Cutting mathematicians some slack

I think I finally get it – why my mathematical colleagues who teach service courses for science students don’t like to include units. For years I have been trying to convince my mathematician friends to be more careful with units.

Occasionally, I would run into an irritating example. Once I walked into the lecture hall before my class to set up a demo. On the board, left over from a calculus class in the previous hour, was an integral of “rho + 1” over an area of the plane. Next to it, the lecturer had written “rho = mass density.” If “rho” is a mass density, you can’t add a pure number to it. It’s like saying “I stayed for a certain amount of time – and, oh yes, for one more.”  One more what? Hour? Minute? Second? You have to specify units, and then you wouldn’t write just the number “1”.

I’ve written about a second example [1] that I found on a calculus exam in a class for scientists.

The population density of trout in a stream is

        

where r is measured in trout per mile and x is measured in miles. x runs from 0 to 10.

(a) Write an expression for the total number of trout in the stream. Do not compute it.

(b)…

If “x is measured in miles” then the “1” in the numerator of the fraction is a distance (1 mile) and the “1” in the denominator of the fraction is an area (1 square mile). Using the same symbol for two different kinds of quantities is not good scientific practice. We do it sometimes – consider all the different things that k can mean in physics – but to do it with just numbers is particularly bad -- and having two different "1"s seems particularly egregious.

For a number of years now, I’ve thought that the reason that mathematicians didn’t want to use units was because they wanted to be consistent about the level of the mathematics they used. Unit checks are in a sense “advanced” mathematics corresponding to group theory – a course many science students never take and that mathematicians and mathematical physicists typically only take as advanced students. The idea of group theory is to classify how mathematical structures change when something else is changed. For example, what happens to geometrical objects if you look at them in a mirror? Do they stay the same? If not, can the result be rotated so it’s the same as the original? Another example is, “What happens if you rotate your coordinate axes about the origin?” This analysis of this last explains why we use vectors and what we mean by a vector. And it’s group theory that underlies the properties of angular momentum in quantum mechanics, and therefore is responsible for the structure of the periodic table.

The reason that unit checks have to do with group theory is because units are about how a quantity changes when the standard that is used to assign a number to a physical quantity changes. If I measure a length with the unit of inches, what I mean is that I see how many times the standard size of “inch” fits into my length. I get some number. If I measure it with centimeters (a different and smaller standard size), I get a different (and a bigger) number. The same thing happens with area, but the number assigned to length increases by a factor of 2.54 while the number assigned to area increases by a factor (2.54)² = 6.45.

So making something into “not just a number” but into a quantity that may transform into something else is dealing with a kind of quantity that is more complex than might be appropriate to talk about in an intro math class.

But thanks to a discussion I had last week (see the post “Units and stoichiometry”), I now get that it’s really even worse than that.

What I discussed there is that when we are measuring something with units, what we are doing is mapping some aspect of the physical world into a mathematical structure – the real number line (or in the case of stoichiometry, the positive integers). What I pointed out was that we were actually modifying the math by blending it with our physical concept. That although it looks like we’re mapping something into the real number line, we limit the math that we keep. With distances, if we define a fixed origin, we can add and subtract our numbers freely, getting positives and negatives – and being able to interpret them with a conceptual correlate – something that makes sense to us and we can interpret in the world in which we live. For lengths, this works for multiplying two lengths together – or even three, as we connect that result to the mental concept of an area or a volume. We can go beyond three powers – sort of – by using metaphor and analogy, talking about multiple dimensions and hypervolumes.

With masses, we have to be more circumspect. Negative numbers are meaningless for mass and we typically don’t use them. (Modulo some issues in quantum field theory.) We sometimes multiply masses – for example when considering the gravitational force between two masses. But in most other cases where the product of two masses appears, the result can be rearranged to be a mass times a function of the ratio of masses. We don’t have a conceptual correlate for a “square mass”. (Though I wonder whether we couldn’t come up with one.) “Square times” appear in physics calculations, such as in accelerations or kinetic energy, but again, we don’t have a direct conceptual correlate.

