## Is math an art or a science?

I left academic math in 2007, but I still identify as a mathematician. That’s just how I think about the world, through a mathematician’s mindset, whatever that means.

Wait what *does* that mean? How do I characterize the mathematician’s mindset? I’ve struggled in the past to try, but a few days ago, a part of it got a little bit easier.

I was talking to my friend Matt Jones – an historian of science, actually – about the turf wars inside computer science surrounding functional versus object oriented programming. It seems like questions about which one is better or when is one more appropriate than the other have become so political that they are no longer inside the scientifically acceptable realm.

But that kind of reminded me of the turf war of the bayesian versus frequentist statisticians. Or the fresh water versus salt water economists. Or possibly the string theorists versus the non-string theorists in physics.

What’s going on in all of those fields, as best I can understand, is that different groups within the field have different assumptions about what the field may assume and what it’s trying to accomplish, and they fight over the validity of those sets of assumptions. The fights themselves, which are often emotional and brutal, expose the underlying assumptions in certain ways. Matt told me that historians often get at a fields assumptions through these wars.

Here’s the thing, though, math doesn’t have that. I’m not saying there are no turf wars at all in math, there certainly are, but they aren’t political in nature exactly. They are aesthetic.

In the context of mathematics, where nothing can be considered truly inappropriate as long as the assumptions are clear, it’s all about whether something is beautiful or important, not whether it is valid. Validity has no place in mathematics per se, which plays games with logical rules and constructs. I could go off an build a weird but internally logical universe on my own, and no mathematician would complain it’s invalid, they’d only complain it’s unimportant if it doesn’t tie back to their field and help them prove a theorem.

I claim that this turf war issue is a characterizing issue of the field of mathematics versus the other sciences, and makes it more of an art than a science.

To finish my argument I’d need to understand more about how artistic fields fight, and in particular that their internally hurled insults focus more on aesthetics than on validity, say in composition or painting. I can’t imagine it otherwise, but who knows. Readers, please chime in with evidence in either direction.

I’m not sure there’s an absolute right answer.

Take Cantor’s set theory as an example.

Are the mathematical / philosophical objections to Cantor’s theory raised by Brower, Poincare and others more similar to (i) the objections that the French art establishment had to the early impressionist painters or to (ii) the objections such as “god does not play dice” that were raised by some in the physics establishment about quantum mechanics?

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I think you are making a philosophically unfair comparison, a sort of apples versus oranges thing; I’ll focus of math and physics. You are taking a formalist view of math, and comparing it against a realist view of physics. I would argue that this is unreasonable, you have to either compare Platonic to realist or formalist to operationalist. Let me decompress that a bit.

When you write:

You are taking a formalist stance towards mathematics. Not all mathematicians would agree. For contrast, a Platonists would say that there is one true reality out there (a privileged model), and we pick formal systems as lenses on that true reality. Similar to how a realist in physics would say “string theory really describes the world; it has the one true metaphysics”.

Of course, the Platonist knows Godel’s theorem (in fact, Godel was a beautiful example of this Platonism done very carefully) and so he knows that any lens he picks is imperfect in that it is incomplete. That being said, he still can believe that some formal systems simply don’t capture mathematical reality but are just fictions, just like a string-theorist would say that loop quantum gravity can accidentally get you some right calculations but its metaphysics is not the “true way the world is”. And I think you actually see this in math, for instance when some set theorists discuss the axiom of choice, or CH. It isn’t always about a formalist: anything goes and pick the one that is funnest. For another example, I discuss this in the context of the Church-Turing thesis.

Similar, there are people in science (I sometimes count myself among them) that simply say that theories are just little fictions we tell to make predictions from experimental set up to measurement. They would treat different theories in the same way as you describe mathematicians treating formal systems; well, what does this theory let me predict? Any theory is good to go as long as it gets me to building bridges.

