Your habit of snipping replies into tiny segments and replying shortly to each makes the discussion much harder to follow. Try and collect your thoughts into something coherent, if you can.

If you have 1 2 litre bottle of milk, and 4 3 litre bottles of milk, even a 3rd grader can count up and tell you how many litres there are, and that any other answer is wrong. 🙄 2+3x4=2+3+3+3+3=14 correct 2+3x4=5x4=5+5+5+5=20 wrong See how the Maths doesn’t work regardless? 😂

So this is the most interesting thing you've said.

In mathematical notation with BODMAS order of operations, you can write your 14 litres of milk as 2 + 3 x 4, sure. But if you had right-to-left order of operations you could just write 2 + (3 x 4). So why is 2 + 3 x 4 the correct way to describe the situation? Writing out a real-life situation in mathematical notation is a question of correctly using the notational conventions to express reality.

Consider another scenario where you have two three litre bottles of milk and two three litre bottles of orange juice - how much liquid do you have in total? With BODMAS order, you could not write this as 2 + 2 x 3 = 8 litres; you'd have to insert brackets: (2 + 2) x 3 = 12 litres. But with left-to-right order you could write this as 2 + 2 x 3 = 12.

So what we have are two scenarios, where one translates readily to BODMAS order without brackets, and the other translates readily to L2R order without brackets. Neither tells you which is the superior or correct order. Neither leads to a contradiction, or problems, or incorrect results, as long as it is interpreted correctly. Yes, if you incorrectly translate my scenario as 2 + 2 x 3 with BODMAS order, you get the wrong answer. But the problem is that you translated the problem into mathematical notation using L2R order, then evaluated the expression using BODMAS order.

I'll certainly agree that translating the problem with one convention then evaluating that with another is wrong! It leads to answers that don't reflect reality! But of course, if you translate the problem into mathematical notation with L2R order, then evaluate it with L2R order, you get the right answer, and all is fine.

Nope, I’ve proven it myself - that’s the beauty of Maths, that anyone at all can try it for themselves and find out.

This should be easy for you to verify: pick your axiomatisation and write the proof! Or link it; that's fine too. But you'll have a struggle given that order of operations is about notation and that is not a (first-order) mathematical concept.

Unfortunately I suspect you think that your scenario above constitutes a proof. It does not. Here is the mathematical definition of a proof in a first order theory: It is a finite sequence of formulae in the theory, where each formula in the sequence is either an axiom of the theory or follows from one or more previous formulae by some rule of inference. The proof is then said to prove the last formula in the sequence.

There is no room for milk and bottles in a proof, unless they have first order definitions in the language of your theory. But the language of arithmetic only has the symbols for addition, multiplication, successor and zero, plus the logical symbols (quantifiers, and, or, brackets).

So I can specify these fake brackets to always wrap the left-most operation first: (2 + 3) x 5

No you can’t, because you get a wrong answer 🙄

You're trying to establish that it's wrong. You're still begging the question. Maybe you're referring back to the milk, in which case, see above. Either way though, this is an example of a pointless comment; it's adding nothing beyond restating what you're already saying.

No we can’t. Even a 3rd grader who is counting up can tell you that 🙄

Count up how many litres we have

Yes it does. Again ask the 3rd grader how many litres we have, and then try doing Addition first to get that answer 😂

No we can’t. Ask the 3rd grader, or even try it yourself with Cuisenaire rods

Yes it is! 😂 Again, ask the 3rd grader to count up and tell you the correct answer

Your imaginary third-grader would be quite capable of looking at the milk and orange juice and writing down 2 + 2 x 3 = 12 and get the correct answer, if you taught him or her the right-to-left convention.

