Asaf Karagila
I don't have much choice...

Back in the days of yore, if one wanted to know mathematics, one would have to go to the university and take a course; or hire a tutor; or go to the library and open a book and learn on their own.

And that was fine. All three options are roughly equivalent, in the sense that they present you the material in a very structured way (or they at least intend to). You don't reach the definition of $$\aleph_0$$ because you defined what is equipotency and cardinality. You don't reach the definition of a derivative before you have some semblance of notion of continuity. Knowledge was built in a very structural way. Sometimes you use crutches (e.g. some naive understanding of the natural numbers before you formally introduce them later on as finite ordinals), but for the most part there is a method to the madness.

Fast forward to the information age. Everything is one Wikipedia, every entry tries to be self-contained with respect to at least a short introduction. You can now learn about Hilbert's Grand Hotel (and his shrewd business acumen), without learning what it means for two sets to have the same cardinality. And that is an essential gap. Yes, the point of the Grand Hotel is to demonstrate that infinite sets can have different properties than finite sets when it comes to cardinality. And yes, depending on the teacher, this can be a segue into the definition of cardinality (although in my opinion not as good as the usual "do I have the same amount of fingers on each hand without counting them?" approach). But nevertheless, in an unstructured learning environment there is a high risk—which is actual reality, as witnessed by the many confused questions on the internet regarding infinity and the Grand Hotel—that the reader is not going to follow through with the definition of cardinality, since this example will already be confusing enough, or distracting enough from being just an example.

Another terrible example is the old Numberphile video about $$1+2+3+\ldots=-\frac1{12}$$. Yes, this can be found in many books and so on. But in all these books, I am sure, it will be mentioned explicitly that this manipulation is not the standard definition of summation, but rather obtained through other mathematically valid methods that have been subjected to abuse of notation. Stripping the context from all this, and just presenting this summation as a magic trick, is a surefire way to confuse everyone who is not already familiar enough with these topics. And of course that it has, I even had students of mine asking me about that back when the video first hit tsunami sized waves across the web.

What's the problem, you might ask? Let those people go online and ask experts! Well, it turns out that there is a reason you don't talk about Ramanujan summation or zeta regularization in the first semester of undergrad. And people come with an honest question, and they expect an easy answer to quickly dispel the dissonance they have between this weird summation and what they know (or think they know). And there are no quick answers which are clear, simple, and not entirely condescending. There is a reason why one has to work through several years of set theory before gaining the actual and intuitive understanding why you need the axiom of choice to prove there is an injection from $$\omega_1$$ into the real numbers. These things are complicated.

Dangerous knowledge usually refers to knowledge that is considered dangerous for other people to have. Like how at some point terrorist organizations realized that if you just teach everyone to make homemade bombs, it's going to be a lot harder to actually stop the bomb production and hinder the organization (and even caused people who just self-identified with the cause of the organization to pick up arms and commit terrible acts).

But in the context of education, I think that a dangerous knowledge is knowledge which you obtain without a structured set up. You are not ready for that sort of knowledge, and you do not have the means of placing it in the bigger picture. I had this problem, through all my life, I have gone to read about things, and I skipped and jumped ahead, and I tried to learn further and better. And every time I jumped and made an unstructured "discovery" I eventually had to go back and correct the err of my ways.

The question, from an educational point of view, is how can you fight this? How can you make sure that dangerous knowledge is kept to a minimum?

One way is to instil into students from a very young age the sense of curiosity and wonderment. I remember reading somewhere about someone who as a kid opened up a book, and read about some problem, then started to work backwards to obtain all the knowledge necessary for understanding it better. It could have been Feynman or Wiles, I am not sure, and it doesn't matter. The point is that when coming across dangerous knowledge, the protagonist of that story "defused" the danger by starting to go backwards and learning the necessary framework.

In today's modern era, where everything needs to be a click-bait-bite-size-immediately-satisfying thing, the above is difficult. It is hard to make sure that people actually sit down to read. People want the information they feel is missing, and not a long list of information they are actually missing. And not to mention that re-educating the whole planet seems like a fairly Herculean task.

But I do think that at least in academia this is possible. It should be possible to try and educate students about this. I think it is important, especially in natural sciences, where there are good chances that the students will go on to research later (either in academia, or elsewhere) to remember this. Because having dangerous knowledge can affect the way you perceive your actual knowledge. It can re-frame your knowledge incorrectly, or shift the importance of something you are currently learning (or about to) from one side of the picture to another, and not necessarily in a good way.

Another option is to educate people about the existence and dangers of dangerous knowledge. Once you are aware that learning something in an unstructured way can be problematic, you can put this knowledge in check automatically, reminding to yourself that you need to know more in order to fully appreciate some anecdotal piece of information that you read online, and heard about. This can also motivate you to go and actually study more about something, which is always a good outcome.

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