Why bending some rules is meaningful, while bending others is fruitless?

Question

This might be a purely philosophical question, but still...

Up to some points in geometry, Euclid's axioms were accepted, even though the 5th one was causing headache.

Then someone comes up and say "hey, what if we consider it's wrong ?", and bam!, here come spherical and hyperbolical geometries. Such theories ave a meaning and can even be used in physics to describe our universe.

In another fiels, mathematicians have known for quite some time that we can't make the square root of a negative number.

Then someone comes up and say "hey, what if we consider $\sqrt{-1}$ has a solution ?" and bam!, here come complex numbers, very useful in describing a lot of things, in particular waves in physics.

What is so special about these "rules" (5th axiom of Euclid and "no squared real number squared is negative") that we can suddenly consider them as false and gets something useful out of it ?

It seems to me, if I say "hey, let's consider $\pi=4$" or "hey, let's consider $x=x+1$ has a solution in some new set of numbers", it might bring some new theories, but they will be a complete waste of time.

Is there a reason why it is so ?

Answer 1

The reason these rules were bend was because the original set of rules did not suffice for every purpose. Euclid's postulates were designed and deducted specifically for plane geometry. It is then reasonable that if one wants to do geometry on a curved space, the set of rules have to be adapted to suit this new situation.

As Peter mentions, it is mostly useful to extend an existing structure in such a way that there are no contradictions in the extended structure. However, one can also modify the existing structure in order to obtain a new structure which has maybe a bit other implications and philosophical interpretations.

It is hard to know beforehand bending which rules gives you meaningful insights, and which not, but experimenting and trying new things without knowing the outcome has been characteristically of science since day one.

Answer 2

An extension makes sense (and can in the best case turn out to be very useful), if it respects the structures we currently have and creates no contradictions. Consistency is necessary to create anything meaningful.

The complex numbers are , for example , useful because they form an algebraically closed field.

Another very important example is the system ZFC , which is a much stronger theory than the PA that are contained in ZFC.

Because you mentioned $\pi=4$ , actually in the USA, this was defined by law (!) but I do not exactly remember where and when. But such ideas , as you correctly pointed out, have no merit whatsoever.