Imagine a large-scale mathematical society whose aim is to work together to broaden the scope of demonstrated mathematical truths. The way they would go about this is by building upon the theorems that have already been proven: finding new relations and isomorphisms between existing theorems, and proving new ones. They wouldn’t all work on the same problems, of course; there would be a division of labor, with different individuals or groups tackling a wide variety of questions and projects. What would make this distributed effort possible is that they will all be building on the same foundation of previously established theorems — so they know that as they strike out in new directions, whatever results they come back with will, if they have done their work carefully, will be consistent and coherent with each other’s work, and with the great edifice of mathematics that already exists.
The reason this sort of distributed cooperation is possible is that the entire framework is built, brick by brick (i.e., theorem by theorem), upon a consistent set of axioms. These axioms are few in number, and they are by definition unprovable. (If they could be proven, then they wouldn’t be axioms, but theorems, and they would in turn have to rest on even more fundamental, and ultimately unprovable, assumptions.) In other words, the regress has to stop somewhere, and where it stops is with a bedrock of postulates whose truth, although unprovable, is self-evident.
This means that for our imaginary mathematical society to be able to hang together and do useful work, its members have to agree on its axioms. (This is also what makes it possible for them to check and correct each other’s results.) If they cannot agree on axioms, they might as well split up, because they will disagree about everything else they try to accomplish, and the whole thing becomes at best a waste of time, and at worst an arena of bitter conflict.
The same is true of societies. If there is a shared framework of moral and civic axioms, then policy-making can proceed in a generally orderly and productive fashion. There will of course be disagreements, sometimes lively ones, about balancing priorities and choosing methods, but in broad terms the goals of the work will be in general alignment, because everyone involved is working from the same set of axioms.
In our ongoing conversation over at the Maverick Philosopher’s website, commenter Joe Odegaard offered a short list of aims that, throughout American history, have been our sturdy, and axiomatic, foundation:
• Strengthen the family
• Return self-governance to the people, and reduce the administrative state
• Defend national sovereignty and borders
• Secure liberty and freedom.
He then directed our attention to an article at Salon that decries each and every one of them as a wickedness to be resisted, as horrifying manifestations of “Christian nationalism”.
Lincoln, quoting Matthew 12:25, said “A house divided against itself cannot stand.” The truth of that postulate should be self-evident as well — and if isn’t already, I believe it soon will be.
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There is also this from Justice Clarence Thomas, speaking a few years ago at the dedication of Christ Chapel at Hillsdale College. He began by quoting John Adams’s address to the Massachusetts militia in 1798: