1/
Since 2+2=5 took over twitter, I've watched as "academics" and "educators" used rhetorical tricks to blow smoke at people and make them feel stupid.
It is unfair, unkind, and it ends today.
If you felt stupid, don't feel bad. You can understand, and I'll help.
A thread:
Since 2+2=5 took over twitter, I've watched as "academics" and "educators" used rhetorical tricks to blow smoke at people and make them feel stupid.
It is unfair, unkind, and it ends today.
If you felt stupid, don't feel bad. You can understand, and I'll help.
A thread:
2/
The explanations will be slow so NOBODY gets left behind! Everyone's smart enough to get what's going on, but we have to go very slowly so no one gets confused. The math experts here may find this a little slow, but we'll go slowly so everyone can make sure they understand 🙂
The explanations will be slow so NOBODY gets left behind! Everyone's smart enough to get what's going on, but we have to go very slowly so no one gets confused. The math experts here may find this a little slow, but we'll go slowly so everyone can make sure they understand 🙂
3/
We will look at the examples in the order that gives us the best way to understand things. They might not be in the order of when they were given.
Let's begin.
We will look at the examples in the order that gives us the best way to understand things. They might not be in the order of when they were given.
Let's begin.
4/
Argument 1 comes from Tim Gowers winner of the Fields Medal (basically the nobel prize of math). please read his argument.
This is hard, let's break it down: The part underlined in red say we have to decide what the symbols in math mean, and the rules of the system. So...
Argument 1 comes from Tim Gowers winner of the Fields Medal (basically the nobel prize of math). please read his argument.
This is hard, let's break it down: The part underlined in red say we have to decide what the symbols in math mean, and the rules of the system. So...
5/
This means
1. We have to give definitions for each of our symbols. For example the symbol for addition (2+2) is defined differently than the symbol for subtraction (2-2).
2. We have to have rules for how each symbol is used in that system.
what he is pointing out here is...
This means
1. We have to give definitions for each of our symbols. For example the symbol for addition (2+2) is defined differently than the symbol for subtraction (2-2).
2. We have to have rules for how each symbol is used in that system.
what he is pointing out here is...
6/
Sometimes the symbols in math can mean different things depending on which kind of math we are doing (there are many kinds of math) and sometimes math systems use different rules then the ones we are used to.
He gives us an example underlined in blue:
Sometimes the symbols in math can mean different things depending on which kind of math we are doing (there are many kinds of math) and sometimes math systems use different rules then the ones we are used to.
He gives us an example underlined in blue:
7/
To understand his example we need to know what "base 2" means.
A base 10 system has 10 symbols that represent numbers: 1,2,3,4,5,6,7,8,9,0
That's 10 symbols: 1-9 and the 0 is the tenth symbol
A base 2 system has two symbols: 0,1
A base 2 is often called a "binary" system
To understand his example we need to know what "base 2" means.
A base 10 system has 10 symbols that represent numbers: 1,2,3,4,5,6,7,8,9,0
That's 10 symbols: 1-9 and the 0 is the tenth symbol
A base 2 system has two symbols: 0,1
A base 2 is often called a "binary" system
8/
This is tricky.
In base 10 we count like this:
0
1
2
3
4
5
6
7
8
9
there is 1 per line right? When we go above 9:
10
11
12
13
14
15
16....
see what happens? above 9 it "rolls over" so we have 2 symbols per line. above 99 we hit 100 and have 3 symbols.
That's base 10.
This is tricky.
In base 10 we count like this:
0
1
2
3
4
5
6
7
8
9
there is 1 per line right? When we go above 9:
10
11
12
13
14
15
16....
see what happens? above 9 it "rolls over" so we have 2 symbols per line. above 99 we hit 100 and have 3 symbols.
That's base 10.
9/ That's normal counting.
In base 2 we only use two symbols 1 and 0.
Lets count to 2 in base2 (with normal number so we can keep track :
0=0
1=1
Remember base 2 only uses 1 and 0, so how do we write 2? Well it "rolls over" and adds a place so we get :
0=0
1=1
10=2
In base 2 we only use two symbols 1 and 0.
