I spent a number of years developing what is called a “three-dimensional Eulerian photochemical grid model,” aka the “Urban Airshed Model.” I was one among many, of course, but I did make some significant contributions to the effort.
The “three-dimensional” part of the name says that a volume was divided up into a lot of compartments, “grid cells” in the jargon, and the “Eulerian” part says that the grid didn’t move around, although there was a bit of cheating on that one in that the top of the modeling region rose with the “mixing layer” in some versions of the model. The alternative to “Eulerian” is “Lagrangian” where the model volume itself moves around, usually with the fluid flow, which is to say, the wind. That’s a “trajectory model” and it usually had only a single box, although we developed some multi-box trajectory models to handle plumes like those from power plants. A single line of boxes is “one-dimensional;” a “moving wall” of boxes is “two-dimensional.” A single box, therefore, is “zero-dimensional.”
So-called “box models” are common in air pollution, and other areas of environmental modeling. They can be really simple, especially if you are dealing with pollutants that don’t react. Then all you have to do is have a source input for emissions, a “ventilation rate” for the combination of wind and diffusion that’s removing material from your box, and boundary conditions for what kind of air is replacing what’s in the box. This is the sort of model that you get on first year chemistry or physics courses; it can be expressed in a single differential equation.
You can make the box pretty big, too, provided you’re willing to take these big honking averages of everything. For either non-reactive or “first-order” (those that just decay all by themselves, without reacting with other things) pollutants, your average result for the single box calculation is the same as if you’d done the multi-box calculation and then averaged all the boxes. That’s what’s called “linear” in the biz.
I did a lot of work with box models, partly because it was easy to test chemical mechanisms with them, and the results are easy to understand also. And I got to thinking about that “ventilation rate.” And wind power.
See, if you extract energy from the wind, it slows down, and that will have an impact on the ventilation rate of any area whose air is passing by the windmills. So I did some box model calculations on the amount of energy that was being extracted from the wind at Altamont Pass near San Francisco, plus the degree of pollution that was in the air that went through the Pass. That allowed an estimate of the increase in air pollutants that would occur in San Francisco due to the decrease in ventilation.
Okay, it was a weird calculation to make in the first place, but the results weren’t that deranged. There was an effect, the largest of which was equivalent to the amount of nitrogen oxides that would have had to be emitted in order to generate the excess of ozone seen at the pass. On a per kilowatt basis, it turned out to be a little less than the amount of nitrogen oxides that would be emitted by a natural gas-fired plant, such plants being the cleanest of all fossil fueled power plants. Of course the result depended on the amount of pollution already in San Francisco; a totally clean area would see no pollution equivalent at all, and since I made those calculations, SF has reduced pollutant levels.
I wrote up my results, sent the paper off to a journal, and then received some of the most flagrantly wrong referee comments I’ve ever received on a paper. One of them showed that I was “wrong” with a calculation that was itself off by five orders of magnitude, assuming, among other things, that wind speeds are constant all the way up to the stratosphere. I think he managed to calculate the wind kinetic energy over the entire Bay Area also, rather than just through the Pass.
Well, I know when I’m licked, and it was obvious that I wasn’t going to get anyone to pay attention to that wacky idea. Even in science, sometimes I’m too clever by half, and that’s a rueful comment, not a brag.
Showing posts with label modeling. Show all posts
Showing posts with label modeling. Show all posts
Saturday, February 23, 2008
Monday, May 7, 2007
Playing the Rent II – Tolls
Of all the alternatives to the jargonizing of the term “rent” by economists, I think that the most useful is probably “toll.” People generally know what it means, and they know how it differs from rent. Rent is what you pay for the temporary use of some material good. A toll is what you pay for access, usually to a transportation or communication system.
The key factor with tolls is that it is not necessary to control all of the things that may be accessed by the transport system that takes the toll. In fact, one does not even need to control the entire transport system, just a “choke point.” This is the epitome of “positional advantage,” an easily defended position that can extract value from a much larger area that need not be under the toll taker’s control or ownership.
