Peter Krautzberger · on the web

Why there is need to train bad mathematicians

Felix Breuer has asked me to repost an old my very first public blog post from way back when originally posted on 2008-08-06 over at scivee [Wayback Machine]. Felix wanted to link to it in his interesting post on excellence and creativity which I want you to read right away (albeit, only if you read German…).

Why there is need to train bad mathematicians

Recently, the Olympic games are the hottest topic on the news. I don’t want to talk about the games as such, rather I want to start my blog by talking about a parallel I noticed yesterday while listening to the German public radio station Deutschlandfunk (

They aired a feature on financial support for German athletes. Apparently, one major factor is the support offered by the German Olympic sports union (DOSB). The feature criticized that the DOSB’s support focuses too much on high level athletes and too little on low-level sports, i.e. local clubs, regional networks etc. Even without an understanding of the latter, special part of German culture I think the problem is easy to understand: The lack of support for everyday, John Doe kind of sports clubs of all inclinations (think kayaking, gymnastics, archery, curling) can reduce the amount of highly talented people that get in touch with such activities. A probable consequence is that in the long run, less people will have a chance to discover their talents in these not very popular sports, which in turn will naturally decrease the amount of high level athletes that could have a chance to compete on an international level.

So what does this have to do with bad mathematicians? Well, obviously I think that something similar is true when it comes to training mathematicians. The mathematics community (in Germany) seems to be so focused on producing the highest level of researchers (even when they are not), that they completely forget to train “bad” mathematicians, that is “low level” mathematicians. So what could “bad” mathematicians be good for? In a negative description, a bad mathematician is a mathematician that never added “significantly” to the field, never published in great and important journals, never impressed the greatest minds of the field. Indeed, a bad mathematician might not even know all that much about any current hype in research, might not even know the whole of mathematics all that well, might in fact never have published at all, never proven a publishable result, never even considered trying. So what good could a bad mathematician possibly be?

Positively described a bad mathematician has studied mathematics, yet a degree has nothing to do with it. Instead a bad mathematician learned what practicing mathematics is like, learned the difficulties of the mind that delves into a mathematical problem without yet knowing what a solution could even be like, learned to overcome these difficulties (though maybe “only” at a basic level), practiced enough mathematics to comprehend the depth and artistic worth of this truest art of the mind and has glimpsed at a universe unkown to most. Above all, a bad mathematician shares an enthusiastic interested in mathematics, maybe because it is beautiful, maybe because it is challenging, maybe because it is ancient and intricate. And I believe to be a bad mathematician you don’t even have to know very much about mathematics — at least not in the way this is statement is commonly understood today.

I have often encountered that many find the following idea to be a cardinal sin: to train people consciously in such a way that they will not know enough about “real” mathematics, not know enough to understand the complex development of modern results, not have the technical expertise to follow a complex proof, not have the experience of trying to create unique, independent results, in short not to train them towards an academic career. But let me abuse the image of sports once more. It is not important to be even close to performing a sport on the highest level to develop a deep love and fascination for it (though let me remind you that I am talking about active amateur sport here, not pure fandom). I think one relevant aspect is the ability to recognize, understand and relate to what high level sportspeople actually accomplish — and as anyone watching some unfamiliar sport during the Olympic games will easily remember: this is quite a feat on its own.

So what does that mean for “bad” mathematicians? I am not saying they should only know lots of calculations or lots of logic puzzles or lots of fun applications (this won’t hurt — it’s just not that interesting, really). No, “bad” mathematicians need to know what the work of a mathematician is like — they do need to know some real, modern mathematics. But they do not need to know a lot of areas of mathematics in great detail — since frankly neither does the greater part of the scientific community. Instead bad mathematicians have learned to appreciate mathematics for what it really is — neither blindly loving it nor excessively criticizing the often either arrogant or mothering presentation of mathematics by mathematicians.

Now a lot of people I have met claim that the German (and maybe European) academic system is actually really good at the general level of education, e.g. french Bac compared to a US high school degree, a German Diplom with an anglo-saxon master etc. However, I think this is off the point. The fact is, that when it comes to mathematics, only a few people study it, even fewer get to appreciate it, hardly anyone falls in love with it, so that we are in danger of loosing have lost the large base of people generally interested in mathematics, people who know that mathematics is not just numbers, people that can appreciate the complex work (of art) that mathematicians try to accomplish everyday, people, in short, that have a clue about mathematics.

These are the only people that could make mathematics really popular — well, maybe not actually popular, but they certainly would turn it into a subject like any other, a subject “of the people”, a subject that will seem sensible to most people, even though they don’t know it themselves, a subject that everybody finds plausible to study, a subject to be interested in just as it is interesting to play a game of football with a couple of friends in a public park — it’s not high level, but it certainly is fun to do every now and then (and really, couldn’t we use the exercise more often?) — and then, if you really want to, just go ahead and try to go pro! If you’re lucky there might be a “bad” mathematician in your neighbourhood and somebody’s able to tell you: “I know just the guy you need to talk to”.