A Historical Introduction to the
Philosophy of Science

Ch. 11: Mathematical Positivism and Conventionalism

Book cover: A Historical Introduction to the Philosophy of Science by John Losee

The following is a summary of the eleventh chapter of John Losee's book, A Historical Introduction to the Philosophy of Science (fourth edition), with some ancillary notes.

Philosophers discussed in this chapter: George Berkeley (1685–1753); Ernst Mach (1838–1916); Pierre Duhem (1861–1916); Henri Poincaré [pronounced 'Enri Poyn-car-ay'] (1854–1912); Karl Popper (1902–94)

[LA: This chapter pulls together around a common theme: Mach's hope that scientific theories as instruments of prediction are refutable, Duhem and Poincaré's denial of that confidence and their solution based on scientists' conventional acceptance of some theories, Popper's unease with this lack of refutability and his positing another form of conventionalism that saves the falsifiability of theories.]

Berkeley's Mathematical Positivism

(p. 144) Berkeley pointed out Newton's hypocrisy in his suggesting what forces are 'in themselves' instead of seeing them as useful mathematical constructions for calculating the motions of bodies. For Berkeley, 'material substances' and forces have no real existence.

(p. 145) Instrumentalism is the view that scientific laws are nothing but computational devices for the description and prediction of phenomena. The terms in these laws and their functional dependencies need not refer to anything that exists in nature.

For Berkeley, only minds and their ideas exist ('Idealism'). Only minds have causal power, not material forces.
[LA: For a critique of Berkeley's Idealism, see my 'The Existence of Mind-Independent Physical Objects'. For a critique of a contemporary version of Idealism argued by Donald Hoffman, see my 'Hoffman's Conscious Realism: A Critical Review'.]

Berkeley rejected Galileo's, Descartes' and Newton's distinction between primary qualities of material bodies (e.g., extension, position, motion) and secondary qualities of subjective perceptual experience (e.g., heat, brightness) as the former are also only given in perceptual experience.

Berkeley also rejected Newton's Absolute Space as spatial interval is meaningless without our perception of bodies and their motion relative to one another.

(p. 146) Berkeley criticized Newton's bucket experiment for not demonstrating absolute circular motion in a bucket, suggesting instead that terrestrial motion be made relative to the fixed stars. For Berkeley, 'Absolute Space' is a useless fiction and should be eliminated from physics.

Mach's Reformulation of Mechanics

Mach mimicked Berkeley's instrumentalist critique of Newton. For Mach, Snel's law of refraction is a 'compendious rule' for the mental reconstruction of the various instances of refraction.

Mach proposed a Principle of Economy as a regulative principle in science for summarizing the greatest numbers of facts using comprehensive theories to deduce empirical laws.

(p. 147) Like Berkeley, Mach eschewed the reality of primary qualities, atoms and electric charges, allowing only phenomena (hence, 'Phenomenalism').

Mach divested Newtonian mechanics of 'metaphysical' presuppositions by reformulating as:

  1. three contingent empirical generalizations
  2. a priori definitions of 'mass-ratio' and 'force'

(pp. 147–8) For Mach, confirmation of his empirical generalizations requires procedures for measuring spatial intervals against the background of 'fixed' stars and temporal intervals by physical processes.

(p. 148) Losee objects that Mach's reformulation of Newton's empirical generalizations is not subject to experimental falsification, as Mach had claimed. Mach's first generalization about 'contrary accelerations in the direction of their line of junction' may be saved from disconfirmation by claiming the problematic experiment was not conducted within a closed system.

Duhem on the Logic of Disconfirmation

Duhem emphasized how empirical generalizations can be made true by convention ('Conventionalism').

For Duhem, scientific predictions (E) are logically deduced conjointly from:

  1. (L) statements of the relevant empirical laws
  2. (C) statements of the antecedent conditions

For Duhem, even where antecedent conditions C is taken to be true by scientists, any one of the hypotheses stated in empirical laws L may be rejected while saving the others. Which hypotheses to save by convention is decided by the objective judgment of scientists.

One reason for saving a hypothesis from disconfirmation is that it is essential in a number of other confirmed theories.

(pp. 149–50) Duhem criticized Francis Bacon's notion of a 'crucial experiment' (e.g., Foucault's supposed experiment falsifying the corpuscular theory of light). Adjustments could be made elsewhere in the Newton/Laplace corpuscular theory and wave proponents failed to prove that the wave theory is the only possible alternative.