There are a great many economic models that have been developed for the specific cases of transportation and communications tolls. In fact, there are so many such models that it is difficult to tell if there are examples of the use of toll models for more general processes, i.e. as a more specialized model for “rents.”
Consider, for example, that every major port city represents a choke point on a transport system, and hence represents some tolling potential. Some of that tolling potential is certainly captured by dock fees, local taxes on businesses, even by actual rents of property connected to the port itself.
But the situation is broader than that. Although the U.S. Constitution forbids tariffs between states, various sorts of “local” taxes still have the effect of capturing some tolls from transport. For example, diesel fuel taxes in California, a state that receives a great deal of foreign shipping, represent some monetary input from what is basically interstate (and international) commerce. Similarly, gasoline prices are observably higher near interstate highways than are prices at some greater distance, which represents a toll on highway drivers, especially those who are less familiar with the local area, where there may be cheaper gasoline sold in filling stations known to locals. It would be interesting to see a study of how gasoline prices cluster in sparsely served areas. Intuitively, large service areas, having major economies of scale, depress prices to some distance, with state taxes, etc. further modifying the economics of the situation.
On a more abstract level, controlling “barriers to entry” can probably be better considered as a toll than a rent. The vesting period for pension qualifications is a toll that is extracted from new employees—a toll that is forfeited by those employees who leave before vesting, and thereby passed on to either the remaining employees, the firm, or both.
This feature, incidentally, is one of those places where, in the past, unions have shown their darker side. A long vesting period allows a union to claim to have obtained great benefits from negotiations, but those negotiations result in greater privileges to current employees at the expense of those newly hired. In recent years, this tendency has grown, as union contracts have sometimes resulted in “multi-tier” pay and benefit schemes, with current employees grandfathered into the greater benefits. This has also often led to the selective firing of employees who near the time of greater benefits, as they become more expensive to the firm. One might suggest that this shows the folly of allowing wedges to be driven into group and class divisions, but such divisions are, sadly, typical when large groups such as employees, deal with smaller groups such as professional managers.
The academic/professional/guild model of employment also has elements of toll taking. The current professionals (tenured professors, physicians, guild members), benefit from the labor of would-be entrants (doctoral candidates, interns, apprentices), who are paid what would be a less-than-market wage, with the promise of later ascending to the ranks of the privileged. Added to this toll taking is a generous dollop of risk; many of the candidates for entry fail to achieve their goal, and the ongoing labor of their class fuels the much smaller class of those who have achieved entry.
Thus, we reach the next method in which wealth is accumulated and transferred from the efforts of the many to the hands of the few: lotteries and gambling. And that is a big subject, so I’ll stop for now.
The key factor with tolls is that it is not necessary to control all of the things that may be accessed by the transport system that takes the toll. In fact, one does not even need to control the entire transport system, just a “choke point.” This is the epitome of “positional advantage,” an easily defended position that can extract value from a much larger area that need not be under the toll taker’s control or ownership.
There are a great many economic models that have been developed for the specific cases of transportation and communications tolls. In fact, there are so many such models that it is difficult to tell if there are examples of the use of toll models for more general processes, i.e. as a more specialized model for “rents.”
Consider, for example, that every major port city represents a choke point on a transport system, and hence represents some tolling potential. Some of that tolling potential is certainly captured by dock fees, local taxes on businesses, even by actual rents of property connected to the port itself.
But the situation is broader than that. Although the U.S. Constitution forbids tariffs between states, various sorts of “local” taxes still have the effect of capturing some tolls from transport. For example, diesel fuel taxes in California, a state that receives a great deal of foreign shipping, represent some monetary input from what is basically interstate (and international) commerce. Similarly, gasoline prices are observably higher near interstate highways than are prices at some greater distance, which represents a toll on highway drivers, especially those who are less familiar with the local area, where there may be cheaper gasoline sold in filling stations known to locals. It would be interesting to see a study of how gasoline prices cluster in sparsely served areas. Intuitively, large service areas, having major economies of scale, depress prices to some distance, with state taxes, etc. further modifying the economics of the situation.