So what I’ve realized as a result of this discussion is that we use units not just to keep track of “different kinds of numbers” as I previously thought. Rather, we use them as a warning to check for “physical reasonableness” in a way that permits some kinds of legitimate mathematical calculations and forbids others in a way that might depend on physical context. So if I’m calculating a kinetic energy, I don’t care that “time squared” occurs in the denominator. If I were calculating a time interval, I would not want to carry out a calculation that resulted in a “square time.”

The use of units is therefore subtle and requires a blending of physical and mathematical knowledge in a way that constrains mathematical manipulations that the mathematics by itself does not. This puts unit analysis outside the realm of what many mathematicians want to be doing in a math class – teaching an “honorable” description of the mathematics that remains true to the math.

I now have more sympathy for their objections so I’m willing to cut them some slack, but still insist that students need to know how to deal with units to use math in science. I have two resulting messages for the two groups:

Mathematicians – Don’t try to “paste in” units into your math classes in a casual or sloppy way. If you must do it (and I would like it if you would), be careful – and realize that you may have to get students to bring in knowledge that goes beyond math, building on their understanding of length, time, and mass from their everyday experience.

Scientists – Don’t assume that units are trivial or even simple or that the mathematicians have handled it for you in their classes. Set aside some time for a careful discussion of why you care and why it’s important – and use units carefully in your lectures and lecture notes.

I have found that even when I am Draconian about units in my teaching – taking off lots of points for wrong units on exams – students still don’t believe that it’s important and that I really care -- and they seem not to understand what's going on. In my future classes, I’m certainly going to try to share this more extended discussion and justification of units with my students. I’ll let you know if it works better than what I’ve done in the past.

[1] Introducing Students to the Culture of Physics: Explicating elements of the hidden curriculum, E. F. Redish, in Proceedings of the Physics Education Research Conference, Portland, OR, July 2010, AIP Conf. Proc. 1289 (2010) 49-52.

Units and stoichiometry

As part of my group’s current work for HHMI (Project NEXUS) to build a physics course for biology majors and pre-meds, physicists and biologists at the University of Maryland are holding extensive discussions. One of the things we like to talk about is the way we each look at the world – what feels like “real physics” or “real biology” to us and what just looks fake. (“Fake biology in physics” is just using biological organisms as the physical objects in a physics example, but not learning anything useful for biology – like using a spherical cow as a projectile.)  In that context, my colleague Wolfgang Losert and I had an interesting discussion about the nature of units and dimensional analysis.

In thinking about how we use math to model the world I like to use a little diagram that helps me make explicit some of the features we often take for granted.[1]  Here is it.

We start on the lower left with some physical system we want to model mathematically. We have some property in mind that we want to describe. We choose an abstract mathematical structure to map onto, creating a mathematical model of the system. From that math, we inherit lots of generative and processing tools that allow us to do things that we can’t easily do with purely conceptual thinking. Once we’re done processing, we have to interpret our results back in the world and then evaluate whether it works – or whether we have to modify our model. [Note this this is meant neither as a description of the cognitive processes of how we actually use math in science nor as a normative way of teaching how to model with math in science. We actually do things in a much more blended and integrated fashion. The modeling model is more of a philosophical analysis rather than a cognitive one.]

Applying this model to units and dimensional analysis is interesting. When we decide how to quantify something in the physical world via measurement, we are modeling. For example, if I decide that a length or distance can be assigned a number by counting the number of times a standard unit fits into that length, I have decided to model length and distance using positive real numbers. With this, I inherit addition, subtraction, multiplication, and division. Addition and subtraction lets me create coordinate systems and vectors and interpret negative distances. Division lets me create fractions of my standard and imagine – at least for a bit – that distance is “exactly” described by the line of real numbers. (We often forget that this is a model.) The fact that I inherit multiplication lets me generate numbers that I can assign to areas and volumes. We use the resulting model both to describe objects (their linear dimensions, areas, volumes) and positioning in space.

