Monday, May 23, 2011

Advances in Baconian Induction: John Herschel (Part 3 of 3)

(Previous posts:
Advances in Baconian Induction: John Herschel (Part 1 of 3)

Advances in Baconian Induction: John Herschel (Part 2 of 3))

John Herschel’s theory of induction is a kind of empiricist epistemology rooted in analogies, from which we can generalize to hypotheses, theories, and the laws which are the foundations for theories. This essay will present Herschel’s views on the higher-stage inductions he believes comprises true scientific theorizing.

Analogical Reasoning in Induction

“Analogy” is perhaps the most important term in Herschel’s philosophy of science and theory of induction. It’s so crucial that it wouldn’t be an exaggeration to say that his entire theory of induction is a progression of generalizations made mainly on the strength of analogous cases or analogous causes.

Before I discuss this any further, however, I’ll briefly explain what Herschel means when he says “analogy.” As Dr. Gildenhuys states in his essay, “[a]nalogies between phenomena are what allow us to group them together as sharing the same explanation.” An analogy is a kind of inference in which a comparison of things that share certain features is used as a basis for inferring that these things share another aspect as well. In Herschel’s time, an argument from analogy would consist in showing similarities between two phenomena and from that inferring that they share another feature, in which it wasn’t already known that one of the two phenomena possesses that feature. Herschel’s view of analogy applies when different phenomena share a common cause, or an analogous cause, even if everything else about the phenomena is too different to warrant more analogies. (See page 149, aphorism 142 of his Preliminary Discourse for more on analogies.) What this means is that Herschel’s isn’t adopting an argument from analogy as the composition of scientific, inductive argument. (See Peter Gildenhuys, Darwin, Herschel, and the role of analogy in Darwin’s origin, Studies in History and Philosophy of Biological and Biomedical Sciences, vol. 35, 2004, p. 596-97. Letters in brackets mine.)

When we seek to explain some phenomena, Herschel remarks, we first check our knowledge of real causes (vera causa) to determine if an analogous phenomena is produced naturally—if it is, then we have some reason to assign that real cause as the cause of the phenomena. (p. 148, aph. 141) If the cause of analogous phenomena doesn’t allow us to discover the phenomenon's explanation, then we’re allowed to propose a cause analogous to the cause of the analogous phenomena (that is, a similar effect warrants the suggestion of an analogous cause). (pp. 148-149) In the case of the analogous cause, how it causes one of the analogous phenomena isn’t obvious and requires argument, whereas it’s obvious or already known how the real cause causes one of the other analogous phenomena. (See Arnould Duwell, Philosophical Influences on the Large Scale Structure of Darwin’s Origin, p. 4, unpublished manuscript.)

An example will help:

Let’s say that you knew the function of a 18th century “manual fire pump,” which distributed water to be expelled from a hose towards an ongoing fire by a fire brigade. Most notably, you knew that it works by the operation of valves, including “one-way valves” that kept water circulating in only one direction. And let’s suppose that you knew this, but didn’t know how blood circulated in the body. If you tied off a living person’s arm, he would immediately feel the lack of blood flow, and any observer could begin to see the effects (like the pale pigment of the skin, swelling on the side that isn’t tied off, etc.), just as you could stop the flow of water in fire pump by closing off a valve. Further experiments on how blood circulates would lead you to conclude that the heart is a “blood pump,” analogous to the fire pump that you know much more about. The “pumping” action would be an analogous cause of two analogous phenomena, the circulation of water and the circulation of blood.

(For more on “manual fire pumps,” see the informative “In the Line of Fire,” and the Wikipedia page on William Harvey, the scientist who proved that the heart is the cause of the circulation of blood.)

