“Wrong answers” only according to our current order of operations, math still works if you, for example, make additions come first (as long as you’re consistent about it).
OFC it is a convention and to change it you would have to change all expressions ever written all at the same time, to avoid confusion between competing standards. I’m not arguing that it should be changed, only that there is no ‘high truth’ behind it.
“Wrong answers” only according to our current order of operations
No, according to arithmetic.
math still works if you, for example, make additions come first
No, it doesn’t - order of operations proof. The only way it could work with addition first is if we swapped the definitions of addition and multiplication around… but then we still have the same order of operations, all we’ve done is swapped around what we call addition and multiplication!
That it’s wrong. If I have 1 2 litre bottle of milk, and 4 3 litre bottles of milk - i.e. 2+3x4 - how many litres of milk do I have? Without even doing the arithmetic, just count it up and tell me how many litres there is.
If we change how equations are parsed so addition comes before multiplication, 2+3x4 is not the equation required to solve that problem. 2+(3x4) is the equation needed. You can’t change how equations work and then expect all equations to work the same after the change.
If your argument is that this will add parentheses where we didn’t need them before, that’s valid and its the reason we do it this way in the first place. But that doesn’t mean there is anything fundamentally wrong with having a different system of writing equations in which operations are executed in a different order.
Our whole system of writing equations is just a convention, and yes, it is a good and easy to understand and use way of writing math. But there is no fundamental truth behind it, only that it is simpler for the majority of use cases.
Noted that you didn’t answer my question - the answer is I have 14 litres of milk. 2+3+3+3+3=14 litres. When you did “arbitrary addition first”, you got 20, which is wrong, which is why no other order of operations rules work than the ones we have.
You can’t change how equations work and then expect all equations to work the same after the change
In actual fact the point is that they will except for what ever your new notation is. e.g. if we instead defined + to mean multiply, and x to mean add, then we would do + before x, and again, that would be the only order of operations which works. i.e. the only order which gives us 14 litres.
that doesn’t mean there is anything fundamentally wrong with having a different system of writing equations in which operations are executed in a different order
No, and if you did that, you would again arrive at only one order of operations rules which works, cos I still have 14 litres, and the Maths in this new system still has to give an answer of 14 litres, not 20.
Our whole system of writing equations is just a convention
Nope, it’s all rules, found in any Maths textbook, and if you don’t obey the rules you get wrong answers (like you did when you got 20).
But there is no fundamental truth behind it
Yes there is - I have 14 litres, and only 1 set of order of operations rules gives that answer.
only that it is simpler for the majority of use cases
If you follow the rules of Maths then it is correct for every use case. That’s why they exist in the first place.
I think you misunderstand my argument. I could use still math to solve a real-world problem with an altered order of operations. You could still do anything you can do with regular math, if you had a different order of operations. You could make a programming language that parses your inputted expressions with a different order of operations and still use it to calculate collisions or render a 3d scene or do anything else that involves math. Do you need me to calculate something, to prove it to you?
The order of operations is just part of a system of notation and any system of notation that exists in the world is inherently arbitrary. The same way the way that how we draw the number 3 or the number 5 has no inherent meaning behind it other than the convention of how we interpret it, the order of operations is nothing more than a standard part of the notation. Again, I’m not saying that we should or could change it, as there would be no way to indicate which convention we are using and the standard order of operations works perfectly fine.
No, you demonstrably didn’t understand mine, which is, what you are saying is impossible, but you’re still saying it’s possible.
I could use still math to solve a real-world problem with an altered order of operations
No, you can’t. You already tried to do addition first in 2+3x4 and found out why it doesn’t work. Ever since then you’ve been ignoring that result and pretending that there’s some other way to make it work. No, there isn’t. As long as multiplication is defined in terms of addition (i.e. 3x4=3+3+3+3) then it’s impossible to get a right answer unless you do multiplication before addition.
You could still do anything you can do with regular math, if you had a different order of operations
No, you can’t. Again, you already proved you can’t.
Do you need me to calculate something, to prove it to you?
Go ahead - I’m not holding my breath. I already told you why it literally can’t work. But note that adding brackets isn’t changing the order of operations - brackets are already part of the order of operations. Writing 2+3x4 as 2+(3x4) is exactly the same thing.
