Why Mathematics is Boring (2007) [pdf]

I do not disagree at all with you. Arnold is my mathematical hero and his advice and insight is invaluable. I understand that math is done starting with proofs, and ending with definitions and axioms.

Yet, I have witnessed many young mathematics students that could not write a concise, self-contained proof, nor understand its value. I certainly was one of those, and this advice helped me. For these people, it is helpful to learn how to organize your thoughts in an over the top, nearly bourbakist, formal way. Also, the correctness of proofs is much easier to check this way, and any incorrect or illogical stuff sticks out immediately. Then, once you have written your stuff in that dry style, you can add some glimpses of discourse that become much more valuable than if you had started with some informal hand-waving. This is pretty much the writing style of Arnold: his proofs are breath-takingly concise and elegant, and there is an insigthful discourse around them. The proofs without the discourse stand on their own, but the discourse alone would be worthless.

I like your analogy of climbing the cliff and pulling the ladder. But there is another cliff that goes even higher and you needed the ladder for that one! Of course you need to help others to build their own ladders.

> Somewhere not too far away, a student in the class you're teaching cries.

Maybe, maybe not. In any case, I agree that you cannot teach math in a purely bourbakist style. I prefer a "visual" style like that inspired by the books of Arnold, Strang, Needham, and I am the sole teacher in my lab that seriously uses the word "amplitwist" to refer to the complex derivative :)

Reminds me of my general principle of the best solutions being "square" in the sense that the effort you expend on them is quite similar on any axis. By the time you're past abstraction's diminishing returns for the given problem, you're spending way more effort on that than anything else (which is understandable - it's fun!) and your efforts aren't "square" any more.

(A toy example of the 'square' thing - imagine you have to make 10,000 widgets for 1 dollar each. Making them all by hand for $1 without optimizing the process would be inefficient. So would spending $9,999 to build a machine which could make widgets for $0.0001. But spending $100 to optimize production so you can make the widgets for $0.01 each is a massive win ($200 cost vs. $10,000 for the other alternatives.)

Imagine you are trying to explain this to a 15 year old.

If math is going to make sense to kids we can't resort to explanations that sound like "and then a miracle occurs".

BTW, I am not being critical of your answer. What I am saying is that there are these corners in seemingly simple math that have me scratching my head when it comes to explaining the concepts to a kid in a manner that makes sense and isn't circular. I have yet to find good answers to these questions.

Kid: What does the 10th. root of n mean?

Dad: It's the number, let's call it x, that, when raise to the 10th power is equal to n

Kid: So: n = x * x * x * x * x * x * x * x * x * x?

Dad: Yes! You got it!

Kid: How do you calculate it?

Dad: Well...

Kid: What if it is the 10.1 root of n?

Dad: Well, that's a little different...

Kid: How?

Dad: It's the number than when raised to the p-1 power times the base raised to the fractional portion of the power is equal to n

Kid: What's the fractional portion?

Dad: For the case of p = 10.1, it's 0.1

Kid: x * x * x * x * x * x * x * x * x * x^(p - int(p)) then?

Dad: Yeah.

Kid: How do I calculate x to the 0.1 power?

Dad: Well, you could use your calculator...(now starting to sweat)

Kid: How does the calculator do the math. You know, like when the math teacher says "Show your work"

Dad: Well, you could use logarithms...

Kid: What are logarithms?

Dad: A better method could be to use Newton's method. Here:

https://en.wikipedia.org/wiki/Newton%27s_method

Kid: It says: "start with an initial guess which is reasonably close to the true root, then to approximate the function by its tangent line using calculus, and finally to compute the x-intercept of this tangent line by elementary algebra"

Dad: Yes...

Kid: I don't know calculus. Is that the only way? I just wanted to understand how to calculate the 10th root of a number?

Dad: OK, let's try this. I just threw it together:

    # Calculate the exp root of n using a binary search
    #
    def root_binary_search(n, exp):
        # Return b, which is the exp root of n
        # b**exp should be equal to n
        #
        min = 0
        
        # For exponents  n:
                    max = b
                else:
                    min = b


    # Tests
    print(root_binary_search(4, 2), f"  result should be: {4**(1/2)}")
    print(root_binary_search(16, 2), f"  result should be: {16**(1/2)}")
    print(root_binary_search(5, 0.1), f"  result should be: {5**(1/0.1)}")
    print(root_binary_search(2, 10), f"  result should be: {2**(1/10)}")
    print(root_binary_search(4, 0.25), f"  result should be: {4**(1/0.25)}")

Kid: So...you are telling me to guess?

Dad: Yeah...? (looking embarrassed)

Kid: And to accept an error? 4-squared is 256, not 255.998046875?

Dad: Well, you have to understand that with a binary search...

Kid: And, did you see what happens if I run this case?

    print(root_binary_search(4, 1), f"  result should be: {4**1}")

Kid: Dad?

Dad: I have to get back to work. Why don't you ask your math teacher tomorrow?

I think you missed (or at least aren't building off of) the point of my comment.

I'm not questioning the pedagogy in the original comment, just the specific math. x^(1/10) takes a value of dimension [length^10] to a value of dimension [length].

Interestingly, I think you could take this in a few aesthetic directions. From a pure math perspective, this is where you can start talking about set theory, cardinality, etc. Irrational numbers are infinite sequences of digits we can only approximate. From a computer science perspective, you can talk about Newton's method, and also make the argument than an algorithm which converges to a number is a quite meaningful way to describe that number. Some would also add a caveat of 'efficiently' converging. And combining the two perspectives together, you can discuss that the set of computable numbers are of a lower cardinality than the set of reals -- aka 0% of real numbers are computable. You could also look at things from a geometrical perspective, and show how roots higher than square roots are tied to higher dimensions are are nonconstructible in the plane (this might be very hard to show!).

>I understand what you are saying, believe me.

Do you? Almost nothing you've said has any relevance to my original comment.