I said earlier that Herschel only requires one similar trait for phenomena to be analogous: either the same cause or both phenomena having causes so similar that they are analogous to each other. However, he does note the benefits of having close or strong analogies among phenomena (meaning multiple, noteworthy analogies between two classes of things), as it makes generalizing and scientific classifying easier for scientists. The lack of such analogies doesn’t defeat the goal of scientific investigation, but does increase the work needed to discover the proximate cause or law of nature. Herschel advises generalizing based on the points of agreement of the analogous phenomena that are the subjects of investigation towards the construction of a law of nature. (p. 98-99) Strong analogy is not only useful for constructing a generalization, but it is also a good reason to not deny that something is a cause, a priori. This was explicitly advocated by Herschel in his third method of philosophizing, what I termed the “method of strong analogy,” that cautions us to avoid denying that something is the cause of a phenomenon “a priori,” without experience, when that thing has many strong analogies in support of it being the cause. So for Herschel, analogies play a role in reaching proximate causes and a role in inductive constraints on what we can properly deny and assent to, namely something with strong analogies supporting it (or the absence of this criterion).

The inductive search for proximate causes being the first stage of induction, the discoveries of more fundamental laws of nature and theories are the second and highest stage of induction. In both, we’re investigating in order to discover the simplest phenomena, the most fundamental laws, and the highest theories possible to us, and the process by which we do both stages bears a close analogy to one another. (p. 191) “Hypotheses, with respect to theories, are what presumed proximate causes are with respect to particular inductions : they afford us motives for searching into analogies ; grounds of citation to bring before us all the cases which seem to bear upon them, for examination.” (p. 196) When enough analogies and probable evidence have been amassed, Herschel claims that we must accept one of two things: either (1) that the hypothesis is an actual statement of what really happens in nature or (2) that the hypothesis has named something that applies to both the phenomenon described within it and the actual reality of the situation, to the extent that we know it. In the forming of theories, the causal agents being referred to must create a change in the phenomena that is analogous to the change we’re attempting to explain by forming the theory. The difficulty of testing the laws referred to in a theory practically precludes the method of forming an hypothesis, fleshing it out, and then testing it against reality, unless an analogy or some other reasoning convinces us that the attempt will work out (or earlier partial inductions carried out lead us to this hypothesis). (p. 200)

For Herschel, analogical reasoning plays a crucial part in the construction of inductions and in the progression of scientific knowledge more generally. It affords us the ground to investigate potential shared causes, and to halt any a priori rejections of purported proximate causes. It gives scientists the grounds for framing an inductive hypothesis, while also providing constraints on what hypotheses may and may not be tried. And analogies ultimately decide the veracity of a theory by how close the phenomena predicted by the theory run parallel with the actual facts.

Hypothesis and Its Value

Hypotheses, according to Herschel, are universal inductive principles that include several particular inductions or laws within them. Just as a well confirmed proximate causal law becomes a particular induction in the first stage of inductive reasoning, a well confirmed hypothesis becomes a theory in this second and highest stage. (p. 196)

Hypotheses are valuable for a number of reasons, in Herschel’s view. Through inductive considerations of general laws, an inferred hypothesis will practically guarantee that we can generalize a level beyond those laws, thus subsuming them under a more universal formulation. (ibid.) More than sketching out a universal law to encompass general laws, however, hypotheses, when a sufficient amount of analogies and probability is gathered up on their side, convince us of one of two things: that it really explains what happens in nature, or that it’s very similar or approximates what really happens and thus expresses both the reality and the proposed reality of the situation. By allowing us to create theories that lead to us reaching general laws, hypotheses, Herschel remarks, “often have an eminent use”:
And a facility in framing them, if attended with an equal facility in laying them aside when they have served their turn, is one of the most valuable qualities a philosopher can possess; while, on the other hand, a bigoted adherence to them, or indeed to peculiar views of any kind, in opposition to the tenor of facts as they arise, is the bane of all philosophy. (p. 204)
Verae causae, true causes, are what hypotheses must propose, or in Herschel’s words, “[i]n framing a theory which shall render a rational account of any natural phenomenon, we have first to consider the agents on which it depends, or the causes to which we regard it as ultimately referable.” (p. 197) As Laura Snyder notes in one of her essays, there are two requirements for something to be a “vera causa” for Herschel. “First, it must be a cause whose existence in other cases is already known. Secondly, it must be a cause whose ability to produce a similar –or analogous – effect is known independently of it putative responsibility for causing the phenomena in this case.” Despite the name, a vera causa might not be the literal, actual cause at work in a given case, but is rather a true cause in the sense that it is “a causally efficacious agent at work in analogous instances.” Another important relationship between analogies and Herschel’s view of vera causa is that a true cause can allow a person to hypothesize or infer a theoretical or unobservable cause, as long as this cause is analogous to other known causes. Snyder notes that, “indeed, this is the way in which Herschel allowed for theoretical science while still endorsing an empirical, inductive method.” (See Laura Snyder, “Hypotheses in 19th Century British Philosophy of Science: Herschel, Whewell, Mill,” pp. 11-12.)

