I don’t understand why you invoke probability theory in a situation where it has no rhetorical value. Your conversation was a rhetorical situation, not a math problem, so you have to evaluate it and calibrate your speech acts accordingly—or else you get nowhere, which is exactly what happened.
Your argument to your friend was exactly like someone justifying something about their own religion by citing their bible. It works great for people in your own community who already accept your premises. To anyone outside your community, you might as well be singing a tuneless hymn.
Besides that, the refuge available to anyone even within your community is to challenge the way that you have modeled the probability problem. If we change the model, the probabilities are dramatically changed. This is the lesson we get from Lord Kelvin’s miscalculation of the age of the Sun, for instance. Arnold Sommerfeld once remarked that the hydrogen atom appeared to be more complex than a grand piano. In a way it is, but not so much once quantum mechanics was better understood. The story of the Periodic Table of Elements is also a story of trying different models.
Mathematics is powerful and pure. Your only little problem is demonstrating—in terms your audience will value—that your mathematics actually represents the part of the world you claim it represents. That’s why you can’t impose closure on everyone else using a rational argument; and why you may need a few other rhetorical tools.
Your confidence in your arguments seems to come from a coherence theory of truth: when facts align in beautiful and consistent ways, that coherence creates a powerful incentive to accept the whole pattern. Annoyingly, there turn out to be many ways to find or create coherence by blurring a detail here, or making an assumption there, or disqualifying evidence. For instance, you consistently disqualify evidence from spiritual intuition, don’t you? Me, too. How can we be sure we should be doing that?
Why not learn to live with that? Why not give up the quest for universal closure, and settle for local closure? That’s Pyrhhonian skepticism.
Great essay!
But, how can a set of ideas be a closed system? It’s ridiculous. If someone were to tell me that Objectivism is closed, I would say, Ha! I just reopened it. Ha! Try and stop me from calling myself an Objectivist if I feel like it! Oh, they can trademark it, I supposed, but if they do, I could rename my system as Reasonablism and explain it as an improved form of what-Ayn-Rand-was-talking-about.
A community of people can close itself off, but ideas are helpless to resist whatever buccaneering minds may prey upon them. This harkens to Buckminster Fuller’s cry that “true wealth only increases”, because true wealth is knowledge and knowledge is infinitely replicable and shareable.