BTW just to FURTHER prove your “addition first” doesn’t work, look at this example…
3x4+2=3x6=18. But earlier you did 2+3x4=5x4=20 - not even the same answer in an “addition first” world! Welcome to why it’s impossible to make addition-first work. But knock yourself out - you’re welcome to try! 😂
The order of operations is just part of a system of notation
No, it isn’t. It’s part of the rules of Maths. Notation is how you write it - underlying that is how Maths actually works. This is embodied in the rules of Maths.
is inherently arbitrary
Completely fixed, and a result of the way the operators are defined - that was the only “arbitrary” bit, deciding what the operators were and what they were going to mean, but once you did that then the order of operations rules were already written for you (having already been determined as soon as you made the definitions of the operators in the first place).
number 5 has no inherent meaning behind it other than the convention of how we interpret it
Again, not a convention, a rule of how to interpret it. You can’t just decide to interpret 5 as four, or again, you end up with wrong answers. The rules of Maths prevent you from getting wrong answers. You found that out yourself when you tried to do addition first in 2+3x4.
the number 5 has no inherent meaning behind it other than the convention of how we interpret it
Again, not a convention, a rule of how to interpret it. You can’t just decide to interpret 5 as four, or again, you end up with wrong answers. The rules of Maths prevent you from getting wrong answers. You found that out yourself when you tried to do addition first in 2+3x4.
It’s only a wrong answer if you use the same expression you would with the standard order of operations. And I’m not saying we can randomly start interpreting 5 as four, just that there is no law of the universe that makes 5 look like that, and we could theoretically (not practically ofc) switch the definitions of the symbols 5 and 4 if we did it all at once and revised old math expressions to match the new standard. Just as there is no reason the letters “bike” mean what they do other than that’s what someone decided to call it, there is no reason the order of operations is what it is other than that is how someone decided to write it.
Scratch doesn’t even have an order of operations. You can still do math in it.
I’m not saying you can take any expression and get the same answer by doing addition before multiplication. I’m saying you can take any problem and get the correct answer by doing addition before multiplication. In your milk example, that means I would use the expression 2+(3x4) because 2+3x4 is no longer the correct expression needed to solve the problem.
(For an example of my distinction of the words “expression” and “problem”, “(4x)+2” is an expression, and “I start with 2 litres of milk. For every dollar I spend, I get 4 more liters of milk. How much milk do I have?” is a problem.)
My argument also relies on a distinction between the language of modern math and the concept of doing math, defining math as the dictionary definition of “The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols”. As you can see, this makes no mention of the notation commonly used in math. All I am saying is that you can still use numbers to solve problems with an altered order of operations, or by altering any part of the system of notation.
Perhaps seeing how I could solve a problem with a different order of operations will help illustrate my argument:
Problem:
2 cars approach an interchange at a 90 degree angle to each other. Car A approaches the station from 15 meters away at 30 meters/second and Car B approaches the station from 50 meters away at 20 meters/second. How fast is the distance between the cars decreasing?
Answer: the rate of change of the distance between the cars is approximately -27.777 meters per second.
As you can see, I used my altered math notation to find the correct answer. I can still solve a real-world problem with this notation, but the same expressions you would use before may not work now.
“Wrong answers” only according to our current order of operations, math still works if you, for example, make additions come first (as long as you’re consistent about it).
OFC it is a convention and to change it you would have to change all expressions ever written all at the same time, to avoid confusion between competing standards. I’m not arguing that it should be changed, only that there is no ‘high truth’ behind it.
No, according to arithmetic.
No, it doesn’t - order of operations proof. The only way it could work with addition first is if we swapped the definitions of addition and multiplication around… but then we still have the same order of operations, all we’ve done is swapped around what we call addition and multiplication!
There is when it comes to order of operations.
Let’s assume for a minute addition comes first. We know 2+3 is 5, and 5x4 is the same as 5+5+5+5=20. What is the issue with that?
That it’s wrong. If I have 1 2 litre bottle of milk, and 4 3 litre bottles of milk - i.e. 2+3x4 - how many litres of milk do I have? Without even doing the arithmetic, just count it up and tell me how many litres there is.
If we change how equations are parsed so addition comes before multiplication, 2+3x4 is not the equation required to solve that problem. 2+(3x4) is the equation needed. You can’t change how equations work and then expect all equations to work the same after the change.
If your argument is that this will add parentheses where we didn’t need them before, that’s valid and its the reason we do it this way in the first place. But that doesn’t mean there is anything fundamentally wrong with having a different system of writing equations in which operations are executed in a different order.
Our whole system of writing equations is just a convention, and yes, it is a good and easy to understand and use way of writing math. But there is no fundamental truth behind it, only that it is simpler for the majority of use cases.
Noted that you didn’t answer my question - the answer is I have 14 litres of milk. 2+3+3+3+3=14 litres. When you did “arbitrary addition first”, you got 20, which is wrong, which is why no other order of operations rules work than the ones we have.
In actual fact the point is that they will except for what ever your new notation is. e.g. if we instead defined + to mean multiply, and x to mean add, then we would do + before x, and again, that would be the only order of operations which works. i.e. the only order which gives us 14 litres.
