Monday, November 4, 2019

William Whewell's "Discoverer's Induction" (Part 5)

William Whewell's "Discoverer's Induction" (Part 3)
William Whewell's "Discoverer's Induction" (Part 4)


This series' penultimate post will cover two of William Whewell’s three steps of induction. These steps are also his general theory of the generation of scientific hypotheses and theories. Whewell believed that these steps of induction are what scientists have followed in some form throughout history to discover and create conceptual knowledge and propel scientific inquiry. This progress in the creation and use of conceptual knowledge impacted all of the various, interconnected fields of science.

The Three Steps of Colligation

As mentioned previously, Whewell holds that a true colligation—his version of induction—consists of three steps. He notes that colligation progresses through the following steps: (1) the selection of the idea, (2) the construction of the conception, and (3) the determination of the magnitudes (Philosophy II, 380). He also describes these three steps as the argument, the law, and the amount of the change (Philosophy II, 383). When applying mathematical theories and calculations to the expression of a phenomenal/causal law and its relevant facts, Whewell mentions that these three steps can be described as (1) the selection of the independent variable, (2) the construction of the formula, and (3) the determination of the coefficients (Philosophy II, 382).

As previously mentioned in Part 3, philosopher of science Malcolm Forster noted that Whewell’s second description of colligation above applies to our post-modern conception of “curve-fitting” (“The Whewell-Mill Debate in a Nutshell,” 2006, 5). In curve-fitting, we construct a mathematical function/curve to determine the best fit for a given series of data points. This process usually includes determining the specific coefficients and the testing constraints to more precisely fit the curve to the available data.

Selection of the Idea

The first step of a true colligation is the selection of an abstract, broad idea that will subsume all of the facts under investigation. Depending on the state of a given science or a given scientist’s/investigator’s knowledge, the idea that is ultimately selected for the colligation may be what Whewell considers a fundamental idea such as space or time.

Whewell notes that sciences may never get off the conceptual ground or can stagnate without someone conceiving of the appropriate, more abstract idea, or fundamental idea, that serves to connect all of the data and facts being examined (Philosophy II, 383–385). Given his previous labors recounting the histories of the sciences of his time, he notes that “we have seen how each science was in a state of confusion and darkness till the right idea was introduced” (Philosophy II, 385).

Unfortunately, Whewell does not know of a way of improving our methods of discovering the right general idea to apply to the facts under investigation. Importantly, he remarks that these discoveries of the right broader idea tend to occur when certain individuals approach the issue who have cultivated a prepared mind:

Such events [selecting broader ideas] appear to result from a peculiar sagacity and felicity of mind;—never without labour, never without preparation;—yet with no constant dependence upon preparation, or upon labour, or even entirely upon personal endowments. (Philosophy II, 386)

Whewell points out that past philosophers and scientists often figure out the right general idea long before a pioneering scientist discovers the proper formula to explain some phenomena. He also notes that in his History that he called this series of identifications the prelude of a discovery (Philosophy II, 387). This sequence of the development of a problem-solution is similar to the history of philosophical problems: the original problem being proposed and its subsequent solution is carried out by different people, often separated across many years, even centuries.

One rule that Whewell believes may help a discoverer determine the right general idea is that the idea and the facts must be homogeneous (Philosophy II, 387). If we analyze a set of facts by reference to time, for instance, then we should colligate the facts by employing time in some form, with a time-related conception or scientific law.

A second and final rule that Whewell holds to be useful is one that he regards as obvious: “the Idea must be tested by the facts” (Philosophy II, 388). Considering the ideas leads to certain conceptions, certain modifications, and applications of the general ideas. These conceptions must be applied to the facts to test if they hold true or not. “The justice of the suggestion cannot be known otherwise than by making the trial,” Whewell reminds us (Philosophy II, 388).

The Construction of the Conception 

Supposing the Idea which we adopt, or which we would try, to be now fixed upon, we still have before us the range of many Conceptions derived from it; many formulae may be devised depending on the same Independent Variable, and we must now consider how our selection among these is to be made. (Philosophy II, 388)

Whewell’s instructions for constructing the proper conception to reach a scientific law are numerous, easily taking up an essay or even a series of essays themselves. The rest of this section will be a relatively short summary of his views on this step of colligation.

