Something I’d love to see, though I have no idea if it’s even possible, is a piece of interactive computer software able to represent the sorts of world models that physicists have in their heads.

That is, a model that has a bunch of “scale-specific compartments,” each consisting of some equations of motion or equilibrium or whatever, and each linked to one or more of the others by taking or relaxing a limit in one or more dimensionless parameters.  These limits could be added in either direction, and perhaps even searched for over some space of equations.

One might have the ability, for instance, to have a single thing in the software’s ontology which has the property “obeys the ideal gas law in the appropriate limits,” and the property “behaves like a collection of many randomly moving corpuscles under the appropriate limits (not the same as the other ones),” and which knows how to derive the former from the latter, but can still use the simplified encoding of the former when doing other calculations that share the same limit.

Each part of a physical system would look either classical or quantum depending on which limit the user was using at the time, and they would do so together.  Statistical mechanics would be (automatically) applied when, and only when, the relevant large-N limit was compatible with the rest of the calculation.  That sort of thing.

To make things work out without needing too much pedantic formalism (or to extend it to areas that no one has rigorously grounded), one might need the ability to stipulate an asymptotic relation without proving it.  But, in the cases where the relation came together with a proof, the program would be able to automatically compute and propagate some information about the leading order error terms in the various asymptotic approximations, which could be useful and interesting in a few ways – as a automatic check for common mistakes when making many approximations, as an automatic way to quantify the relative sizes of different effects on deviations from various equilibria, etc.

This would be nothing like a numerical simulation (which is what I usually think of when I think of software that represents physical laws interactively), and although much of the underlying backend would be something like a computer algebra system, the focus and design would be very different from existing computer algebra systems I’m aware of.

disconcision replied to your post “ There’s something a little mysterious to me about the usage of “the”…”

aliens guy: “category theory”

i feel like what you’re getting at is that we can fall into an intuitive sense that equality of objects lives in some kind of objective arena. what we find out in math (or just thinking precisely about anything, i guess) is that this is abjectly not the case. equality is always with respect to some underlying category/type/whatever, and which arena we choose as our ground determines which things are equal and hence what is unique

also reminscent (though tangentially): http://math.ucr.edu/home/baez/qg-spring2004/s04week01.pdf

Ooh, yeah, those Baez notes seem to be talking about the same thing as my “constraints” and “structure” (except he says “properties” instead of “constraints”).

Anyway, your second reply feels right, but I think there is a little more here than just “equality of objects is relative” – that sounds like an observation about some independently defined things called “objects,” as if we have a good handle on what an “object” is but not necessarily on when objects are equal.  But it’s actually that way of looking at things that (in various specific cases) tends to feel wrong to me.

It feels like there is a tension between two ways of thinking which are both supposed to be hallmarks of modern/higher math: formalism and abstraction. Formalism tells you that the explicit capital-D Definition of an object is the ultimate source of truth about it, and closer to “what the object really is” than the set of the motivating examples you keep around in your head and use for intuition. But abstraction tells you to care only about intrinsic patterns/structures and not the contingent ways they may happen to be encoded. From the perspective of abstraction, a formal definition is just a way of expressing a pattern, and our intuitions can get at aspects of the pattern that the definition misses. (E.g. if we move away from set-theoretic foundations, no one is going to say that the word “group” can’t be used anymore because a group just is “a set equipped with (etc).”)

To continue on this riff: formalism has this problem where it allows you to start with definitions that have more structure than you really want, and then happily carry it around with you forever, expressing its irrelevance by saying “two objects that differ only in that way are isomorphic” — as if this is some further fact you’ve happened to learn about the pattern you are studying, when it’s really a fact about your (bad) encoding. For example, for any kind of object based on a set, we could imagine forming a stupid variant of that object where the set is ordered (a tuple), and then all of the results would be the same except we’d pointlessly act like there were multiple copies of each instance (one per ordering) that “just happen to be” isomorphic. I don’t think anyone does exactly this, but there’s this uncomfortable feeling that the same kind of error could be happening in fancier ways without us noticing.

Two things I think are kind of interesting about this:

(1) I feel like programming computers has shaped the way I think about this stuff. I’m used to drawing the distinction between the data I want to store and the data structure / encoding I use to store it, since the former is usually fixed by the problem at hand but the latter can be chosen and matters for speed, etc. So, in programming, formalism — taking one way of encoding the data and saying it is the data — is recognizably a bad habit, which will prevent you from finding better encodings. (The set vs. tuple thing from the last paragraph is a common practical issue in my everyday work, that’s why it came to mind!)