As for the politics of everything. I think that is fine. It is good to have people with different base assumptions because that leads to a plurality of methods. As long as people don’t go about persecuting others (or denying them jobs) for not believing in the God-given metaphysics or formal system.

Finally, I don’t think art and science should be treated as two opposed categories, for me both are just different sorts of narratives. Art is a narrative that is constrained by human nature, and Science is a narrative that is constrained by empirical observation. I guess I would say Math is a narrative that is constrained by proof (i.e. formal justification).

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I second your comments. I think mathbabe just accepts the Hilbert description of mathematics as a given. I will add another example from Foundations. (Not every mathematician studies Foundations, much less the history thereof), There was a significant dissident movement, the Intuitionists, which fundamentally objected to pure existence proofs. They maintained that only constructionist proofs should be accepted.

A second, more recent example would be the battles commencing in the mid-70s over Catastrophe Theory.

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David Aubin’s catastrophe theory history has a map showing the history http://residencyprograms.eu/read-more/david-aubin-a-cultural-history-of-catastrophes-and-chaos-around-the-institut-des-hautes-etudes-scientifiques-france-1958-1980-1998

Is this what you mean by battles?

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That paper looks interesting but I am not familiar with it. From its abstract it doesn’t seem to be exactly what I meant. Prompted by this discussion I am reading http://view.officeapps.live.com/op/view.aspx?src=http%3A%2F%2Fcob.jmu.edu%2Frosserjb%2Fcat.rise.and.fall.doc which seems to be covering what I meant by battles better.

It occurs to me that I may be using a broader definition of mathematics than mathbabe. I recall a wayback seminar in which someone whose field was logic made a distinction between physics and mathematical physics, a difference those working in the field were largely unaware of*.

*Not entirely unaware. For example, mathematicians who came at problems in for example fluid dynamics or differential manifolds tend to use different terminalogy and techniques than physicists working of the same problems.

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This was a good paper on Zeeman’s stock market model, and the ups and downs of catastrophe theory’s applications. I thought the pre-history was illuminating, since Aubin doesn’t touch on it, even in 800 pages!

Part of the problem is the technical barrier with the math — I was studying Morse Theory at about the time this was happening, and could only follow it up to a point. Cobb, for example, takes the mathematical aspect to another level, and success is predicated on how applicable it is, and how useful the applications are and for how long.

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I think the turf war in computer science is primarily the result of computer science being very young and that there is simply a lack of understanding of what factors improve programming speed and quality.

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Although I largely agree, there are some exceptions. Some people are uncomfortable with infinity/infinities for any number of reasons, so for example they argue often about the axiom of choice. Others (the intuitionists) dislike the law of the excluded middle or issues of computer-verifiability, so for example they consider what theorems hold in topoï other than the one of sets. The original arguments were genuinely about what was acceptable in mathematics, and were not perceived as aesthetic (in the sense of your post) at the time.

Over time, the methodology of mathematics itself transformed both sides into valid but separate mathematical systems, thereby partly subverting the argument into aesthetics. (The fact that mathematics does this is pretty amazing.) On the other hand, the aesthetic choice is still somewhat rooted in questions of underlying assumptions. To explain Socratically, how do you yourself seriously explain to someone why, say, the axiom of choice is *actually* true/false?

By the way, is it possible for there to be exceptions that are *not* of an essentially logical character?

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I agree with RS here, and this gives me a chance to plug Amir Alexander’s excellent book INFINITESIMAL, about the pitched battles over the foundations of calculus. When I go to machine learning conferences, people yell at each other a lot, in a way you’d never see at a math conference, and I think that’s because the field is so new that lots of foundational disciplinary questions — what is this subject actually about, what are our goals? — are up for grabs. In math, by this time, I think there is rough agreement, if not total consensus, about this kind of question, and so our arguments are about how to achieve those goals. I think arguments of this kind are typically less yelly.

I know a lot of poets and if anything the poets are more like the physicists than they are like us.

