User blog comment:Primussupremus/Illogical numbers/@comment-29915175-20170415070717/@comment-5982810-20170420231322

Firstly, this isn't something new. The idea that "mathematics" is just an arbitrary set of rules invented by people, and therefore is no "better" than any other set of rules, is an old one. In fact, this criticism is commonly raised by people who tend not to like mathematics in the first place.

The problem is, you can "come up" with whatever you like, but it doesn't necessarily mean it's interesting. The limitations in mathematics, the fact that not everything can be done and not everything is correct IS what makes it interesting.

Think of it like this, Chess is a game that has very precise rules that force players to consider their positions and options very carefully and consider their inevitable consequences. This leads to interest on both sides in trying to outmanuever and unthink their opponent. This is kind of like mathematics. Then there is two kids arguing about whose action figure wins. Since there are no rules, nothing said can have any meaningful consequence. Nothing said actually matters or is of any consequence, because each kid is incapable of convincing the other kid to "lose" since no one is bound by any rules and there is nothing to be gained in doing so. The result is they both eventually lose interest. This is kind of like what you're doing.

If every statement was "true", in mathematics, simply because we declared it "true", then proof becomes irrelevant. No theorem is of any interest because it's a given for everyone, and it holds no special place. No knowledge is hard one, and no deep result is possible. Fermat's last theorem becomes trivial, like every theorem, and it's negation is also trivial. Since everything is "true", no "truth" provides any insight into anything.

In short, to have anything of consequence you can not have unlimited freedom. Meaning is created through limitation.

That being said, Mathematics is not a monolith. It is better thought of as an infinitely branching tree, where you can always add more branches. But in order to qualify as mathematics there must be some actual order and some internal consistency to it. Sure you can create a mathematics where "2+2=5", but unless (1) you can find the sum for any other pair in your "system" and (2) each has a unique answer which doesn't lead to contradictions, you won't actually have something of interest.

Most of the people who have attempted to create some kind of "alternate" mathematics of this variety have failed at this.

Cranks still try though and often claim that their "anything is possible" wonderland is somehow superior to standard forms of mathematics. However, to me at least, it is painfully obvious that it is actually vastly inferior, precisely because nothing gleaned from it can ever be of any consequence. Everything is, after all, a given.