positive numbers.
Let epsilon=1/24, then there exists an N such that for all n>N, abs(x_n+1/12)<1/24, so x_n<-1/24<0, but none of the partial sums are negative, contradiction. So the limit of the partial sums is not -1/12.

And yes, mathematicians sometimes say sum instead of  Ramanujan sum, but that should only be done where it is clear from context what they really mean. Unfortunately Numberphile did not make it clear

So its okay to break the rules when it suits you, or convention?

So, show me that 0.999... exists in reality, and I'll concede 1.999...9923 is invalid due to it not being valid for giving directions across a pond.

So its okay to break the rules when it suits you, or convention?

So, show me that 0.999... exists in reality, and I'll concede 1.999...9923 is invalid due to it not being valid for giving directions across a pond.

The topic is, "Help me understand" and it seems to me that you are doing your best to confuse everybody for no other purpose than to show what smart fellow John Davis is.

OK, I'll grant that John Davis is a smart fellow, and a self made man who worships none but his maker, so what?

Every number you could ever string together can be found in pi somewhere.

Every number you could ever string together can be found in pi somewhere.

Including every set of Lotto - $275M MegaMillions Bets winning numbers. What are we waiting for?
Hold it, I feel a new scam coming on.
Any takers?

Nothing about what you described, or reality, prohibits this action. You didn't defend - you moved the goalpost.

Redefining addition doesn't magically make 2+2+2=4.
You are now simply doing something else.

It is quite simple, you have a table, you place 2 apples on it (the first group), and then you place 2 more apples on it (the second group).
How many apples in total? 4.
This is a simple fact of addition, 2+2=4.
You magically counting two of them twice to pretend it is 2+2+2 doesn't actually make it 2+2+2. It is still just 2+2.
If you want it to be 2+2+2, then put another 2 apples up, then you end up with 6.

Every number you could ever string together can be found in pi somewhere.

Do you have any proof of that?