It’s just a question of words

I wrote this piece sometime ago but for some reason never got round to posting it, possibly because I think it really needs expanding. However the years I have spent studying both the philosophy of mathematics and the philosophy of science have convinced me that an expanded version of this answer would become a fairly substantial book; a book I have no real desire to write. I have chosen to post this piece now because the Irish student of philosophy Cathy who inhabits my twitter stream as @Cathyby recently posed the question dealt with here, so Cathy my answer to your query.

I want to turn my attention to a question that has bothered me whenever I have met it in one form or another in the intertubes, “is mathematics a science?” Usually one meets the question in the form of a denial, “but that’s mathematics and that’s not a science”. I have deliberately not chosen this question as title for this post because I don’t think it’s actually a legitimate question as it’s based on a mistaken idea of what science is.

The question that I shall be considering here cannot be asked in German or rather it would be rather strange. In German each academic discipline is a Wissenschaft. Wissen is the German for knowledge and the suffix “shaft” is equivalent to the English suffix “hood” as in neighbourhood or brotherhood and functions as a collective for everything that falls under the concept, so Wissenschaft is everything that falls under the concept knowledge. It is interesting in this context to remind ourselves that both the Latin word scientia and the Greek word mathema also originally meant knowledge. German differentiates between the different types of knowledge so the closest it gets to the common English understanding of the word science is Naturwissenschaft, which however can be translated as the natural sciences and this brings us to what I consider to be the crux of the problem.

All those who vehemently deny the status of science to mathematics have a very limited concept or view of what constitutes science. They believe there is one thing called science that employs something called “the” scientific method. What they actually mean is physics or the physical sciences. Contrary to what these people think there is no monolithic scientific method but rather a fairly large set of related and similar methods that are employed in different branches of the sciences. What constitutes a test for a hypothesis in biology is not necessarily the same as that which constitutes a test for a hypothesis in physics. This is actually tacitly recognised in that we group the sciences according to the subject matter that they investigate and the methodology that they use. We differentiate between the physical sciences, the life sciences, the earth sciences, the social sciences and so on and so forth. Each group delivers its own form of knowledge conform to the subject matter of its investigations.

Mathematics belongs together with informatics (computer science) and formal or symbolic logic to the formal sciences or in German die Formalwissenschaften. These sciences are distinguished by the fact that their true statements are true as a result of their form and not their content, in all three we talk about well-formed statements. Like physics or biology mathematics delivers knowledge and as such is without question a science but it is a different type of science to physics or biology, which in turn also differ from one another.

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8 responses to “It’s just a question of words

  1. The distinction (in English) between math and science reflects a difference between mathematicians and other knowledge enterprises that can be marked in other ways in other languages, e.g. the distinction between die Formalwissenshaften and die Naturwissenshaften, Geisteswissenshaften, usw. The mathematicians I’ve known all had a very distinct notion of what counted as doing math and what counted as doing physics, even or especially if they were individuals who occasionally changed hats and did physics themselves—in his Road to Reality tome, Roger Penrose regularly identifies when he is talking physics and when he is talking mathematics and obviously believes that it matters a great deal. The difference between math and science identifies something important about the self-definition of mathematicians and scientists. And you don’t have to take the native’s word for it. At least in America, the formal distinction between math and science has a sociological reality visible in the way academic departments are set up. That may be different elsewhere, and it may be becoming muddier in these parts because of the emergence of computer science as a kind of third thing—something like a math/science split is visible inside hybrid fields such as informatics or operations research.

    On a purely formal (!) level, it may be possible to put the argument about math and science to bed by identifying the former, along with informatics and symbolic logic, as sciences that “are distinguished by the fact that their true statements are true as a result of their form and not their content,” although there are logicians and philosophers of mathematics that would dispute the neat Humean dichotomy that underlies this view of things. On the ground, however, there is a tension between math and science, hence the math-isn’t-really-a-science meme. Maybe this is all based on an illusion, but it’s a pretty durable illusion and not just a matter of semantics.

    • Jim, I would not dispute anything that you have said but you are actually making exactly the point that I’m criticising. The claim by physicists and others science = physics! This is and has always been rubbish and should be stopped. The result is, as you point out, when people put the question “is mathematics science” what they think they are asking is “is mathematics physics”, correct answer “of course not” However I repeat “physics≠science”. If people persist with the claim science = physics then the word science becomes redundant.

      • Thony, I almost added some sentences saying something like “of course there are folks who want to say that mathematics is not a science in order to bad mouth mathematics, usually to the greater glory of physics. I’m not endorsing that variety of epistemic imperialism. Hell, shameless or obsolete as I am, I even think the Geisteswissenshaften are Wissenshaften! I do think, however, that there is a different and more interesting issue relating to the tension between math and the other natural and social sciences, a tension that surfaces in economics and population genetics and many other disciplines that operate near the line.”

        I admit I have a bad habit of trying to change the subject.

      • Jim, when are you going to write a guest post for the Renaissance Mathematicus? We would regard it as an honour.

  2. Chris

    All those who vehemently deny the status of science to mathematics ….

    It isn’t denying the status of science to mathematics, it is denying the status of mathematics to science.

    Modern science is distinct from mathematics.

  3. All those who vehemently deny the status of science to mathematics have a very limited concept or view of what constitutes science. They believe there is one thing called science that employs something called “the” scientific method.

    Really? I don’t think maths is a science, but not because of methodology (I thought Feyerabend torpedoed that approach in Consolations for the Specialist). I don’t think maths is a science because it doesn’t study the natural world: it studies an ideal world which can be reached through legal drugs like caffeine.

  4. Terry Stancliffe

    Perhaps this interesting topic can also gain a little light from a historical point of view, by considering the evolving usages in English of words such as science, mathematics, physics, philosophy &c.

    For example, in the US in the 1780s when the US constitution was written, it seems that the word ‘science’, which occurs in the Constitution, had a much wider connotation than only natural sciences or analogous fields of study. Studies of the Constitution indicate that for those who wrote it the author of any book might be taken as contributing to ‘science’ in its broad sense. With this kind of usage, it seems there could have been no question then of distinguishing mathematics from ‘science’. On the other hand, experiments and other incidents of what would now be called science or engineering were then sometimes called ‘philosophical’, and ‘natural philosophy’ was still very much alive as a descriptor for what we now call physics.

    Such issues from the history of the use of terminology can perhaps come to fog discussions of a quite different (modern-philosophical) kind, when it comes to questions about concepts and reality in mathematics and modern science.

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