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What is purity? June 14, 2008

Posted by Ben Webster in math life.
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So, maybe I’m a little late to this party, but I wanted to comment on a recent XCKD (yes, Isabel blogged it first, though I added it to our del.icio.us before she blogged it). This arranged several scientific fields, by order of purity. Of course, the mathematician is standing at the far end of the scale (saying “Oh hey, I didn’t see you guys all the way over there”). The same point has circulated in saying form as

“Biologists defer to chemists. Chemists defer to physicists. Physicists defer to mathematicians. Mathematicians defer only to God”

(does anyone have an attribution for this quote? My Google-fu was insufficient).

Now, I think it’s clear that I love XKCD (one of these days, I’ll get my shit together and drive out to the Boston meet-up. Who knows, maybe next Saturday), but I can’t help but disagree a bit with the premise of this comic. I think it misses something pretty important about mathematics.

I spend a reasonable amount of time thinking about what mathematics is, and how to explain this other people. You see, unlike (I believe) all my other co-bloggers, I’m single, which leads to me periodically dating, and thus having to explain to women who are not mathematicians, but rather, say, journalists or librarians or public-health types what mathematics is and why it is an endeavor a reasonable person would undertake. (One might suggest that even romantically-attached mathematicians have non-mathematical friends that they need to explain this stuff to. I would reply that there’s a bit less, ahem, biological imperative involved there).

And I think one of the key points here is this: mathematics is not science. Mathematics is often lumped in with science, and is often used by scientists. Mathematicians often know more science than normal people, and certainly scientists know more mathematics. But mathematics and science are fundamentally different activities, as different as making a gun and fighting in a battle. I mean, no one would claim there are no links between those occupations, or that gun-makers don’t pay a lot of attention to how guns are used, but not even a child would mistake one for the other. Putting mathematics on a continuum of purity with sciences is like putting it on a continuum with disciplines of art ordered by “highbrow-ness” (actually, I would argue that the latter captures the nature of mathematics better).

Consider the inane saying I quote above; anyone mathematician who’s met a physicist doesn’t really believe that. Physicists don’t defer to mathematicians, they defer to data. As do chemists, biologists, psychologists and sociologists, irrespective of their social relations with each other. Mathematicians, on the other hand, have escaped the tyranny of data. We are often guided in what we do by examples from other fields, but mathematics by definition is universal, rather than science, which is constrained by actual observations of the world. Now, of course, scientists would argue that the actual important questions about this universe, the one we live in, the world around us, we need data, and of course, they are right. You will never here me putting down science. But that’s not what I do, and personally I find that liberating. Not everybody agrees, and that’s fine, they can spend years hunched over a lab bench, or anxiously awaiting the results of terrifying expensive experiments at CERN. I’ll be happy staring at my chalkboard, scratching my head.

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Comments

1. John Armstrong - June 14, 2008

Agreed. I’ve drawn a sharp distinction between mathematics and science myself. The analogy I use is linguistics and rhetoric, but the same point is there.

2. Josh - June 14, 2008

I have a feeling there will be much discussion about this post.

I disagree with you, but, because this matters, how do you define “science” and “mathematics”.

3. Alf - June 14, 2008

There was a russian book from the 60-ies, “Physicists Joke”, which contained a lot of quasi-serious articles, poems, anecdotes, etc. about math and physics, some of which were translated from English
and some russian ones.
There was an article called “Physics as an art and science” written by C.Darrow on the occasion of the 20-th aniversary of API, and it discussed the topic “applied vs pure” in detail… For those who read russian, the book is on the web.

In any case, I think “purity” should be defined as self-sufficiency, so with that definition the diagramme seems OK to me. Although probably it would be better if math were represented as an initial object, and if there were arrows and dots instead of just an axis.

Without being sarcastic, I thought it was clear that math is not a natural science (natural is important) and that mathematicians are platonists. Still, the most interesting questions come from physics, but why this is so is already metaphysics.

4. Nathan Dunfield - June 14, 2008

To follow up to Josh’s comment: yes there are serious philosophers and historians of science who argue that mathematics is indeed a science. I think the point is that they view proof as a very exacting form of experiment, differing quantitatively but not qualitatively from physics. See e.g. here for a discussion.

Myself, I think mathematics is not a science but for reasons having more to do with the very differing cultures of the two groups (c.f. the book Beamtimes and Lifetimes for an anthropological take on physics).

5. Josh - June 14, 2008

When you’re reading a textbook or journal article (blue Rudin particularly comes to mind), it’s easy to see why some don’t like math being classified as a science. It’s all laid down very neatly–axioms, definitions, theorem, proof, rinse and repeat.

Clean, tidy, exact and not very much like science. But think about how we do research. Supposing we even know the right way to frame the question, the work we do bears little resemblance to how it’s presented in texts. We come up with a problem, do a literature search, we “experiment” with ways to prove something and by looking for counterexamples. If we find one, we adjust our hypothesis and try again.

How is this different, in any fundamental way, from how a biologist or chemist works? They collect data and measure it against their theory. We collect “data” and compare it to the statement of our theorem.

The best definition of “science” I heard was an undergrad entomology class where science was defined as “what scientists do”.

6. Kea - June 14, 2008

LOL! Er, physicists used to defer to mathematicians, but now with scale duality they can look down on God.

7. Mathlog - June 15, 2008

Reine Wissenschaft…

“Fields arranged by purity” (Fächer nach Reinheit geordnet) ist der Titel des heutigen xkcd-Komiks…….

8. Michi’s blog » Blog Archive » On purity and essence of mathematics - June 15, 2008

[...] has, recently, written a post inspired by the xkcd comic on purity in the sciences. The comic is funny, and rings true, but Ben [...]

9. Mikael Vejdemo Johansson - June 15, 2008

Josh: the point is less the process and more the epistemology of the field. In the empirical sciences, we arrive at Knowledge by failing to falsify it. In mathematics, we arrive at Knowledge by deduction.