The Co-Occurrence of Steps 2 & 3 of Colligation

Whewell brings to our attention the general fact that the construction of the conception (step 2) and the determination of the coefficients (step 3) may be theoretically separable as colligative steps but practically occur at the same time (Philosophy II, 389). A hypothesis’s theoretical rule may lead to results that match the observed data, a circumstance which would confirm the veracity of the proposed formula/law as well as the formula’s coefficients. Reaching scientific truths requires aligning one’s hypotheses to the facts and also multiplying facts to test out hypotheses (Philosophy II, 389).

When it comes to determining the right mathematical formula, a discoverer may find themselves preoccupied with thoughts and experiments to not only satisfy the conditions for solving the original problem but also striving to draw out and clarify pieces and elements of the hypothesized formula. Whewell notes that when the observations of a scientific problem involve a progressive series of numbers, an incorrect formula will begin to have enormous deviations from the facts as experiments are carried out with that formula (Philosophy II, 392). Many observations and experiments across a broad range should be conducted: anything less would generally lead to scientific failure.

Here, Whewell reminds us that the true version of induction is not a matter of blind guesses/hypotheses or of merely collecting and comparing facts. “In such cases, the mere comparison of observations may long fail in suggesting the true formulae” (Philosophy II, 394). Once again, he mentions that the great inductive discoverers of the sciences held a combination of dedication, perseverance, good fortune and sagacity to some degree. Whewell cites times when Kepler correctly hypothesized planetary orbits to be elliptical in their motions but failed to derive the proper optical refraction formula. He also cites Étienne-Louis Malus’ (1775–1812; French officer, mathematician, Rumford Medal winner) failure to discover the formula between the polarizing angle of reflection and the reflecting material’s refractive index, even though it was Malus who originally discovered how light is polarized by reflection.

Special Methods of Induction for Quantities

In volume 2 of Philosophy of the Inductive Sciences, Whewell notes that the methods of induction applicable to matters of quantity or of resemblance lead the investigator to the discovery of only phenomenal laws. Induction methods that rest on other ideas, such as of cause and of substance, lead to more fundamental, causal laws (Philosophy II, 425).

When the phenomena under investigation can be numerically measured and may be subsumed under a numerical law, then Whewell suggests these mathematical methods to produce a more accurate formula. Of the seven methods/laws mentioned below, Whewell notes that the Methods of Curves, Means, Least Squares and Residues are the most important and prevalent.

The Method of Curves — When a quantity goes through numerous changes that depend on the progress of another quantity, this state of dependence can be mathematically expressed as a kind of curve. The rising and falling forms of any given curve will represent the relative increase or decrease of the quantity under study, with this occurring at intervals of space that represent the intervals of the quantity which is regulating the observed change, such as time.

Whewell gives the example of a curve describing the top heights of all high waters caused by tides at a specific place in a given year. This curve would have a pattern of ascending and descending forms that fit the annual progression of high and low tides (Philosophy II, 396). This method is immensely helpful, as it can express very complex figures, tables of calculations and numerical relations in an easily perceptible line shape.

It can be very effective at visually representing regular patterns of changes in a quantity, which can be observed as a pattern of rising and falling lines on a given drawn curve (Philosophy II, 397). The Method Curves is also a highly effective way to correct errors of observation, which may have remained unnoticed without all of the data points being graphed out and a line is drawn among them to expose the error (Philosophy II, 398–399). The results of clearing out erroneous observations are that we can gain data which are “more true than the individual facts themselves,” and wind up with separate facts “corrected by their general tendency” (Philosophy II, 399).

