Belief systems versus rationality

Page last updated 11 Jan 2026

Many years ago in my youth I became captivated by born again christianity. I was enticed by the members of a group who struck me as charmingly friendly, charismatic and joyful. They seemed to have something special and I wanted to feel on the inside what they seemed to have as seen from the outside. But as I became more deeply immersed, I began to read the textbook that they said was the rock that their system of beliefs was based on - the bible. It didn’t take long for that reading to start to reveal contradictions within it, contradictions that gnawed at the roots of the beliefs that I had been told were the foundation stones of the house of the one and only existing god - a god whose overriding goal was the happiness of both the human race as a whole, but also of each and every individual human being.

And so, after a time I had to admit to myself that the inherent contradictions showed that the bible could not possibly be an authoritative basis for the beliefs of my group. It became apparent that they simply latched onto the parts of it that they wanted to agree with while at the same time managing to ignore those parts that did not. I came to see that they preferred to live a life based on a lie than a life based on reason. What had been vague shadows became transformed into the realization that they preferred to choose a life that was based on a set of ideas that made them feel comfortable, that made them feel part of a community with the same ideas, with the same beliefs. That they were able to hold beliefs with such conviction that they would refuse to contemplate in any depth any challenge to those beliefs. That they had chosen that over a life based on a continual search after truth. That they preferred the dogma that the truth had been unearthed, eternal, perfect, untarnished, instead of an acknowledgment that one can only ever continue to dig for truth, that one can only ever make a rational evaluation of the evidence that one is able to obtain, and that one must always be prepared to re-evaluate.

I do not deny that the edifice of faith that they inhabited imbued them with what one might call happiness. But such an edifice that imparts such unquestioning happiness on one set of people can, and often does, result in the trampling on the happiness of others. I see the long history of religious conflicts and despair.

Years later I encounter the same problem in a very different guise. I find a community that presents itself to the casual observer as being open to new ideas. However, if you probe a little bit below the surface, you will find that it is only open to new ideas provided that they do not question the underlying ethos. You then see that it operates in exactly the same way as christianity; there are deep-rooted contradictions in their system of beliefs, but they choose to ignore these contradictions, and refuse to even allow discussions about the possibility of such contradictions within their community. In exactly the same way as christians, they hold steadfast to their conviction that they have already found the truth, preferring to uphold a lie than to countenance any deep exploration of the basis of their beliefs.

This community started over a hundred years ago as a small group of mathematicians extolling new foundational ideas, but for some strange reason these foundational ideas have now become almost completely unchallenged as the “correct” foundations for all of mathematics. Now, when a logical flaw in these theories is pointed out, the fundamentalists believe that they can counter it by simply quoting one or more statements from the theories they espouse - in precisely the same way that a bible-thumper will quote a verse from his bible. It does not seem to occur to them that simply relying on authority rather than logic is completely contrary to the basic ethos of mathematics.⁠ For more on this see the appendix below.

The sense of belonging to a select group

After analyzing the Banach-Tarski paradox and discovering that its argument has a crucial error (see The Root Cause of the Banach-Tarski Paradox), the fact that most set theorists believe that Banach-Tarski is a genuinely correct mathematical theorem and that it is not contradictory leads to an observation. There are certain groups of people who share certain characteristics; groups like believers in a 6000 year old earth, believers in intelligent design, conspiracy believers, and so on. They start off by accepting a notion that is not supported by logic or evidence, usually because they have been presented with a set of “facts” that does not reflect the true picture. And here’s a key point: when they come round to accepting the notion they come to feel that they are part of a select group of people who have a special insight into certain things that the ordinary person fails to see. This emotional feeling of belonging to an exceptional select group makes the person have a sense of superiority which blinds them against any rational argument that runs counter to their beliefs. They start to believe in an alternative reality; their picture of the world does not correspond to actual reality. While one night say that no-one has a completely perfect undistorted internal picture of reality, such people are at a completely different level of removal from reality.

When faced with further evidence against their ideas, instead of applying a logical analysis to the evidence, they will invent tortuous convolutions to make it seem that this new contradictory evidence is actually evidence that supports their crazy notions - but the only people who believe their convoluted claims are themselves.

We can see the same sort of mindset in believers in conventional set theory; they first encounter the contradictory argument that one limitlessly large set can have fewer elements than another limitlessly large set. They are fed a collection of arguments that have been carefully crafted to avoid certain facts and which make it appear that the contradictory notion has actually been proved - it hasn’t, see The Diagonal Proof.

Eventually they suppress their initial reservations; once they take that first step to accepting the contradiction they have started on a road parallel to the road dictated by strict logic. They become one of a group of believers that believe that they have a special insight that others fail to see. And like believers in conspiracy theories, believers in a 6000 year old earth and believers in intelligent design, once that first step is taken, it then becomes so much easier to accept further absurdities. It can even be the case that they welcome new absurdities and take them under their wing, since that serves to strengthen the feeling of membership of an elite club that looks down scornfully on the hoi polloi. And when faced with further evidence that contradicts their cherished ideas, they hunker down and construct ever more convoluted arguments to try to prop up their ideas. When they encounter the Banach-Tarski argument, and the claim that one unit is equal to, not only two units, but any finite number of units, they hunker down and assert that, not only is it not contradictory, it is a true result that has some sort of exceptional significance; they will compare it to some monumental discovery such as Einstein’s discovery of relativity, or quantum mechanics.

Conclusion

I would note that it has to be said that many mathematicians continue do their work without any need to reference these ideas, and they seem to prefer to keep their heads below the parapet instead of questioning things. Unfortunately, in so doing, they allow the fundamentalists, as in many religions, to proclaim the substance of the apparent beliefs of the community, and force the direction of that community. The details of these contradictions in these foundational ideas are explored on other pages on this site.

Perhaps I have been over optimistic about the capacity and desire of humans, both collectively and individually, to tread a path towards greater rationality, where both thoughts and actions become ever more based on reason rather than ignorance. Today, looking at the prevalence of absurd beliefs in things such as a 6000 year old earth, a complete worldwide flood, the denial that human activities are irrevocably crushing the possibilities that we leave to our children, and so many other examples, I am forced to the conclusion that I have been naive.

Ultimately, it doesn’t matter; whatever happens, it seems likely that the earth will continue to support life of some sort for many more years, and then, like all things, it must cease to be as it is. The existence of change seems to be the one thing that we can rely on to never cease to exist. It would have been nice to think that one was part of some sort of progression, a progression that humans might almost universally agree to be something desirable, something that one might even say was noble and uplifting. But perhaps all human aspirations are destined to be no more than the dust that time scatters as it marches resolutely onward.

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