But we often forget the limitations of the model. First, lengths of objects are not perfectly modeled by real numbers when you look closely. No object has a perfectly sharp edge. Most are “fuzzy” – if you measured to high resolution you would get slightly different numbers by measuring at different places. At one level this comes from the process to construct the object (How carefully were the boards the table is made of sanded?), at another from the fundamental physics of the structure (Atoms and molecules are discrete – and quantum fuzzy.). If we use distances in space, inheriting the math of the real number line means that we are assuming we are measuring in a Euclidian line, plane, or volume. If that turns out not to be the case (e.g., 2-D maps on the surface of the earth), we need to choose a different mathematical model.

Each time we choose a different concept to measure (and typically an operational definition to go with it), we have a new set of numbers to decide about. If, for example, we are thinking about mass, we map to the positive real numbers since (so far) we don’t have an interpretation for what negative mass would be. (To fit into our current mathematical structures consistently, it would have to have anti-gravity when interacting with positive masses.) We inherit addition and subtraction, but multiplication doesn’t do us much good (except when calculating a gravitational force). We don’t have any physical quantity that we map onto “square mass” so we have to treat the unit of mass differently than we treat a unit of length. (See Solomon Golomb’s cute little problem, “Proving a penny equals a dollar”.) Also, when we put two masses together, they don’t quite add. Their gravitational potential energy reduces their joint mass somewhat (using E = mc² on the negative gravitational potential energy) yielding a “gravitational binding energy” that can be significant for astronomical sized objects. It's usually small and we can ignore it in almost all of physics, but it speaks to the limitations of the “real number” model of mass.

I find it particularly interesting that although each of these two models maps properties in the real world into real numbers, they behave differently in the way the real numbers can be used correctly (squares are meaningful in one and not in the other, negative values are meaningful in one and not in the other). This is one of the points students often don’t get. They think math in science is just about assigning numbers and they prefer to drop the units until the end. They miss that the way you can use those numbers changes depending on which physical dimension you are talking about. And we almost never tell them.

This is why we physicists think that dimensional analysis and units belong to us and why learning to think with units is so much to the essence of “thinking like a physicist;” because with the units, you are supposed to bring along your knowledge of the physical nature of the measurement being considered and constrain your use of the number assignments appropriately.

But the comment that started me thinking about this again and how it fits with my model for analyzing modeling, was Wolfgang’s comment that “Stoichiometry is chemistry’s dimensional analysis.” I thought that was a really exciting insight.

Now I think many of my readers are probably physicists and, if you are like me, you might not have studied much chemistry. (I only studied chemistry in high school when there were a lot fewer elements in the periodic table!) I had to ask him to remind me what stoichiometry was. What he meant was what Wikipedia calls “reaction stoichiometry” – the way the ratio of the numbers of atoms in a chemical reaction need to be adjusted so they react together properly. Basically it’s the statement that in a chemical reaction the number of each kind of atom doesn’t change – they just rearrange. If I’m building water molecules out of hydrogen and oxygen molecules, since hydrogen and oxygen atoms come in pairs (usually, as H₂ and O₂), and since water is H₂O, you need to combine two hydrogen molecules with one oxygen molecule and you wind up with two water molecules:

2H₂ + O₂ ⟷ 2H₂O.

From the point of view of my model of modeling, the chemists are mapping each kind of atom onto the set of positive integers. Wolfgang's insight is that this is like introducing a “unit check” for every distinct atom – using integers rather than the real numbers. We inherit addition and subtraction (you can move things from one side of the reaction arrow to the other) but multiplication, division, and fractions don’t associate with new physical quantities.