The actual day-to-day work can be very experimental, very much about setting up hypothesis and looking for evidence one way or another; but the way this fiddling and figuring out what we think should be true then transforms into actual new knowledge is by finding a way that derives the new fact from previously known principles completely deductively.

Which is to say in a manner completely different from the empiric way of setting up hypotheses, setting up experiments that falsify hypotheses, and failing to falsify them in the experiments.

I warmly recommend pozorvlak’s take on it:

http://pozorvlak.livejournal.com/107454.html

10. Nathan Dunfield - June 15, 2008

To my mind, a good piece of evidence for the distinction that Mikael is advocating is that there are plenty of conjectures in mathematics for which there is more solid data in their favor than many “established facts” in the sciences (e.g. the Riemann Hypothesis), yet we continue to regard these questions as open.

In other words, experiment is not the final arbiter of truth in mathematics. Of course, one could imagine mathematicians having a different internal culture where non-falsification was the gold standard, or which accepted the looser deductive standards favored by theoretical physicists…

11. Alex - June 15, 2008

I think the “theorem” as the end point in discussing the debate isn’t completely fair. Mathematicians have “programs”, basically a fancy way of saying, “I think this approach will apply to solve some class of problems, or shed light onto some class of structures.” There one collects evidence as to the validity of the program: some of it is experimental (i.e., computer, or hand examples), and some of it theoretical (theorems that support the general principle in special cases, etc). I think there’s a reasonable analogy between this and
the scientific method.

12. Vyacheslav Egorov - June 15, 2008

Anyone trying to explain what the mathematics is should definitely read a chapter “Insanity but not without a method” from Summa Technologiae by Stanislaw Lem (still I am not aware of any English translations).

The way Lem envisions mathematics and mathematicians is really insightful.

13. Greg Kuperberg - June 15, 2008

Darn, I thought that this post would be about purity theorems, such as the one in intersection cohomology.

14. Terence Tao - June 15, 2008

I would pretty much agree with xkcd, actually: maths and science do form a (widely separated) spectrum rather than a dichotomy. Certainly some disciplines in science have far more mathematical content than others, and conversely some of the more speculative, experimental, or numerical parts of mathematics do have scientific aspects to them. (See also the debate arising from the Jaffe-Quinn thesis: http://arxiv.org/find/all/1/all:+AND+Jaffe+Quinn/0/1/0/all/0/1 ).

There is also a second dimension to the maths/science axis, though, namely the pure/applied axis. In particular, the distance between applied science and applied maths (and engineering) is somewhat less than between pure science and pure maths. The discipline of computer science, for instance, arguably encompasses all three types of activity.

15. Terence Tao - June 15, 2008

I should also mention a short article by Atiyah exploring the position of mathematics in a third dimension, the art/science axis:

http://www.ams.org/bull/2006-43-01/S0273-0979-05-01095-5/home.html

A truly accurate map of the sciences is likely to be very high-dimensional (and have non-trivial topology and geometry).

16. Ben Webster - June 15, 2008

I remain unconvinced. My take away point from what Alex and Terry have said is that there are analogies between how some mathematicians and some scientists work (“experimental mathematicians” behaving more like what we think of as “scientific” and some scientists, like theoretical physicists, operating in a mode we think of as “mathematical”), which is a valuable and important point. I just don’t think it’s a powerful enough tie to justify putting science and math on a single continuum.

What I was trying to get at was Mikael’s point that math and science are epistemologically different; they are trying to create different kinds of knowledge. Ultimately, science is about the external world. It is trying to create the sort of knowledge that you can ultimately only check by observing it. Mathematics is about the sort of knowledge you can create sitting at a desk with a piece of paper. Even “experimental mathematics” is about the sort of things you could do with pen and paper, if you were patient and careful enough. Mathematics is self-contained by definition; science, by definition, isn’t.

Of course, reasonable people can disagree about what mathematics is and what science is, but I see the gap in the sort of knowledge produced as a pretty fundamental one.

17. boris - June 15, 2008

mathematics as experiments of the mind. Proofs give the step to reproduce. See:

http://www.maths.manchester.ac.uk/~avb/micromathematics/downloads

18. Josh - June 15, 2008

To me the difference isn’t one a qualitative one but a quantitative one. All the sciences have different bars of acceptance before something is accepted as scientific fact. (I seem to remember a physicist telling me that quantum mechanics has been confirmed to a higher degree than aeronautics.) Mathematicians just have the highest bar of all–absolute certainty.

I still say that a science is simply what scientists do. And the method of obtaining knowledge by mathematicians is in bijective correspondence to the method scientists employ.

A question: would you classify theoretical physics (e.g., string theory) as science?

19. Terence Tao - June 15, 2008

Ben, I would describe this distinction as yet another dimension: the concrete/abstract axis. The physical and life sciences are, as you say, mostly rooted in the physical universe (though one could argue that the most theoretical and speculative aspects of those sciences, such as xenobiology or certain parts of cosmology or string theory, are not), but at the other extreme, computer science (and to some extent, economics) are of the type which could, in principle, be studied entirely with pen and paper, as is the case with pure mathematics and large portions of applied mathematics. (The social sciences are perhaps in the middle; they could be studied with pen, paper, and a large number of humans or other intelligent beings, but the presence of the physical universe is not fundamentally important for this; one could for instance study the social aspects of a virtual network such as the internet.)

The discussion here reminds me a little of the classical debates as to the distinction between Euclidean geometry and non-Euclidean geometry, with the former being claimed to have a different ontological and epistemological status than the latter due to its perceived connections to physical reality (special and general relativity notwithstanding). There is some substance to this (for instance Euclidean geometry (or more precisely, the near-Euclidean geometry which is physical space) could, in principle, be studied experimentally and scientifically, to an extent that non-Euclidean geometries cannot, at least not without computers) but not enough to separate the two types of geometries into totally disjoint categories. A modern parallel might be with the study of the cosmology of the actual universe, and theoretical cosmologies of other universes; they are ontologically and epistemologically different subjects, but are still part of a single discipline, and the dividing line between them is not completely clear-cut.

