Abstract
“Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascod, so pages of formulae will not get a definite result out of loose date” (Thomas Huxley (1869) Geological Reform, Presidential Address to the Geological Society). Reasoning in science is a rich and complex phenomenon. On one hand, we find detailed, sophisticated and rigorous calculations. But on the other, we encounter a multiplicity of models and approximations whose status has been the subject of extensive debate (See [6] How the Laws of Physics Lie (Oxford, New York, Oxford University Press) and [7] The Dappled World (Cambridge, Cambridge University Press)). Detailed and demanding calculations give the appearance of mathematical rigour, and from a practical perspective, inferences and calculations based on successful models have proven to be reliable guides to our world, predicting the results of many measurements and suggesting interventions in the world that produce startling and impressive novel phenomena ranging from laser light to transistors to monoclonal antibodies and new types of sub-atomic particles. But the logical incompatibility of different models, each making different assumptions and approximations, together with the application of distinct, conflicting models in the course of deriving important results, raise serious questions about the nature and status of the both the premises and the conclusions of scientific reasoning.
...all models are wrong, but some are useful.
(G E.P. Box and N.R. Draper, 1987)
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Notes
- 1.
See [6], p. 119f., where Cartwright argues that the order in which a series of approximations each of which appears to be justified is to be made is chosen not on a principled basis (such as a theoretical argument for which ordering will produce more exact results), but instead on the basis of which order captures the observed Lamb shift in the excited state. Though in this case we can show one order produces a more precise approximation, in general this is not the case, and extra precision is not always required either. The subsequent discussion of the Lamb shift in the ground state provides an even more difficult case of a choice between approximations grounded in the result obtained rather than the mathematical precision of the approximations. Finally, a more general problem in the neighborhood is the lack of mathematical justification for the (brilliantly) successful calculational practice of re-normalization.
- 2.
Old quantum theory is a case in point: it developed via a series of strange proposals, beginning with the quantization of energy exchange between matter and the radiation field (Planck), and continuing with Bohr’s hydrogen atom and its refinements and extensions (including ionized helium atoms), Ehrenfest’s adiabatic principle (applying thermodynamics to connect stationary states of different physical systems), and more. See, for instance, Rechenberg, H. “Quanta and Quantum Mechanics,” [19], pp. 143–248 in Laurie M. Brown, Abraham Pais, and Sir Brian Pippard, Eds., Twentieth Century Physics, Vol. 1, Bristol and New York: Institute of Physics Publishing and American Institute of Physics Press, for a helpful discussion of OQT.
- 3.
A famous cartoon by S. Harris in American Scientist provides an ironic illustration of this expectation—a scientist’s long calculation on a blackboard is interrupted, at one point, by the words, “and then a miracle occurs,” after which further calculations continue to a conclusion. Another comments drily, “I think you should be more explicit at this point”.
- 4.
See Norton, “The Force of Newtonian Cosmology: Acceleration is Relative” Philosophy of Science, 62, [16], pp. 511–22, [17] “Classical Particle Dynamics, Indeterminism and a Supertask,” Synthese 115: 259–265, and [9], ‘Comments on Laraudogoitia’s “Classical Particle Dynamics, Indeterminism and a Supertask”, British Journal for the Philosophy of Science, 49: 123–133.
- 5.
- 6.
See Brown, B. and Priest, G., “Chunk and Permeate II: Bohr’s hydrogen atom,” European Journal for the Philosophy of Science, [3], doi:10.1007/s13194-014-0104-7.
- 7.
[10] “Inconsistency in Classical Electrodynamics,” Philosophy of Science 11/2004; 71:525–549. doi:10.1086/423627.
- 8.
Morgan, M. and Morrison, M. (eds.) [15], Models as Mediators:Perspectives on Natural and Social Science, Cambridge, Cambridge University Press.
- 9.
See [18], “Two Dogmas of Empiricism,” The Philosophical Review 60: 20–43.
- 10.
See Brown and Apostoli, “A Solution to the Completeness Problem for Weakly Aggregative Modal Logic,” Journal of Symbolic Logic, 60, 3, September 1995, 832–842 for the original completeness proof, and Brown and Schotch, “Logic and Aggregation” Journal of Philosophical Logic 28: 265–287 (June, 1999) [5] for a generalization in the context of hypergraph colourings.
- 11.
See “Ambiguity Games and Preserving Ambiguity Measures,” in On Preserving: Essays on Preservationism and Paraconsistent Logic, Schotch, Brown and Jennings, eds., University of Toronto Press (2009).
- 12.
In fact, even where rigorous general accounts are available, they typically emerge from careful reflection on scientific practices applicable to particular cases which were quite successful long before those rigorous accounts emerged. Consider as an example the development of quantum mechanics leading up to von Neumann’s formal account.
- 13.
We will say a description is satisfactory if what it says about the system is approximately true in the innocent sense of agreeing, within contextually determined limits, with reports of various observations of the system, expressed in terms of the same concepts.
- 14.
Consider the impact of Brahe’s observations on astronomy and Kepler’s efforts to improve on Copernicus’ model for planetary motion; the shift to ellipses governed by Kepler’s laws allowed for a much better fit with the observations. Another example is Einstein’s focus, as he was working towards his theory of General Relativity, on capturing the precession of Mercury’s perihelion.
- 15.
See Cartwright on ‘phenomena’ in [6], p. 100ff.
- 16.