(One could imagine a parody of formal mathematics in which every definition starts by telling you that the object is stored as JSON.)

(2) It’s kinda cool how this tends to connect very abstract and very down-to-earth ways of looking at something, distinguishing them from some middle ground. You start out with an intuition that something can be abstracted from some examples, then you write a formal definition of the abstraction, but then as you prove more equalities / isomorphisms, you find more and more ways that your “naive” original examples are completely representative of other things, while the formalism can get more and more (but not less) misleading.

Arnold mentioned a few cases like this in “On Teaching Mathematics”:

What is a group? Algebraists teach that this is supposedly a set with two operations [two?? -nost] that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? “Oh, curse this maths” - concludes the student (who, possibly, becomes the Minister for Science in the future).

We get a totally different situation if we start off not with the group but with the concept of a transformation (a one-to-one mapping of a set onto itself) as it was historically. A collection of transformations of a set is called a group if along with any two transformations it contains the result of their consecutive application and an inverse transformation along with every transformation.

This is all the definition there is. The so-called “axioms” are in fact just (obvious) properties of groups of transformations. What axiomatisators call “abstract groups” are just groups of transformations of various sets considered up to isomorphisms (which are one-to-one mappings preserving the operations). As Cayley proved, there are no “more abstract” groups in the world. So why do the algebraists keep on tormenting students with the abstract definition? […]

What is a smooth manifold? In a recent American book I read that Poincaré was not acquainted with this (introduced by himself) notion and that the “modern” definition was only given by Veblen in the late 1920s: a manifold is a topological space which satisfies a long series of axioms.

For what sins must students try and find their way through all these twists and turns? Actually, in Poincaré’s Analysis Situs there is an absolutely clear definition of a smooth manifold which is much more useful than the “abstract” one.

A smooth k-dimensional submanifold of the Euclidean space R^N is its subset which in a neighbourhood of its every point is a graph of a smooth mapping of R^k into R^{(N - k)} (where R^k and R^{(N - k)} are coordinate subspaces). This is a straightforward generalization of most common smooth curves on the plane (say, of the circle x² + y² = 1) or curves and surfaces in the three-dimensional space.

Between smooth manifolds smooth mappings are naturally defined. Diffeomorphisms are mappings which are smooth, together with their inverses.

An “abstract” smooth manifold is a smooth submanifold of a Euclidean space considered up to a diffeomorphism. There are no “more abstract” finite-dimensional smooth manifolds in the world (Whitney’s theorem). Why do we keep on tormenting students with the abstract definition?

the-moti:

nostalgebraist:

 There’s something a little mysterious to me about the usage of “the” vs. “a/an” in math.  It seems related to a difference which comes up when we’re characterizing mathematical entities through their properties:

1. Sometimes we want to make statements that apply to every thing that has these properties, even things that also have some other properties we haven’t mentioned (”a/an”)
   
2. Sometimes we’re trying to single out an object characterized by these properties and nothing else (”the”)

After thinking about it for a while recently, I get the sense that you can look at a lot of things in either of these ways, and the standard linguistic choice just reflects the perspective that comes more naturally, not some specific type of property that’s shared across every case.  But maybe I just don’t understand this?

The difference between (1) and (2) is that (1) applies after adding properties to an object.  By “properties” I’m actually thinking of two different kinds of things – I’ll call them constraints and structure.

Constraints are extra equations of the same kind as the original characterizing ones.  When you characterize a group by its presentation, you specify the (cardinality of the) underlying set along with some equations relating an element to another  So, for example (thanks Wikipedia), the cyclic group C_8 has presentation < a | a^8 = 1 >.  But this doesn’t just mean that it has one element, a, satisfying the equation a^8=1 – because there are another groups like C_4 and the trivial group that satisfy this equation.  What uniquely identifies C_8 is that it is the “freest” object fitting this description, i.e. the one that doesn’t satisfy any other equations.

Some of the things that can be said about C_8 would be equally true for any group with one generator satisfying a^8 = 1, and we could imagine having a (similar but not identical) description of these things.  We would call these “C_8s” or some other plural noun, we would say things like “a C_8,” and the specific group now called C_8 would be “the free C_8.”  This is the situation for the relations that characterize Abelian groups, for instance.  (The reverse would be to call the free Abelian group “the Abelian group” and call specific Abelian groups “homomorphisms of the Abelian group” or something.)