It’s also worth noting that in classical times mathematics, science, and philosophy were not as separated from each other as they are now; science was for instance once called “natural philosophy”. It was not too long ago that it was seriously debated whether concepts such as time, space, and number belonged to one or the other of these disciplines.

20. Ben Webster - June 15, 2008

Yes, string theory is science. It could be correct science or incorrect science; that I’m not qualified to judge. But string theorists are aiming at describing the physical world. It doesn’t matter that they aren’t doing the experiments at CERN themselves, all of them (at least the sane ones) believe that string theory should be experimentally verified. String theory has a mathematical component, and quite a bit of that mathematics has been done by people who identify as physicists, but that doesn’t magically make it physics.

More generally, I don’t agree at all with your statement that “science is simply what scientists do.” There’s a stupid objection to that statement (if a scientist goes salsa dancing, that doesn’t make it science), but there’s also a smarter one: scientists often have to do things related to their research, which take the scientific nature of that enterprise into account without actually being science. One of them is mathematics, but there’s also building equipment. I’m sorry, but building a Bunsen burner is not chemistry.

21. a theoretical physicist - June 15, 2008

Oh shit, I have to defer to data? No one told me that!

22. Ben Webster - June 15, 2008

Maybe I’m getting beyond my philosophical depth (perhaps I should call in the philosophical cavalry), but what I see is people arguing that there are grey areas between science and mathematics, activities which combine them, or are hard to categorize as either, or people who identify as practitioners of one, who spend much of their time doing the other.

These are all true, but not the point I was making. Almost any two worthwhile disciplines of human endeavor have some points of contact and grey areas. I mean the fact that one could turn a mathematics lecture into performance art (every once in a while I suspect I am seeing this) doesn’t reveal some kind of deep kinship between mathematics and performance art, just that humans are clever creatures who like to mix and match disciplines and occasionally succeed.

To me, there is a pretty big distinction between mathematics and all branches of science (including social science), which is the distinction between things which are part of physical reality and things which are not. You can argue about the importance of that distinction (some other commenters seem to think the methods of work are more important than the ontological status of the objects of study), but I personally think it is the central one for understanding what mathematics is and isn’t.

23. Ben Webster - June 15, 2008

Darn, I thought that this post would be about purity theorems, such as the one in intersection cohomology.

One of these days I will explain what purity and knot homology have to do with each other, I swear.

24. Nathan Dunfield - June 15, 2008

It’s also worth noting that in classical times mathematics, science, and philosophy were not as separated from each other as they are now; science was for instance once called “natural philosophy”.

Even now, it’s still the case that mathematics is (almost?) always grouped with the the physical sciences within the formal structures of academe. Though sometimes entities end up with names that indicate the tensions apparent here as to whether math really is a science. For instance, at NSF it’s the ” Directorate for Mathematical and Physical Sciences” and there are a number of universities with a “College of Natural Sciences and Mathematics”.

25. Emmanuel Kowalski - June 15, 2008

When I was young, I used to argue that mathematics was art, and tended to be quite smug about it in comparison with other sciences. Nowadays, I find it funny to be strongly on the side of those arguing for a continuity of purpose between the two…

History (as already described by T. Tao in Comment 19) goes strongly in that direction: if there is a discontinuity, it must have happened quite recently. To take a well-known example, where will this put Alan Turing?

And to me, making a distinction between the objects of study seems dubious: the particle physicists who does computations to reach a number that is a prediction, that can then be checked by experimentalists, works in a purely abstract setting, possibly with physical intuition, but also I’m sure with lots of mathematical intuition. Also there remains a strong feeling among many practicing mathematicians that mathematical objects “exist” in a sense that is hardly less real than, say, a black hole. When you’ve turned around an argument for a few days or weeks and feel that you’re almost there, but it remains murky until it “clicks” suddenly in complete clarity, it does not feel like just pencil marks on paper. (Especially as it seems to often happe purely in the brain…)

And as to saying that experimental mathematics can also be done in principle with pen and paper, I think it’s no more correct than saying that probability theory is measure theory with a measure having mass 1, and therefore doesn’t deserve a special study.

26. Greg Kuperberg - June 15, 2008

Having failed to sabotage the subject of this thread, I’ll instead plagiarize my comment on the same topic in another blog:

Let me offer the analogy that mathematics is to science as floating currency is to backed currency. At first glance, backed currency (backed by gold or whatever) is the only credible kind. But as the currency becomes more widely used, its abstract properties become more important, until finally the currency becomes completely reliable without backing.

Likewise, at first science seems more important than mathematics, because it is backed by reality. But as you develop it, the abstract reasoning becomes more and more important, so that eventually it no longer has to be backed by reality to influence science.

To complete the analogy, many people have trouble believing in floating currency. They want to return to a spurious “gold standard”. The same sort of people have trouble believing that mathematics done for its own sake is still much like science.

27. Josh - June 15, 2008

Ben, I think both your objections to the definition of science that I gave (the “stupid” one and the “smart” one) miss the point. Perhaps it’s my fault for not being more clear. I didn’t mean that anything a scientist does is automatically science; I meant that the actions the scientist engages in, that she identifies with “doing science” is exactly what science is. But it’s a lot easier to say “science is what scientists do”. In that respect, I don’t see a qualitative difference between science and mathematics.

But all of this aside, how do you define science and mathematics?

28. Alf - June 15, 2008

I think the question is, when do we consider two sciences equal/close:
if the have equal/close objects of study or equal/close methods?
(I.e., what’s the equivalence relation we’re modding out by. Though probably it’s better to work with the stack.)

Natural sciences study nature, while mathematics studies an internal world of ideas (assuming we’re mostly platonists and not constructivists), so a similarity of methods doesn’t seem to me strong enough to compensate for this difference, as Josh suggests.