It’s interesting to evaluate this integer model and think about why it works. It’s not that the atoms remain inert when they react. The electronic structure changes as atoms undergo chemical reactions. In some sense, the reason you get away with stoichiometry in chemistry is that chemical reactions don’t affect atomic nuclei at all. So what you are counting are the different nuclei and saying you have the same number of each kind throughout a chemical reaction. Despite the fact that we talk as if the atoms were staying fixed in the chemical reaction, it’s really the nuclei that remain conserved. The electron states of the “atom” are shifted around, so although it’s a reasonable approximation to treat the atoms as fixed in a chemical reaction, it’s not quite correct. The stoichiometric equations and the “ball-and-stick” models chemists use are highly symbolic and rely on applying the mathematical counting model described here. The chemists’ “space filling models” are a little more realistic, showing the atoms overlapping and deforming a little. But for some circumstances it must be necessary to keep in mind that the electrons are shared and no longer may belong to a particular atom. We might have to think of some of the electrons are really shared over an entire molecule, producing a kind of “band structure” like we have in conducting crystals.

So even some of the simplest mathematical ideas in physics and chemistry – unit checks and counting of atoms – reveal themselves to be (very) useful models, but to only tell a part of the story.

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[1] Problem Solving and the Use of Math in Physics Courses, E. F. Redish, in Proceedings of the Conference, World View on Physics Education in 2005: Focusing on Change, Delhi, August 21-26, 2005

Species

The idea of “species” is clear and simple in our everyday folk biology. In the Hebrew bible, to save the animals from the flood, Noah brought “two from all” (“shnaim mi-col” = שְׁנַיִם מִכֹּל), male and female onto the ark. (An “amah” = אַמָּה – translated today as a “cubit” – must have been quite a bit bigger than we think, given what we now know about the “all”, even just of all animals.) We have a pretty good idea what is intended by the story. Everyone can tell the difference between a tiger and a lion, even my three-year-old granddaughter. (But what’s a “liger” or a “tiglon”?) Darwin’s On the Origin of Species caused a furor, in part because he suggested that species were malleable.

But from the point of view of a physicist – especially one like me who has studied a little cog sci – the idea of “species” looks more to me like the standard categorization that our brain does. We define two things as the same if they are similar – equivalent for whatever limited purpose we might have in mind at the time. (See “One and the Same”.) Categorization seems to be based largely on our early experience, cultural environment, and practical considerations. It does not seem as if things have a “true essence”, a la Plato.

In physics, over the past century we have built up pretty good evidence that the particles of which atoms are made (and photons) do need to be thought of as identical because of the structure of quantum mechanics (about which more in a later post). But when it comes to living organisms, it’s pretty clear that while some are similar, they are not “identical” in the sense that if you exchanged them there would be no experiment you could do that would tell you that the exchange has taken place.

So if all we really have is a lot of distinct individuals, what do we mean by a “species”?  One thing we like to do in physics, especially when we are getting started in studying a field and don’t have a strong theoretical framework, is to define things operationally – by some measurement process. That’s effectively how a species is defined in biology today – as a breeding population. Saying two organisms are of the same species doesn’t mean that two individuals can breed – they may be of the same sex, for example – but it means that two individuals are of the same species if there is a chain of individuals that can breed and produce fertile young. So Scooby-Doo (a Great Dane) and Ren (a Chihuahua) are of the same species even though a Chihuahua female could not carry a Great Dane’s pup to term and survive. Note that the production of a viable zygote, fetus, or even adult is not sufficient (viz. the liger or tiglon). It has to be able to survive a natural birth and be able to grow to adulthood and be fertile. You can “measure” whether two organisms are of the same species by doing a series of breeding experiments.

Many animals in biology are considered to be different species because different behaviors or choice of habitat makes natural breeding impossible. For example, two species of fly seem to differ only by the sounds they make in courting. (Of course there is a hidden assumption here – that we are talking about beings with two sexes. Things get all messed up when an organism can basically clone itself by reproducing by parthenogenesis, or when, like bacteria, they may exchange genetic material with very different organisms via viruses. But we’ll leave that discussion for another time.)

The definition of species as a breeding population is interesting since it means that you might not be able to tell if two individuals are members of the same species by experiments on them alone. (You might … if they could actually breed directly.) It might seem the opposite might be more obvious – that we could show that they are not members of the same species, but this raises interesting questions.

