If you bring Matrices into the picture, then ax(bxc)=(axb)xc does not hold true.

As you get into higher physics you need higher order math....physisist use mathematicians to prove their theories and engineers distort mathematical decorum to use physic theories

You can multiply numbers in any order you wish and get the same results. This is called the commutative property of multiplication. Addition has the same property. What you cannnot do is mix adding/ subtraction with multiplication/division in any order and expect the same result.

I put both socks on first, then my pants, then the shoes. Seems sorta natural. And usually the right foot first for sock and shoe, and the left for first through the pant leg.
Thanks for having me think about that.

(a x b) x c does equal a x (b x c).

Cool article! Makes sense considering the lack of positioning in electrons. Quantum theory has lacked an explaination or two regarding that particular query. I think considering disassociative Algebraic expressions as a route makes sense. Thanks for the the share.

It is the Associativity axiom, M1, which holds true if a, b and c are elements of the field.

That was a fascinating article, as far as I could understand it! I think the gist is that the brackets make no difference when multiplying "real" numbers, but they do when working with these higher dimensional (?) numbers. Moving forwards 2 feet, then left a foot, then up 6 foot gets you to the same place whichever order you do it. But going forward 6 feet, rotating left 35 degrees, going that way 3 feet, leaning right 20 degrees then moving to the new left 10 feet....change the order and you'll end up in a completely different place. I'm making this up as I figure it out so apologies if I get it wrong ?

The commutative property of multiplication necessitates that a ( b ( c ) )=( a ( b ) ) c=b (a ( c ) )=c (b ( a ) ). It’s numerically demonstrable.

The universe of octonions seems very strange ... to me at least!

(2 x 3) x 4 = 6 x 4 = 24
2 x (3 x 4) = 2 x 12 = 24

(2 x 3) + 4 = 6 + 4 = 10
2 x (3 + 4) = 2 x 7 = 14

(axb)xc does equal a x(bxc)??
It's the commutative property of multiplication.
On further reading, I guess it's the associative property.

Tomatos and oranges girl.... The sequence of spatial events (socks & shoes) it's topology, the association of operations that's arithmetic. Of course my comment is just a top level perspective

Well in your example, they would equal the same since it's all multiplication.