You don’t, because the second 2 is associated with a Division that has to be done before the addition. Maybe go back to school and learn how to do Maths 🙄

Right, so you cannot derive precedence order from the definition of the operations. Your argument based on the definition of multiplication as repeated addition is wrong.

or you get wrong answers

This is begging the question. We are discussing whether the answers are flat wrong or whether there is a layer of interpretation. Repeating that they are wrong does nothing for this discussion, so there’s no need to bother.

You have nothing to say that I can see about why the different interpretations are impossible, or contradictory, or why they ought to qualify as “wrong” even though maths works regardless; you’ve just heard a school-level maths teacher tell you it’s done one way and believe that’s the highest possible authority. I’m sorry, but lots of things we get taught in high school are wrong, or only partially right. I see from your profile that you are a maths teacher, so it’s actually your job to understand maths at a higher level than the level at which you teach it. It may be easier to to teach high school maths this way, but it’s not a good enough level of understanding for an educator (or for a mathematician).

Left to right is a convention (as is not writing the brackets in RPN). Brackets before Multiplication before Addition are rules.

OK, let’s try a different tack. When I hear the word “rules”, I think you’re talking either about a rule of inference in first order logic or an axiom in a first-order system. But there is no such rule or axiom in, for example, first order Peano arithmetic. So what are you talking about? Can you find somewhere an enumeration of all the rules you’re talking about? Because maybe we’re just talking at cross-purposes: if you deviate from the axioms of Peano arithmetic then we’re fundamentally not doing arithmetic any more. But I contend that you will not find included in any axiomatisation anything which specifies order of operations. This is because from the point of view of the “rules” (i.e. the axioms) the addition and multiplication operations are just function symbols with certain properties. Even the symbols themselves are not really part of the axiomatisation; you could just as well get rid of the + symbol and write A(x, y, z) instead of “x + y = z”; you’d have the exact same arithmetic, the exact same rules.

If you’re able to answer this, we can get away from these vague terms which you keep introducing like “notation definition”, and we can instead think about what it means to be a convention versus whatever it is you mean by “rule”. (For example, Peano arithmetic has a privileged position amongst candidates for arithmetic because it encompasses our intuition about how numbers work: you can’t just take an alternative arithmetic, like say arithmetic modulo 17, and say that’s an “alternative convention” because when you add an apple to a bowl of 16 apples, they don’t all disappear. But there’s no such intuition about how to write mathematics to express a certain thing. I contend that is all convention.)

It works because it treats every operation as bracketed without writing the brackets. Also that’s only a Maths notation, not the Maths itself.

So, you understand that a notation can evaluate things in a certain order with what you call “treating every operation as bracketed without writing brackets.” What does it mean to be “bracketed without writing brackets”? There are exactly two aspects to brackets:

1. the symbols themselves - but we’re not writing them! So this isn’t relevant.
2. the effect they have - the effect on the order of evaluation of operations

So what you’re admitting with these phantom brackets is that a notation can evaluate operations in a different order, even though there are no written brackets.

So I can specify these fake brackets to always wrap the left-most operation first: (2 + 3) x 5 and hey look, this notation now has left-to-right order of evaluation, not the usual multiplication first. If you prefer to think of there being invisible brackets there, go right ahead, but the effect is the same.

So, how do we decide whether our usual notation “has bogus brackets” or not? Convention. We could choose one way or the other. Nothing breaks if we choose one or the other. Symmetrically, we could say that left-to-right evaluation is the notation “without bogus brackets” and that BODMAS evaluation is the notation “with bogus brackets”. Which choice we make is entirely arbitrary. That is, unless you can find a compelling reason why one is right and the other wrong, rather than just saying it once again.

They don’t actually. Welcome to most e-calcs give wrong answers because the programmers failed to deal with it correctly.

What problems does it cause? Are the problems purely that they don’t have the order of operations you expect, and so get different answers if you don’t clarify with brackets? Because that, again, is begging the question.

To re-iterate, you are in a discussion where you’re trying to establish that it’s a fundamental law of maths that you must do multiplication before addition. The fact that you’ve written a post in which you document how some calculators don’t follow this convention and said that they’re wrong is not evidence of that. It’s just your opinion. Indeed, it’s really (weak) evidence that your opinion is wrong, because you’re less of an authority than the manufacturers of calculators.

On calculators, there’s something important you need to realise: basic, non-scientific, non-graphing calculators all have left-to-right order of operations. You can test this with e.g. windows calculator in “standard” mode by typing 2, +, 3, x, 5 (it will give you 25, not 17). Switch it to “scientific” mode and it will give you 17.

Why is it different? Because “standard” mode is emulating a basic calculator which has a single accumulator and performs operations on that accumulated value. When you type “x 2” you are multiplying the accumulator by 2; the calculator has already forgotten everything that you typed to get the accumulator. This was done in the early days of calculators because it was more practical when memory looked like this:

Now, you can go on about your bogus brackets until you’re blue in the face, but the fact is that this isn’t “wrong”. It has a different convention for a sensible reason and if you expect something different then it is you who are using the device wrong.

From your other comment, since having two threads seems pointless:

So if you have one “notation definition” as you call it which says that 2+2*3 means ”first add two to two, then multiply by three” and another which says “first multiply two by three, then add it to two”, why on earth do the “rules” have anything further to say about order of operations?

No we don’t. We have another notation which says to do paired operations (equivalent to being in brackets) first.

What do you mean “we don’t”? I just made the definition. It exists. This is why terms like “notation definition” are not actually helpful IMO, so let’s be precise and use terms that are either plain english (like “convention”) or mathematical (like “axiom”, “definition”, etc).