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Theories and models -- Joe's version

Page history last edited by Julia Gouvea 9 years, 2 months ago

[In this collection of materials a number of different authors have written about the complicated and interesting topic of "what's a theory".  Here's my take on it.  For some background, I was a researcher in theoretical nuclear physics for 25 years, exploring a variety of theories that were being developed in the 1960's, '70s, and '80s. The phenomena I studied included the force between neutrons and protons, the interaction of fast protons  with nuclei, and how one developed a theoretical framework for describing the complex variety of nuclear reactions that take place when you use accelerators to shoot nuclei at each other.  Since about 1990 I have been working on physics education, studying how to think about students learning (and not learning) physics. For this, I have had to study and explore theories in psychology, sociology, anthropology, and linguistics.]


In science in general and in this class in particular we use the terms "theory" and "model" a lot. There's a lot of confusion about what these mean, and they're not used consistently in different areas of science. In some places different scientists in the same area use them in different ways.  (Psychology is particularly bad about this!)


What's a theory?


In physics, biology, and chemistry, the use of the terms theory and model are fairly consistent.  "Theory" is a big stable thing; "Model" is a smaller thing that usually lives inside a theory. Another way of phrasing this is to say that within each theory there resides a family of related models. Let me be more specific.


In science, a theory typically means a large scale framework that specifies the structures and tools that help you to think about a particular class of phenomenon.  The theory may let you make sense of  what is going on; it may help you make new hypotheses about what might happen if you did something; it might lead you to suggest new experiments and things to look for; and in some cases, it can allow you to make testable predictions.


In this class, our big theories will be Newton's theory of motion and Maxwell's theory of electricity and magnetism (or at least parts of it).  Each of these describes a large class of phenomena.  Both are broad, well-tested, and have given rise to powerful technologies that have profoundly changed the human experience.


Note that both of these are also limited in their scope.  Neither correctly describes what is happening at sub-molecular scales (or at very high frequencies for electromagnetic radiation).  Even more interesting is that they don't quite match. There are mathematical inconsistencies between them having to do with moving observers. Einstein's special theory of relativity is a unifying theory that resolves the discrepancies between them -- but it makes the combined more complicated. Some systems that we could describe fairly easily in Newton's theory become too hard to handle in Einstein's theory. So even though we know that Einstein's theory of motion is "better" than Newton's, in most cases (where objects are not moving really really fast) we just use Newton.  We know where Newton fails and we know when we need to be careful and move to a more complete theory.


A theory tells us the framework in which it is useful to talk about something: What kinds of "things" should we be identifying and talking about?  What kind of processes should be pay attention to?  So if we want to talk about motion of something in size between an atom and a galaxy that is not moving at a significant fraction of the speed of light, Newton's theory of motion is the correct tool.  It tells us that what we should be looking at are objects and what we should be paying attention to are their masses and their interactions, which may be described either with the concept of forces or the concept of energy (often both).


In biology, one of the most important theories is the theory of evolution.  This theory provides a language and a framework for understanding and organizing our knowledge about a wide range of phenomena from the structure of DNA to the responses of ecosystems to changing climate. Modern versions of evolutionary theory combine ideas from Darwin, Mendel, and other more modern scientists. Just like Newton's theory of motion, Darwin's theory of evolution by natural selection, while broad and powerful, was not complete. In particular, Darwin's theory did not include a correct understanding of the mechanisms of inheritance. A synthesis of natural selection and genetics occurred years later during the Modern Synthesis. Today, the theory of evolution extends beyond Darwin's ideas, just as Einstein's theories expanded on the ideas of Newton.


No theory is perfect and and no theory describes everything.  That isn't even their purpose. The question in testing a theory is whether a particular theory it is the best tool you currently have to help you make sense of something you are trying to understand. To challenge a theory by saying "here's something the theory doesn't explain" is just silly. The phenomena of the world are so complicated that seeing whether a theory  explains something well or does not may be the work of dozens of scientists working for decades. Quantum theory was well established by 1932 when Slater showed that it could explain molecular bonding. But it took half a century to develop the tools to be able to explain and predict the properties of complex organic molecules.


A theory should always be judged in comparison with alternative explanations. Given something that a theory doesn't explain, the proper response of a scientist is to ask first, "Given the theoretical framework, do I expect to be able to explain this phenomenon, or does the theory warn us that it is so complicated that we aren't ready to solve it yet."  Every theoretical framework always has lots of such "not ready" problems.  In some cases, a theory is "almost" or "just now" ready in that new tools make more analysis and understanding possible that weren't possible before.  Finding such problems is called having good scientific taste -- knowing what's ready to figure out that no-one has figured out before. Sometimes, choosing such a problem is a route to a Nobel Prize!


