According to John Robbins:
In the laboratory the scientist seeks to determine the boiling point of water. Since water hardly ever boils at the same temperature, the scientist conducts a number of tests and the slightly differing results are noted. He then must average them. But what kind of average does he use: mean, mode, or median? He must choose; and whatever kind of average he selects, it is his own choice; it is not dictated by the data. Then too, the average he chooses is just that, that is, it is an average, not the actual datum yielded by the experiment. Once the test results have been averaged, the scientist will calculate the variable error in his readings. He will likely plot the data points or areas on a graph. Then he will draw a curve through the resultant data points or areas on the graph. But how many curves, each one of which describes a different equation, are possible? An infinite number of curves is possible. But the scientist draws only one. What is the probability of the scientist choosing the correct curve out of an infinite number of possibilities? The chance is one over infinity, or zero. Therefore, all scientific laws are false. They cannot possibly be true. As cited above, the statement of Karl Popper is correct: "It can even be shown that all theories, including the best, have the same probability, namely zero."
Many popular Scripturalists have taken this quote and run with it (link, link). For the sake of this post, I am not interested in the argument so far as it criticizes empiricism. My interest lies in the idea that if one must choose from an infinite number of alternatives, his chance of correctly choosing is zero. Let's apply this to the following statement made by Vincent Cheung, a Scripturalist who agrees with the above reasoning:
For every truth, there is logically an infinite number of possible falsehoods related to it or deviations from it. For example, if the truth is 1 + 1 = 2, then, we can deviate from this by saying 1 + 1 = 3, or 4, or 5, or 6, and so on to infinity. This is the case regarding any truth. (link)
If there are an infinite number of possible falsehoods, then consider this analogy:
In the search for truth, an epistemologist will encounter an infinite number of possible worldviews. But the epistemologist chooses only one. What is the probability of the epistemologist choosing the correct worldview out of an infinite number of possibilities? The chance is one over infinity, or zero. Therefore, all epistemological choices of worldviews are false. They cannot possibly be true. As cited above, the statement of Karl Popper is correct: "It can even be shown that all theories, including the best, have the same probability, namely zero."
A few implications from this line of reasoning:
I don't think it's the case that just because one chooses one alternative out of an infinitude of alternatives, his choice is necessarily false. I disagree with Robbins argument, or at least the reasoning he used to reach it. Perhaps a scientific law cannot be know to be true, but certainly not all scientific laws would be false simply due to the presence of an infinite number of false scientific laws.
Furthermore, in the context of epistemology, if it can be shown that the unique characteristics of a given worldview are necessary preconditions for knowledge, then that worldview would be both true and knowable. And assuming a certain view of mathematics, one can use transcendental arguments to refute an unlimited number worldviews, as I point out here.
Speaking of mathematics, the more I study divine omniscience and epistemology, the more striking its importance appears. I'll admit that it's hard for me to wrap my head around it.
Clark would probably avoid this whole discussion by arguing that there are only a finite number of possible worldviews, and this of course has significant implications regarding mathematics. But it doesn't seem to me that Clark had a very good grasp of mathematics. This is just a guess based on an anecdote in Gordon Clark: Personal Recollections, but I think he probably rejected the concept of infinite knowledge soon after he was shown that not all infinite sets are countable.
At any rate, I suspect that a form of mathematical induction - which is actually deductive (link) - could possibly be used to refute what might be called trivially similar and impragmatic worldviews, especially ones centered on [a] number(s).