Extraordinary vs impressive

When people put forth extraordinary claims, this evokes in me a very specific feeling. It's hard to put this feeling into words, but I might express it as "What do you take me for?" or "Is this supposed to impress me?" It is a feeling of disappointment. I am disappointed if there is not enough evidence for an extraordinary claim. I am even more disappointed when people seem to think there doesn't need to be any evidence. The claims become ever more fantastic, as if this would make up for the decreasing likelihood.

But I am rambling. Perhaps this feeling is best conveyed not through words, but through math. Math is the highest art form, or at least the highest art form accessible through Mathematica. The following graph can express my feelings.

The important thing to notice is that the more extraordinary a claim is, the more impressed I am, but only up to a certain point! After that point, further extraordinariness merely strains credulity. The more extraordinary a claim, the less impressed I am. Because then it's just less likely to be true. In fact, I would say that it's exponentially less likely to be true.

Why is this so hard for people to understand?

But I think my graph is missing something. Evidence! Evidence is pretty important. After all, extraordinary claims can be impressive if they have extraordinary evidence. In fact, you might say that I'm more impressed by an extraordinary claim than an ordinary claim if it has extraordinary evidence supporting it.

This can only be expressed in a 3-d graph!

The height of this curve represents the how highly impressed I am, while the two horizontal axes represent the extraordinariness of the claim and the extraordinariness of the evidence.

These graphs, by the way, are based on the function I(x,E) = x/(e^(x-E)+1), where I is the impressiveness, x is the extraordinariness, and E is the evidence. One interesting feature of this function, is that even if the evidence is zero, there exists a particular nonzero value of extraordinariness which gives you maximum impressiveness. This particular value would be a fundamental constant of skepticism! But then again, my function is just a model, and I could have picked any number of other curves to serve the same purpose. So I guess we shouldn't read too much into the details.

What's that you're telling me? You're saying that normal people don't spend their free time modeling skeptical expressions with math equations?