When a theory is capable of describing a phenomenon, we usually need to develop a model within the theoretical framework.


What's a model?


Suppose that we've chosen a certain phenomenon we want to explain. For example, in the physics of everyday life we might consider how an object shot into the air moves; in the physics of biology we might consider how a bacterium infecting an animal cell moves through that cell. Both cases are very complicated. In order to make sense of what's going on, we have to start by figuring out what are the aspects of the problem that matter most -- and what we can safely ignore, at least until we understand the problem at a basic level.


A physical example


  1. Let's be specific.  For a projectile shot in the air, imagine punkin'-chunkin' -- the competition where people build devices to throw a pumpkin as far as possible. At the right are shown pictures of a punkin-chunker (a trebuchet) and a chunked pumpkin. If you watch a video of this (or go to a punkin-chunkin competition), you'll see that this is pretty complicated.  The trebuchet has a long arm which is wound down with a lot of effort. When it is released, the long arm of the trebuchet swings around, moving faster and faster. Near the top of its swing the pumpkin is released and it flies through the air, first rising, then falling. It may travel a long way. The world record for chunkin' a pumpkin is more than a mile (using an air cannon, not a trebuchet). 


If we wanted to describe what's going on here (and to design a better chunker) we have to first decide what matters most. There are many pieces to what's going on. Probably the best way to start is to think about the flight of the pumpkin.


Source: Wikipedia Commons

We'll work within the theoretical framework of Newton's Laws of Motion -- which says that what matters for describing motion are the velocities, accelerations, and forces each bit of an object feels.


Clearly the initial speed of the pumpkin matters and the point of release (the angle at which it leaves the launcher) -- since if we shot it straight up it wouldn't go very far, and it we shot it toward the ground it wouldn't go very far. Somewhere in between it will go the farthest. To figure out where it should be released and how the speed affects the distance, we might start with a very simplified model. Since the distance we expect it to go (about a mile) is MUCH larger than the pumpkin and even than the launcher, we might consider treating them both as very small and ignoring their structure. We might take our model as consisting of three objects: the launcher, the pumpkin, and the earth (which provides the gravitational force that changes the pumpkin's velocity).  We'll treat the launcher and the pumpkin as small -- so we can ignore their structure -- and the earth as large -- so we can treat it as flat (and therefore that the gravitational force as constant pointing straight down). The only interactions we need to consider are between the launcher and the pumpkin -- which gives the pumpkin an initial velocity -- and between the pumpkin and the earth -- which pulls on the pumpkin throughout its flight.


This is a pretty simple model and it is one we can solve, once we know Newton's theory of motion. But it's pretty oversimplified. We are ignoring air resistance, which is likely to be important, and we are ignoring the structure of the launcher, which is what is going to determine the initial velocity of the pumpkin.


But the best approach to this problem is to analyze it in a series of increasingly complex models.

  1. Treat the problem as a simple model with launcher (structure ignored), pumpkin (structure ignored), and flat earth. Only relevant factors in this model are the initial pumpkin speed, its launch angle, and its mass. Figure out how the pumpkin's distance depends on its velocity and angle.
  2. Add in another factor -- air resistance treated as a simple drag force. Now we need to know the size of the pumpkin and the density and drag coefficient of the air. Figure out how this changes the results from the previous model. If the pumpkin is a small one and the velocity isn't too large the previous model is pretty good -- so it gives us a good starting point in thinking about the situation. But if we want a world record, we need to do better so this more complicated model is needed.
  3. If we want to get more sophisticated, the details of the pumpkin's surface produce variations from the simple drag force of a sphere moving through a low density fluid. You might need to model the surface of the pumpkin and its interaction with the air in more detail.

The first model simply says that the faster you go the farther you will go and you have to launch an angle of 45o to go the farthest.  But as the velocity gets faster, air resistance becomes more significant. Each of these models will give you an idea of what the best velocity is. And of course at some point you might want to model the launcher to see how to achieve higher velocities without destroying the pumpkin -- for which you will need to model the structural strength of the pumpkin.


For this problem, the first simple model is pretty good. If you looked at all the pieces of the final model we discussed, you might just give up before starting. Good science starts by figuring out "what matters most" and trying to make and understand the simplest possible model that gives us insight into what's going on. Then, by comparing against observed data, the model is refined and improved to the point that it works well.


A biological example


But the best approach to this problem is to analyze it in a series of increasingly complex models.

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