In The Ancestor’sTale: A Pilgrimage to the Dawn of Evolution, Richard Dawkins tells the interesting tale of a “ring species.” (I gather from conversations with biologist friends that this is a standard example in intro bio classes.) Apparently there are two species of salamanders that live together amiably in a region in California near the south end of a lake. Let’s call them groups A and B. They are clearly different species since they look different and don’t interbreed. But … if you travel up along the side of the lake on one side, you will find apparent variants of salamander A that do breed successfully with it. Similarly, if you travel up along the other side of the lake you find variants of salamander B that breed with it. And at the north side of the lake? You find salamanders that breed with both variants establishing the chain that prove that salamanders A and B are actually of the same species! Although this is called a “ring species” in biology, I prefer to call it a horseshoe species – because it’s open at the bottom.

This is neat – but a bit creepy. Suppose Harry was a salamander of group A, and Sally was a salamander of group B. Since there is a chain of friends through whom they can exchange genes they are of the same species. (To get in the right frame of mind, listen to Tom Lehrer’s, “I got it from Agnes”.) But now suppose a mining company was granted mineral rights on the north side of the lake by an eco-unfriendly administration. They destroy the habitat on the north side of the lake and all of the northern salamanders are wiped out. The chain is broken. Harry and Sally can no longer share genes and therefore they are now members of different species – despite the fact that they haven’t done anything or changed in any way!

Now here’s where it gets interesting. Physicists like myself who have studied relativity and quantum field theory are used to space and time being considered variations of the same thing. To see the implications of doing this, let’s consider a map of the lake as a 2-D graph of the different members of our horseshoe species of salamanders. But that’s at a particular instant of time. If we want, we can stack up 2-D maps of space, each one representing a later (or earlier) time. This makes our 2-D maps into a 3-D space-time graph, with the vertical axis (perpendicular to the maps) representing time. Let’s now imagine rotating our 2-D horseshoe downward, keeping Harry and Sally fixed in place, but taking our north-side salamanders down into an earlier time. The joining point of the two legs of the horseshoe now represents the common ancestor of the two groups of salamanders A and B. To see the implications of this for the concept of species, let’s switch to dogs where it’s easier for most of us to get a picture.

Analogous statements are true of Scooby-Doo and Ren. A plague wiping out all dogs except Great Danes and Chihuahuas would have the same effect on them. We would no longer consider them to be of the same species than we would tigers and lions to be the same species.

Let’s start with a “generic” dog – something like a Tamaskan (picture below) that looks a lot like a wolf – and imagine it as the common ancestor of all the dogs today. Clearly this is an oversimplification, since we should think of a breeding population as the ancestor of a population, not a single animal. (Even in the Bible, Seth, the third son of Adam and Eve, had to find a wife.) The descendants of that population were selected for by people from the natural variation the original breeding population showed – and variations that developed later (by recombination and mutation). The result is the variety of current dogs we see today. Now the ancestors of various breeds produced a range of variations in their pups from which the ones with desired characteristics were selected.

(Source, Wikipedia Commons)

Suppose now that all of the ancestors had not only pups with the variations that eventually led to Chihuahuas and Great Danes, but also ones that bred true. We’d then have not just a continuous breeding population of current dogs connecting our two extremes, but a breeding population connecting all current dogs to dogs who are identical genetically to all their ancestors. 

So this suggests that what looks to us to be a “species” is not defined by the individuals involved, but by the gaps – which ancestors and intermediates happened to die out before today. If the right ancestors had “bred true and through”, we would not separate lions from tigers nor horses from donkeys (nor humans from chimps). These are all pretty close, though. What about more significant gaps – like lizards from birds?

This is what makes The Ancestor’s Tale so interesting. Dawkins traces our ancestry back, considering close relatives and showing what our common ancestors might have looked like. He goes further and further back and attaching to our family tree species that seem farther and farther from us.

So if I take the time-rotated picture seriously, it suggests to me that we are all part of one grand continuum of life, with our definition of species being (certainly) a convenience to us, but one that we might view as accidental if we had a more complete historical record. It’s a definition that depends on which intermediates happened to die, and not on anything intrinsic to the individuals we are considering.