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Are simpler models inherently better?

October 4, 2011

It is common for cranks to be motivated in their denunciation of the mainstream or in their development of a pet theory by a specific phenomenon, X. More often than note this phenomenon is one which the mainstream community has an incomplete understanding of, allowing the crank to make unjustified or eroneous claims with less fear of being smacked down by experimental observations contradicting it. Alternatively their mind set is akin to that of creationist; by making the flawed rational that since mainstream understanding of X is not well developed then any other tenative explanation is equally worthy of consideration. In the case of creationism many creationists think that if they disprove evolution then somehow their position is more valid.

Regardless of why a crank might pick a particular X by narrowing the part of physics they consider the above claim is flawed. By narrowing attention to X the number of different things the model must describe are vastly reduced and the construction of the model is bottom-up. Much of mainstream physics works in the same manner initially but given sufficient time and development top-down models are often constructed to unify the various seperate results obtained by a bottom-up approach. In its most naive form bottom-up approaches are little more than ‘curve fitting’, given a set of experimental results a few of ad hoc equations are found which output those results as closely as possible and in some lucky cases it might be possible to spot a pattern or particularly nice closed form expression for the relevant formulae. With only a ‘small’ (as all phenomena other than X are ignored) amount of data to explain the equations obtained might be quite simple. Unfortunately this tells you very little about the nature of the system, obtaining a mathematical formula does not automatically provide someone with physical insight, nor is there any apriori reason to think the results will apply to any related but dissimilar phenomena. A top-down construction of a model is generally done by making some physical statements about the system, converting them into mathematical expressions and then deriving results from them. If the physical statements apply to other phenomena aside from X it would be reasonable to think the model can be applied to other phenomena too.

Two examples can be used to illustrate this rationale; dark matter in gravitational models and the construction of electromagnetism.

Dark matter and gravity

General relativity (GR) surplanted Newtonian gravity (NG) in the 1910s and 1920s when it successfully explained the precesion of Mercury, which NG couldn’t, and the deflecting of light by the Sun, which NG gave the wrong value for and since then its applications have spread to cover the entirety of cosmology. Anyone who has looked at textbooks on GR will know it is vastly more complex than the simple gravitational force formula Newton gave but is applies to things NG cannot explain. However, observations of galaxy rotation rates cannot be explained by GR unless the galaxies are also filled with dark matter and this addition to an already complex theory is viewed by some, particularly cranks, as the last straw and thus believe GR should be done away with. Thus Modified Newtonian Dynamics (MOND) was born, to explain the rotation rates of the galaxies without dark matter being needed and it is this phenomenon which makes up the majority of MOND development.

MOND has failed in two ways; it has failed to explain the rotation curves of observed galaxies in a way which works for all of them and it has failed to correct the issues standard NG has with other phenomena. MOND’s development was/is guided almost entirely by “We must explain galaxy rotations” with little attention given to the question of if MOND applies to anything else and it didn’t. Even though MOND is mathematically simpler than GR it has nowhere near the predictive power of GR nor does it have the same pseudo-top-down origins of GR. MOND is not free of somewhat ad-hoc assumptions either and thus Occam’s Razor applies; both NG and GR require extension in order to even vaguely explain galaxy rotation curves accurately but in the case of GR it then explains other phenomena like the Bullet Cluster. A model must be as simple as possible but no simpler than that and typically cranks make their models simple by not worrying if they apply to anything other than X.

Electromagnetism

The second example is how electromagnetism might be developed. Faraday realised through a great many experiments that electricity and magnetism are related to one another and through his careful measurements people such as Maxwell were able to develop a series of equations to formalise the relationship which now bear his name. These equations received extensive interest and it was seen that they possessed an invariance under a set of transformations we now call Lorentz transformations and these transforms took centre stage in special relativity. The nature of the Maxwell equations prompted much important research in physics but their origin was heavily based on experimental results. However later development in differential geometry and Lie theory provided tools which allowed not only for a more fundamental top-down construction of electromagnetism but also its generalisations, including their application in quantum mechanics, now known as gauge theory.

The underlying principle of gauge theory is that the predictions of a model for a given system should not depend on arbitrary choices we have in how we describe it, in a manner analogous to how special relativity follows from the postulation that all inertial frames are equivalent in their descriptive power and predictions. The formalism in gauge theory is much more advanced than the basic vector calculus needed to understand and apply Maxwell’s equations to a given physical problem so would undoubtedly be regarded as more complex but the insight into other areas of mathematical physics gauge theory provides cannot be obtained by a bottom-up approach based on ‘curve fitting’ experimental results.

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