# Are geometric proofs less reliable than others?

When I submit a homework with a proof that uses a graph, ball, shape etc., most of the time the professors are not happy with them. They respond with a statement like:

"The proof you made seems very true but why don't you just make a usual proof without drawing anything?"

Of course this is something I can do, but I don't like proving something without any visualization.

So, is it because geometric proofs are more likely to be misleading?

Edit: For example: An open ball $B(x,\epsilon)$ is open.

• Can you give a concrete example? – mrf Apr 11 '14 at 18:52

Sometimes visualization is indeed misleading, check this post:

https://math.stackexchange.com/a/743458/136544

This is a very deep question.

A proof in terms of numbered formulae and various $\Rightarrow$, resp. $\Leftrightarrow$-signs could be checked by an automated proof checker. On the other hand, a "figure" is just a bitmap, or a pixel heap, and I doubt that an automated proof checker would ever be able to make out what this figure is telling us.

In other words: Figures are viewed at and interpreted by humans. Sometimes these humans consent in accepting such a figure as proof of some statement, but sometimes they are in error in doing so. When a figure mainly serves to explain a certain concept, say, the derivative of a function $f:\>{\mathbb R}^n\to{\mathbb R}^m$ at some point $p\in{\rm dom}(f)$, then there is not much harm possible, but as soon as there are "cases" involved, say in a geometric proof of $\sin(x+y)=\ldots\$ for arbitrary angles, the question arises whether the power of $1$ (one) figure is sufficient to prove the general statement. To put it differently: A general statement might involve very different morphologies, only one of which is captured in a single figure.

Concerning your example, it is certainly not sufficient to draw a point $x$ and a circle of radius $\epsilon$ around $x$. But inserting another point $y$ into this circle and drawing a very small circle around $y$ would make the idea of the intended proof clear. Nevertheless, in a course & homework situation it is expected that the idea so obvious in the figure is "verbalized" in a coherent argument.

I honestly have no idea why your professors would react in that way. My experience has been the opposite, that drawing pictures demonstrated that you weren't just mindlessly deriving consequences from a definition. My topology professor, for example, loved proofs where we would actually draw an open ball and identified an important element graphically, or something. It is hard to think of examples.

The only risk in a geometric proof is relying completely on an image that you may have accidently drawn in such a way that what you were trying to prove was immediately true, and then accidentally using results from the picture that you did not actually have. However, this should not happen for an experienced prover-er.

If you draw a picture to show why a ball $\{x | d(x,x_0) < \epsilon \}$ is open, then you are almost certainly appealing visually to properties of the 2-dimensional euclidean distance, by drawing balls as discs in 2-d. But the real reason why the ball is open is because of the triangle inequality, which holds for all distances. So you have to be careful with picture proofs. Drawing balls as discs instead of squares or other weirdly shaped regions, etc. will typically assume more than what's stated in the problem.

• I don't see how graphically representing any arbitrary open ball as a 2d disc could assume more than what is stated. Could you give an example? – Jonathan Hebert Apr 11 '14 at 19:10
• @Doop Euclidean 2-d distance arises from a norm $L_2$ on finite-dimensional $\mathbb{R}^2$, and so for example any property that holds for finite-dimensional norm-based distances but not arbitrary distances would lead you to be fooled by a 2-d picture using 2-d Euclidean distance. E.g. maybe you could help show that any closed ball in 2-d with euclidean distance is compact with a picture, but it wouldn't be true for a closed ball in an arbitrary metric space. – user2566092 Apr 11 '14 at 19:32
• I'm not really sure how one would do that by just drawing discs and lines, unfortunately. For the OPs example, I think my proof would be nearly completely geometric with circles and lines. I'm not doubting you, I've just personally never been able to come up with something that wasn't true for any open ball by just drawing circles and lines. I've always considered that to be an accurate representation of any ball, independent of what it "really" looked like, because the properties of a metric seem to ensure a basic generalization of the notions provided by lines. – Jonathan Hebert Apr 11 '14 at 21:48

Let's not get into formal proofs. I don't think that is what you or your instructor are talking about.

The picture above can be used to prove that the angle bisctor at $A$ intersects side $\overline{BC}$ in such a way that $\dfrac{AB}{BE} = \dfrac{AC}{CE}$. The proof involves showing that $\triangle AEC \sim \triangle DEB$ and showing that $\triangle ABD$ is isoscelese.

The proof is clever yet very straightforward and there is a big problem with it. The proof assumes that the ray $\overrightarrow{AD}$ intersects the interior of side $\overline{BC}$ at point $E$. Just because it is so very obvious does not mean that it is true.

Proofs using pictures are like that. They make some things so obvious that the presenter does not bother justifying them. Students almost by definition do not have the experience necessary to get away with this.