Structure, as opposed to constraints, means properties of a different kind which are invisible from the perspective of the original characterizing properties.  With a group, you can turn it into a ring by adding another operation, but this is not related to group-level properties (i.e. not relevant for group isomorphism): you can’t look at a group and say whether it’s “currently being a ring operation” rather than “just being itself,” the way you can say whether or not something is the freest group of some description.

The justification for collapsing all objects fitting a description into a single object, worthy of “the,” usually involves some particular isomorphism.  All of the objects satisfying the (absolute) presentation of C_8 are group isomorphic, so from the group perspective, it feels like there’s just this one thing, C_8.  But you can of course exhibit two different versions of C_8 with some extra structure, and don’t have that structure’s isomorphism.  In this way, you can make any one thing plural by adding some extra distinguishing variables.  So “the” is always at risk of turning into “a/an” if you find some companion structure you want to talk about a lot.

Like, why do we say “a vector space of dimension n over R,” rather than “the vector space of dimension n over R,” since they’re all the same thing (isomorphic)?  This was the thought that led me into this – that’s always felt off to me somehow.  And it seems like the reason is that these objects (pretty boring by themselves) are mostly used with extra structure added, so it’s natural to think of there being many different versions of each one.  (This is equally true whether it’s something like an inner product, or something about what the vectors “really are,” e.g. the polynomial vector space P_2 and Euclidean 3-space have the same dimension, but you can evaluate polynomials at points in R, you can’t do that with Euclidean points.)  This is very different from the situation in group theory, where you are thinking of groups abstractly and isomorphism feels like identity.

This perspective also seems to illuminate why I always found descriptions of vector space duality weirdly offputting.  They’re talking about these two “different” vector spaces, but they’re isomorphic, so in pure vector-space-world, what difference could they have?  I guess the answer is, each one of them is actually given some (different) extra structure.  But this extra structure is described entirely in terms of vector spaces and it’s easy to get the sense it is something intrinsic.  (I guess I am saying that once you are talking about V and V*, these objects are no longer quite as generic as pure vector spaces.)

I think your intuition can be made completely precise here. Saying “a vector space of dimension n” and therefore pretending that there are many different vector spaces of dimension n is a hack to help our brains avoid thinking about non-canonical isomorphisms. If you say “Let V be a vector space and let V^v be the dual vector space” then anything you say after will be invariant under the action of GL_n on V. If you consider a vector space R^n, you may view it as both a space and the dual space, and thereby construct something that is only invariant under O(n).

So some mathematicians will believe that we should only say “the” when the object in question is unique up to unique isomorphism.

(via the-moti)

 There’s something a little mysterious to me about the usage of “the” vs. “a/an” in math.  It seems related to a difference which comes up when we’re characterizing mathematical entities through their properties:

1. Sometimes we want to make statements that apply to every thing that has these properties, even things that also have some other properties we haven’t mentioned (”a/an”)
   
2. Sometimes we’re trying to single out an object characterized by these properties and nothing else (”the”)

After thinking about it for a while recently, I get the sense that you can look at a lot of things in either of these ways, and the standard linguistic choice just reflects the perspective that comes more naturally, not some specific type of property that’s shared across every case.  But maybe I just don’t understand this?

The difference between (1) and (2) is that (1) applies after adding properties to an object.  By “properties” I’m actually thinking of two different kinds of things – I’ll call them constraints and structure.

Constraints are extra equations of the same kind as the original characterizing ones.  When you characterize a group by its presentation, you specify the (cardinality of the) underlying set along with some equations relating an element to another  So, for example (thanks Wikipedia), the cyclic group C_8 has presentation < a | a^8 = 1 >.  But this doesn’t just mean that it has one element, a, satisfying the equation a^8=1 – because there are another groups like C_4 and the trivial group that satisfy this equation.  What uniquely identifies C_8 is that it is the “freest” object fitting this description, i.e. the one that doesn’t satisfy any other equations.

Some of the things that can be said about C_8 would be equally true for any group with one generator satisfying a^8 = 1, and we could imagine having a (similar but not identical) description of these things.  We would call these “C_8s” or some other plural noun, we would say things like “a C_8,” and the specific group now called C_8 would be “the free C_8.”  This is the situation for the relations that characterize Abelian groups, for instance.  (The reverse would be to call the free Abelian group “the Abelian group” and call specific Abelian groups “homomorphisms of the Abelian group” or something.)

