The 300-Page Proof that One Plus One Equals Two

With all of life’s complexities, it’s nice to know that some things are so obvious that anyone can recognize them. When the world is falling to pieces everywhere you look, you can bask in a moment of tranquility by reflecting on the certainties of life.

Most of us can see undeniable facts in all areas of human knowledge. There are self-evident truths in the Declaration of Independence. Euclid gave us his first axiom that two things that are equal to the same thing are equal to each other. And, of course, every sane person can build a whole way of life around the indisputable reality that pineapple has no business being anywhere near a pizza.

Then you have people like Alex Gronsky. Alex moved into the neighborhood at the beginning of second grade. He thought he could win any argument by challenging his opponent with the words, “Prove it.”

Alex’s family moved during the summer between the second and third grades. We always wondered what happened to him. Now we know. He was sucked into a wormhole, only to be vomited out at the beginning of the 20th century. He evidently overheard mathematicians Alfred Lord Whitehead and Bertrand Russell say that 1 + 1 = 2 and responded with the words, “Prove it.”

That’s the only explanation that makes sense of the massive work that went into producing the three-volume opus Principia Mathematica. You can read it online here. Well, you can access it here. It would be easier to read the alien scratching allegedly found on the wreckage at Roswell. Consider, for example, page 378:

Admittedly, it’s still more compelling than the William Faulkner book we had to read in Senior College English. Even so, it is evident at a glance that this is not a book to pick up if you are looking for some light reading during your lunch hour.

Principia Mathematica was published between the years 1910 and 1913. It was written with the goal of using pure logic to unify the various proofs for the different fields of mathematics. Ultimately, it turned out that this wasn’t possible, so the whole undertaking was largely a waste of time.

Speaking of wasting time, Principia Mathematica rose to the challenge we suspect was raised by Alex Gronsky, by proving conclusively that 1 + 1 = 2. You may think that is a simple enough task. Whitehead and Russell took 300 pages to do it.

The precursor theorem appears at 54.43 on page 360:

You may have difficulty understanding this. So does everyone else. Part of the problem is that the authors use a lot of mathematical symbols that are no longer used. Some of the symbols are referenced elsewhere in one of the three volumes. Lastly, the writing seems to be intentionally designed to cause headaches, irritability, and an irrepressible urge to track down the person who suggested that we devote an article to this topic, lock him in a small warm room with Michael Moore, and force him to listen to an endless loop of William Shatner singing “Mr. Tambourine Man.”

The entire proof takes 300 pages, but this introductory theorem is the CliffsNotes version of it. Let’s take it line-by-line:

This asserts that if you have two sets of numbers, Set A and Set B, each of which contains one and only one element, they have nothing in common with each other if and only if their union has exactly two elements.

The proof of the above is demonstrated with the lines that follow. The subsequent line states that if Set A = {x} and Set B = {y}, then Set A joined to Set B has 2 elements if and only if x is different from y.

This says that x and y are different from each other only if x and y are different from each other. OK, technically, it’s saying that they are different from each other if the numbers contained in the numerical set that contains only x has nothing in common with the numbers contained in the numerical set containing y, but c’mon…. It’s saying two numbers are not the same as each other if those two numbers happen to be different from each other.

If the above is true, then that means that the entire Set A must be different from Set B, but that is true only if the above is true.

The first conclusion (helpfully labeled Conclusion (1)), with theorems ∗11.11 and ∗11.35, implies that if there exists x and y so that Set A is {x} and Set B is {y}, then Set A added to Set B is equal to 2 if and only if Set A and Set B have nothing in common. This conclusion is labeled (2).

Finally, Conclusion (2), together with theorems ∗11.54 and ∗52.1, implies the theorem that we set out to prove.

This, of course, is the highly-condensed version. If you want more detail, we refer you to the full 300-page proof. We also refer you to this helpful book:

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