Lets count to 2 in base2 (with normal number so we can keep track :
0=0
1=1
Remember base 2 only uses 1 and 0, so how do we write 2? Well it "rolls over" and adds a place so we get :
0=0
1=1
10=2
10/
Base 2 CHANGES THE RULES for the symbols we use to count. So, now let's count to ten in "base 2":
0=0
1=1
10=2
11=3
100=4
101=5
110=6
111=7
1000=8
1001=9
1010=10
See that?
Base 2 is a different system of counting governed by different rules. It only uses 1's and 0's
Base 2 CHANGES THE RULES for the symbols we use to count. So, now let's count to ten in "base 2":
0=0
1=1
10=2
11=3
100=4
101=5
110=6
111=7
1000=8
1001=9
1010=10
See that?
Base 2 is a different system of counting governed by different rules. It only uses 1's and 0's
11/
Now Mr. Gowers says in 'base two' that "1+1=10" is true.
Yes, BECAUSE THE RULES OF BASE 2 ARE DIFFERENT. All this shows is answers look different when we change the rules for what symbols to use.....
Now Mr. Gowers says in 'base two' that "1+1=10" is true.
Yes, BECAUSE THE RULES OF BASE 2 ARE DIFFERENT. All this shows is answers look different when we change the rules for what symbols to use.....
12/
So in response to 2+2=4, Gowers points out that in base 2 we would write 1+1=10 instead of 1+1=2.
And in french they say 'le ciel est bleu' instead of 'the sky is blue' but this doesn't change the color of the sky
Different systems, different ways to write the same thing.
So in response to 2+2=4, Gowers points out that in base 2 we would write 1+1=10 instead of 1+1=2.
And in french they say 'le ciel est bleu' instead of 'the sky is blue' but this doesn't change the color of the sky
Different systems, different ways to write the same thing.
13/ Mr. Gowers next claim claim is it is acceptable to use 2+2=5 as an abbreviation. His example, is an event with a 2 break session, followed by a break, followed by a 2 break session. In this case there is 5 total breaks. Now, follow what he does here:
14/ Mr Gowers says you might abbreviate that situation as 2+2=5 (accounting for the break between sessions without mentioning it explicitly in your equation) and this is fine if you admit to redefining the terms.
This is muddying the waters. Let's get clear about it...
This is muddying the waters. Let's get clear about it...
15/
First, note we can write 2+2=4 many ways:
1. 2+2=4 (symbols)
2. Two plus two is four (English)
3. Deux plus deux font quatre (French)
That's three different sentences, but it's the same PROPOSITION. All three of those express the same the underlying proposition...
First, note we can write 2+2=4 many ways:
1. 2+2=4 (symbols)
2. Two plus two is four (English)
3. Deux plus deux font quatre (French)
That's three different sentences, but it's the same PROPOSITION. All three of those express the same the underlying proposition...
16/
Saying that sometimes 2+2=5 because we can redefine the terms, is like saying we can turn peanut butter into jelly by relabeling the jar.
And like relabeled Jars, Mr. Gowers redefinition of terms serves only to confuse and frustrate those who come across them.
Saying that sometimes 2+2=5 because we can redefine the terms, is like saying we can turn peanut butter into jelly by relabeling the jar.
And like relabeled Jars, Mr. Gowers redefinition of terms serves only to confuse and frustrate those who come across them.
17/
Lets move on.
Our next entry is from Kareem Carr his example here uses what is referred to as modular math, or, "clock" math. This also can be confusing, so I will explain clock math, and then what Kareem is up to.
Lets move on.
Our next entry is from Kareem Carr his example here uses what is referred to as modular math, or, "clock" math. This also can be confusing, so I will explain clock math, and then what Kareem is up to.