My definition of science is roughly a system of knowledge with extra restrictions; not all sciences are natural sciences (= natural philosophy).

29. Ben Webster - June 15, 2008

Josh-

I’m hesitant to try to give a firm definition, but I’m not really willing to go to what seems to be your position that “science is whatever scientists say it is.” I would say science is “Attempting to describe/understand the natural world based on observation” and mathematics is “deriving the consequences of sets of axioms.” That certainly what I feel like I’m doing when I wear my scientist and mathematician hats respectively.

30. Alf - June 15, 2008

I definitely agree with Ben.
The natural scientist tries to “fit” the world with some consistent system of (mathematical) axioms, a model. The “fit” is usually “local” and approximate. Even the assumption that there exists some mathematical law/model fitting the reality is a kind of religious belief (or if you want, a philosophical paradigm).

For the matematician the “world” is the model, so there is no question of approximation/fitting.

31. Alex - June 15, 2008

Ben,

<>

I find it difficult to see how this definition separates math/science. Sometimes one needs to develop (paper and pencil) math just to make the observation (one can imagine data whose structure is difficult to see just from “staring”), so the development of that math is an attempt to describe the natural world.

32. Alex - June 15, 2008

(I was referring to ben’s definition
“I would say science is “Attempting to describe/understand the natural world based on observation””, but inproperly tried to
include it.)

33. Kenny - June 15, 2008

I think I should put up a post in response to this. But in general, I do agree more with the side that counts mathematics as a science. (Perhaps you should take this with a grain of salt though, as I’m also one of those people that think philosophy is science too, at least when done right.) I generally also agree with the claim that “science is what scientists do” (qua scientists, i.e., not salsa dancing) though this is very far from “science is whatever scientists say it is”. Note the above people who say that science is the attempt to falsify theories – this was a popular philosophical thesis about science in the mid-20th century, but I think this has been pretty conclusively refuted. I also note your claim that mathematics is “deriving the consequences of sets of axioms” – how much of your day do you spend deriving consequences of axioms, and how much do you spend looking for the right definitions, the right axioms, the conjectures you want to work towards, and perhaps more importantly, just thinking about the objects you study more directly, rather than reasoning about them in a purely formal system?

Also, regarding the distinction that you correctly point out between the kinds of knowledge produced by mathematicians and other sciences, I think you can draw equally sharp distinctions between the kinds of knowledge produced by a syntactician, a theoretical physicist, and an organic chemist. Now sure, you can find enough people in between all of those three to show the continuity between them, but I think the same thing is true with the mathematicians. As far as I can tell, any mathematics that’s worth doing, is only worth doing because it either helps us understand the physical world, or helps us understand other mathematics that’s worth doing. If that’s right (and the claim is made sharp, so that the mathematics that’s worth doing is the smallest set closed under that operation), then there’s even an inductive proof that all mathematics is connected to the other sciences, even if the connection is quite remote.

34. Josh - June 15, 2008

Just wanted to comment that Kenny’s said what I meant to say better than I said it, wrt, “science is what scientists do.”

…trying to think more ways to use variations of “says”…

35. Alf - June 15, 2008

Maybe I didn’t read it carefully, but that Wikipedia article
(the “pretty conclusively refuted” link) essentially says that to check a theory means to check a theory, i.e. you shouldn’t forget your assumptions.

In any case, I don’t see how trying to patch the physical world with axiomatic systems/models is the same things as to study one axiomatic system in detail.
And how can one identify the two totally diferent truth criteria: “If it doesn’t agree with the experiment it’s wrong” and the mere (he-he) self-consistency.

These are just two different worlds which are non-trivially overlapping. Identifying the two seems to me like a religious belief or
a philosophical paradigm.

36. Urs Schreiber - June 15, 2008

I see math in one sequence with the other sciences as follows:

In math you study the consequences, in principle, of all imaginable axiom systems. In theoretical physics the focus is on certain axiom systems called “physical theories”.

In physics you study the consequences, in principle, of all possible realizations of those axiom systems called theories. In chemistry the focus is on certain theories describing molecular behaviour.

In chemistry you study, in principle, the behaviour of all molecules. In biology only those forming “organic compounds”.

And so on.

The point being: much of research in theoretical physics is nothing but pure math (if maybe done in a different style), differing from math research only in that the focus is on very particular mathematical structures.

That doesn’t imply that “math is science”. Just as it is not true that “biology is social studies”.

37. Serge - June 16, 2008

Vladimir Arnold wouldn’t agree
“Mathematics is the part of physics where experiments are cheap.”

38. Chris Brav - June 16, 2008

When I first saw the `Purity’ comic I was a little annoyed for a few reasons. First because purity carries a moral connotation and I don’t think of the difference between various sciences primarily in such terms (though part of why I like the culture of mathematics is that it is painfully honest). Second because, like a lot of other people who have commented here, I don’t think the comic accurately portrays the relation between mathematics and the sciences.

On the one hand, like Urs Schreiber above, I think that the sciences are (or maybe should be) a lot more like mathematics than we sometimes think. Ultimately I don’t think that by doing science we can give a `true’ description of the material world. We simply propose a model or tell a story about the world that allows us to make predictions that we hope agree with our observations, at least in a certain set of restricted cases. If our predictions don’t agree with our observations, then we make a new model.

Would we say that classical mechanics is a true description of the world? No, but it a very accurate model of the world on a certain scale. And I suspect the same is true of quantum mechanics or any other theory yet to be proposed.

I see the model-making aspect of science as basically mathematical. The model itself is a mathematical object, which in simple cases (like classical and quantum mechanics) is well-defined and in complicated cases (like QFT) has yet to be completely defined.

