Induction: inducing the law of cure

Induction, Deduction, Medicine and Scientific Method

It has been argued recently that the HPV vaccine is safe because so many millions of doses have been administered, therefore it is safe. Not so, say philosophers of science. The reason is because the argument is based on induction: because the sun rises every morning is no guarantee it will rise tomorrow morning. Thus, because the vaccine worked or was safe in 3,000,000 people is not a guarantee it will be so in the 3,000,001st

Induction and Scientific Method

Geoffrey Gorham in his Philosophy of Science (p. 53) says that science uses empirically testable theories and explanations. But when has a theory failed or passed and at what point can we conclude this? How do we come to a theory? Historically, he says, there are two methods, deduction and induction.

Frederick Copleston explains further in A History of Philosophy (p. 289): science has two sides: the observational and inductive and secondly the mathematical and deductive.

Induction, by definition, examines a particular event and draws conclusions based on that experience and empirical observation. From this particular knowledge it extrapolates to a general rule or law.

Deduction is the opposite: working from a general rule it follows a certain logic leading to knowledge of a particular.

Logically, in induction, the premise can be true but the conclusion false. With deduction, if the premise is true the conclusion has to be true.

Samir Okasha gives examples of induction in Philosophy of Science: a Very Short Introduction (pp. 18-19):

The first five eggs in the box were rotten
All eggs have the same sell-by date
—————————————————
Therefore, the sixth egg will be rotten.

With deduction, the logic is as follows:

All French men like red wine
Pierre is a French man
——————————–
Therefore, Pierre likes red wine

In the last case, there is an appropriate relation between the premises and conclusion. So, if the premises are true, it follows the conclusion is.

With the previous example of inductive logic, the premises can be true but the conclusion false. We move from an object we’ve examined (empirically) to an object we haven’t examined.

“In effect”, says Okasha, “scientists use inductive reasoning whenever they move from limited data to a more general conclusion which they do all the time.” (p. 22)

As mentioned, inductivists favour a philosophical position of Empiricism – knowledge from experience – while deductivists prefer Rationalism, whereby knowledge is justified by rational ideas. Rationalism seeks a priori knowledge, Empiricism a posteriori.

Descartes, the French philosopher and mathematician who was famous for seeking certainty, said all knowledge should be deductive. For Newton we generalise, he believed, from finite experience to universal law which is uncertain and risky but is “exactly or very nearly true”, for a grounded experience-based science rather than a science based on metaphysical hypotheses, Gorham tells us. (pp. 56-57)

Yet neither of them adhered purely to their own methodology, according to Gorham.

For Francis Bacon (see The Unprejudiced Observer), direct and unprejudiced observation was the way to sure knowledge, based on sense experience and rationale controlled by observation (Copleston, 300ff). This is the method that persisted until now, according to Bacon, and has been unsatisfactory as it ended up with hasty abstraction.

So, to improve the quest for truth, Bacon insisted we must go from sense perception of particulars to attainable axioms, then gradually to more general axioms, which is how the mind interprets nature.

Examples of Induction

By now it should be clear there are problems obtaining certainty of knowledge, particularly with regards to induction. This is despite countless examples of induction that led to many discoveries. Okasha cites some such examples: splitting the atom, landing men on the moon and inventing computers etc. Likewise, not learning from induction can be costly: because arsenic killed someone, am I going to swallow it without proof it will kill in all cases?

Okasha further explains: scientists’ claim that genetically modified maize is safe is, in reality, saying it has only been proven safe in those on whom it has been tested. Similarly, the claim that a drug works because it was tested on a large group of people and was shown to work doesn’t mean it will work in every case. Again, Newton, he says, inductively inferred his principle of universal gravitation. But, “He couldn’t have examined the gravitational attraction on every body. The fact his principle holds true for some bodies doesn’t guarantee it holds for all bodies.” (p. 22)

Okasha concludes: “The word ‘proof’ should strictly only be used when we are dealing with deductive inferences. In this strict sense of the word, scientific hypotheses can rarely, if ever, be proved true by the data.” (p. 23) “Yet”, says Okasha, “it’s very hard for scientists to get by without some sort of inductive reasoning.”