Think here of efforts to produce ‘effects’ predicted by simple theoretical models—masers and lasers, transistors and many other basic components of modern electronic and optical technology are concrete demonstrations of phenomena that theoretical results had identified as potentially realizable.
- 17.
For example, we may sum the first few terms of a series whose terms quickly become quite small, and accept the result as an ‘approximation’ to the sum of the series as a whole without a formal proof that the series actually converges to a limit. Successful application of such results are often counted as a positive result for the theory in question, even though it’s possible that they are not in fact good approximations to what the theory would predict if a more rigorous calculation were performed.
- 18.
What I have in mind here is related to Cartwright’s account of phenomenal laws (How the laws of physics lie), which describe reliable regularities that hold of phenomena we either find or learn to create. Such laws are not strictly derived from the theory’s principles—instead, they invoke concepts drawn from the theory, reasoning with them in ways that are not logically rigorous, but which might (at least approximately) capture results that could, in principle, be rigorously derived; one might say they are inspired by the theory rather than derived from it.
- 19.
Consider Bell’s work on non-locality in QM (see [1], “On the Einstein Podolsky Rosen Paradox,” Physics 1 (3), 195–200): his discovery that the statistics of such QM observations would differ in an experimentally testable way from those of a hidden-variable theory demonstrated the possibility of settling experimentally what had, up to that point, been widely thought of as a metaphysical issue.
- 20.
Brown and Priest, “Chunk and Permeate II: Bohr’s Hydrogen Atom,” European Journal for Philosophy of Science, Jan 2015 [3]. http://link.springer.com/article/10.1007/s13194-014-0104-7.
- 21.
Ibid.
- 22.
Schilpp, Paul Arthur, editor. Albert Einstein: Philosopher-Scientist, pp. 19, 21, Open Court, La Salle, Illinois, [1949; 1951] 1969, 1970. ISBN 0-87548-286-4.
- 23.
In fact some had thought no such results could be expected, since, if these lines characterized ‘resonant’ frequencies of a tiny, complex system, the mathematics of determining the structure of the system responsible for them (as in calculating the shape of a bell from its sound) seemed beyond solution.
- 24.
- 25.
- 26.
This went beyond spectral data to include a satisfactory estimate of the typical size of a hydrogen atom, and an explanation of why lines due very high-energy states were missing from known spectral studies, due to the orbital radius of such states and their consequent instability/absence at normal pressures.
- 27.
On this topic see, for example, [11]. “Observation of Inhibited Spontaneous Emission”. Physical Review Letters 55 (1): 67–70, in which inhibition of emission is used to narrow spectral line-widths.
- 28.
[8], Coverage bias in the HadCRUT4 temperature series and its impact on recent temperature trends. Q.J.R. Meteorol. Soc. doi:10.1002/qj.2297.
- 29.
Of course for navigational purposes, earth-centered astronomy is still a convenient tool.
- 30.
Sellars, W., “Philosophy and the Scientific Image of Man,” in Frontiers of Science and Philosophy, Robert Colodny (ed.) Pittsburgh, PA: University of Pittsburgh Press, (1962), 35–78; reprinted in Science, Perception and Reality, London: Routledge and Kegan Paul, New York and The Humanities Press [20], 1–40.
- 31.
(see T.S. Kuhn The Structure of Scientific Revolutions, 2\(^{\text {nd}}\) ed., Chicago, University of Chicago Press, 1972).
- 32.
Laudan, Larry. “A Confutation of Convergent Realism”, Philosophy of Science, Vol. 48, No. 1, (Mar. [14]): 19–49.
- 33.
The Scientific Image, Oxford, Clarendon Press, 1980.
- 34.
Though examples of “Kuhn loss,” The Structure of Scientific Revolutions, Chicago: University of Chicago Press ([13], 2nd edition, with postscript) 99–100), i.e. the surrender of what seemed to be successful explanations in the transition to new theories, have been proposed including the purported explanation of the similarities of different metals (shininess, ductility etc.) on the basis of their containing phlogiston, I do not pursue this issue further here.
- 35.
[2], “The Pragmatics of Empirical Adequacy,” The Australasian Journal of Philosophy, 82, 2, 242–264.
- 36.
This is not an easy account to follow through on. It is difficult to see how we can determine what observations humans can make in the language of a theory when we don’t know how to describe ourselves in the language of the theory without the help of a very substantial body of observations.
Setting aside this worry about epistemic circularity in hopes that a more pragmatic approach could dissolve the problem, we still need to assume that a pragmatic approach to observation would not entirely undermine the assumption of a privileged epistemic status for observations made using unaided human senses. And at that point we would still need work out how to categorize the many things that are, intuitively, humanly observable (that we believe would trigger a distinct sensory response in a human being to whom they were present), but which occur in situations such that a properly trained human being who was present would not survive long enough to actually recognize and interpret her sensory response. See the discussion in [2], op. cit.
- 37.
[7], op. cit.
- 38.
[6], op. cit.
- 39.
Sellars, op. cit.
- 40.
See [12], Thinking, Fast and Slow (Doubleday) for a rich overview of some of this work.
- 41.
For climate simulations and nesting, http://www.ouranos.ca/en/scientific-program/climate-science/climate simulations/, http://www2.mmm.ucar.edu/wrf/users/tutorial/200807/WRFNesting.pdf.
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Brown, B. (2016). On the Preservation of Reliability. In: Andreas, H., Verdée, P. (eds) Logical Studies of Paraconsistent Reasoning in Science and Mathematics. Trends in Logic, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-319-40220-8_4
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