A Historical Introduction to the
Philosophy of Science
Ch. 8: Newton's Axiomatic Method
The following is a summary of the eighth 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: Isaac Newton (1642–1727)
[p. 72] In two years (1655–57), Newton had:
 formulated the binomial theorem
 developed the calculus ('method of fluxions')
 constructed the first reflecting telescope
 discovered the universal nature of gravitational attraction
[LA: Note that Newton appeared at the tail end of the giants, Galileo Galilei, Francis Bacon and RenĂ© Descartes.]
Newton appointed Professor of Mathematics at Cambridge in 1669.
Newton elected fellow of the Royal Society in 1672 and President in 1703.
Robert Hooke accused Newton of appropriating his theoretical explanation of elliptical planetary motion.
Leibniz quarrelled with Newton on who developed the calculus.
The Method of Analysis and Synthesis
[p. 73] In his Opticks, Newton opposed Descartes' derivation of basic physical laws from metaphysics, insisting on experiment and observation instead.
Newton improved upon Aristotle's inductive–deductive procedure and the 'Method of Analysis and Synthesis' advanced by Grosseteste, Roger Bacon, Galileo and Francis Bacon by emphasizing the importance of:
 confirmation of deduction from the Synthesis stage, and
 deduction of consequences beyond the original inductive evidence [novel prediction]
Newton's method was vindicated with his experiments on light and prisms. By the Method of Analysis, he induced the theory of light as composed of different refracted colors.
This was not Newton using induction by simple enumeration, but theorizing about the nature of light.
[p. 74] By the Method of Synthesis, Newton deduced further consequences of his theory: that red light from a prism will be bent at a certain angle through a second prism. This was confirmed by experiment.
[pp. 74–5] Newton claimed in his Mathematical Principles of Natural Philosophy that he formulated his three laws of motion using the Method of Analysis; a process of induction of general laws from particular propositions.
[p. 75] But there are two senses of 'induction':

intuitive insight about ideal bodies (qua Aristotle) –broad sense

simple enumeration and methods of agreement and difference –restricted sense
Newton could not have derived his laws using induction in the restricted sense (e.g., no bodies described by the first law exist or can be observed on Newton's own account).
So, Newton's laws of motion are obtained inductively only in the broad sense; by abstraction from particulars.
Newton's concepts of Absolute Space and Absolute Time are also abstractions from their physical measurement. For Newton, Absolute Space and Absolute Time are ontologically prior to bodies and their motions.
[p. 76] Newton conceded that it may not be possible to measure Absolute Time as there may be no equable motion to use as a basis of measurement. He recommended using the eclipses of Jupiter's moons and the vibrations of pendulums as standard measures.
Against Descartes, Newton advanced a theological argument for the existence of Absolute Space as a receptacle for creation ex nihilo.
Newton also advanced his rotating bucket experiment as a physical argument for the existence of Absolute Space as there is no fixed correlation between acceleration and deformation of the water (see table on p. 77).
[p. 77] Newton concluded from his rotating bucket experiment that the acceleration of the water is with respect to Absolute Space. Ernest Mach and others objected that the acceleration may be with respect to the fixed stars.
Newton conceded that even if his experiment demonstrated absolute motion with respect to Absolute Space, he could not specify a system of coordinates for locating bodies in such space.
Newton's discussion of this problem illustrates his axiomatic method of analysis instead of his propounded inductive method.
An Axiomatic Method
Three stages in Newton's Axiomatic Method:

formulating an axiom system with axioms, definitions, and theorems

specifying a procedure for correlating theorems of the axiom system with observations

confirming the deductive consequences of the empirically interpreted axiom system
[pp. 77–8] Stage 1: Formulate axiom system
In Newton's theory of mechanics, the three axioms are:

Every body is at rest or rectilinear motion unless acted upon by a force.

Change of motion is proportional to the force impressed and is in line with that force.

