Everyone would understand math if it was fully reduced to its underlying simplicity.
Sometimes it's badly taught; you alerted us to the scandalous fact of professors who ignore certain
'other solutions' to equations.
John Cramer's work pays attention to the 'backwards in time' solutions.
There is still the question of the following, though:
The situation regarding "what is maths?" is addressed in detail by John Barrow. I found a long extract from his book "THe World within the World" in another book.
He shows that since there can be several different solutions to a physical problem, some can not apply to reality. An additional 'correspondence principle' is needed to judge reality-math applicability.
5 men shipwrecked on an island. 1 monkey. Lots of coconuts. They agree: split coconuts 5 ways equally, remainder for the monkey.
During the night; 1 man wakes up and decides to make the share-out. A coconut is left over, and given to the monkey. He then hides his 5th share and puts the other 4 shares back together as if nothing happened. Later at night a 2nd man wakes up, thinks nothing has happened, does the same thing. As do the 3rd, 4th, and 5th men. Always one was left and given to the monkey.
In the morning, no one admits their independent actions overnight; they all just split the coconuts 5 ways equally (not knowing of the shares stashed away by each during the night). And there's one left- given to the monkey. Find the initial number of coconuts.
They say that there are an infinite number of solutions to this problem, but the minimum number of coconuts is 15621. But Paul Dirac gave another solution: -4 coconuts. Each man finds -4, the monkey gets 1 so that's now -5. He takes a fifth ( -1) and then there's -4 again!
So math (eg. also 'toy universes' and math-fantasy-scenarios) does not necessarily correspond with reality. Although Julian Barbour rightly exposes the importance of not neglecting previously ignored solutions; the discrepancies between some math correct solutions, and actual reality, show that 'comparing and matching patterns' is more basic than math. Toy universes of math-play might only be real as logical constructs.