You are currently browsing the tag archive for the ‘euclidean geometry’ tag.

The spectre of common usage is one of the greatest bugaboos for amateur grammarians, who fear that if we accept a usage because everybody’s using it, we’re weakening the language. For example:

We have often noted that often repeated language and grammar errors seem to become “correct” usage. Wouldn’t it be weird if math used that philosophy? When enough people said 2+2=5, it would! It would still equal 4, of course, but it would also equal 5.

I’ve heard this “2 + 2” kind of argument many times. It’s a false analogy, a virulent argument that seems reasonable but is wrong at its very core, and is wrong in multiple ways. It misrepresents both language and math, and that makes me mad, because the two things I’ve spent substantial portions of my academic life on are math and language. So let me tell you why this argument is rubbish in both aspects.

**Grammaticality isn’t Truth.** Math and language are different in a lot of ways. Duh, of course; don’t writers claim to not be “math people” and mathematicians claim to not be “language people”? Well, yeah, but it runs deeper than that. Language comes from our minds and our cultures. There isn’t some true, verifiable, Platonic version of language floating out in the ether that we’re trying to use. It is a social construct. And it’s a nebulous social construct, because we don’t know where it came from, or how it’s evolved over long periods of time. Hell, we don’t even have a clear view of the proper theoretical framework to analyze language (see, e.g., the debates between the Minimalists and HPSG or LFG syntacticans).

If everyone suddenly decides that the word *flartish* describes an object that is orange-brown, then the word means that. If everyone later decides that *flartish* describes an object that is wider than it is tall, then the word means that. There is no physical, no Platonic, no **real** meaning for a word. The meaning of a word is nothing more or less than what the speakers of its language believe it to be. That doesn’t mean that a meaning can’t be wrong; if you start re-assigning definitions to words, like Humpty Dumpty did to *glory*, you’ll be wrong in that no one will understand you.

The same is true of a language’s grammar. There is no English outside English, nor Telugu outside Telugu. When a language changes — and they do, constantly — that changes what is standard and non-standard in that language. The English you speak now is the results of billions of changes that took place over thousands of years from Proto-Indo-European. The reason that English and Albanian and Urdu aren’t the same language is that they all have undergone changes through common usage. It’s what happens. Math doesn’t change in this same way. Correct proofs can’t become incorrect in the way that grammatical sentences can become ungrammatical.

**Sometimes 2+2 doesn’t equal 4.** For comparison’s sake, how does mathematical truth work? Well, it works like this.* Before you do anything in math, you have to first lay down a set of axioms, a set of statements that you take as true and cannot prove. The most famous set of these is probably Euclid’s “4 + 1” postulates of plane geometry, which state the existence of line segments, lines, and circles, as well as the equivalence of all right angles and the uniqueness of parallel lines. If you want to prove something in Euclidean geometry, you build up from those axioms. If you want to define something (a triangle, for instance), you define it in terms of those axioms or in terms of things built up from those axioms. So when you prove something like “the sum of the angles of a triangle is 180 degrees” — one of the rudiments of Euclidean geometry that we learn as kids –, it’s true only as long as the axioms are true.

Now, here’s the rub: a true theorem under one set of axioms is not necessarily true under other axioms. For instance, a triangle has to have 180 degrees worth of angles in plane geometry because Euclid’s axioms hold on a plane. A sphere breaks the Euclidean axioms, though, because parallel lines don’t exist on its surface.** This causes the mathematical truth to become untrue, as shown by a thought experiment, pictured below.

Suppose you decide to go to the North Pole. You start out heading due north from your current position. You get to the North Pole, and take a 90 degree turn to the right, and then head due south until you’re at the same latitude you started at. Now you’re hungry, so you decide to head back to your starting point. So you take another 90 degree turn to the right and head west until you’re back at your starting point. Then you turn 90 degrees one last time to face north and return to your starting alignment.

You’ve made a triangle, but you’ve turned a total of 270 degrees. The “true” theorem that a triangle’s angles sum to 180 degrees isn’t true if its axioms aren’t valid. Returning to the seemingly stronger example of 2+2=4, it’s also true under certain arithmetical axioms and groups, but not all. If we’re talking about the group of natural numbers, then yes, 2+2 equals 4. But if we move from base 10 to base 3, 2+2=11. And if we’re talking of the cyclic group of order 3, then 2+2=1, and 4 doesn’t exist.

Mathematical truth is only as true as its underlying axioms, and these examples show that when those axioms are changed, the “truth” falls apart. The claim that common usage could make two plus two not equal four isn’t scary; it’s obvious. We only take this as truth because in common usage, we’re usually talking about the infinite set of natural numbers.

Well, that’s sort of how different languages work. It’s as though English has a rule that says 2+2=7, Malay has a rule that says 2+2=5, and so on. But because the languages have different systems, those different rules can each be valid.

**Languages aren’t “right”.** There’s a point I really want to hammer home with all of this: who says the English we have now has two and two equalling four? The quote at the beginning of this post presupposes that the present form of English is “right”, and that new deviations from it must therefore be wrong. But our modern English is different not only from other contemporary languages but also from its earlier forms. How do we know that Old English didn’t have two and two equalling four and our modern version has it wrong?

Well, we do know that’s not the case, and that’s because languages aren’t inherently right or wrong. By various measures, one can argue that a specific change to a language brought on by common usage is helpful or harmful, but change itself is not inherently bad — or good, for that matter. So let’s stop deifying the language we currently have and demonizing the changes. It might well be that we’ve got the roles reversed.

—

*: I want to note here that I only have a Bachelor’s degree in math, and we really didn’t go too far into the philosophy of mathematics. I might have some mistakes or controversial interpretations with the details of this section (and if I do, please point them out in the comments), but I believe the core points of this section are accurate.

**: “Lines” on a sphere are restricted to circles that span a diameter of the sphere, so-called “great circles”. Any two great circles on a single sphere will inevitably intersect, so no non-intersecting lines exist in this geometry.

## Comments Recently Made