On a more abstract level, controlling “barriers to entry” can probably be better considered as a toll than a rent. The vesting period for pension qualifications is a toll that is extracted from new employees—a toll that is forfeited by those employees who leave before vesting, and thereby passed on to either the remaining employees, the firm, or both.
This feature, incidentally, is one of those places where, in the past, unions have shown their darker side. A long vesting period allows a union to claim to have obtained great benefits from negotiations, but those negotiations result in greater privileges to current employees at the expense of those newly hired. In recent years, this tendency has grown, as union contracts have sometimes resulted in “multi-tier” pay and benefit schemes, with current employees grandfathered into the greater benefits. This has also often led to the selective firing of employees who near the time of greater benefits, as they become more expensive to the firm. One might suggest that this shows the folly of allowing wedges to be driven into group and class divisions, but such divisions are, sadly, typical when large groups such as employees, deal with smaller groups such as professional managers.
The academic/professional/guild model of employment also has elements of toll taking. The current professionals (tenured professors, physicians, guild members), benefit from the labor of would-be entrants (doctoral candidates, interns, apprentices), who are paid what would be a less-than-market wage, with the promise of later ascending to the ranks of the privileged. Added to this toll taking is a generous dollop of risk; many of the candidates for entry fail to achieve their goal, and the ongoing labor of their class fuels the much smaller class of those who have achieved entry.
Thus, we reach the next method in which wealth is accumulated and transferred from the efforts of the many to the hands of the few: lotteries and gambling. And that is a big subject, so I’ll stop for now.
Wednesday, February 28, 2007
This Years Model IV
To recap a few things:
I attended Rensselear Polytechnic Institute from 1968 to 1974, graduating with a Master's degree in Engineering Science. Engineering Science was at that time and remains to this day a funny program, a "roll your own" sort of thing. Engineering Science degrees at RPI require that you convince the curriculum chairman that the course of study that you had designed was an appropriate course of study for an engineer. Then, of course, you actually have to complete the program, which is not as easy as you might think (and certainly not as easy as you thought when you first thought up the idea).
One buddy of mine studied urban planning, transportation, and architecture, so now he designs airports around the country and the world. Another took courses in electrical and biomedical engineering, and he's now a hospital management consultant. All in all, it worked out well, I think, at least sufficiently well that the Engineering Science program at RPI continues to this day.
As for me, my course of study centered on the simulation modeling of urban and environmental systems. These days, it's hard for me to even write that sentence without marveling at youthful hubris. What were we thinking? Well, grand notions were in the air. The Cybernetics movement of the 40s and 50s has flowed into what was called General Systems Theory: the idea that science and engineering has developed a set of analytical tools that were so powerful that they might even be able to handle the social and biological sciences.
I put together a course of study that included linear systems and control theory, voice and image processing, urban analysis, with a solid chunk of statistics and operations research for trying to get the data that most people agreed would be necessary to validate and calibrate these huge models that we were going to build. I'm not sure how coherent the course of study, but I will say that for a while it seemed that every course eventually would up with us trying to invert some damn matrix or another.
Finally, I did my graduate work on rewriting a simple simulation model to compare to the very large model that the Lake George Ecosystem project at RPI had prepared. Then I graduated, moved to California, and began to look for a job. Eventually a dream job fell into my lap (literally, as one of the guys I was living with at the time tossed the phone number into my lap, saying, "We decided I wasn't the right guy for this, but it sounds right up your alley), and I became a smog scientist. Why this was a perfect next step will now require some simple math.
Conceptually, the most general model of a dynamic (changes with time) system is the state variable formulation. It pretty much goes, “Here is a series of variables that describe some phenomenon. Each variable is linked to the other variables such that the state of that variable at an instant in time is a function of the other variables at some previous time.” Often, the “previous time” is the instant immediately preceding (for a differential equation), or a discrete time step back (for a difference equation). Sometimes, however, the functional relationship looks back some period of time, although this can be turned into an instant/single time step formulation just by creating more state variables which then contain a time lag link to previous variables, i. e. “memory variables.”