At the stage of hypothesis, we are dealing more with the elements of our reason than the first stage of induction, which deals mainly with the realm of sense. The phenomena of hypotheses and theories are general phenomena, “creatures of reason rather than of sense.” Herschel elaborates on this point:
In raising these higher inductions, therefore, more scope is given to the exercise of pure reason than in slowly groping out our first results [that is, the “first results” were particular inductions]. The mind is more disencumbered of matter, and moves as it were in its own element. What is now before it, it perceives more intimately, and less through the medium of sense, or at least not in the same manner as when actually at work on the immediate objects of sense. (p. 190, words in brackets added)
Because of his advocacy of hypotheses, it has been supposed that Herschel was an early advocate of the modern “hypothetical-deductive method,” in which a person forms a hypothesis without any requirement for inference or reasoning, but simply as a matter of conjecture or guessing, and then tests the hypothesis’s consequences to determine if they are true. But it should be clear from everything I’ve said about Herschel’s methodology, his intellectual mentor Francis Bacon, and his theory of science, that he was a proponent of induction, not of creating hypotheses without inference requirements. After discussing how the higher inductions are more abstract and more in the element of reason, he goes on to say:
But it must not be therefore supposed that, in the formation of theories, we are abandoned to the unrestrained exercise of imagination, or at liberty to lay down arbitrary principles, or assume the existence of mere fanciful causes. The liberty of speculation which we possess in the domains of theory is not like the wild licence of the slave broke loose from his fetters, but rather like that of the freeman who has learned the lessons of self-restraint in the school of just subordination. (pp. 190-191)
There are inductive constraints on Herschel’s account of hypotheses. Hypotheses are inferred from an inductive consideration of laws (laws that are less fundamental than what the hypothesis proposes), or are inferred from some other source, such as from analogy. (p. 196 and 200)

Such is Herschel’s view of hypotheses.

Theories

A theory is a hypothesis which explains a natural phenomenon which is in the process of being verified, or has been verified.

In Herschel’s view, we first make particular inductions to explain proximate causes we presume, and then go to create hypotheses to explain general, elementary laws which also serve as their foundation. These particular inductions lead to the positing of a general law. Deducing the particular cases of such a general law, and determining if what a given hypothesis asserts stands to scientific trial is what Herschel believes “constitutes theory in its largest sense.” (90) He gives a helpful example:

Particular inductions were given by inferring the motions of the planets around the Sun, and of the satellites around their respective planets (by Newton) led us to the law of gravitation: an attractive force which is exerted by every particle of matter on every other in the universe. To verify such an induction, Herschel notes, we assume the law, and assume that the universe really operates according to it, and deduce previously unknown consequences. This allows us to notice that the planets must all attract each other, and pull each from the orbit they would have if it was only the sun were acting on them. For the theory of universal gravitation to be considered true, scientists would have to calculate the deviations in the projected orbits this new insight would lead to, and to determine if these new results match with the facts of observation.