18/
To simplify, lets use a 12 hour clock. Suppose it's 9:00. You work an 8 hour day. What time will it be when you are done? Well look at picture I included. If we go to 9:00 and count 8 hours we end up at 5:00. and this makes sense. When we work an 8 hours day we work 9-5.
To simplify, lets use a 12 hour clock. Suppose it's 9:00. You work an 8 hour day. What time will it be when you are done? Well look at picture I included. If we go to 9:00 and count 8 hours we end up at 5:00. and this makes sense. When we work an 8 hours day we work 9-5.
19/
So on a clock, 9:00 plus 8 hours gives us 5:00 this does NOT mean that 9+8=5. It is a clock and as you know, once a clock gets to 12:59 it "resets" back to 1:00. That's how clock math works. And our "clock" can be 12 or 24 (a 24 hour clock) or we can define it another way...
So on a clock, 9:00 plus 8 hours gives us 5:00 this does NOT mean that 9+8=5. It is a clock and as you know, once a clock gets to 12:59 it "resets" back to 1:00. That's how clock math works. And our "clock" can be 12 or 24 (a 24 hour clock) or we can define it another way...
20/ in clock math we can set the limit to what ever we want. On a clock it is either 12 or 24, but we can make it whatever we want. We can use clocks or, if we want, race tracks!
a 4 turn track. If I am on turn 3, and I add 5 more turns, where on the track will I be?
a 4 turn track. If I am on turn 3, and I add 5 more turns, where on the track will I be?
21/
Well lets count (counterclockwise on the diagram, follow the turns)
If we are on turn 3, and add 5 turns it would go
1. turn 4
2. turn 1
3. turn 2
4. turn 3
5. turn 4
And we would end up on turn 4!
See, it's that simple. That,s clock math
Well lets count (counterclockwise on the diagram, follow the turns)
If we are on turn 3, and add 5 turns it would go
1. turn 4
2. turn 1
3. turn 2
4. turn 3
5. turn 4
And we would end up on turn 4!
See, it's that simple. That,s clock math
22/
This is what Kareem is on about.
Now, in clock math the way that we do out notation is a slight bit tricky but we can do it!
what we do is treat them like fractions then we divide and find the remainder. that sounds hard, lets walk through it...
This is what Kareem is on about.
Now, in clock math the way that we do out notation is a slight bit tricky but we can do it!
what we do is treat them like fractions then we divide and find the remainder. that sounds hard, lets walk through it...
23/ Our example of starting at 9:00 and adding 8 hours, goes like this: We express each number as a fraction out of 12 (cause the clock has 12 hours) so
1. 9:00 becomes 9/12
2. 8 hours becomes 8/12
3. we add 9/12 + 8/12 and get 17/12
4. we divide 17 by 12 and find the remainder
1. 9:00 becomes 9/12
2. 8 hours becomes 8/12
3. we add 9/12 + 8/12 and get 17/12
4. we divide 17 by 12 and find the remainder
24/
5. 17 divided by 12 is 1 with a remainder of 5
6. The remainder tells us where on the clock we are. In this case 5.
is that correct? well let look on our clock. We start at 9, now count ahead 8 hours....where do we end?
That's right. 5
5. 17 divided by 12 is 1 with a remainder of 5
6. The remainder tells us where on the clock we are. In this case 5.
is that correct? well let look on our clock. We start at 9, now count ahead 8 hours....where do we end?
That's right. 5
25/ That's how clock math works. So, we explained that clock math does not mean 9+8= 5. It just means that if you are at 9:00 and add 8 hours it will be 5:00.
So what does Kareem mean when he says that a clock can prove that 2+2=5?
So what does Kareem mean when he says that a clock can prove that 2+2=5?
26/ there is another trick we can do with clock math. We can count total rotations.
lets look at a 24 hour clock.
Well we can do this lets start at 24:00 and lets add 24 hours. where do we end up? Well, right back where we started. That's 1 rotation.
lets look at a 24 hour clock.
Well we can do this lets start at 24:00 and lets add 24 hours. where do we end up? Well, right back where we started. That's 1 rotation.