Having said all that, Keep It Simple, Stupid is a good rule to live by, and it’s a pretty good rule in science and engineering. So let me write a simple equation:
dC/dt = kAB
In case anyone here has math nausea, let me emphasize how simple this is. It just says that the rate that C is changing with time depends on the product of A and B with k as a rate parameter (just multiply A and B and k). A, B, and C represent state variables, while k is a parameter, and t is time.
C can be anything, but it is most interesting when C has an effect on A and/or B. For example, suppose
C = -A-B+D+E or
This is like a chemical reaction, where A and B react to form D and E. Another way of writing it is
A + B => C + D
That’s your basic chemical shorthand.
Or suppose you are dealing with a predator/prey relationship:
Wolf + Deer => (1+∆)Wolf (i. e., a well fed wolf)
Or an aquatic ecosystem:
Phytoplankton + Zooplankton => (1+∆)Zooplankton
Phytoplankton + light +phosphate => (1+∆)Phytoplankton –phosphate
Now why is this so interesting?
The most interesting thing about such equations is how generally applicable they are. As I’ve just shown, you can use them for chemical compounds or ecosystems with equal abandon. Why? Because the setup is similar; the rate of change of the state variables depends upon how often the individuals (molecules, plankton, wolves) come in contact with each other. That is a very common situation.
The equations are non-linear but they can come close to being linear when either A or B is much larger than the other. Sometimes, this is called "well-behaved" which means "I think I sometimes understand how it works." Nevertheless, large, "well-behaved" systems often are "counter-intuitive," which means, "Okay, so I was wrong at first, but this time I'm sure I'm right, maybe."
This is just the chemistry part, of course, and the rest of photochemical modeling also has a lot of physics and mechanics in it. Whitten and I used to joke about how academic lectures of smog modeling would usually begin with somebody writing the diffusion equation on the board, a really daunting looking three-dimensional partial differential equation describing fluid flow, followed by a couple of single letters that represented “emissions” and “chemistry.” The joke was that there were well-established ways of solving the diffusion equation numerically, and the process itself (fluid mechanics) has been pretty well understood for generations. In other words, all the really hard work, preparing the emissions inventory and developing the chemistry, were compressed into two humble little letters.
A while back, in “The Right Formula” (May 7), I wrote a bit on “the stiffness problem” that often occurs when you’re trying to solve dynamic state equations that have widely varying time scales. In that essay, I briefly noted the “Gear-Hindemarsh” routines that we used in simulating smog chamber chemistry, before we hard coded the chemical kinetic mechanisms into urban smog models. When doing the latter, we used various tricks that can only be used if you already know the chemistry you’re dealing with. Obviously this isn’t very good when you’re developing the chemistry, but that’s okay, because we had the Gear routines.
The Gear-Hindemarsh codes were developed at Lawrence Livermore Labs, in order to deal with equations like this:
Li6 + n => He4 + H3
H3 + H2 => He4 + n + 17.2Mev
H2 + H2 => He3 + n
H2 + H2 => H3 + H1
In case you didn't notice, the 17.2 Mev means that if you do this to a substantial amount of Li6 and H2 (deuterium), you get a lot of energy. If the pressure and density of the material is right, you get a very large bang, i.e. a thermonuclear detonation. Hence, Livermore's interest.
Gear-Hindemarsh and similar schemes solve stiffness problem but at a cost: every time the systems see an input with discontinuities (step discontinuities even at the 4th or 5th derivative), the solver drops to a lower order predictor corrector and takes very small steps. This is fine for smog chamber experiments, not so good for urban simulations, where inputs and boundary conditions keep changing by the hour.
