Q&A: On Structures of Perception and Thought
On Structures of Perception and Thought
Question
Hello, honored Rabbi,
My name is A., and I am a student at Yeshiva M. in G.
First of all, I have to thank you for writing the book; it has contributed enormously, beyond measure, to my knowledge and understanding of many important topics.
I have a question about the content of something you said. In the second edition of the book, on page 401 (chapter 2 of the 11th section), you spoke about Wittgenstein and how he shows, for example, that in the arithmetic sequence 1, 5, 11, 19, 29—a sequence for which there are infinitely many possible formulas (for example, a(n)=n²+n-1)—the expected answer would specifically be 41, because the most natural and simple continuation is that the difference between each two numbers increases by 2. The question then is: why assume that this is specifically the most natural and simple continuation? The assumption is such because a person has a certain form of thinking, and that is what defines what a “natural continuation” is, and he assumes that all human beings have the same cognitive structure. This assumption is possible because we all share a common structure of thought, and it is innate (not acquired), among all human beings from all cultures.
I’m not sure I understood the jump from the conclusion that we all have a shared structure of thought to the claim that it is innate and not acquired. After all, it seems very reasonable to suppose that since the study of mathematics began in one place and spread throughout the world (I assume that even in China they know the Pythagorean theorem and learn mathematics in the same way), the method by which we study mathematics—which for that reason is the same—is what causes us to think mathematically in the same way. Conversely, it doesn’t seem so implausible to me that a person who learned mathematics differently would regard different things as natural.
Now one could ask why the original method was specifically that one, and one could answer that the earliest mathematical sages had a shared mode of thought that is not necessarily shared by all the inhabitants of the world, and since they developed the study of mathematics in their own way, that form is the only one we know.
I would appreciate a prompt reply.
With thanks and blessings for a good year,
Answer
It isn’t the mathematics we learned that shapes our perception; rather, our perception precedes the way we learn mathematics. After all, we learn mathematics in light of the concepts already within us, and that itself is Wittgenstein’s claim. He argues that unless all of us already had an innate and uniform way of generalizing, it would be impossible to teach us mathematics, because learning always takes place through generalizing from examples. If we did not all generalize in the same way, those first mathematicians would not have been able to teach all of us their method.