But we should not confuse the model with the thing that we are trying to model. In particular, the actual details of the model and its interpretation are not necessarily essential. One could certainly have two very different looking models of the same thing which turned out to be mathematically equivalent. Think for instance of Schrodinger’s formulation of quantum mechanics in terms of wave functions. The intuition here may be very useful, but I think it is unwise to insist that there `really is’ a wave a function of a particle that collapses when we make an observation. In the end what matters more is the formal properties of the theory captured by Hilbert spaces and operators, however we choose to intuit and interpret them.

So while I’d like to think of the sciences (or at least physics) as more mathematical than I was led to believe by my science classes,
I still think that there is a real difference between mathematics and science, or at least our reasons for doing them.

You can see one aspect of this difference by comparing, for instance, our attitudes toward ancient Greek mathematics and astronomy.

While hardly anyone today actually reads Euclid’s Elements, my high school geometry book was more or less plagiarized from the Elements. Almost two thousand years later its content is still considered useful and valid.

On the other hand, who would even consider plagiarizing Ptolemy’s Almagest? It certainly provides a mathematical model for the solar system which for over a thousand years was quite useful for making accurate calendars. Nowadays in high school we teach Newton’s mathematical model for astronomy. While also incorrect, Newton’s model is more accurate and efficient than Ptolemy’s.

In mathematics, if something is logically valid and also interesting and useful (like Euclidean geometry), then it will still be around thousands of years from now. In science, on the other hand, a model, however logically valid and elegant, is only used until we find something more accurate.

So I suppose the difference is that in mathematics we are interested in the models themselves, while in science we are interested how useful the model is in understanding the world around us.

39. mdreid - June 16, 2008

Just to confuse the issue some more, where does philosophy of mathematics fit into the picture?

I’ve seen one answer: as a comic.

40. Thomas Riepe - June 16, 2008

Conc. Stanislav Lem: He once mentioned to have written, but not published, a “Golem-Lecture” on mathematics.

41. Thomas Larsson - June 17, 2008

In science, unlike in math, axioms can be wrong, even if self-consistent. There is nothing wrong with ether theory, except that it is, well, wrong.

42. dcorfield - June 17, 2008

To say axioms in mathematics can’t be wrong is to set yourself against understanding it as a historically-situated discipline wherein the growth of knowledge is possible. If you take axiomatisation to be the codification of fundamental ideas in a field at a certain time, then evidently you can choose wrongly. If Eilenberg and Steenrod had written their axioms specifying that the homology of a point was zero in all dimensions, I’d be happy to say they had gone wrong. They would have failed badly to capture existing (co)homology theories.

Sometimes it easy to perceive differences where it might be useful to think also of similarities – the whale and the mouse, for example. Certain chapters of my book couldn’t have been written without reading the philosophy of science literature.

43. Terence Tao - June 17, 2008

Consider the following collection of papers:

* Paper 1 collects astronomical data about the universe, and compares it to the standard cosmological model of spacetime.
* Paper 2 analyses the standard cosmological model of spacetime, and compares it to existing astronomical data.
* Paper 3 analyses a rival cosmological model of spacetime, and compares it to existing astronomical data.
* Paper 4 analyses a rival cosmological model of spacetime, and compares it to the standard cosmological model.
* Paper 5 analyses a class of spacetime models that includes the standard model and its rivals. The author intends to use this analysis subsequently to compare the standard model with its competitors.
* Paper 6 analyses a class of spacetime models that includes the standard model and its rivals. However, the author has no intention of specialising this analysis to the standard model or its competitors.
* Paper 7 analyses a class of solutions to the Einstein equations, which turn out to be equivalent to the class of spacetime models used in cosmology. The author emphasises these connections in the paper.
* Paper 8 analyses a class of solutions to the Einstein equations, which turn out to be equivalent to the class of spacetime models used in cosmology. The author does not emphasise the connections to cosmology.
* Paper 9 analyses a class of solutions to a general class of nonlinear wave equations, of which the Einstein equations are a special case. The author highlights the applications to the Einstein equations.
* Paper 10 analyses a class of solutions to a general class of nonlinear wave equations. The Einstein equations are covered by this work, but the author does not emphasise this fact.

The papers early in this sequence are very clearly science papers, and more specifically papers in astronomy and cosmology. The papers late in this sequence are very clearly mathematics papers, and more specifically papers in mathematical relativity and nonlinear PDE. But I would say that the papers in between are a mixture of maths and science, or perhaps an interdisciplinary combination of both.

Here is another example. Consider the following two empirically verified phenomena:

1. The spectrum of large nuclei exhibit a spacing distribution in striking agreement with the GOE or GUE distributions.

2. The spectrum of zeroes of the Riemann zeta function exhibit a spacing distribution in striking agreement with the GUE distribution.

Is an experimental verification of phenomenon 1 science or mathematics? What about a numerical verification of phenomenon 1 (using standard physical models to perform a computer simulation of the nucleus)? What about a numerical verification of phenomenon 2? A theoretical verification of phenomenon 2? A theoretical verification of phenomenon 1?

To complete the set of hypotheticals: suppose that theoretical physicists manage to design a substance, Riemann zetonium, whose spectral lines are predicted to match the zeroes of the Riemann zeta function exactly. Experimental physicists then create this material and verify that the first tens of thousands of spectral lines do match perfectly. Further experiments then assume this matching continues indefinitely, and then performs further experiments on the spectral distribution to provide _experimental_ evidence towards phenomenon 2. Is this maths or science?

44. Mikael Vejdemo Johansson - June 17, 2008

Terence: I do not think anyone in this debate disputes the existence of a significant grey zone and interdisciplinary area between mathematics and sciences. As I view it, the interesting question for starters is less what happens in the grey zone, and more how the extremes – the blacks and whites – compare. In the grey zone, your questions challenging us to try (in vain) to draw clear boundaries, become legitimate, if not essential.

