The essence of mathematics lies precisely in its freedom. This statement from Georg Cantor is quoted so very often, and perhaps this is because of the surprise coupling of the words mathematics and freedom, or because of the implications of the word essence, which calls to mind other words like intrinsic, inherent or something that is by nature. Various biographical accounts of Cantor will explain that, for him, the statement had meaning on more than one level. It does characterize his own experience with mathematics but it is also an admonition to traditionalists who might thwart the creativity of new ideas. I found a very nice discussion of Cantor’s battles with his contemporaries in a paper by Joseph W. Dauben on The Battle for Transfinite Set Theory. He says about Cantor’s statement:
This was not simply an academic or philosophical message to his colleagues, for it carried as well a hidden and deeply personal subtext. It was, as he later admitted to David Hilbert, a plea for objectivity and openness among mathematicians. This, he said, was directly inspired by the oppression and authoritarian closed-mindedness that he felt Kronecker represented, and worse, had wielded in a flagrant and damaging way against those he opposed.
There is no room in mathematics or an ‘authoritarian closed-mindedness.’ The history of this intellectual conflict, as Dauben among others presents it, is very interesting and shows us something about the hurdles mathematics had to clear before it could take us further. The philosophical message is consistent with my own early experience with mathematics, where I found the boundless reach of its conceptual possibilities invigorating and reassuring. I continue to wonder about how we find those useful generalities from particulars– like the real number from quantity, equivalence from equality, manifold from space, n-dimensional from 3-dimensional. The process can be very tedious and laborious, involving many minds and many centuries. The exploration of relations among ideas is enough to capture the full attention of the pure mathematician. As Alain Connes has said:
Investigating these, one truly has the impression of exporing a world step by step….
Their utility is the focus of the applied mathematician. Mathematics seems to carve out a path that leads from the world of substances (where thoughts find relationships among things we experience) to the world of purely thoughtful relationships (which often reveal something new about the things we experience).
Cantor’s observation about the difference between a countably infinite collection and a non-countable infinite collection is grounded in a very thoughtful analysis of something that is not found in our physical experience, yet is completely tied to the very concrete experience of counting. For many of his contemporaries his work was laughable or, worse, destructive. But his analysis of the infinite relies on something fundamental and at the same time submits to the requirements of logic. This is probably the source of its utility.
In the book Labyrinth of Thought, Jose Ferreiros Dominguez looks at the history of set theory and gives us this from Cantor:
By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law…
This is the notion of a class of things, an aggregate, that has penetrated and influenced many fields of mathematics. These collections of objects, also objects themselves, describe a new starting point and make an analysis of the infinite possible. The idea that the collection is bound up into a whole through a law refers, I think, to the logical necessities that give the theory of sets meaning.
The essence of mathematics is freedom perhaps because it formalizes thought processes with logic and deduction, while letting the intuition play freely with metaphors. This has the remarkable affect of allowing the imagination to perceive beyond the limits of the senses. This sounds much like something Hermann Weyl, one of Cantor’s critics, once said:
We stand in mathematics precisely at that point of intersection of limitation and freedom which is the essence of man himself.