We have seen that drugs aren’t proven in the strict sense of the word, despite what many would have us believe, so where does this leave homeopathy and is there proof of Hahnemann’s law of cure?

Homeopathy and Induction

A quick search of a homeopathy data base reveals hundreds of references to “induction” and “inductive”, particularly in late nineteenth century journals. For example, in the proceedings of the American Institute of Homeopathy, 1900, we read: “Induction led to other theories, even cholera treatment without having seen a cholera case.”

The Homeopathic Herald (1941, Vol. IV, no. 8), in an article “Medicine in the Two Elizabethan Eras”, claims “Harvey arrived at the inductive proof that the heart acts as a muscular pump – although proved by experiment was the subject of much bitter opposition.”

In “Common Ground Reached at Last” (Homeopathic World, 1946, Vol. VII) we read: “Cullen rejected the inductive method of inquiry which would have led them to the cause and reasons but adopted the deductive method which led to nothing but effects.”

In The Homeopathic Physician, (1881, Vol I, no II) in an article “Isopathy a Fatal Error” we are told:

The deductive method of isopathists is wrong and Hahnemann’s inductive method is correct. Isopathists cure a man made sick by cucumber with cucumber. So, are we to treat all allergies with that food (i.e. deductive)? No! Take for example “Alumina for aggravation from potatoes”. The inductivist uses Alumina for individuals who have the particular reaction to potatoes as shown in the provings of Alumina only.

Isopaths haven’t shown isopathy works well in all cases.

The Homeopathic Physician (1881, Vol I) comments:

The observations then made are just as valuable today, will be as valuable for all time: as are now and ever will be the inductive reasoning to which this and other observations led this indefatigable philosopher [Hahnemann].

Again homeopathic literature, appealing to examples from the history of medicine, shows how induction has worked:

John Hunter showed by studying inflammations by inductive research that rapid processes followed physiological laws. Broussais (1728-93) assumed most diseases had their seat in the stomach and intestines, therefore requiring the withdrawal of blood by means of leeches and venesection. (cf. Richard Haehl, Vol II, Ch. XXI, p. 391)

But the law of cure, that like cures like, was discovered inductively, we can say, by observing cured cases. So, the law is a general and the application of that law is by means of deduction. Also, regarding the application of that law, a drug is proven to elicit the symptoms it can produce, its pathogenesis, and from these symptoms we prescribe for similar symptoms in a sick person. Carroll Dunham in his The Science of Therapeutics (p. 30) quotes J.S. Mill to this effect:

If, for instance, we try experiments with mercury upon a person in health, in order to ascertain the general laws of its action upon the human body, and then reason from these laws to determine how it will act upon persons affected with a particular disease, this may be a really effectual method, but this is deduction. (A system of logic, ratiocinative and inductive: being a connected view of the principles of evidence and the methods of scientific, ch. X, section 8)

The eliciting of symptoms as Mill describes here – unwittingly the homeopathic method – is deductive and therefore sound logic so long as the premise is true. The premise being that Mercury causes certain symptoms and will remove similar symptoms. On this point, Dunham says:

A homeopathic prescription, as we have defined it, is a deduction from a generalisation, which has been established by induction from a multitude of cases. This is the law similia similibus curantur, in accordance with the remedy is selected, under the three requirements that have been specified. So well established is this law of nature that if we are so fortunate as to be able, in any given case of disease, to comply closely with those requirements, and particularly with the second, we may with certainty predict, and in confidence await, the favourable result of our prescriptions. Such certainty of foresight and such confidence it is our great object to attain, and nothing but a scientific method can afford them. (p. 185)

One may ask how this is different to medicine. Why homeopathy generalises in a deductive way but medical prescriptions aren’t applied deductively is explained by Dunham as follows:

The generalisation to which Hahnemann objected was to that of disease in general upon nosological hypotheses [i.e. putting a disease name on a group of symptoms] made on theoretical grounds, and then applied a priori to individual cases. (p. 33)

(Many homeopaths, using the words “induction” and “inductive” may have misinterpreted the concept and applied it as if it meant treating individually but which Dunham has clarified; taking the individual case and treating individually from a general law.)