However, much of the debate above, and what the original xkcd and blog post deal with are not the boundary issues. This debate deals, rather, with trying to figure out how the extremal points relate. And as far as I can tell, there seem to be at handfull of major standpoints there:
1) Mathematics is essentially similar to science, because the mindset in exploring ideas and results is similar. The contrast between laboratory experiments and blackboard experiments is one of format, not one of essence.
2) Mathematics is essentially different from science, because the epistemology is fundamentally different. In mathematics, we prove statements. In science, we try to falsify statements, formulate hypotheses, and if we fail enough falsifications, we can view the statements as good.
3) Mathematics is different from science, because science deals with Reality, whereas mathematics is essentially platonic.

Certainly, there are more standpoints than these three expressed, and they are done so with much more eloquence. However, they still remain statements about the blacks and whites, not the shades of grey.

45. David Corfield - June 17, 2008

Re 41, I think it’s preferable to view axiomatisation in the context of real historical mathematical work, as the codification of a certain body of existing theory. When Eilenberg and Steenrod gave their axioms they could certainly have gone wrong. Had they stated that the homology of a point is zero in all dimensions, they would have been wrong.

In general, looking for the similarities between science and mathematics has been beneficial to me. I relied heavily on the philosophy of science literature when I wrote several of the chapters of my book.

46. Not Even Wrong » Blog Archive » Is the Universe Actually Made of Math? - June 17, 2008

[...] I noticed two interesting blog postings related to the topic. The first was Ben Webster’s posting inspired by a recent XKCD comic. The discussion in the comment section is well worth reading, [...]

47. Is Mathematics a Science? « Combinatorics and more - June 17, 2008

[...] (June 1 A related very successful discussion titled “What is Purity” is taking place over “Secret Blogging Seminar”. Ben Webster made the point: [...]

48. Gil Kalai - June 17, 2008

Emmanuel wrote: “When I was young, I used to argue that mathematics was art, and tended to be quite smug about it in comparison with other sciences. Nowadays, I find it funny to be strongly on the side of those arguing for a continuity of purpose between the two…”

My opinions and sentiments on the matter, and the way they evolve are quite similar. At present I simply think that mathematics is science.

49. Tim McKenzie - June 18, 2008

Some people seem to be saying that mathematics involves only deductive reasoning, while science permits both deductive and inductive reasoning. Others disagree.

In Terence’s sequence of hypothetical papers, none of them after number 3 necessarily involve any inductive reasoning.

The verification of his two phenomena differ in this way: sufficient experimental or numerical verification of number 1 might be accepted as conclusive, and suitable as a basis for future deductive reasoning; however, numerical verification of number 2 would not be accepted as conclusive, and no mathematical result would be considered proven if its “proof” relied on an unproven assumption that an observed pattern continued forever.

50. Thomas Larsson - June 18, 2008

David C., I don’t know the content of the Eilenberg-Steenrod axiom you talk about, but there are really only two possible problems:

1. The axioms are not self-consistent.
2. The axioms are barren, since they don’t lead to anything interesting.

In the natural sciences, there is a third possibility:

3. The axioms are consistent and fertile, but in disagreement with experiment.

51. Diskussion “What is purity?” « mathematik, bücher & meer - June 18, 2008

[...] Diskussion zu “What is purity?” ist einfach köstlich, lehrreich und unterhaltsam. Angeregt durch den xkcd-Comic “Fields [...]

52. Alessandro - June 18, 2008

” however, numerical verification of number 2 would not be accepted as conclusive, and no mathematical result would be considered proven if its “proof” relied on an unproven assumption that an observed pattern continued forever.”

What about the four color theorem in graph theory?

53. t8m8r - June 18, 2008

I see why mathematics can be thought of as being not science, but math is not completely reality-independent, since brains are physical objects and also our brain is evolved so that the mental processes are beneficial to us. It is possible that in another universe (or in a remote part of this universe) math will be completely different.

54. andreask9 - June 18, 2008

I suppose that even accepting the hypothesis that the above described ‘grey-zone’ points towards a deeper correspondence between both methods and objects of ‘natural sciences’ and mathematics, which would qualify mathematics as a science dealing with mental objects and propositions about these objects being deeply rooted in the physical reality of the human brain, there still remains Wittgenstein’s question, if any proposition about mathematical (logical) objects is in fact more than a tautology (and of course: producing tautologies cannot be a science).

As possibly became clear in meta-mathematics (Goedel etc.), mathematics is always more than a mere combination of arithmetic rules under consideration of a set of axioms, still Wittgenstein, as opposed to Russell (if I am not wrong in this) continued his aim to formalize even ‘meta-mathematics’ and to my knowledge, the ‘war’ between Wittgenstein and Russell regarding the actual status of what a proof actually is never was completely resolved. In our context one could possibly pose the question: is mathematics, provided we accept it a a science, pointing towards a ‘tautological’ structure of any natural science, i.e. do physicists uncover ‘real phenomena’ as a set of tautological propositions (provided one choses an appropriate set of axioms) in the same way as mathematicians uncover tautological relations between mental objects, or is mathematics beyond tautology in the same way as physics is beyond what by physicists is commonly considered as mathematics, namely algorithms? These questions are not-so-new, nevertheless they seem to indicate that pure mathematics and the natural sciences started to reflect the ontological status of their findings already long ago.

55. monado - June 18, 2008

I thought it was “God wishes She were a mathematician.”

56. Tim McKenzie - June 18, 2008

Alessandro: What about the four color theorem in graph theory?
As I understand it, they proved that the four-colour theorem was equivalent to a finite list of lemmas that could each be checked algorithmically. Any one of the lemmas could be checked by a human in reasonable time, but the list was so long that no human could be expected to check them all. So, they wrote a computer program to check the whole list for them.

The four-colour theorem is still a deduction from the truth of the lemmas—no inductive reasoning is required there. However, in accepting the truth of the lemmas, they were assuming that the computer that executed their program was functioning properly when they ran it. This is inductive reasoning from the observations of the form: the computer hardware usually functioned correctly in the past; similar computers aren’t known to have significant hardware problems; and so on. Perhaps there was even inductive reasoning involved in making assumptions like: the program was correct; the compiler worked properly; and so on. However, software can (in principle, at least) be deductively proven to be correct; hardware can’t.

