The structure of Euclid's Elements, the most important work of mathematics ever

Euclid, who flourished around 300 BC in the generation following the death of Aristotle (384 - 322 BC), wrote the Elements, a magisterial work of geometry. The level of rigor that he achieved in this book was not surpassed until the 19th century, over two thousand years later. And even then, the Elements served as a north star that guided mathematicians.

The level of rigor achieved by the work was in large part a result of its structure. And that structure wasn’t unprecedented or completely Euclid’s innovation. It is likely that mathematicians did something similar to Euclid in their own work before him (but their works didn’t survive the passage of time), and Aristotle draws upon those mathematicians when, in the Posterior Analytics, he outlines how a science should be structured. I’ve talked about that in another post, but Euclid’s Elements is probably the best example of someone living up to Aristotle’s ideal that we have from antiquity.

Let’s talk briefly about Aristotle’s view before we jump into how Euclid structured his mathematical research.

Aristotle thinks that sciences should be organized deductively. Sciences should be laid out as premises whose conclusions follow necessarily from them. So, here would be a very simple example:

1. All humans are mortal.

2. Socrates is a human.

3. Therefore, Socrates is mortal.

This argument is way too simple to actually feature in any interesting science in the 21st century, but it illustrates the idea of arguments laid out such that their premises necessarily yield their conclusions. (If you’re confused by the word ‘premise’, we can think of premises as statements from which conclusions are inferred in an argument.)

You might wonder where the premises of these arguments are supposed to come from, and the answer is this: they should be the conclusions from other arguments. Since the conclusions follow necessarily from the premises, that means that if the premises are true, so is the conclusion — and if the premises are conclusions of other arguments, then that means that the premises are true (assuming that the premises of the other arguments were true, too).

Obviously, this means that there has to be a stopping point, a kind of logical “bedrock” that you can hit. The premises of argument T are the conclusions of argument S, and the premises of argument S are the conclusions of argument R, and so on. So, what are the premises of the very first argument?

Aristotle thinks that those are undoubtable, entirely foundational statements/premises that we know in some immediate way. It’s just not possible to doubt them.

Here’s an example: no contradiction is or could be true. This is something that all of our arguments assume. You can’t prove that contradictions are always false. Any argument you give just assumes that if a statement is false, then it isn’t also true.

When Euclid writes the Elements, he lays things out in exactly this way, and he makes it very clear what the logical “bedrock” of geometry is. These are the things that can’t be doubted and that geometers just need to assume as they go about proving the interesting geometrical truths that they need to.

Euclid thinks that geometrical arguments should be made from three kinds of starting points, initially. Once you’ve inferred or deduced things from these starting points, you can infer things from the results of those arguments.

That’s exactly like what Aristotle thought: if the initial starting points are rock-solid, and you infer things from them, then the conclusions are rock-solid certain. And if those conclusions yield other conclusions in turn, then those conclusions will be rock-solid certain. Truth is transitive like this and is preserved through the argument, assuming that we are properly identifying arguments in which the premises necessarily yield those conclusions.

So, Euclid identifies three kinds of starting points:

1. Definitions.

2. Postulates.

3. Common notions.

Definitions are the meanings of key terms that allow everyone to be on the same page.

For instance, here are the first two definitions of the first book of the Elements:

1. A point is that of which there is no part.

2. A line is a length without breadth.

Postulates, meanwhile, are instructions for construction processes in geometry.

Here is the first postulate of the Elements:

1. Let it be postulated to draw a straight line from any point to any point.

Common notions are premises for more than one science. They are common in the sense that they are shared between different sciences. In the context of the Elements, these notions are shared between geometry and arithmetic.

Here are some examples:

1. Things which are equal to the same thing are also equal to one another.

2. If equals are added to equals, the wholes are equal.

3. If equals are subtracted from equals, the remainders are equal.

Once we have these three kinds of starting points, we can begin to do math and prove things. Our proofs will refer to these definitions, postulates, and common notions.

Once we have more conclusions, we can use those conclusions just as securely as the definitions, etc. Is that unjustified? Not at all! So long as the definitions, etc., were certain, and we made good, substantiated inferences from those definitions, etc., our conclusions will be just as certain. The certainty is transitive and preserved through the arguments.

The crucial point, then, will be ensuring that our definitions, postulates, and common notions actually warrant the conclusions that we draw from them. If we mess up at all and introduce something that doesn’t necessarily follow from the premises we’ve set up for ourselves, then everything else in the argumentative chain is going to be that much weaker.

But the starting points that Euclid has chosen do seem rock-solid and certain, at any rate. It’s hard to doubt his definition of point, for instance. And if we asked him to provide an argument for it, it wouldn’t really seem possible to do so. It’s logical “bedrock.”

In this way, Euclid has set up his Elements in accordance with Aristotle’s restrictions and requirements in the Posterior Analytics. He has conceived of geometry as a body of knowledge that is governed by rock-solid starting points, from which we can infer conclusions that will feature as premises in other arguments.