Story

The context of mathematics can be a human context.

In mathematical legend, this result -- that \(\mathbb{R}^4\) and \(\mathbb{R}\) are the same size -- is taken as an example of how mathematical truth can be logically indubitable, while still surprising and counterintuitive. It seems to demonstrate that mathematicians, sometimes unwittingly, can stumble upon and discover truths that no one could have anticipated or imagined.

As the story goes, Cantor proved that \(\mathbb{R}\) and \(\mathbb{R}^n\) have the same cardinality in the summer of 1877, and immediately wrote about it to his mathematical colleague Richard Dedekind (1831-1916), stating "I see it, but I don't believe it!"

(He must have been very impassioned at the time, because he wrote it in French, "Je le vois, mais je ne le crois pas", although both he and Dedekind were German and the rest of the letter is in German.)

This story fits into the larger story of mathematical Platonism, in which mathematical truths exist eternally and universally, independent of humans, and mathematicians are brave explorers and discoverers.

The counter-story, of humanistic mathematics, instead focuses on the context of mathematical results: the communities of practitioners -- with their established methods and ceremonies, their modes of thought, and their tacit knowledge -- and the larger societal context in which these mathematicians work. In this story, mathematics is a part of culture, like music or politics, and it is created by humans.

Both stories capture valid aspects of mathematics, and it is interesting that the story of Cantor and this particular result can be fit into either. The traditional telling, as mentioned above, threads Cantor's discovery into the Platonic story of mathematics. But a close reading of Cantor's correspondence with Dedekind (see, for example, Fernando Gouvea's 2011 article Was Cantor Surprised?), allows for a humanistic interpretation, as we now explain.

Cantor and Dedekind had been corresponding for several years, discussing the question of establishing one-to-one correspondences between different infinite sets. In the summer of 1877, Cantor had sent Dedekind a proof that \(\mathbb{R}\) and \(\mathbb{R}^2\) have the same cardinality (or rather that \((0,1)\) and \((0,1)\times(0,1)\) do, but as discussed in the third module this is an equivalent assertion).

But two days later Dedekind wrote back, pointing out a mistake (having to do with repeating digits of nine, as discussed in our In-Depth explanation section). The way to fix the flaw pointed out by Dedekind, as we did in our In-Depth explanation, requires using the Cantor-Schroeder-Bernstein theorem, which in 1877 had not been proven.

Instead, to get around this Cantor responded, after three days, with a completely different proof, involving several elaborate and technical steps. After four days Cantor was impatient, and wrote again, saying about the tricky steps of the proof, "I can have no peace of mind until I obtain from you, honored friend, a decision about their correctness. So long as you have not agreed with me, I can only say: Je le vois, mais je ne le crois pas."

"Seen in its context, the issue is clearly not that Cantor was finding it hard to believe his result. He was confident enough about that to think he had rocked the foundations of the geometry of manifolds.

"Rather, he felt a need for confirmation that his proof was correct. It was his argument that he saw but had trouble believing."

— Fernando Gouvea

And so we could read in this a type of mathematics where logical rigor is not enough to establish a feeling of certainty. Instead, one needs a colleague or a community to come to agreement and establish validity.

Mathematical truth is established by proofs, but proofs are only valid if they are convincing to the right people.

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You can read a short explanation of why \(|\mathbb{R}^4| = |\mathbb{R}|\), or an in-depth explanation (a rigorous proof), or you can take our word for it, and go read the story or the meta-discussion.

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