Structure, as opposed to constraints, means properties of a different kind which are invisible from the perspective of the original characterizing properties.  With a group, you can turn it into a ring by adding another operation, but this is not related to group-level properties (i.e. not relevant for group isomorphism): you can’t look at a group and say whether it’s “currently being a ring operation” rather than “just being itself,” the way you can say whether or not something is the freest group of some description.

The justification for collapsing all objects fitting a description into a single object, worthy of “the,” usually involves some particular isomorphism.  All of the objects satisfying the (absolute) presentation of C_8 are group isomorphic, so from the group perspective, it feels like there’s just this one thing, C_8.  But you can of course exhibit two different versions of C_8 with some extra structure, and don’t have that structure’s isomorphism.  In this way, you can make any one thing plural by adding some extra distinguishing variables.  So “the” is always at risk of turning into “a/an” if you find some companion structure you want to talk about a lot.

Like, why do we say “a vector space of dimension n over R,” rather than “the vector space of dimension n over R,” since they’re all the same thing (isomorphic)?  This was the thought that led me into this – that’s always felt off to me somehow.  And it seems like the reason is that these objects (pretty boring by themselves) are mostly used with extra structure added, so it’s natural to think of there being many different versions of each one.  (This is equally true whether it’s something like an inner product, or something about what the vectors “really are,” e.g. the polynomial vector space P_2 and Euclidean 3-space have the same dimension, but you can evaluate polynomials at points in R, you can’t do that with Euclidean points.)  This is very different from the situation in group theory, where you are thinking of groups abstractly and isomorphism feels like identity.

This perspective also seems to illuminate why I always found descriptions of vector space duality weirdly offputting.  They’re talking about these two “different” vector spaces, but they’re isomorphic, so in pure vector-space-world, what difference could they have?  I guess the answer is, each one of them is actually given some (different) extra structure.  But this extra structure is described entirely in terms of vector spaces and it’s easy to get the sense it is something intrinsic.  (I guess I am saying that once you are talking about V and V*, these objects are no longer quite as generic as pure vector spaces.)

Let me try to explain my own view of the difference between geometry and algebra. Geometry is, of course, about space, of that there is no question. If I look out at the audience in this room I can see a lot; in one single second or microsecond I can take in a vast amount of information, and that is of course not an accident. Our brains have been constructed in such a way that they are extremely concerned with vision. Vision, I understand from friends who work in neurophysiology, uses up something like 80 or 90 percent of the cortex of the brain. There are about 17 different centres in the brain, each of which is specialised in a different part of the process of vision: some parts are concerned with vertical, some parts with horizontal, some parts with colour, or perspective, and finally some parts are concerned with meaning and interpretation. Understanding, and making sense of, the world that we see is a very important part of our evolution. Therefore, spatial intuition or spatial perception is an enormously powerful tool, and that is why geometry is actually such a powerful part of mathematics—not only for things that are obviously geometrical, but even for things that are not. We try to put them into geometrical form because that enables us to use our intuition. Our intuition is our most powerful tool. That is quite clear if you try to explain a piece of mathematics to a student or a colleague. You have a long difficult argument, and finally the student understands. What does the student say? The student says, ‘I see!’ Seeing is synonymous with understanding, and we use the word ‘perception’ to mean both things as well. At least this is true of the English language. It would be interesting to compare this with other languages. I think it is very fundamental that the human mind has evolved with this enormous capacity to absorb a vast amount of information, by instantaneous visual action, and mathematics takes that and perfects it.

Algebra, on the other hand (and you may not have thought about it like this), is concerned essentially with time. Whatever kind of algebra you are doing, a sequence of operations is performed one after the other and ‘one after the other’ means you have got to have time. In a static universe you cannot imagine algebra, but geometry is essentially static. I can just sit here and see, and nothing may change, but I can still see. Algebra, however, is concerned with time, because you have operations which are performed sequentially and, when I say ‘algebra’, I do not just mean modern algebra. Any algorithm, any process for calculation, is a sequence of steps performed one after the other; the modern computer makes that quite clear. The modern computer takes its information in a stream of zeros and ones, and it gives the answer.

Algebra is concerned with manipulation in time and geometry is concerned with space. These are two orthogonal aspects of the world, and they represent two different points of view in mathematics. Thus the argument or dialogue between mathematicians in the past about the relative importance of geometry and algebra represents something very, very fundamental.