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I’m old enough to remember when machine learning was more or less part of mathematics. The battles between the schools were not just political, they were practically religious. This was fueled in no small part because the controversies involved esoteric fragments of philosophy, psychology and mathematics. It would take you several years of dedicated study to truly understand any party’s position and then, well, you were committed.

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I too think you’re mixing a bit of apples and oranges here. And both “science” and “art” are actually broad, somewhat ambiguous terms open to interpretation anyway. But I still think xkcd essentially nailed it (most of you have no doubt seen this, but in case any haven’t):

http://xkcd.com/435/

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I agree with RS too: all the disciplines you mention – statistics, economics and computer science – are young, at least compared to math. And I’m not sure that it’s just that the foundations of the discipline are still in play, but rather that the people who do it are still caught up in battles for independence and legitimacy of their subject. Stats and CompSci needed — fairly recently — to establish that they were proper, independent branches of inquiry; to address questions like “Why isn’t Stats just part of applied math?” or “Aren’t computers best handled by mathematicians and electrical engineers?”

In the case of Stats I think the battle is more or less won and young statisticians are now sufficiently far removed from those early struggles that they don’t need to carry such big chips on their shoulders: my anecdotal experience is that they seem much less argumentative than previous generations (though this may just reflect my ignorance). In CompSci the living memory of the first generation is just beginning to fade and echoes of those early struggles still colour current practice in the field. Economics is of course much older and maybe their issue is different: my outsider’s view is that much theoretical dispute in economics arises from either (a) an unwillingness to face a certain irreducible uncertainty about human behaviour or (b) masks deeper, genuinely political (in the sense of real, how-to-organise-society politics, rather than academic politics) disputes about who should have power and what they should be able to do with it.

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It all depends where the observer is located — outside, inside that social networks.

All this is handled adeptly by Stephan Fuchs, Against Essentialism. http://books.google.com/books?id=5tSc4uLimrkC&lpg=PR13&ots=VwBL5IrUaT&dq=%22against%20essentialism%22&lr&pg=PR13#v=onepage&q=%22against%20essentialism%22&f=false

The closer one is to the “mature core,” then the more rigid and the more real the essentials look and feel, including essential differences.

All this is a sociology of knowledge question, for the broader context.

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Wikipedia has an article on the sociology of knowledge, but leaves out Karl Marx for some reason. http://en.wikipedia.org/wiki/Sociology_of_knowledge

Notice the ties with Science and Technology Studies, etc., at the very bottom of the article.

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Jackson Pollock is a good example of how artistic fields fight — through art dealers, gallery owners, museums, and Lee Krasner, his curator. There was also the Peggy Guggenheim, the colorful nymphomaniac, whose entryway mural became the stuff of legend.

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I tend to think most things are composites of art and science, and the unique blending of the two by a particular practitioner. From the standpoint of sports psychology and performance (my field) you start with a unique ability to handle a basketball – dribbling, passing, shooting. Then you develop the skill – what is your shot percentage from the free throw line? How do you increase it – can you increase it in critical moments, specifically, or do you have a high shot percentage but you still struggle to hit the critical shots? Part of setting goals for measurable increases in skill is the art of assessing the intrinsic make-up of that persons structure – physically, emotionally, mentally. But – can you create ability? Can you fail to develop skills even though you have ability? If you have met one athlete, you have met one athlete. So, I would say I do create a universe with a logic matrix to solve these “problems.” And of course, I haven’t yet added in the human dynamics of groups, competition, building skills vs winning, etc. Not sure if this explanation fits in with your query, but I know I enjoy your columns because they make me think in different terms and if, and how, I can apply mathematics structured thinking into the art and science of sports performance. So, thanks for that!

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This. I also think we are still struggling to understand CS at a fundamental level itself. The OO vs FP is the perfect example it not only shows we don’t yet understand how to solve problems in a repeatable fashion or what about the problems cause the different methods to fail or was it even the problems. Though these maybe just different shades of your answer.