Some mathematicians were happy to accept inductive reasoning at the level of the computer hardware; others weren’t. You can argue that all deductive reasoning carried out by humans relies on inductive reasoning about the correct functioning of their brains.

57. Alex Lopez-Ortiz - June 18, 2008

Until very recently math was as grounded on the real world as physics. Math developments were inspired by the real world and driven by their application in daily life. The Egyptians took on geometry not to prove theorems from axioms, but to measure actual physical plots on the Nile’s flood plain. Newton designed calculus not to prove axioms from the yet-to-be-developed Dedekind axioms, but to solve vexing questions from physics.

Young researchers are driven astray by a really neat property of mathematics which is that once we have properly modeled the real world, we know math results hold and there is no need to go back and recheck them against reality. This is not an accident , since not only are the axioms grounded on the real world, but so are the logic rules we use to derive theorems.

So to sum up, we have designed a set of operations (called logic) which are derived from our day to day experience, which we then apply to a collection of axioms which are also constructed to reflect our day to day experience, and lo-and-behold, it describes reality!

To followup on Greg’s analogy, since math has time and time again proven to reflect reality we have stopped asking to see the gold before we accept its currency.

58. Thomas Riepe - June 19, 2008

Why should “purity” refer to a specific content or method, but not to an attitude of doing someting intrinsical as opposed to extrinsic motives? So the question is how attitudes and motivations are distributed among mathematicians.

59. Henry Cohn - June 19, 2008

You can argue that all deductive reasoning carried out by humans relies on inductive reasoning about the correct functioning of their brains.

Yes, and long or complicated human proofs often even make additional technological assumptions, for example that when you write something down on paper and then read it later, it hasn’t changed in the meantime. This is generally a safe assumption, if your handwriting’s not too bad, but it’s easy to imagine fanciful scenarios in which it fails completely (such as selectively disappearing ink, or angry elves who insert sign errors in your scratch paper during the night). This is really just a mild version of the same assumption people make about computers, so every calculation that’s complicated enough to require scratch paper is in principle suspect. Of course, worries about computers are much more realistic, but it’s a difference of degree rather than kind.

60. KM - June 19, 2008

In response to post 3. from Alf – could you post a link to that book?

Thanks

61. Thomas Riepe - June 20, 2008

Bourdieu’s “Pascalian meditations” analyses the mindset related to ideas of “purity” of research:

62. Tim McKenzie - June 20, 2008

@Henry Cohn: Thanks for giving me an opportunity to revisit my comment. I think I’ve clarified my thoughts now.

In the case of the four-colour theorem, they used inductive reasoning about the computer to conclude that (with very high probability) there exists a purely deductive proof of the theorem. Notice that it was the deductive proof they were really interested in; they used inductive reasoning only because it helped them learn something about the purely deductive proof.

As you pointed out, that situation isn’t very different from the proof of, say, Fermat’s last theorem. The proof is so complicated that we have to use inductive reasoning when we refer back to the statement of a previous lemma, whose proof we think we remember accepting some time last week. I suppose the difference is that the proof of Fermat’s last theorem is more “open for inspection”, in a way.

In any case, our use of inductive reasoning doesn’t detract from the fact that the proof we’re actually interested in is a purely deductive one. I think the distinction between inductive and deductive reasoning is an important one, especially for science and mathematics. Once we’ve made this distinction, do we use the word mathematics to refer only to purely deductive proofs? Does the word cover every purely deductive proof? These are just questions of terminology, though.

63. Top Posts « WordPress.com - June 20, 2008

[...] What is purity? So, maybe I’m a little late to this party, but I wanted to comment on a recent XCKD (yes, Isabel blogged it [...] [...]

64. Jose Brox - June 28, 2008

I strongly agree with the statement that math is not a science, based on the same epistemological reason mentioned before.

When we do: Math
We start from: A set of totally invented axioms
And apply: Deduction
With the objective of: Finding the less-trivial theorems possible, or the “most elegant” ones, or indeed, whatever objective we want to set. The mathematician craft its art in his own way, he just have to apply some rules (and he can even change them if he wants!).

When we do: Science (physics)
We start from: A bunch of raw data collected from the real world
And apply: Induction
With the objective of: Finding the smallest/simplest/most elegant set of axioms that describe all the events we know and predicts all the future ones. Great and good ideas and models could and will go down the sewers if they don’t keep up with the reality check.

So, in math we create several axioms and try to go as far of them as we can (“down the ladder”), applying several inference rules and asking only for self-consistency.

In contrast, in science we start with non-processed information extracted from the world (“anywhere in the ladder”) and try to figure out the inverse problem, to go as near as possible (“up the ladder”) to finding a set of axioms that describe all the world and only the world, therefore asking for self-consistency AND consistency with the outside reality.

Epistemologically, they are not only different paradigms, they are completely opposite!

Terry, when you described your papers with their gray areas, you said yourself that the first ones were obviously physics, while the last ones where evidently mathematics. You couldn’t do that distinctions if it weren’t for their epistemologic differences. And since those differences clearly exist and anyone can see them, it’s for the better if we make our classification pointing them out.

Finally, the arguments about similar “mindsets” when doing science and math are not very strong: I also have a similar mindset when writing science-fiction, making puns, studying engineering, extemporizing on the piano and designing boardgames, but I wouldn’t classify all these activities on the same group (without any finer structure) just because they are related because they involve a mixture of creativity and logic – essentially, all of them require to follow some definite rules while trying to surpass them in order to achieve novel, surprising results. It’s the “twisting of the brain” produced by the confrontation of immutable rules (strict, logical thinking) with the wish for innovation (lateral, free thinking) what creates that familiar mindset. But we shouldn’t classify our activities only because of the particular mindset we are (or can be) in when we execute them.