As the final bit of something a little like irony, I’ll note that I once revisited my Master’s Degree work and used a Gear-Hindemarsh solver on it instead of DYNAMO, which was a simulation language developed at MIT and used by Jay Forrester for his Urban Dynamics and World Dynamics models. The Gear-Hindemarsh results were substantially different from the DYNAMO results, indicating that the (pretty crude) numerical solver in DYNAMO was still sensitive to step size in my simulations. Oops. I have no idea if Forrester’s results suffered from the same problem, though I don’t actually think it matters that much. Forrester’s work had substantially worse problems than a bad number cruncher.
I attended Rensselear Polytechnic Institute from 1968 to 1974, graduating with a Master's degree in Engineering Science. Engineering Science was at that time and remains to this day a funny program, a "roll your own" sort of thing. Engineering Science degrees at RPI require that you convince the curriculum chairman that the course of study that you had designed was an appropriate course of study for an engineer. Then, of course, you actually have to complete the program, which is not as easy as you might think (and certainly not as easy as you thought when you first thought up the idea).
One buddy of mine studied urban planning, transportation, and architecture, so now he designs airports around the country and the world. Another took courses in electrical and biomedical engineering, and he's now a hospital management consultant. All in all, it worked out well, I think, at least sufficiently well that the Engineering Science program at RPI continues to this day.
As for me, my course of study centered on the simulation modeling of urban and environmental systems. These days, it's hard for me to even write that sentence without marveling at youthful hubris. What were we thinking? Well, grand notions were in the air. The Cybernetics movement of the 40s and 50s has flowed into what was called General Systems Theory: the idea that science and engineering has developed a set of analytical tools that were so powerful that they might even be able to handle the social and biological sciences.
I put together a course of study that included linear systems and control theory, voice and image processing, urban analysis, with a solid chunk of statistics and operations research for trying to get the data that most people agreed would be necessary to validate and calibrate these huge models that we were going to build. I'm not sure how coherent the course of study, but I will say that for a while it seemed that every course eventually would up with us trying to invert some damn matrix or another.
Finally, I did my graduate work on rewriting a simple simulation model to compare to the very large model that the Lake George Ecosystem project at RPI had prepared. Then I graduated, moved to California, and began to look for a job. Eventually a dream job fell into my lap (literally, as one of the guys I was living with at the time tossed the phone number into my lap, saying, "We decided I wasn't the right guy for this, but it sounds right up your alley), and I became a smog scientist. Why this was a perfect next step will now require some simple math.
Conceptually, the most general model of a dynamic (changes with time) system is the state variable formulation. It pretty much goes, “Here is a series of variables that describe some phenomenon. Each variable is linked to the other variables such that the state of that variable at an instant in time is a function of the other variables at some previous time.” Often, the “previous time” is the instant immediately preceding (for a differential equation), or a discrete time step back (for a difference equation). Sometimes, however, the functional relationship looks back some period of time, although this can be turned into an instant/single time step formulation just by creating more state variables which then contain a time lag link to previous variables, i. e. “memory variables.”
Having said all that, Keep It Simple, Stupid is a good rule to live by, and it’s a pretty good rule in science and engineering. So let me write a simple equation:
dC/dt = kAB
In case anyone here has math nausea, let me emphasize how simple this is. It just says that the rate that C is changing with time depends on the product of A and B with k as a rate parameter (just multiply A and B and k). A, B, and C represent state variables, while k is a parameter, and t is time.
C can be anything, but it is most interesting when C has an effect on A and/or B. For example, suppose
C = -A-B+D+E or
This is like a chemical reaction, where A and B react to form D and E. Another way of writing it is
A + B => C + D
That’s your basic chemical shorthand.
Or suppose you are dealing with a predator/prey relationship:
Wolf + Deer => (1+∆)Wolf (i. e., a well fed wolf)
Or an aquatic ecosystem:
Phytoplankton + Zooplankton => (1+∆)Zooplankton
Phytoplankton + light +phosphate => (1+∆)Phytoplankton –phosphate
Now why is this so interesting?
The most interesting thing about such equations is how generally applicable they are. As I’ve just shown, you can use them for chemical compounds or ecosystems with equal abandon. Why? Because the setup is similar; the rate of change of the state variables depends upon how often the individuals (molecules, plankton, wolves) come in contact with each other. That is a very common situation.
