Art vs. science in mathematical discovery

Is mathematics science or art? Mathematics resembles science. In math, data and examples are collected and hypotheses made. There is a difference in the hypotheses of math versus science: in the former they can be proved, but that could be considered a small point. But I have met mathematicians who don't consider themselves scientists, and I think most mathematicians could probably understand this point of view. The underlying motivation of mathematics is often more artistic than scientific: that practitioners may seek out results they find elegant and beautiful, and not yield to justifications about understanding the world. However, mathematics sometimes struggles against its artistic roots and I have a feeling this is becoming a problem.

Science is beneficial to mathematics in numerous ways. But there is also a dangerous aspect to this relationship as well. If mathematics is too closely viewed as a science, it is treated as such in practical ways: how new results are supported, funded, and published. This danger is perhaps more relevant to pure mathematics, but it has serious implications for applied mathematics as well.

I'd like to start with problems in science. Here, by 'problems', I mean the interesting questions that scientists ask, or natural phenomenon that scientists study. Problems motivate the advancement of science and thus the human understanding of the natural world. When a scientific study is undertaken, it is centered around such a problem or phenomenon and work and money is put into collecting the data and analysing it in the context of the problem. This is one of the greatest pillars of strength of science: the world is presenting us with endless new puzzles to solve and it seems that every bit of understanding of the world is useful for our progress as human beings.

In this regard mathematics is quite similar, but it is also a little different. Mathematics still has endless new problems and progress in mathematics also gives us a new understanding of the world, though the 'world' isn't always the natural world. In this regard, mathematics doesn't have that same solid set of goalposts present in science.

That's by no means a bad thing. But it is where the artistic element enters the picture. In mathematics whether pure or applied, a large part of new results is new technique and ideas, rather than application of existing ideas to solve new problems. It is the intuitive choice of avenues to explore than can lead to elegant results or ugly results. G.H. Hardy said that "There is no permanent place in the world for ugly mathematics," and he was commenting on how important it is for the development of mathematics that we keep an eye on beauty as a prime motivator.

This is where the influence of science can be confusing. Mathematics is treated a lot like the sciences these days. In fact, grant proposals with massive projects and endless detail are becoming more common in mathematics, akin to the process of science where a highly complex phenomenon is slowly made understandable by years of research. This sort of "massive project" approach seems to work well in science when there are complex phenomenon that many people want deciphered.

And this approach can work well in mathematics, also. But often times it is pushed too far, justifying endless generalisations and abstractions that become so specialised that very few people will actually read the research. Building vast cathedrals of abstraction is no doubt a useful activity, but so is simplification and random directions as well. By taking mathematics and emphasising the behemoth approach, we constrain the artistic element, even though aesthetic appreciation also can thrive in huge projects.

This is a relevant and recent problem. A hundred years ago, there were problems in mathematics that were of wide appeal and naturally interesting to many mathematicians. The basics of set theory and its clarification pretty much helps everyone. The fundamentals of modules and homological algebra is used everywhere in algebra and needed creation and development. The period of 1900-1980 (and this is a rough estimate) was truly a golden age of mathematics where the fundamental pillars of our subject were created.

Today we are facing a radically different kind of mathematics. Today's mathematics is much more developed. That doesn't mean we should stop exploring. On the contrary. There are still all kinds of problems that need to be understood. But it does mean that we should take a look at the kind of mathematical activity that is being encouraged by the academic system:

  1. The scientific tradition to publish lots of papers encourages further generalisations without regard to artistic merit or quality. That can also happen in science, but it is made worse in mathematics where arbitrary generalisations can be made all over the place in random directions.
  2. The tendency to give resources to huge projects, while often interesting and beneficial, leads to a lack of resources for research on more self-contained results

One might ask: so what? Perhaps this general direction of increasing technicality is what everyone wants, and so there's no harm in letting it proceed. I don't believe that's correct. For one, many new graduate students get discouraged about the massive enterprises in mathematics. Most of this is due to the arbitrariness of their thesis problems. That's not true of everyone: there are still great problems out there and still cool things to do, but there are also a lot of random problems that may not have very much relevance to potential entrants into the field.

I think there is a danger of mathematics losing cohesiveness. For the health of a discipline, there should be a minimum level of collective enthusiasm and energy created by a sufficient number of new students into the field, thereby producing results that interest many people. This might involve new directions in research that are not so technical, and in turn involve a change in how grants are administered to give a chance to new directions or ideas that aren't merely riding the coattails of an existing massive project. Part of this will involve looking at mathematics more as an art rather than a science. As an art, the merit of results should rest more on their aesthetic qualities rather than their relevance to age-old research areas.

Mathematics has reached a new level of development never before seen, and we should pause to consider how we should continue to develop our field for the enjoyment all future generations of mathematicians.

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