Hey, you’re right, Cajori does talk about operator precedence.
Unfortunately, it talks about how the rules, especially for mixed division and multiplication, have changed over time. Supporting my point that these “rules” are not in fact rules of maths, but instead rules of mathematicians.
That is why Cajori includes them in a book about the history of how we write mathematics. No matter how you write multiplication and addition, they must always be commutative, associative relations which obey the distributive law; if they didn’t, they wouldn’t be multiplication and addition. However, you can write them down in different ways, by using different symbols for example. Using different symbols for multiplication changes what a sequence of mathematical symbols means, but it doesn’t change what multiplication is. Doing the operations described by a sequence of mathematical symbols in one order or another order may break one set of rules of precedence, but those are rules made by mathematicians not by the fundamental working of the universe.
How do I know this? Because Cajori says that, at the time he was writing, there was “no agreement” over the order in which to perform divisions and multiplications if both occur in an expression. So here’s a question for you: do you
agree with Cajori that at one time there was no agreement over which order to perform multiplications and divisions, or not?
If you do agree that there was no such agreement, do you then agree that, for there to be agreement now, such as there may be, that change must be through rules created by mathematicians, rather than by rules given to us from the universe itself? Because the universe certainly didn’t change in the meantime, did it?
If you don’t agree then that would rather expose your fetishisation of textbooks as hollow trolling, of course.
especially for mixed division and multiplication, have changed over time
and yet, have not changed since he died. 😂 Keep going - you’re on the right track but the rabbit hole is deeper
Supporting my point that these “rules” are not in fact rules of maths
says person who doesn’t know the difference between rules and conventions, and thus does not support what you are saying 😂
instead rules of mathematicians
who proved them, yes
associative relations which obey the distributive law
Property, not Law, yes
may break one set of rules of precedence
there’s only one set! 😂
those are rules made by mathematicians not by the fundamental working of the universe
says person failing to give a single example of such 😂
How do I know this?
Same way you “know” everything - you just make it up as you go along, but never can produce any evidence to support you 😂
at the time he was writing, there was “no agreement” over the order in which to perform divisions and multiplications if both occur in an expression
Yep, and why was that, or have you already forgotten the assignment? 😂
So here’s a question for you: do you agree with Cajori that at one time there was no agreement over which order to perform multiplications and divisions, or not?
Of course, and I, unlike you, know exactly what he was talking about 😂
do you then agree that, for there to be agreement now
There isn’t, given he was talking about conventions, and now, same as then, different people use different conventions, but all of them obey the rules 🙄
that change must be through rules created by mathematicians
from proof of same
rules given to us from the universe itself?
NOW you’re getting it!
Because the universe certainly didn’t change in the meantime, did it?
Nope, and neither have the rules 😂
If you don’t agree then that would rather expose your fetishisation of textbooks as hollow trolling, of course.
And, yet I did agree, sorry to spoil your fun. 🤣🤣🤣 BTW Cajori isn’t a textbook, in case you didn’t notice 😂
So, you’re saying that, some time after 1928, a mathematician proved that a ÷ b × c = (a ÷ b) × c? Where was this result published? What’s the citation? Who was the author (or authors)? Or maybe you don’t have the citation to hand, but know the proof off the top of your head? Please, let me see it.
Does it not seem weird to you that such a basic aspect of mathematics as multiplication and division remained undiscovered until the 20th century? It’s hardly Fermat’s Last Theorem. If the meaning of a ÷ b × c were not a matter of choice but instead an open question, why were mathematicians using the notation at all when its meaning was not known?
And what of the textbooks like High School Algebra, Elementary Course (1917) which used the convention - err oops, rule - of performing multiplication first? See page 212, example 155. (Would be 810, not 90, if using strict left-to-right priority for division and multiplication)
If it were proved that this convention is wrong, then you will surely be able to find some serious error that flows from doing division in this order. After all, from any contradiction, you can prove anything.
Of course you won’t be able to do this because the question that you are saying was proved is “what does the sequence of symbols a ÷ b × c mean”. The meaning of sequences of symbols is not a fundamental aspect of the universe, is it. They could have different meanings to different people, in different places, or at different times, couldn’t they?
Hey, you’re right, Cajori does talk about operator precedence.
Unfortunately, it talks about how the rules, especially for mixed division and multiplication, have changed over time. Supporting my point that these “rules” are not in fact rules of maths, but instead rules of mathematicians.
That is why Cajori includes them in a book about the history of how we write mathematics. No matter how you write multiplication and addition, they must always be commutative, associative relations which obey the distributive
law; if they didn’t, they wouldn’t be multiplication and addition. However, you can write them down in different ways, by using different symbols for example. Using different symbols for multiplication changes what a sequence of mathematical symbols means, but it doesn’t change what multiplication is. Doing the operations described by a sequence of mathematical symbols in one order or another order may break one set of rules of precedence, but those are rules made by mathematicians not by the fundamental working of the universe.How do I know this? Because Cajori says that, at the time he was writing, there was “no agreement” over the order in which to perform divisions and multiplications if both occur in an expression. So here’s a question for you: do you agree with Cajori that at one time there was no agreement over which order to perform multiplications and divisions, or not?
If you do agree that there was no such agreement, do you then agree that, for there to be agreement now, such as there may be, that change must be through rules created by mathematicians, rather than by rules given to us from the universe itself? Because the universe certainly didn’t change in the meantime, did it?
If you don’t agree then that would rather expose your fetishisation of textbooks as hollow trolling, of course.
and yet, have not changed since he died. 😂 Keep going - you’re on the right track but the rabbit hole is deeper
says person who doesn’t know the difference between rules and conventions, and thus does not support what you are saying 😂
who proved them, yes
Property, not Law, yes
there’s only one set! 😂
says person failing to give a single example of such 😂
Same way you “know” everything - you just make it up as you go along, but never can produce any evidence to support you 😂
Yep, and why was that, or have you already forgotten the assignment? 😂
Of course, and I, unlike you, know exactly what he was talking about 😂
There isn’t, given he was talking about conventions, and now, same as then, different people use different conventions, but all of them obey the rules 🙄
from proof of same
NOW you’re getting it!
Nope, and neither have the rules 😂
And, yet I did agree, sorry to spoil your fun. 🤣🤣🤣 BTW Cajori isn’t a textbook, in case you didn’t notice 😂
So, you’re saying that, some time after 1928, a mathematician proved that a ÷ b × c = (a ÷ b) × c? Where was this result published? What’s the citation? Who was the author (or authors)? Or maybe you don’t have the citation to hand, but know the proof off the top of your head? Please, let me see it.
Does it not seem weird to you that such a basic aspect of mathematics as multiplication and division remained undiscovered until the 20th century? It’s hardly Fermat’s Last Theorem. If the meaning of a ÷ b × c were not a matter of choice but instead an open question, why were mathematicians using the notation at all when its meaning was not known?
And what of the textbooks like High School Algebra, Elementary Course (1917) which used the convention - err oops, rule - of performing multiplication first? See page 212, example 155. (Would be 810, not 90, if using strict left-to-right priority for division and multiplication)
If it were proved that this
conventionis wrong, then you will surely be able to find some serious error that flows from doing division in this order. After all, from any contradiction, you can prove anything.Of course you won’t be able to do this because the question that you are saying was proved is “what does the sequence of symbols a ÷ b × c mean”. The meaning of sequences of symbols is not a fundamental aspect of the universe, is it. They could have different meanings to different people, in different places, or at different times, couldn’t they?