The War on Knowledge in Mathematics

War on Knowledge

(April 19: this is a bit an extended version of a remark added here)

Oliver Knill

While it has changed recently, math education used to be very "anti knowledge", especially in the US. The mantra was "YOU SHALL NOT MEMORIZE STUFF", "DO NOT LEARN ALGORITHMS". It was a disaster and we still live the consequences. Some of the brightest minds of our time do not know how to find the solution of a quadratic equation or do not know the definition of trig functions because they were taught "LEARNING BY HEART IS EVIL". No other science has adopted this devastating philosophy. It is maybe only topped only by general religious or ideological anti science movements. So, why is it important to know algorithms and know definitions, and to know some tricks? Because all of science is made of "knowledge" (it is in the name of it even). The dream of AI of "figuring out things on your own" was also not that effective for machines. It also did not work for education. If "Jenny and Jonny" do not know stuff, then even doing the most basic things become frustrating. It might disappoint to see this, but much of modern AI's success comes through "knowledge". Alexa, Siri, Wolfram Alpha, Cortana, IBM Watson etc, KNOW a lot of stuff or KNOW to look up stuff and they have become so good at it, that they will soon pass the Turing test with flying colors. It has been known since the early AI research that "knowing how to solve a problem is the best and fastest way to solve it" (as pointed out by Marvin Minsky) See this working paper, where we worked for one year to teach a machine mathematics (the bot had access to various computer algebra systems and web etc)). Not that teaching to "figure out things is bad, what happens is that:

Creativity can only blossom, if there is a fertile ground of knowledge available already. Knowledge is the soil on which creative plants can grow!

It will become scary if AI will figure that "meta knowledge" out (they will) and learn and learn. It is an old fear about AI: You certainly have seen the cult movie "Wargames" with Matthew Brotherick, where the machine learns.)

But it is important to know, not just being able to "look up". There has to be a balance: of course, nobody should learn the 100x100 multiplication table by heart, that is non-sense, but knowing the 10 x 10 multiplication table is a useful "prototype", like TIC-TAC-TOE is an important prototype in Game theory. It is a fertile ground to discover laws about prime numbers or understand the multiplication table in an abstract group. A chemist who does not know a wide variety of compounds and mechanisms will not be able to figure out a new one. Knowledge is a complex network and it becomes more powerful if more nodes are available. It can grow faster and discover new nodes faster if it is larger. More knowledge also helps to consolidate things. Knowing category theory in mathematics for example allows to see many special cases as a general phenomenon and many collapse knowledge nodes to one, making the knowledge network more effective. I learned PASCAL in college from a professor who would write down the specifications of the language on the blackboard, I think it is still the programming language, I understand best, maybe because of a bit of a "mindless" approach, but feeding the specifications with insight and examples and general principles, it was quite effective together of course with lots of programming beside that lecture). To wrap it up, not realizing the simple fact that "daemonize knowledge is evil" is in my opinion one major reason why K-12 education in the US does not do so well internationally in tests. An other reason of course is the low pay for teachers, especially in K-12. It should at least double. In Switzerland for example the pay is much higher. A third reason is the diversification of knowledge. Computer science and statistics for example have entered earlier in the classrooms (which is good), but that of course takes away time from actual mathematics. Time is bounded and the game of choosing subjects is a constant sum game. If you add more calculus for example, then this hurts geometry. And we see that. Geometry knowledge for example is in general much lower than 20 years ago. But students also know already more calculus than 20 years ago when they come to college.]

Updates:

April 11: 2019: An article in the German "Die Zeit" has the title Knowledge beats intelligence It is what AI researchers have observed since decades. It makes the ``war on knowledge" even more questionable. Of course, one has to have a balance between ``knowing" and ``deriving" or ``understanding". Apropos German wisdom: David Hilbert, one of the giants said ``Wir müssen wissen, wir werden wissen". (We must know, we will know). Of course, Hilbert had an other meaning in mind, but still, the key is "to know". The war on knowledge can be rephrased as ``Ignorarmus et ignorabimus", (we do not know and we will not know). Also this expression, coined by the German Emil du Bois-Reymond was muttered with an other meaning as he wanted to express that our scientific knowledge is limited. But the two sentences mark the topic.

Update April 15: 2019: as just somebody pointed out to me, there are pedagogical perils in allowing cheat sheets in exams (which is often done today): but teachers who give tests in which one can look things up often make them so long and hard that it is impossible to do them by looking things up. One is just slow in looking things up.

An other remark: one can see the perils of ignorance also in programming: if one masters a programming language so well that one does not have to look up templates for everything, the coding is much more pleasant, the code is better and better structured. It is not a copy-paste mess of various programmers but done from the ground up and with insight.