The equations are non-linear but they can come close to being linear when either A or B is much larger than the other. Sometimes, this is called "well-behaved" which means "I think I sometimes understand how it works." Nevertheless, large, "well-behaved" systems often are "counter-intuitive," which means, "Okay, so I was wrong at first, but this time I'm sure I'm right, maybe."
This is just the chemistry part, of course, and the rest of photochemical modeling also has a lot of physics and mechanics in it. Whitten and I used to joke about how academic lectures of smog modeling would usually begin with somebody writing the diffusion equation on the board, a really daunting looking three-dimensional partial differential equation describing fluid flow, followed by a couple of single letters that represented “emissions” and “chemistry.” The joke was that there were well-established ways of solving the diffusion equation numerically, and the process itself (fluid mechanics) has been pretty well understood for generations. In other words, all the really hard work, preparing the emissions inventory and developing the chemistry, were compressed into two humble little letters.
A while back, in “The Right Formula” (May 7), I wrote a bit on “the stiffness problem” that often occurs when you’re trying to solve dynamic state equations that have widely varying time scales. In that essay, I briefly noted the “Gear-Hindemarsh” routines that we used in simulating smog chamber chemistry, before we hard coded the chemical kinetic mechanisms into urban smog models. When doing the latter, we used various tricks that can only be used if you already know the chemistry you’re dealing with. Obviously this isn’t very good when you’re developing the chemistry, but that’s okay, because we had the Gear routines.
The Gear-Hindemarsh codes were developed at Lawrence Livermore Labs, in order to deal with equations like this:
Li6 + n => He4 + H3
H3 + H2 => He4 + n + 17.2Mev
H2 + H2 => He3 + n
H2 + H2 => H3 + H1
In case you didn't notice, the 17.2 Mev means that if you do this to a substantial amount of Li6 and H2 (deuterium), you get a lot of energy. If the pressure and density of the material is right, you get a very large bang, i.e. a thermonuclear detonation. Hence, Livermore's interest.
Gear-Hindemarsh and similar schemes solve stiffness problem but at a cost: every time the systems see an input with discontinuities (step discontinuities even at the 4th or 5th derivative), the solver drops to a lower order predictor corrector and takes very small steps. This is fine for smog chamber experiments, not so good for urban simulations, where inputs and boundary conditions keep changing by the hour.
As the final bit of something a little like irony, I’ll note that I once revisited my Master’s Degree work and used a Gear-Hindemarsh solver on it instead of DYNAMO, which was a simulation language developed at MIT and used by Jay Forrester for his Urban Dynamics and World Dynamics models. The Gear-Hindemarsh results were substantially different from the DYNAMO results, indicating that the (pretty crude) numerical solver in DYNAMO was still sensitive to step size in my simulations. Oops. I have no idea if Forrester’s results suffered from the same problem, though I don’t actually think it matters that much. Forrester’s work had substantially worse problems than a bad number cruncher.
This Year’s Model I
I’ve been asked to write a few things about models and modeling, as that is (in theory at least) my core area of expertise. Before I get to the cool stuff, the computer simulation models that I’ve spent at least 25 of the last 35 years working with, I’m going to “go meta” for a bit.
The word “model” in the most general sense in science is very, very broad, and can be applied to almost anything. In fact, any abstraction is a “model” of something else, as is any analogy, simile, theory, or description. This almost, but not quite, makes the general sense of model so general as to be useless.
At this high level of abstraction, even the idea of “fact” becomes a theory. Indeed, it’s pretty hard to find an instrument or even a sensory experience, that doesn’t depend upon some theoretical construct for its meaning and interpretation. So “meaning” becomes a model, as does “language,” and “thought.” That’s what Plato was getting at when he was babbling about shadows on the cave wall. Reality itself is a theoretical construct.
