6 upvotes, 1 direct replies (showing 1)
View submission: Ask Anything Wednesday - Engineering, Mathematics, Computer Science
This is probably more a question about the history of algebra than anything else, but it's worth pointing out that mathematical statements are just assertions of truth:
3 + 5 = 8
x +7 = y
When we make changes to an equation, like 'cancelling', or resolving for a given variable, we're making new statements of truth, justified by the previous statement being true, and adhering to the rules of logic when making the change.
That means, when we change, say 5+3 = 8 to 5 = 8-3 or even 8=8 (or 5 = 5), we're not just saying the same thing over and over, but making new statements that we're confident are true. 5=5 is different to saying 8=8, but we're confident both are true because of the original statement 5+3 = 8, and making sure that any adjustment we make to one side of an equation is balanced out by making the same on the other side.
The concept itself is intuitive enough that it probably pre-dates the idea of equations in the first place - kids understand it, say, when sharing things equally (if we have six things to share, we both start off with 0 and if give you three things, I need to give myself three things to even it out). Guaranteed the ancient Greeks understood it, and certainly the Persians understood it with the development of algebra.
Comment by OpenPlex at 14/12/2023 at 02:35 UTC
0 upvotes, 1 direct replies
When we make changes to an equation, like 'cancelling', or resolving for a given variable, we're making new statements of truth, justified by the previous statement being true, and adhering to the rules of logic when making the change
Interesting approach. It seems reasonable to assume that people in ancient times understood the parts about stating a truth, and about balancing every adjustment..
What isn't obvious is why make specific adjustments purely for the sake of revealing or discovering an unknown, unsuspected aspect in a mechanical model of how nature works. Who came up with that? How did they figure out that canceling could reveal surprises?
The adjustments almost seem like playing around with the equation to see what unanticipated insights might pop out.