27/
Ok, what if we add 2 days?
same thing, we end up right back where we started.
So if add 24 hour days, no matter how many we add, we end up where we started. Why? because on a 24 hour clock, adding 24 hours is one complete circle.
so then we can do this:
Ok, what if we add 2 days?
same thing, we end up right back where we started.
So if add 24 hour days, no matter how many we add, we end up where we started. Why? because on a 24 hour clock, adding 24 hours is one complete circle.
so then we can do this:
28/
If you start at 24:00 and then add 2 days (48 hours) you get to 24:00. add two more days (48 hours), you still get to 24:00.
so that means 2 days + 2 days gets you to 24:00
Start at 24:00 and add (5 days). You get to 24:00
And what does all that mean?
If you start at 24:00 and then add 2 days (48 hours) you get to 24:00. add two more days (48 hours), you still get to 24:00.
so that means 2 days + 2 days gets you to 24:00
Start at 24:00 and add (5 days). You get to 24:00
And what does all that mean?
29/
Since adding 5 days gets you to 24:00 and adding (2 days + 2days) gets you to the same spot. You can argue that in that case 2+2=5.
To get your head around that lets use a race track example:
If we start at the start/finish line and do excatly one lap we end at...
Since adding 5 days gets you to 24:00 and adding (2 days + 2days) gets you to the same spot. You can argue that in that case 2+2=5.
To get your head around that lets use a race track example:
If we start at the start/finish line and do excatly one lap we end at...
30/
the start/finish line.
If we do 2 laps, then add 2 laps, where do we end up? At the start/finish line.
If we do 5 laps where do we end up?
At the start/finish line
So, in terms of which spot on the track you end up, 2 laps + 2 laps gets you to the exact same spot as 5 laps
the start/finish line.
If we do 2 laps, then add 2 laps, where do we end up? At the start/finish line.
If we do 5 laps where do we end up?
At the start/finish line
So, in terms of which spot on the track you end up, 2 laps + 2 laps gets you to the exact same spot as 5 laps
31/
We are almost home!
I know this is long and hard but math is hard.
We can do it!!!
These last two examples are fairly simple:
We are almost home!
I know this is long and hard but math is hard.
We can do it!!!
These last two examples are fairly simple:
32/ The first example is a $500 flight that gets to the same spot as two smaller flights that cost $200 each. In that case the $200 flight plus another $200 flight is the same as a $500 flight. So he says this is an example of 2+2=5.
33/
What he is saying here is you get the same value for $200 + $200 that you get for $500 because the 2 small flights get you to the same spot as the long flight. So what you get for 200+200 is worth what you would get for 500.
but this doesn't mean 200+200=500 or 2+2=5...
What he is saying here is you get the same value for $200 + $200 that you get for $500 because the 2 small flights get you to the same spot as the long flight. So what you get for 200+200 is worth what you would get for 500.
but this doesn't mean 200+200=500 or 2+2=5...
34/ In the same way here is an example:
Milk at Store A costs 4$
Milk at Store B costs 3$
This does not mean 3=4, it means that the price of milk is better at store B
Milk at Store A costs 4$
Milk at Store B costs 3$
This does not mean 3=4, it means that the price of milk is better at store B
35/
The next example so say we are going to round to the nearest whole number. suppose we have 2.4 +2.4 =4.8 well the closest whole number to 2.4 is 2. but the closest whole number to 4.8 is 5. Which could be symbolized as 2+2=5.
But this is a flaw in the system. because we...
The next example so say we are going to round to the nearest whole number. suppose we have 2.4 +2.4 =4.8 well the closest whole number to 2.4 is 2. but the closest whole number to 4.8 is 5. Which could be symbolized as 2+2=5.
But this is a flaw in the system. because we...
36/ know rounding answers are approximate. "approximately 2 + approximately 2 = approximately 5" is not the same as 2+2=5.
Lots of men on Tinder are approximately 6 feet tall and make approximately a million dollars a year. That doesn't add up either.