In the recent past, Steve Gillette and I have had a few back and forths that touch on this area when we were discussing “brute facts” vs “institutional facts.” That division is similar to the “analytic” vs “synthetic” distinction suggested by the Positivists (who rewrote a page out of Kant). It’s also possible to map “analytic” to “deductive” and “synthetic” to “inductive.” The basic idea is that there are certain sorts of propositions (or facts) that are derived from a priori definitions, using logic to derive propositions of greater and greater complexity. These are analytic/deductive/institutional propositions/facts. Then there are other sorts of propositions/facts that must take the sensory world into account in their formulation. These are synthetic/inductive/brute facts.
One of the big arguments that this sort of talk produces is that it suggests that mathematics is an invention, similar to, for example, the law, or musical theater. Some mathematicians claim instead that mathematics has objective reality. I’ll only note that the best mathematician I know personally is of the strong opinion that mathematics is a human invention.
Grind this grist fine enough and you get back to basic epistemology and questions about the nature of reality. Fine. I consider my position on the matter to be entirely defensible, but recognize that it seems to be hard to get across to some people. I once had a fairly lengthy online exchange with someone where I was trying to get across to him the difference between an abstract concept that the concrete reality that the concept might cloak. Consider a three-sided load bearing structure that we call a truss. One might very well locate a fossil, or a geological formation that can be said to be a three-sided load bearing structure. Was it a truss a million years ago? He insisted that it was, completely oblivious to my suggesting that “truss” is a human invention, and without humans, no truss exists.
I know, I know, a pointless argument, except maybe not, because it is a trap to believe that concepts exist apart from the conceiver. The sentence “If a tree falls in the forest, does it make a sound?” is a Zen koan, actually, and, contrary to common belief, koans often have answers. In this case, the answer is, “The one I’m thinking of did.”
The word “model” in the most general sense in science is very, very broad, and can be applied to almost anything. In fact, any abstraction is a “model” of something else, as is any analogy, simile, theory, or description. This almost, but not quite, makes the general sense of model so general as to be useless.
At this high level of abstraction, even the idea of “fact” becomes a theory. Indeed, it’s pretty hard to find an instrument or even a sensory experience, that doesn’t depend upon some theoretical construct for its meaning and interpretation. So “meaning” becomes a model, as does “language,” and “thought.” That’s what Plato was getting at when he was babbling about shadows on the cave wall. Reality itself is a theoretical construct.
In the recent past, Steve Gillette and I have had a few back and forths that touch on this area when we were discussing “brute facts” vs “institutional facts.” That division is similar to the “analytic” vs “synthetic” distinction suggested by the Positivists (who rewrote a page out of Kant). It’s also possible to map “analytic” to “deductive” and “synthetic” to “inductive.” The basic idea is that there are certain sorts of propositions (or facts) that are derived from a priori definitions, using logic to derive propositions of greater and greater complexity. These are analytic/deductive/institutional propositions/facts. Then there are other sorts of propositions/facts that must take the sensory world into account in their formulation. These are synthetic/inductive/brute facts.
One of the big arguments that this sort of talk produces is that it suggests that mathematics is an invention, similar to, for example, the law, or musical theater. Some mathematicians claim instead that mathematics has objective reality. I’ll only note that the best mathematician I know personally is of the strong opinion that mathematics is a human invention.
Grind this grist fine enough and you get back to basic epistemology and questions about the nature of reality. Fine. I consider my position on the matter to be entirely defensible, but recognize that it seems to be hard to get across to some people. I once had a fairly lengthy online exchange with someone where I was trying to get across to him the difference between an abstract concept that the concrete reality that the concept might cloak. Consider a three-sided load bearing structure that we call a truss. One might very well locate a fossil, or a geological formation that can be said to be a three-sided load bearing structure. Was it a truss a million years ago? He insisted that it was, completely oblivious to my suggesting that “truss” is a human invention, and without humans, no truss exists.
I know, I know, a pointless argument, except maybe not, because it is a trap to believe that concepts exist apart from the conceiver. The sentence “If a tree falls in the forest, does it make a sound?” is a Zen koan, actually, and, contrary to common belief, koans often have answers. In this case, the answer is, “The one I’m thinking of did.”
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