The Windows calculator is an e-calc which was written by a programmer who didn’t check that their Maths was correct. 🙄 Now try it with any actual calculator 🙄

Demonstrably not 😂

No they don’t! 😂

Instead of using the stack*, to store the Multiplication first, like *all actual calculators do

No, the dumb programmer is. All actual calculators did the Multiplication first and put the result on the stack

But actual calculators have put that result on the stack

No it wasn’t. All calculators “in the early days” used the stack

And even then the stack existed 🙄

Wow, 8 separate replies from you all expressing the exact same thing, and all confidently incorrect.

https://en.wikipedia.org/wiki/Order_of_operations#Calculators

Note especially the phrase: "Many simple calculators without a stack"

https://en.wikipedia.org/wiki/Calculator_input_methods#Chain

Here is an example of a calculator manual from the 70s showing (in Example 6) that the order of operations is left-to-right: https://www.wass.net/manuals/Sinclair%20Executive.pdf

And the successor, one of the first affordable pocket calculators (bottom of page 8): https://www.wass.net/manuals/Sinclair%20Cambridge%20Scientific.pdf

I have no idea how you have forgotten these old, basic calculators.

So, now we've established that you're confidently incorrect about "all actual calculators" having a stack, and about Windows calculator being "wrong" in its emulation of stackless calculators, let's bring this back to the point: calculators are perfectly usable even though their order of operations is left-to-right. As I said before: it had a different convention for a sensible reason, and if you expect something different it is you who are using the device wrong. How to use the device is written in the manual, so every user of it can use it correctly.

By the way, if you want to continue this discussion, please acknowledge that you were wrong about this. This is a simple, verifiable matter of fact that you've been shown to be wrong about, and if you can't cough to that then you certainly won't cough to something more nebulous.

wrong answers

Nope! proven rules as found in Maths textbooks 🙄

As per Maths textbook

says person ignoring the Maths textbooks I quoted and the actual calculators giving the correct answer

So, as above, the different calculators have different conventions. But let's stick with textbooks. Because you are saying all through this that order of operations is not merely a convention, but a rule. So, it's not actually about textbooks, is it? Yet they are, in fact, the best resource you have: your spilled milk establishes the opposite of what you want it to, so textbooks are all you have.

So consider, if all the textbooks were edited overnight to teach L2R order of operations, what would happen? Children would learn that to add 2 litres of milk to 3 bottles of 4 litres, they ought to write 2 + (3 x 4), which they would calculate and get 14. They would learn that to add the volume of two three litre bottles to two three litre bottles you would write 2 + 2 x 3 and get 12.

The textbooks are, in fact, how you can see that this is just a convention. If the textbooks changed, only what people write would change. The answers would stay the same.

But the textbooks you've been linking haven't been about order of operations, but about the "primitive meaning of multiplication". Yet, here are the axioms of arithmetic:

1. For all x (0 = S(x))
2. For all x, y (S(x) = S(y) -> x = y)
3. For all x (x + 0 = x)
4. For all x (x * 0 = 0)
5. For all x, y (x + S(y) = S(x + y))
6. For all x, y (x * S(y) = (x * y) + x)
7. The axiom schema of induction

There is no "definition of multiplication" here because when you get down to it, definitions are things for human beings, not mathematics. Axiom 6 no more (partly) "defines multiplication" than it (partly) "defines addition."

You know we have Mathematical definitions of the difference between conventions and rules, right??

There's a mathematical definition of an axiom in a first order theory, but there's certainly no mathematical definition of a convention, because a convention is a social construct.

What don’t you understand about “we don’t”?

The definition exists. Saying "we don't have it" doesn't make sense. I've told it to you, so now you have it; you can choose to ignore it, but that's just making the choice of convention I'm saying you're making.

Nope, neither.

So... what is it then?

Same thing as we’re adding the 2 in 2+3 without writing a plus (or a zero) in front of the 2 - all Arithmetic starts from zero on the number-line. Maths textbooks explicitly teach this, that we can leave the sign omitted at the start if it’s a plus.

In first-order arithmetic, the + symbol is a binary operation. We're not "leaving it out" in front of the 2, because it would make no sense to put it there.

Your repeated talk of "wrong answers" makes it sound like you're a slave to the test. A test has right and wrong answers, after all, and if you read 2 + 2 x 3 on a test and answer 12 you'd be marked wrong. But your job is to establish not that the answer is wrong in this situation, but that if you changed the test then it would be wrong. How are you going to do that? So far you have not even tried to write down what it would mean for the test to be wrong. But I can lay out my definition of "it's a matter of convention" easily: It's a matter of convention because humans have agreed to do it one way even though all of maths, all totalling of milk and orange juice, everything could be done another way, and be consistent with itself and with physical reality. Maybe you can say what you find defective with that.