Lots of men on Tinder are approximately 6 feet tall and make approximately a million dollars a year. That doesn't add up either.
37/
Finally we have this. Kareem says you might have a rooster and a hen, and they might have a baby, and so you might have 1 Chicken + 1 Chicken and end up with 3 chickens.
He also says that if someone say 2+2= 5 that they might mean that they have 4 chickens and on gave birth.
Finally we have this. Kareem says you might have a rooster and a hen, and they might have a baby, and so you might have 1 Chicken + 1 Chicken and end up with 3 chickens.
He also says that if someone say 2+2= 5 that they might mean that they have 4 chickens and on gave birth.
38/
These are my families chickens, and I assure you this is not how chickens work. If you have a chicken, and buy another one, you have two chickens. 1+1=2. If they have 1 offspring. You have 3 chickens. 2+1=3.
And we teach this for a very good reason....
These are my families chickens, and I assure you this is not how chickens work. If you have a chicken, and buy another one, you have two chickens. 1+1=2. If they have 1 offspring. You have 3 chickens. 2+1=3.
And we teach this for a very good reason....
39/
We don't know how many offspring a chicken will have, but we do need to keep track at all time. So what do we do? If we have one chicken, and we buy another chicken we have 2 chickens. 1+1=2. if there is a baby that means we had 2, and added one. 2+1=3. And this matters
We don't know how many offspring a chicken will have, but we do need to keep track at all time. So what do we do? If we have one chicken, and we buy another chicken we have 2 chickens. 1+1=2. if there is a baby that means we had 2, and added one. 2+1=3. And this matters
40/
because if you start acting like you have more chickens or cows or sheep than you do really, you can lose track of how many animals you have quite quickly. So it is important to have and exact count of the animals, especially on family farms where margins are thin....
because if you start acting like you have more chickens or cows or sheep than you do really, you can lose track of how many animals you have quite quickly. So it is important to have and exact count of the animals, especially on family farms where margins are thin....
41/ This is so important we actually have a saying about this:
"Don't count your chickens before they hatch."
So no, it is never true that 1+1=3. That is simply wrong, and it really doesn't map on to the world that well.
"Don't count your chickens before they hatch."
So no, it is never true that 1+1=3. That is simply wrong, and it really doesn't map on to the world that well.
42/
to conclude, all the 2+2=5 nonsense revolved around exploiting quirks of language and math notation to create the impression that 2+2=5 sometimes. It doesn't. If you see 2+2=5 either someone has no clue what they are talking about, or they are using specialized definitions...
to conclude, all the 2+2=5 nonsense revolved around exploiting quirks of language and math notation to create the impression that 2+2=5 sometimes. It doesn't. If you see 2+2=5 either someone has no clue what they are talking about, or they are using specialized definitions...
43/ These people are seeking to make things less, not more clear. There is not excuse for using specialized definitions to create confusing equations that work because to the technicalities of the way an equation like 2+2 has been redefined. That is not fair.
No more.
No more.
44/
Finally, I know there are some cases where it isn't clear how to count things. IE: I have 8 fingers, but I have 10 if you count my thumbs. Those sorts of difficulties don't mean 8=10, it means we have to be careful when deciding how we count things. What we don't do is use
Finally, I know there are some cases where it isn't clear how to count things. IE: I have 8 fingers, but I have 10 if you count my thumbs. Those sorts of difficulties don't mean 8=10, it means we have to be careful when deciding how we count things. What we don't do is use
45/
obscure definitions of math to redefine common equations, call it nuance, and then laugh when normal people get confused and frustrated. That's not playing with math, that's being an intellectual bully. it isn't fair it isn't right. I hope I was able to clear up at least...
obscure definitions of math to redefine common equations, call it nuance, and then laugh when normal people get confused and frustrated. That's not playing with math, that's being an intellectual bully. it isn't fair it isn't right. I hope I was able to clear up at least...
46/ a little bit of this for some people.
Thank you.
/fin
Thank you.
/fin