Of course it does not pay to think of this as an argument in which one side loses and the other side wins. I like to think of this in the form of an analogy: ‘Should you just be an algebraist or a geometer?’ is like saying ‘Would you rather be deaf or blind?’ If you are blind, you do not see space: if you are deaf, you do not hear, and hearing takes place in time. On the whole, we prefer to have both faculties.

In physics, there is an analogous, roughly parallel, division between the concepts of physics and the experiments. Physics has two parts to it: theory—concepts, ideas, words, laws—and experimental apparatus. I think that concepts are in some broad sense geometrical, since they are concerned with things taking place in the real world. An experiment, on the other hand, is more like an algebraic computation. You do something in time; you measure some numbers; you insert them into formulae, but the basic concepts behind the experiments are a part of the geometrical tradition.

One way to put the dichotomy in a more philosophical or literary framework is to say that algebra is to the geometer what you might call the ‘Faustian offer’. As you know, Faust in Goethe’s story was offered whatever he wanted (in his case the love of a beautiful woman), by the devil, in return for selling his soul. Algebra is the offer made by the devil to the mathematician. The devil says: ‘I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.’ (Nowadays you can think of it as a computer!) Of course we like to have things both ways; we would probably cheat on the devil, pretend we are selling our soul, and not give it away. Nevertheless, the danger to our soul is there, because when you pass over into algebraic calculation, essentially you stop thinking; you stop thinking geometrically, you stop thinking about the meaning.

(Michael Atiyah, Mathematics in the 20th Century)

This uses terms in a non-standard way, or at least a non-standardly broad way, and it’s coming from from a brilliant geometer who’s clearly a partisan in this “divide” insofar as it has partisans, but I still found it striking enough to quote it in a big long block.

Reminiscent of intuitions I’ve tried to express elsewhere, and – well, I was going to say it reminded me of @drossbucket‘s post on coupling / decoupling, except then I went back to that post and realized it was a riff on this other post of theirs in which they cite the same Atiyah quote – as well as Gowers’ “Two Cultures of Mathematics,” which led me to the Atiyah essay in the first place.  Everything is connected, man!!

(Also made me think of Wittgenstein’s saying vs. showing.)

Incidentally, reading this stuff last night also led me to re-discover that weird bit of unrest-in-the-math-fandom where a pseudonymous mathematician with an owl avatar and a blog titled “Stop Timothy Gowers! !!!” started some shit over whether the subfield of “Hungarian combinatorics” was … insufficiently deep?  Not #valid?  Something like that? ???  The internet is always very much the kind of thing that it is.

Dijkstra's in Disguise →

arxiv.org →

ordinalitis:

“[B]ig databases /have/ regularities, but they are mostly ‘‘spurious’’, a notion that we will define using algorithmic randomness. Mathematically, we will show [using Ramsey theory] that spurious correlations largely prevail, that is, their measure tends to one with the size of the database.”

—

The Deluge of Spurious Correlations in Big Data - Cristian S. Calude & Giuseppe Longo

https://doi.org/10.1007/s10699-016-9489-4, pdf here

(via eka-mark)

(via ordinalitis)

(This is another one of my promised nervous-energy nonsense posts)

I’ve had a lot of bad things to say about academia in this space.  I’m much happier in my current non-academic job than I was in grad school, and than I expect I would have been as a postdoc.

However, I do have to (grudgingly?) admit that grad school was a very useful educational experience for me.  It makes sense to me that employers in my line of work see Ph.Ds as important qualification, because I can tell I’m applying specific skills at work that I picked up in grad school.

What interests me here is whether these skills could be taught without grad school.  Usually when people talk about the transferable benefits of grad school, they talk about the years and years spent working on one subset of one sub-field, the way that it changes someone to immerse themselves in a single subject for that long.  @4gravitons recently said things along these lines:

Grad school is a choice, to immerse yourself in something specific. You want to become a physicist? You can go somewhere where everyone cares about physics. A mathematician? Same deal. They even pay you, so you don’t need to try to fit research in between a bunch of part-time jobs. They have classes for those who learn better from classes, libraries for those who learn better from books, and for those who learn from conversation you can walk down the hall, knock on a door, and learn something new. You get the opportunity to surround yourself with a topic, to work it into your bones.

And the crazy thing? It really works. You go in with a student’s knowledge of a subject, often decades out of date, and you end up giving talks in front of the world’s experts. In most cases, you end up genuinely shocked by how much you’ve changed, how much you’ve grown. I know I was.