He notes that theories can be formed not only to explain laws that apply directly to materials (like gravity), but also in order to suggest a “system of mechanism, or a structure of parts,” through which some natural action becomes observable to us. (202) Sometimes, it is hard for someone to accept the hypothesis of a previously unknown mechanism or structure, thinking that it is too complex to be admitted. Understanding this worry, Herschel nonetheless states that:
[…]yet, if the admission of this or any other structure tenfold more artificial and complicated will enable any one to present in a general point of view a great number of particular facts,--to make them a part of one system, and enable us to reason from the known to the unknown, and actually to predict facts before trial,--we would ask, why should it not be granted? (203)
In any event, our first concern for evaluating a theory shouldn’t be how well or poorly it proves its hypothetical structure or mechanism, something which we could only know indirectly anyway, only by it leading to the same results. The most important thing to know by far is whether or not a given theory “represent[s] all the facts, and include all the laws, to which observation and induction lead.” (204) A theory which did include all relevant facts and laws would provide great support towards the validity of the hypothesized mechanism or structure.

In verifying theories, we can check them against not merely particular facts and cases, but whole classes of facts, general laws that the theories are based on. (206) This fact has a very important application to the case where two theories are both capable of explaining a great wealth of facts: referring to general laws allows for a much greater range of cases to test the two theories against each other, in order to find a case in which a certain effect would occur in a certain way if one of the theories was right and the other wrong, and vice versa.

The best way to verify a theory is to deliberately pick all varieties of cases, even extreme cases that might refute the theory, and a number of these that would be sufficient to reasonably detect any probable errors. Herschel’s view is that this means of verifying the theory will also verify the “whole train of induction” from the lowest, first stage inductions, onto the highest ones that compose the theory. (208)

Lastly, Herschel notes that it shouldn’t matter how a theory was originally framed, or what postulates it contained, if it is capable of holding its own in the face of extensive tests, even if the critic believes that the postulates are strange and inadmissible. If the application of the theory consistently lead to logical conclusions in accordance with numerous observations, under a variety of circumstances that the theory must account for to be considered valid, then “we cannot refuse to admit” it. (208-209) He allows that a skeptic of the theory must at least consider the theory to be a temporary substitute for the truth until the full truth, that is, a better theory, can become known.

Herschel elaborates:
If they suffice to explain all the phenomena known, it becomes highly improbable that they will not explain more; and if all their conclusions we have tried have proved correct, it is probable that others yet untried will be found so too; so that in rejecting them altogether, we should reject all the discoveries to which they may lead.
The “Three Ways” to Know the laws of a primary causal agent

How do reach those elementary laws of nature that our theories will be based upon?

Herschel believes that there are only three ways to do this:
1.
By inductive reasoning; that is, by examining all the cases in which we know them to be exercised, inferring, as well as circumstances will permit, its amount or intensity in each particular case, and then piecing together, as it were, these disjecta membra [scattered members], generalizing from them, and so arriving at the laws desired… (Prelim. Disc., Aph. 210, p. 198, words in brackets mine.)
More specifically, Herschel is referring to reaching inductive laws in which “one quantity depends on or varies with another.” (Consider Newton’s gravitational law, in which the effect of gravity differs depending on the distance between two given masses.) To inductively reach such laws, one must conduct a “series of careful and exact measures in every different state of the datum and quaesitum [fact; inquiry].” Here, the ultimate goal of the induction is to reach the mathematical form of the law.

Herschel says that the greatest attention must be given to the extreme cases of the purported law, and to cases in which there’s a rapid change in one quantity accompanying the relatively small change of another quality. He informs us that the results of the measurements should be copied down on a table in which the fact being investigated increases in magnitude from the lowest to the highest limit that one can achieve. It then depends on our mathematical knowledge to determine if we can include this table with the statement of a mathematical law. (Aph. 185, p. 176-77) Such mathematical laws Herschel terms “empirical laws.” He claims that these same directions apply for the higher inductions of theories, since they connect lower inductive laws, not merely facts:
There is no doubt, however, that the safest course, when it can be followed, is to rise by inductions carried on among laws, as among facts, from law to law, perceiving, as we go on, how laws which we have looked upon as unconnected become particular cases, either one of the other, or all of one still more general, and, at length, blend together in the point of view from which we learn to regard them. (Aph. 217, pp. 204-05)
2. “By forming at once a bold hypothesis, particularizing the law, and trying the truth of it by following out its consequences and comparing them with facts…” (Prelim. Disc., Aph. 210, p. 198-99)

