Are proofs in geometry rigorous?

Solution 1:

Geometry can be axiomatized and thus allow for proofs that are as rigorous as any other proof in a formal system. In fact, Euclid's "Elements" is a very systematic and rigorous account of geometry, starting with axioms and rules of deduction. Using modern notation and style we would have written it differently, but it is, even though thousands of years old, rigorous.

There are in fact various different kinds of geometry, each with different axiomatizations. Famously, Euclid's fifth postulate resisted many attempts to deduce it, formally and rigorously, from the other axioms set by Euclid. It was not until the 19-th century that mathematicians came to realize that in fact two possible alternatives to Euclid's Fifth produce valid models, thus proving that the Fifth can't be proved from the other postulates. This gave rise to hyperbolic geometry. There are also finite geometries and other 'strange' creatures. In all, the proofs can be made as rigorous as one wishes them to be.

Here is an example of a rigorous proof. Let us model the plane as $\mathbb R^2$ with the Euclidean metric $d(x,y)^2=(x_1-y_1)^2+(x_2-y_2)^2$. Once can prove that algebraically that $d$ is a distance function. Now, the definition of circle becomes $\{x\in \mathbb R\mid d(x,p)=r\}$, where $p\in \mathbb R^2$ is the center and $r>0$ the radius. The diameter of any subset $S\subseteq \mathbb R^2$ is defined to be $\sup{d(s,t)}$ where $s,t$ range over $S$. Now (Theorem): the diameter of a circle of radius $r$ does not exceed $2r$. Proof: Given any $x,y$ on a circle with center $p$ and radius $r$, we have, by the triangle inequality, that $d(x,y)\le d(x,p)+d(p,y)=r+r$. (One can improve this theorem to show that the diameter is precisely $2r$).

Another, more interesting, example would be the proof of Pythagoras theorem through the axiomatization of angles and distances by means of an inner product. So, we now model the plane as $\mathbb R^2$, as a vector space, with the standard inner product. The Cauchy-Schwarz inequality tells us we can interpret the relation $(x,y)$ to mean that $x,y$ are perpendicular. A straight forward computation then shows that $\|x+y\|^2=\|x\|^2+\|y\|^2$ for all perpendicular vectors. Viewing the norm as a length and from the parallelogram law for vector addition, this is precisely Pythagoras Theorem.

Solution 2:

I have a suspicion that what's bothering you isn't really a technical issue, but more of linguistic issue, having to do with how geometric proofs (and actually, proofs in general) are worded.

Maybe you've seen proofs that look like this:

Theorem: Property P is true for all circles.

Proof: Let C be a circle. (...) Therefore, property P is true for C. QED.

Looked at superficially, this seems like we've only proven P to be true for that particular circle. But the thing is, C was just a circle. An arbitrary circle. We didn't say "Let C be the circle centered at $(0.5, 6.3)$". In the proof, we only used facts that are true of any circle, and therefore the proof would work just as well no matter which specific circle C is.

This isn't specific to geometry, it's just how proofs are worded in mathematics generally.