This increase is a loss for everyone

Felix Breuer's Blog

Not only beyond Journals, not only beyond Papers. Beyond Theorems.

Feb 27th, 2012

http://blog.felixbreuer.net/2012/02/27/beyondtheorems.html

The mathematical community is discussing how we can leave the journal publishing model behind us. However, putting journals online, making them open access or even leaving them behind us entirely will not solve the challenges the mathematical sciences face in the next couple of decades. A couple of weeks ago, Peter Krautzberger captured this point of view with the simple and beautiful slogan:

Not beyond Journals, beyond Papers.

I agree! And I would like to go one step further. We have to go beyond journals. We have to go beyond papers. And we have to go beyond that which we, as mathematicians, hold dearest:

We have to go beyond theorems!

In this post, I want to make an argument why, from my point of view. This is going to be a long post, but I will try not to rant too much.

The number of new results per year is increasing

The number of mathematical articles - and theorems! - published each year has increased dramatically in the last decades. To give you just one piece of evidence, here is a graph of the number of new research articles entered in the Zentralblatt MATH index each year. This figure is taken from Bernd Wegner’s article in the June 2011 newsletter of the European Mathematical Society.

This trend will continue. The human population on earth is increasing and (on average) becoming more prosperous. So, more people will pursue an academic career and more people will become research mathematicians. And, because new theorems are the currency in which mathematicians currently measure research activity, this means that the number of new (!) theorems proven each year will continue to increase dramatically - unless we change our way of thinking about research.

This increase is a loss for everyone

There are roughly 100,000 mathematical articles on the Zentralblatt index that have been published in 2010. One hundred thousand articles, of which the majority contain (supposedly) new results. I’d go so far as to say that most mathematicians cannot read and properly digest more than 100 articles per year. (For me personally, the number is certainly lower than that.)

This means that most mathematicians cannot even make use of 0.1% of the new results published each year.

This must have at least one of the following two consequences. In my opinion, it has both.

Mathematical research is becoming more and more specific.

Mathematicians have a great talent for inventing ever more particular fields and sub-fields and sub-sub-fields in which we can discover ever more new theorems. If this strategy is successful, then these theorems are truly new and they have never come up in another area before. The flip side of the coin is of course that the majority of these very particular theorems will simply not be relevant for most mathematicians (and they will be utterly irrelevant for the rest of the world). A favorite objection to this argument is that, a couple of decades down the road, these results will become relevant in other fields and that results from today will be made use of then. But how is this supposed to happen if mathematicians can only read less than 0.1% of what is published this year? It is much more likely that mathematicians who, 20 years later, arrive at an equivalent problem in another field will simply solve it again on their own. Which brings us to the other consequence.

Mathematical research is becoming more and more redundant.

Independent rediscoveries of theorems happen all the time. They may happen more or less simultaneously. Or they may happen a couple of years apart. Or they may happen in entirely different fields, in which case it may take a while for people to notice that these results are, in fact, equivalent. Whatever form these redundancies take, their number is going to increase.

One can make the argument, that this is no problem at all. Science is redundant. So if all a rediscovery achieves is to make an old result popular in a different time or a different area, then so be it! But this clear eyed view of the world goes against the grain of our current self-image as mathematicians, where only original (never-seen-before) theorems count.

Also, redundancy becomes a real problem in conjunction with increasing specialization. What is the point of doing foundational research in a narrow area of specialization today, if the people who may need this stuff twenty years down the road have no means of discovering it? The only insurance against this type of disaster is to focus on advertising our small corner of mathematics, in the hope that somebody will remember it when the time comes. But again, our mathematical community is not built on this socio-dynamic view of research.

In the end, both trends, increasing specialization and increasing redundance, have the same effect:

Mathematical research is becoming increasingly irrelevant.

Note that I am not saying that mathematics is becoming increasingly irrelevant! There will always be seminal theorems that have a far reaching impact on both mathematics and the real world. But the work of the average research mathematician is going to become increasingly irrelevant.

Unless we finally stop to believe that the only valid form of “research” in mathematics is proving new theorems.

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