In this case, Herschel informs us, “the law assumes all the characters of a general phenomenon resulting from an induction of particulars, but not yet verified by comparison with all the particulars, nor extended to all that it is capable of including.” (Apr. 213, pp. 200-201) By assuming a hypothesis, one can analyze the circumstances which may “modify the effect of the cause whose laws of action we have arrived at and would verify.” Herschel explains this point in a well-reasoned discussion of the application of the law of gravitation to planetary orbits:
…when we would verify this induction [the law of gravity], we must set out with assuming this law, considering the whole system as subjected to its influence and implicitly obeying it, and nothing interfering with its action; we then, for the first time, perceive a train of modifying circumstances which had not occurred to us when reasoning upwards from particulars to obtain the fundamental law: we perceive that all the planets must attract each other, must therefore draw each other out of the orbits which they would have if acted on only by the sun; and as this was never contemplated in the inductive process, its validity becomes a question, which can only be determined by ascertaining precisely how great a deviation this new class of mutual actions will produce. (Apr. 213, p. 201, words in brackets mine)
Though the method of assuming a hypothesis and testing it had worked successfully on occasions, Herschel points out that the sheer difficulty in tracing out an assumed fundamental law into its consequences prevents this method from becoming common and widespread. The only exception would be if we had some form of inference, some reason, whether from analogy or something else, that convinces us that the attempt will be successful, or having reached some partial induction to particular, “empirical laws” which then point out the assumed fundamental law to be tried. (Aph. 212, p. 200)

3. Lastly:
By a process partaking of both these, and combining the advantages of both without their defects, viz. by assuming indeed the laws we would discover, but so generally expressed, that they shall include an unlimited variety of particular laws; --following out the consequences of this assumption, by the application of such general principles as the case admits;--comparing them in succession with all the particular cases within our knowledge; and, lastly, on this comparison, so modifying and restricting the general enunciation of our laws as to make the results agree. (Prelim. Disc., Aph. 210, p. 199)
This third and last method Herschel believes is the best, because it, as he says, combines the advantages of both without their defects. The inductive process (the first method) uses particular facts and empirical laws and leads to a very specified general law, but is susceptible to missing certain minor details and consequences that could be noticed as a result of using the assumption method (the second method). The second method works well at seeking out modifying circumstances that would change the specifics of the fundamental law being assumed, but we often lack the incentive, the evidence, whether analogical, inductive, or something else, that points towards the assumed law being a hypothesis with a basis, and thus warranting a deductive inquiry.

Herschel observed that the mathematicians of his time found this third method to be the “most universally applicable, and the most efficacious.” He also comments that it is of particular benefit in cases where inductions of empirical laws have already been reached, thus capable of being expressed mathematically; this leads to tracing out the consequences of more fundamental laws by reference to inductively reached, empirical (less-fundamental) laws. The example of this Herschel gives are the inductive, mathematical laws of the elliptic motions of planets, and their relation to the assumed elementary law of force.

Conclusion

John Herschel was the most accomplished “man of science” of his age, and many future scientists would emulate the methods and advice he presents in this monumental work on the philosophy and methodology of science, most notably Charles Darwin.

I do not have the necessary space to present his views of verifying the data of a theory, and the role of probability theory in science. (That discussion includes a response to a “vicious circle” objection regarding the use of observations assumed in a theory that are used simultaneously to verify it.) Those points will have to wait for another essay.

I’ll end with a quote from Herschel, near the close of Part II of his book:
In the foregoing pages we have endeavoured to explain the spirit of the methods to which, since the revival of philosophy, natural science has been indebted for the great and splendid advances it has made. What we have all along most earnestly desired to impress on the student is, that natural philosophy is essentially united in all its departments, through all which one spirit reigns and one method of enquiry applies. (Prelim. Disc., Aph. 231, p. 219)

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