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Actually, there is an art to science as well. When one perceives elegance in method or result, THAT is art.

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While I agree that mathematics is now philosophically “house broken” it wasn’t always that way. For an excellent account of when mathematical disputes involved actual bloody consequences read Amir Alexander’s excellent history Infinitesimal. If your tastes are inclined toward revisionist views C. K. Raju holds there are still “religious” points of view in mathematics. As for art versus science: it’s art until people find uses for it in other sciences.

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An art or a science? Perhaps both or neither — and at different times? I think Math is closer to the arts (or the humanities) than any of the (other) sciences. It is fundamentally about solving problems and understanding ideas — unlike the sciences which aim to understand the real world. The main criteria which turf wars center around are importance rather than truth. This is partially because mathematicians don’t spend forever disagreeing about whether something is true: in Galois’ last letter, he asks that the recipient ensure that his notes are sent to Jacobi to ascertain “not the validity but the importance of my work”.

At the same time, math is not just about aesthetics. Whereas the fundamental rules of art are constantly changing, the principle rules of math don’t — or if they change, they change much much slower.

What math is today is very different than it was 2000 years ago. It’s only in the last couple hundred years — with the modernization of math — that you could say that math is really apart from the sciences. To a modern mathematician, the question of ‘valid’ assumptions as a much narrower meaning than what it would have meant to Euclid or his predecessors.

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A related discussion on the unusually frequent ability of mathematics to reach consensus is at http://m-phi.blogspot.com/2011/10/inconsistency-of-pa-and-consensus-in.html

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p.s. for what it’s worth, Atiyah thinks the answer to the question is “both”: http://www.ams.org/staff/jackson/fea-atiyah.pdf http://www.ams.org/journals/bull/2006-43-01/S0273-0979-05-01095-5/S0273-0979-05-01095-5.pdf

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Maybe it’s just me and I’m nuts but it appears that anything out there that is becoming complex is now becoming “a science” and I don’t mean to take away from this topic at all, but does anyone else see that? I wrote a post the other day asking when health insurance is going to become a science based on the fact it’s so complex?

I still think of “science” as splitting atoms if you will…and yes math is part of science research so is it a tool or a “science of it’s own”??? Maybe we’ll never know:)

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To medicalquack et al. : I think the counter-example to a complex subject qualifying as a “science” is that of financial derivatives based on an underlying asset. The algorithms and related networks can become incredibly complex, but they are not in the least scientific – unless you consider their use of long-solved textbook fluid-dynamic and continuous-function-approximation problems the only requisite to be a “science”. The required assumptions of financial derivatives rapidly become unworkably contradictory and can only be ignored by slight-of-handwaving that is truly breathtaking, which of course can only lead to eventual Black Swans that are stunning.

To me, math is a tool that is in fact a language of abstractable thought, i.e. a way of expressing human thought in symbolic terminology, phrases, sentences, paragraphs, etc. in a way that is far more efficient than verbal exposition to arrive at logical processes and results that would be extremely tedious without the compact symbology of math. Its connection to intuitive linguistics is so strong that it is difficult to classify it as a science that must be empirically validated and capable of falsification. When the fantasy-capable realm of math is actually applied with severe restrictions to the real world, its linguistic tool can be very effective in carrying out such validation and falsification for the particular subject claiming to be a genuine science.

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interesting musings as often. you site the closed stackexchange question. closing on stackexchange is worthwhile of study, its an entire complicated culture that could lead to a significant/complex anthropological, sociological, and psychological analysis. lets just say that closing tends to be somewhat random at times (and you can see that on comments on the post).

turf wars are part of all human endeavors just as politics is an intrinsic aspect of a group. any time you have more than 2 humans you will find disagreements. there is some hope of collective intelligence software maybe mediating some of these conflicts (ala stackexchange) but in some ways they might also just exacerbate it.

you might also look into kuhnian perspectives on science which address some of these aspects in ways that are quite deep/meaningful. (aka paradigm shifts).

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