The Naive and the Mature is didactic in a useful way. The main barrier to intellectual growth is the idea of intellectual progress. Many people, for example, consider Zeno’s paradoxes to be solved by Calculus. One philosopher, the author of the book Zeno’s Paradoxes, considers these many people to be quite far from a good understanding of both Calculus and Zeno’s paradoxes. And children can sometimes come up with Zeno’s questions (they are only paradoxes to the mathematician, For Zeno, they are merely questions put to the mathematicians because they reduce a mathematician’s beliefs into absurdity/paradox.) Children would probably never come up with Calculus. There is a certain attitude of naivetĂ© required to reject the very mature and also dishonest representations of movement and the whole/part relationship taught in a standard Calculus class in college.

Someone who is a pedant will naturally follow the pressure to assent to the representations of Calculus. I remember reading about how one math learner was “helped” to believe in the Axiom of Completeness, even though he was a smart math student studying advanced math, he was having a crisis of faith and needed to be “helped” not because he didn’t understand, but because he wanted to disagree with this axiom. This is how a normal Calculus class is dishonest. Disagreement isn’t really allowed.

There is a strange interplay between naivete and mature thought that is required to attain understanding in mathematics. So I can look at the idea of a limit, or the version of likeness used to build our number systems (equality), and if I criticize these ideas, I can be accused of being naive. In the face mathematical complexity and exactitude, its formal structure and its teachers with fancy salaries and forceful minds, all this pretension about mathematics is actually not complex. The pretense of mathematical complexity is as simple and naive as a simple question in critique of mathematics.

In my research I discovered some significant statistical results that students have an easier time learning logic when they are given alternative logics to compare with Aristotle’s logic. Learning Aristotle’s logic is not the point, or we would follow these results and start teaching alternatives to help them understand Aristotle’s logic. The point is assent, control, and to prevent too many people from having a different mind about things. That is why Aristotle’s logic has to be the only one taught. (even though probability is another fuzzy logic, and statistics follows another logic to reject null hypotheses.) These are labeled “theories” instead of logic, to prevent the inevitable dissonance with the message that there is only one kind of deductive logic.

The skeptic attitude is to learn everything about a certain subject, and then to decide that belief in these theories is too rash a decision. A skeptic, after 20 years of learning mathematics, has to reject it in the end, out of a sense of honesty.

It does take naivete to really see the troubles with mathematics. It requires that you trust yourself and can suspend all your learning to look at a simple idea in mathematics, such as a graph representing mathematics discontinuity or continuity. I would offer that this naive attitude is essential for learning, and anyone who stops learning because they know so much, doesn’t know anything.

Mathematical learning is the most forceful subject for conditioning people out of the skeptical attitude. So I have a written a dissertation on mathematics, and am accused of being naive when after all my education, I reject almost all of mathematical constructions. Not because disbelieving in mathematics is simple: skepticism about mathematics is most certainly a more complex attitude to take after so much learning on the subject. But in a way the pedants are right, I do take the air of the Socratic interrogator, and I daresay that attitude is less naive than acceptance of mathematics.

Full disclosure: I failed my Calculus I class twice, even though I learned some of it in high school, but went on to get an A in Calculus II, and a B+ in the easier Calculus III. Then I learned analysis over and over again. I never passed Calculus I, even though I have taught the subject to students many times.