Regards!

65. Andrew Stacey - July 2, 2008

An experimental physicist is one who thinks that the theory is a good model for reality.

A theoretical physicist is one who thinks that reality is a good model for the theory.

A mathematician is one who thinks that the theory is the reality.

Seriously, I don’t want to speak for anyone else so I’ll just say for myself rather than generalising. When I “do mathematics” in the sense that I do something I think I can get paid for as a mathematician, do I really care about its connection to physics? And, via physics, to the real world? I may use the connections to physics to justify the existence of my job to non-mathematicians but I can honestly say that I have never sat down in the morning and thought “how can this lead to a better model of the real world?”. I say, with a little justification, that what I do is related to string theory. But if string theory is proved wrong tomorrow, will that make the slightest difference to what I do? Not a bit! I’ll still be thinking about loop spaces because I think that they are fascinating and produce some beautiful mathematics.

I completely reject Kenny’s point in comment 33 about “mathematics that’s worth doing” being “mathematics that is connected to other sciences”. I regard mathematics as being worth doing purely because it is mathematics. I have much more sympathy for the artistic link in that maths is worth doing purely because it is there and it is, to my perhaps warped and twisted mind, beautiful.

(By the way, there is a flaw in Kenny’s deduction: one ought to introduce a topology since a piece of mathematics close to something worth doing is clearly also worth doing. Whereupon, it may be that not everything in the closure can be linked by a finite number of steps to something “scientific”; moreover, it may be that not everything in the closure can be linked by a sequence to something “scientific”).

I think that there is a distinction between mathematics and science, but that it is extremely hard to tease it out. Partly, I think, because a physicist can occasionally do mathematics and vice versa. Also, a physicist and a mathematician can be working on exactly the same thing but one is doing physics and the other mathematics. To allude back to my facetious opener, I would say that the difference is in ones response to the “They just found a …” remark.

“They just did an experiment which goes against the predictions of your theory”

Response: Ah well, better start working on a different theory.

Conclusion: physicist.

Response: So what?

Conclusion: mathematician.

“They just found a counterexample to your theorem”

Response: well, but it’s not one that could actually occur in the real world so we can just ignore it and, maybe, put some limits on the range of validity of the theory to be absolutely sure.

Conclusion: physicist

Response: Ah well, better get back to proving the Riemann hypothesis

Conclusion: Mathematician

Hmm, I think the fact that I am completely unable to spell “physicist” correctly first time is what sealed my fate. Doomed to mathematics through being unable to spell.

66. Tim - July 3, 2008

The comic didn’t say scientific fields.

It said fields.

Arguing about an imputed implicit claim of continuity across the “range” ignores some of what is compelling about the comic, for me.

(by the way, when did “math” decide it needed to look plural? and why? is that a semi-secret tribal identifier? if the s is to remind me of a distinction between math and maths, I’m missing it and would sincerely appreciate a quick explanation)

But as for the dis/continuity between math and the sciences listed in the comic:

Is there an interesting discontinuity that has a temporal flavor? Chris Someoneorother above pointed to Prediction as a marker of the kinds of sciences on the left in the comic. I think it is a powerful marker.

I ever-so-naively see math(s) as something like instantaneous, or timeless in an unromantic sense of the word. Whereas, at least insofar as they are “about” prediction, those sciences are not instantaneous/timeless in the sense I’m trying to talk about.

This isn’t unrelated to earlier comments to the effect that ancient geometry retains more value than, say, ancient medicine or something. But it isn’t just that either.

Please forgive the airiness of what’s just above. I’m only trying to point to something I *seem* to see, wondering if someone with sharper eyes or more experience has seen it more clearly or can explain that it is chimerical.

As for the various comments dealing with scientific/mathematical projects which are “worth doing”:

Questions of worth(value) in the sense predominant here strike me as (applied) ethical questions, or anyway entailing ethical questions. Are mathematicians and scientists not engaging in an activity generally described as philosophical as they begin and as they move along with their projects? If they are settling applied ethical questions without attending to the ethic(s) underneath, well, does this impinge on any of the issues focused upon throughout this discussion?

67. Ben Webster - July 3, 2008

math vs. maths is nothing sinister, it’s just a pondian issue: Americans say “math” whereas Britons and Australians say “maths.”

68. anon - July 3, 2008

Is it just me? I preferred Wednesday’s comic.

http://xkcd.com/443/

69. Scott Carnahan - July 3, 2008

I think of the purity described in the comic as more of a lack of complexity in the objects of study. As we move to the left, we have to worry more about the effects of aggregate behavior of large numbers of things that can interact in increasingly nontrivial ways, e.g., integers, particles, atoms, cells, neurons, people. We can derive facts in math from “first principles” because the complexity of our objects is low enough that we don’t have to make many approximating assumptions.

In response to Greg (#13), pure objects are less complicated than mixed ones, so someone who studied for example Weil cohomology of smooth proper varieties would stand a bit to the right of someone who studied families with nasty singularities.

70. Andrew Stacey - July 3, 2008

maths is an abbreviation for mathematicS.

math ith a roman catholic thervice.

71. Ben Webster - July 3, 2008

anon-

Well, maybe that one was funnier. I don’t think I would have preferred to live through it. Also, we try to stay on topic; I wouldn’t blog many of my favorite xkcd cartoons just because they aren’t relevant to math.

72. isomorphismes - July 1, 2013

Right, and it’s not really true that biologists defer to chemists either. Chemistry made great improvements to bio in developing microbiology. Then again biology gives chemistry of large molecules (e.g. computational chemistry) new life because biological chemicals are relevant to problems that people care about.

Randall Monroe himself sent up the superior attitude of the “purer” folks in xkcd.com/793.

73. isomorphismes - July 1, 2013

(oops, within “people care about” I meant ‘end users’ or ‘non-scientists’ for ‘people’, not ‘researchers’)


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