But this isn’t what I’m talking about.  In fact, I don’t think grad school had this effect on me at all.  I had to do an undergrad thesis in college, and I got pretty obsessed with my subject back then; at one point, while writing my PhD dissertation, I looked back at that thesis and was almost startled by the similarity in thought processes and writing style, in spite of the seven intervening years.

And I don’t really feel like I came out of grad school with a deep, mature understanding of my subject.  It kind of felt the other way around, really: as the cliche goes, the more I learned, the more I knew I didn’t know.  My store of knowledge grew monotonically, but my feeling of mastery and comprehensive understanding barely budged, and in some respects even declined (as I lost the arrogance of the newcomer).

No, what I learned in grad school was how to engage with academic literature.

Not specifically with the academic literature on my chosen subfield – I’m doing entirely different stuff now, and the skill transfers just fine.  I learned how to read papers.  How to find papers.  How long to spend looking for papers, before just saying “huh, maybe no one has written a paper about this.”  How to do a lit search in service of a specific goal.

Maybe this, too, can only be acquired through years of practice, while “immersed” in a single little area?  But somehow I doubt that.  After all, one of the things I am so much better at now is leaping into a totally new area I’ve never touched before, looking into the literature on it, and getting a sense of the key landmarks without getting lost.

Some of this is just tacit knowledge acquired from reading a lot of papers – getting a feel for where to look, what’s likely to be promising, based on lots of cues that are hard to articulate.  But I do wonder if a lot of it couldn’t be explicitly taught.

Like, one thing that stands out to me – possibly the biggest single way I’ve changed – is that I’m less deferent to academic literature when I’m reading it.  I used to read papers a lot more closely, more raptly, assuming that anything in there I didn’t understand was either (1) a deep point I was too ignorant to grasp or (2) an error in a published paper, which, omg, such a big deal!

Now I have a much clearer sense of how the sausage gets made, how papers are produced as the awkward net result of multiple often conflicting pressures; how academics are often just bad writers, and how it’s better to “route around” the bad writing (just look at the pictures and equations, say, if it’s that kind of paper) than to scrutinize it for some deeper logic; how in every paper there are usually a few big punchlines, what the researchers actually found as opposed to how they’re framing it; that if the punchlines aren’t relevant to my interests, I don’t have to keep reading; that academics are always under pressure to make their findings look as important as possible, and that one must ignore the hype and boil each paper down to “okay, you tried adding a Fleeble to the Standard Blarnicator and that made one of the Good Numbers go up by 2% and another by 1.2%, in this one special case you tried, and who knows how many others you tried and conveniently didn’t report.”  Or the like.  That each sub-subfield has its own special terminology, sometimes contradictory with that of adjacent sub-subfields, and that I should figure out as quickly as possible what is really being talked about, yet without assuming that the terminology makes sense, since sometimes it doesn’t.  That sometimes academics will just say straight-up wrong things in parts of the paper they don’t care about but have to write anyway.

There is a kind of confidence involved here, and perhaps it can only be acquired – as true justified confidence, and not just the arrogance of youth – by doing a lot of research over the course of years and years.  But then, I wonder how much easier all this would have been to learn if someone had just told me all of it.  But no one did, and so it was a gradual process of unlearning the mindset I learned in 16 years of school and college, the mindset of taking classes and trying to deeply understand the assigned readings and to develop finely honed skills at well-defined tasks.

The usual narrative is that in grad school, you leave this bright comfortable world behind and enter a more muddled, adult world – the world of research – where everything is too deep and complex and incompletely known to be put between the covers of a textbook and mastered step by organized step.  And that’s true.  But having heard that narrative, I took my first steps into the new world with too much trust and too much expectation of awe.  The expectation that everything I read was itself deep and complex – or that if it wasn’t, this was some scandalous failure.

What I eventually learned, and what was immensely valuable, was a certain lack of respect for the world of research (including my own output, of course).  The ability to navigate the literature with a casual, readily dismissive touch, looking for specific things I want and not for any “deep understanding.”  The ability to read all this stuff about “state-of-the-art” this and “fundamental” that, of a thousand figures where the home team’s line soars above the opposing team’s line (unless lower is better, in which case it’s the opposite), and see it with an appropriately mercenary eye, always asking, “what did they really do, and what does it mean for my goals?”

Can this be taught outside of grad school?  Perhaps you do need to be immersed in a world before you can develop the useful kind of cynicism about it.  But what if we tried to teach it directly?