Poincaré's Conventionalism

(p. 150) Poincaré rejected Kant's and Whewell's appeal to necessary a priori scientific truths. For Poincaré, scientists agree by convention that certain physical laws are true by definition.

Poincaré showed how a decisive test of Newton's law of inertia would require an impossibility; that each body in the universe reassume its earlier position and velocity. In fact, scientists assume that bodies undergoing test are 'reasonably isolated' from the rest of the universe. Test discrepancies can be attributed to an incompletely isolated system.
[LA: Why did Poincaré think a decisive test of Newton's first law requires all bodies to return to their intial position and velocity? It seems because doing so allows other unknown causal influences to play out, giving a different test result if there are such influences.]

(p. 151) For Poincaré, 'inertial motion' (Newton's first law) means the motion of a body whose acceleration depends only on its position and the positions and velocities of neighbouring bodies.

However, for Poincaré, Newton's first law was also an empirically significant generalization holding approximately for 'almost isolated' systems.

Poincaré also analysed 'force' and 'mass' in Newton's second and third laws as being both definitional and empirical generalizations for 'almost isolated' systems.

(p. 152) For Poincaré, such conventional definitions are not arbitrary. They are justified by their fruitfulness in future research.

Poincaré thought physical relations will always be described by Euclidean geometry as it is the simplest to apply. Any discrepancy with experimental tests can always be attributed to the bending of light rays.

But Hempel pointed out that the principle of simplicity applies to the conjunction of pure geometry and physical hypotheses. Overall simplicity may be got by adopting a non-Euclidean geometry.

Popper on Falsifiability as a Criterion of Empirical Method

(p. 153) Popper recognized that a theory can always be saved either by rejecting the observational evidence, adding auxiliary hypotheses or modifying the rules of correspondence.

Popper eschewed saving a theory by methodological fiat. He set a meta-criterion for proper methodological rules: that no rule protect a statement in science against falsification.

So, when adding auxiliary hypotheses to save a theory, only those be added that increase the falsifiability of the theory (e.g., allowable Pauli's exclusion principle vis-à-vis bad Lorentz contraction hypothesis).

(p. 154) Popper proposed a clear demarcation between a scientific theory and non-science. A scientific theory:

  1. is exposed to the possibility of falsification
  2. has withstood serious attempts at experimental refutation
Book cover: The Logic of Scientific Discovery by Karl Popper

For Popper, a serious test compares a deductive consequence of the hypothesis (plus initial conditions and auxiliary hypotheses) with a 'basic statement' recording an observation by multiple observers in a specific region of time and space.

As basic statements may record faulty observations, Popper conceded that the acceptance of such statements by the scientific community is by convention.

For Popper, the worth of a physical law or theory is measured by the number, diversity and severity of tests it has passed. Most philosophers of science accept this as a qualitative account of justification.

(p. 155) Popper provided a quantitative measure of justification. For any two comparable theories, one is closer to the truth (more verisimilitude) compared with the other if it either has more truth content than the other or has less false content than the other.

However, Tichy and Miller proved that if both theories are false, then neither condition is satisfied. Popper failed in his attempt to quantify theory acceptability for known false theories and hence failed to show how science progresses.

Popper insisted that passing severe tests does not show a theory to be true or approximately true and resisted appeals to inductive inference from past successes.

Losee complains that Popper's appeal to the evolutionary fitness of theories passing severe tests, then, is misguided. Popper has given us no reason to select for further applications theories that have survived such tests over and above failed theories.

(p. 156) In the end, Popper accepted a 'whiff of inductivism' based on the assumption that reality must be in some respects similar to what our theories tell us it is. He argued that it would be a fantastic coincidence for our theories to make spectacularly unlikely predictions that proved to be true unless there was some truth to them.

Poppers critics claimed that he had abandoned the anti-inductivist programme.

Questions to Consider:

  1. Are scientific theories only instruments for prediction, or do they point to a posited reality behind the phenomena?
  2. What role does convention play in scientific confirmation and falsification of theories?
  3. Is Popper's acceptance of conventionalism with regard to his 'basic statements' a serious objection to his objectivist account of science?
  4. Did Popper succeed in giving a quantitative measure for the comparative worth of theories?
  5. Was Popper's attack on inductivism successful?

Copyright © 2022–3

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