The Method of Means — This method involves determine the statistical mean quantity, the quantity that is equally distant from both a greater and smaller quantity. Through determining the mean of a set of data, we can mathematically remove errors of observation, the co-occurrence of other laws or other accidents that are distinct from the relation that is being investigated (Philosophy II, 404). Whewell here points out the corrections made to a study’s data using the Method of Curves is carried out through the Method of Means, of taking the total mean of the observations. Through the consideration of a great number of cases and observations, the Method of Means can be very effective at determining regularities in quantities that may seem random when only a few cases are considered. Whewell sagaciously notes that:

The effect of law, operating incessantly and steadily, makes itself more and more felt as we give it a longer range; while the effect of accident, followed out in the same manner, is to annihilate itself, and to disappear altogether from the result.” (Philosophy II, 408)

The Method of Least Squares — Whewell notes that the Method of Least Squares is a variant of the Method of Means. The Least Squares method seeks out “the best Mean of a number of observed quantities” or “the most probable Law” derivable from a set of observations that are admittedly imperfect to some measurable degree. This method supposes that “all errours are not equally probable, but that small errours are more probable than large ones” (Philosophy II, 408). What this means is that the best mean is determined not by taking merely the sum of the errors and finding the lowest sum of errors, but by determining the smallest sum of the squares of errors available (Philosophy II, 408).

The Method of Residues — This method is useful in cases when the proposed law appears to explain the vast majority of the progression and changes of the observed facts, but may still leave some quantities unaccounted for. These “Residues” can be treated similarly to how the whole observed set of data was originally investigated, in the hopes of determining a law of change to account for the residual quantity. These researches may lead to a series of investigations into second, third, fourth and more residual inquiries until all observed residual quantities can be sufficiently explained (Philosophy II, 410).

Special Methods of Induction for Resemblances

The Law of Continuity — This law means that a quantity “cannot pass from one amount to another by any change of conditions, without passing through all intermediate degrees of magnitude according to the intermediate conditions” (Philosophy II, 413). Whewell notes several interesting facts about this law: the First Law of Motion depends upon it (ex. zero resistance means zero retardation of an object moving horizontally); Galileo applied the law to the case of a body moving at some velocity from an initial state of rest; Leibnitz described the law more generally in his disproof of the Cartesian laws of motion (Philosophy II, 413–414). He also remarks that the usefulness of the Law of Continuity tends
to be reserved for disproving erroneous hypotheses and correcting faulty propositions, rather than for discovering new truths. “It is a test of truth, rather than an instrument of discovery,” Whewell admits (Philosophy II, 416).

The Method of Gradation — Discovering how phenomena agree, and how they differ, are both very important endeavors in any scientific inquiry. Another important part of practicing science is determining in what ways phenomena differ in degree when they are under comparison. By working with intermediate stages of a given property, it is possible to determine if some phenomena change from one classification to another through a distinct gap or leap, or through a continuous, gradual path. Whewell offers a plethora of examples of the Method of Gradation at work: Faraday’s gradations of conductors and non-conductors (insulators); the underlying similarity of voltaic and Franklinic electricity; LaPlace’s “Nebular Hypothesis”; Macculloch’s disproof of geological trap-rocks being properly classed as sedimentary rocks; Lyell’s geological theory of uniformitarianism (Philosophy II, 416–419).

The Method of Natural Classification — This method classifies phenomena not by their observed properties, but according to their most prominent resemblances. The science of classification relies on this method extensively, Whewell points out, using the example of how the genera of all of the various plants and animals are constructed, and of Mohs’ and Weiss’ crystalline classification based on their degree of symmetry (Philosophy II, 421). Natural classifications are important for science in that they facilitate the generation of general propositions (Philosophy II, 421). Once investigators have established classes through this method, they often seek out the properties that distinguish the different classes, often leading to provisional definitions of the classes known at the time of the investigation (Philosophy II, 424).


Whewell has more to say about this second step of colligation, but I believe I’ve covered the essential points of his theory. Part 6 will be the final part of this series on Whewell. It will cover the scientific tests for hypotheses that he views as necessary for the confirmation of our scientific theories as well as his general perspective on scientific induction.


Forster, M. (2006). “The Whewell-Mill Debate in a Nutshell.” [Online] Retrieved 
Whewell, W. (1847). Philosophy of the inductive sciences, founded upon their history (2nd ed.). 
            London: John Parker.
Whewell, W. (1858). Novum organum renovatum (3rd ed.). London: John Parker.

Next post: William Whewell's "Discoverer's Induction" (Part 6/Final Part)

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