Every action is opposed by an equal and opposite reaction.
[p. 78] Newton distinguished his axiomatic 'absolute magnitudes' from their experimental 'sensible measures'.
[p. 79] Stage 2: Correlate theorems with observations
Newton's Theory of ColourMixing relied on Pythagorean speculation and not inductive generalization for his pie slicing of the 'principal colours'. Granting this nonempirical axiom, Newton failed to provide a procedure for empirically determining the 'number of rays' for each colour.
On the other hand, with the 'Rules of Correspondence' used in his mechanics, Newton did provide links between statements about Absolute spatial and temporal intervals with events measured in the physical world. He used the centre of gravity of the solar system as the Absolute reference point for measuring distance.
[p. 80] For measuring Absolute Time intervals, Newton favoured linking these with the swings of a pendulum as this procedure delivered measurements that are more regular.
[LA: Note how Newton's choice of measuring instrument here presumes that the acceleration of balls down a plane is 'regular', which is precisely what he wished to prove with the use of the pendulum.]
[p. 81] Newton's most important contribution to the theory of scientific method and deductive systematization was his distinction between the axiom system and its application to experience.
Stage 3: Confirm deductive consequences of interpreted axiom system
Newton recognized that the degree of confirmation between the theorems and empirical observation can be increased by progressive modification of the original empirical assumptions (e.g., modification of assumption that earth is a homogeneous sphere to better account for the moon's motion).
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Newton failed to distinguish clearly his two theories of scientific procedure:

Method of Analysis and Synthesis – generalizing from the results of observation and experiment

Axiomatic Method – using creative imagination to create a formal system
"Hypotheses Non Fingo"
[p. 82] Newton agreed with Galileo in restricting the subject of physics to primary qualities ('manifest qualities').
Newton eschewed entertaining 'hypotheses' in his scientific work. For Newton:
hypothesis = statements about 'occult qualities' that cannot be measured
theory = invariant relations among manifest qualities
For Newton, his theory of colours had conclusive experimental evidence about refractive properties while avoiding any 'hypothesis' about the nature of light as waves or corpuscles.
For Newton, his theory of gravitational attraction was established while avoiding any 'hypothesis' about the underlying cause of the attraction. He rejected Descartes' Vortex Hypothesis.
[pp. 82–3] Inconsistently, Newton sometimes entertained hypotheses (e.g., the ether), while insisting that the sole purpose of such hypotheses is to direct future research.
The Rules of Reasoning in Philosophy
[p. 83] Newton nominated four regulative principles ('rules of reasoning in philosophy') for finding fruitful explanatory hypotheses:

Only admit causes of natural things that are both true and sufficient to explain their appearances.

To the same natural effects, assign the same causes.

Assume the qualities of bodies found in experiments to apply to all bodies.

Assume generalizations using induction to be true until contradicted by new observations.
[pp. 83–4] William Whewell [pronounced 'HUGHell'] objected that Rule 1's reference to 'true' cause may be either too restrictive or too vague and suggested that 'true' cause refer to a cause embedded within a theory that had diverse supports.
John Stuart Mill suggested that 'true' cause refer to a cause with independent evidence in its favour.
[p. 84] For Rule 3, Newton accepted the qualities that apply to all bodies as including extension, hardness, impenetrability, mobility and inertia.
Newton's insistence that these qualities also apply to the microconstituents of bodies set off future research programmes in chemistry and electromagnetism.
The Contingent Nature of Scientific Laws
[pp. 84–5] Newton rejected Descartes' deduction of scientific laws from metaphysical principles and their necessity. He saw scientific knowledge as contingent and provisional.
Questions to Consider:

Why do you think scientists regard authorship of a theory as so important?

Did Newton link successfully his axioms with observable quantities?

What do you take as the difference between the Method of Analysis and Synthesis and the Axiomatic Method?

Is Losee's distinction between two senses of 'induction' defensible?

How are the terms 'theory' and 'hypothesis' used by scientists today different compared with Newton's usage?

What do you see as Newton's greatest achievements in the theory of scientific procedure?