to understand vs to understanding

Question

I was wondering why we use "understanding" instead of "understand" in this sentence? can we just use "understand"?

Math is the hidden secret to understanding the world.

Answer 1

understanding the world is a gerund (it doesn't include "to"). The gerund can stand on its own as an object.

If you want to use the infinitive (a different use of "to"), you need an auxiliary verb, for example:

Math is the hidden secret needed to understand the world.

Answer 2

What may be confusing you is that we can say the key to understanding the world.

The reason is the noun key, which can take a to- phrase. So this is

the key [to [understanding the world]]

where the preposition to is followed by the -ing clause "understanding the world". The phrase does not contain "to understanding" except accidentally, because the words happen to come together.

But this option is not available with the word "secret": we don't usually talk about "the secret to [something]". So we cannot say the secret to understanding the world. (People will understand you if you do say this, but it is not something a native speaker would say).

Edit: This ngram shows that I'm out of date. "The secret to understanding" hardly occurred before 1960, but since 1990 has hugely outstripped "the secret of understanding".