No, and if you did that, you would again arrive at only one order of operations rules which works, cos I still have 14 litres, and the Maths in this new system still has to give an answer of 14 litres, not 20.
Nope, it’s all rules, found in any Maths textbook, and if you don’t obey the rules you get wrong answers (like you did when you got 20).
Yes there is - I have 14 litres, and only 1 set of order of operations rules gives that answer.
If you follow the rules of Maths then it is correct for every use case. That’s why they exist in the first place.
I think you misunderstand my argument. I could use still math to solve a real-world problem with an altered order of operations. You could still do anything you can do with regular math, if you had a different order of operations. You could make a programming language that parses your inputted expressions with a different order of operations and still use it to calculate collisions or render a 3d scene or do anything else that involves math. Do you need me to calculate something, to prove it to you?
The order of operations is just part of a system of notation and any system of notation that exists in the world is inherently arbitrary. The same way the way that how we draw the number 3 or the number 5 has no inherent meaning behind it other than the convention of how we interpret it, the order of operations is nothing more than a standard part of the notation. Again, I’m not saying that we should or could change it, as there would be no way to indicate which convention we are using and the standard order of operations works perfectly fine.
No, you demonstrably didn’t understand mine, which is, what you are saying is impossible, but you’re still saying it’s possible.
No, you can’t. You already tried to do addition first in 2+3x4 and found out why it doesn’t work. Ever since then you’ve been ignoring that result and pretending that there’s some other way to make it work. No, there isn’t. As long as multiplication is defined in terms of addition (i.e. 3x4=3+3+3+3) then it’s impossible to get a right answer unless you do multiplication before addition.
No, you can’t. Again, you already proved you can’t.
Go ahead - I’m not holding my breath. I already told you why it literally can’t work. But note that adding brackets isn’t changing the order of operations - brackets are already part of the order of operations. Writing 2+3x4 as 2+(3x4) is exactly the same thing.
BTW just to FURTHER prove your “addition first” doesn’t work, look at this example…
3x4+2=3x6=18. But earlier you did 2+3x4=5x4=20 - not even the same answer in an “addition first” world! Welcome to why it’s impossible to make addition-first work. But knock yourself out - you’re welcome to try! 😂
No, it isn’t. It’s part of the rules of Maths. Notation is how you write it - underlying that is how Maths actually works. This is embodied in the rules of Maths.
Completely fixed, and a result of the way the operators are defined - that was the only “arbitrary” bit, deciding what the operators were and what they were going to mean, but once you did that then the order of operations rules were already written for you (having already been determined as soon as you made the definitions of the operators in the first place).
Again, not a convention, a rule of how to interpret it. You can’t just decide to interpret 5 as four, or again, you end up with wrong answers. The rules of Maths prevent you from getting wrong answers. You found that out yourself when you tried to do addition first in 2+3x4.
It’s only a wrong answer if you use the same expression you would with the standard order of operations. And I’m not saying we can randomly start interpreting 5 as four, just that there is no law of the universe that makes 5 look like that, and we could theoretically (not practically ofc) switch the definitions of the symbols 5 and 4 if we did it all at once and revised old math expressions to match the new standard. Just as there is no reason the letters “bike” mean what they do other than that’s what someone decided to call it, there is no reason the order of operations is what it is other than that is how someone decided to write it.
Scratch doesn’t even have an order of operations. You can still do math in it.
I’m not saying you can take any expression and get the same answer by doing addition before multiplication. I’m saying you can take any problem and get the correct answer by doing addition before multiplication. In your milk example, that means I would use the expression 2+(3x4) because 2+3x4 is no longer the correct expression needed to solve the problem.
(For an example of my distinction of the words “expression” and “problem”, “(4x)+2” is an expression, and “I start with 2 litres of milk. For every dollar I spend, I get 4 more liters of milk. How much milk do I have?” is a problem.)
My argument also relies on a distinction between the language of modern math and the concept of doing math, defining math as the dictionary definition of “The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols”. As you can see, this makes no mention of the notation commonly used in math. All I am saying is that you can still use numbers to solve problems with an altered order of operations, or by altering any part of the system of notation.
Perhaps seeing how I could solve a problem with a different order of operations will help illustrate my argument:
Problem: 2 cars approach an interchange at a 90 degree angle to each other. Car A approaches the station from 15 meters away at 30 meters/second and Car B approaches the station from 50 meters away at 20 meters/second. How fast is the distance between the cars decreasing?
Answer: the rate of change of the distance between the cars is approximately -27.777 meters per second.
As you can see, I used my altered math notation to find the correct answer. I can still solve a real-world problem with this notation, but the same expressions you would use before may not work now.