Are proofs in geometry rigorous?

Question

What I mean by that is, suppose say I have a circle centered at some point in the Euclidean space for which a certain property $P$ is true. How can I conclude from this, that $P$ is true for all circles centered at any arbitrary point in the space. Is there some kind of invariance principle in geometry like that in Physics. We measure the speed of light here on earth and from that we conclude that this is the speed of light throughout the universe. How can I check if the property is invariant under translation. Can you give me a concrete example of this ? In case of a circle, we can rotate it and that property might still be true, I would say then that the property is invariant under rotation for that particular circle.

Answer 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.

Answer 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.

Answer 3

If the property P is proven only for the circle centered at some specific point, then of course it is only true for that specific circle. It is not necessarily true for other circles, centered at other points.

But if P is proven for circles in general, then it does not rely on the position of the center. It is therefore true for all circles, regardless of where they are centered.

In other words, there's absolutely no need for any "invariance principle", if the proof of the property does not rely on it.

Answer 4

Well the thing is you define a circle as "the points at distance R from a point X". If you don't use any particular properties of R and X, the conclusions you get are true for every circle.

Moreover, there is the whole theory of translations, rotations etc.

Answer 5

We absolutely do presume an invariance principle in geometry. In pure euclidean geometry, we're not working in terms of coordinates, but rather in terms of geometric objects like lines, points, circles and so forth.

The presumed invariance principle is that it doesn't matter where these objects are, i.e. that only their relative positions matter. Note that this implies what we also assume that our space extends arbitrarily far in all directions! If we do geometry on a piece of paper of a fixed size, two non-parallel lines can for example fail to intersect on that paper.

Answer 6

To prove a geometry theorem using coordinates, one often 'fix' the coordinate system: you can choose an origin and a direction. You can assume that a given point P has coordinates (0,0). To prove that this reasoning is valid, one first need to prove that the predicates that form the property P you want to prove are invariant by translation and rotation. It can be done rigorously, it can even be machine checked by a software called a proof assistant. See Section 3 of the paper by John Harrison: http://www.cl.cam.ac.uk/~jrh13/papers/wlog.pdf or page 9 of https://liris.cnrs.fr/~jdgeneva/website/papers/cpp2011.pdf