Questioning Induction

David Chalmers in his book What’s This Thing Called Science? asks if laws can be derived from facts. He argues they can’t if “derived” means “logically derived” because logically valid deductions aren’t necessarily true unless the premise is true. If they are true, then everything logically derived from them will be true. He goes on to list three requirements for a good inductive argument: a large number of observations, observations must be repeated under a wide variety of conditions and finally, no observation statement conflicts with the law.

What constitutes a large number of observations and a wide variety of conditions is a moot point but Homeopathy satisfies each of the criteria, doesn’t contradict the criteria but verifies it.

Further problems arise for Chalmers. Much knowledge in science refers to the unobservable such as electrons and DNA; how would this knowledge be accounted for in the inductivist position? Induction only makes generalisations about the observable world so it can’t yield knowledge of the unobservable world it seems, according to Chalmers (p. 49). (See Bronowski’s comment below.)

Many scientific laws take the form of mathematically formulated laws. In comparison we have the inexactness that constitutes the observable evidence for laws; so how can exact laws be justified on the basis of inexact science?

Then we have Hume’s problem with induction, known as Hume’s Fork; how can induction justify itself – either by logic or experience. Logic won’t work because inductive inferences are not logical inferences. Appealing to experience by saying induction works because it worked before means using induction to justify induction which, as a circular argument, is unsatisfactory, Chalmers says. But surely one can demonstrate by experiment that a particular induction experiment has worked? (Cf. Bronowski)

Despite his objections, Chalmers sees one strong benefit of induction; objectivity. He is sympathetic to induction despite doubts because, he says, not every principle can be proven and even logic can’t prove itself without begging the question.

He comes closest to approving of induction by approving of Deborah Mayo’s approach called The Experimentalism which requires testing all other possibilities. He explains: “A key idea underlying her treatment is that a claim can only be said to be supported by experiment if the various ways in which the claim could be at fault have been investigated.” (p. 199)

Hahnemann satisfied this by showing that every possible alternative to homeopathy does not cure. He disproved allopathy (cf. Organon: aphorism 22, footnote 12, § 37, fn. 19, § 52-55), antipathy (Organon: § 55-60, 67, 69, 70, § 7, fn. 4, fn. 23) and isopathy (Organon: § 56, fn. 3).

Defending Induction

Ed Feser, in a lengthy comment on Hume’s problem of induction, critiques Hume from two points of view. He says:

First, the initial premise of Hume’s argument is an application of Hume’s Fork, the principle that all knowable propositions concern either relations of ideas [logical connections e.g. bread will always nourish] or matters of fact [it worked before so it will work again]. But Hume’s Fork – which is itself neither true by virtue of the relations of its constituent ideas, nor true by virtue of empirically ascertainable facts – is notoriously self-refuting. It is as metaphysical a principle as any Hume was trying to undermine with it, and its very promulgation presupposes that there is a third epistemic point of view additional to the two Hume was willing to recognize. In that case, though, Hume’s celebrated “problem of induction” cannot even get out of the starting gate. Its entire force depends on a dichotomy that is demonstrably false.

He concludes with another view which the Humean needs to address…

For the non-Humean, then, it is simply not the case that all propositions are either necessary but mere conceptual truths (“relations of ideas”), or empirical but merely contingent truths (“matters of fact”). There are also truths which are empirical but nevertheless necessary. That bread will nourish the body could be a necessary truth even if we can know that it is true (if it is in fact true) only by empirical investigation of the natures of bread and of the body.

Naturally, the Humean will disagree with all of this, but the point is that, unless he offers an independent argument against these alternative ways of understanding the nature of concepts, necessary truth, etc., he will not have given us any non-question-begging reason to believe that there is a “problem of induction.”

The real problem, then, is not the problem of justifying induction. The real problem is justifying the claim that there is a “problem of induction” that remains once we have put aside the false or otherwise problematic philosophical assumptions that Hume himself deployed when arguing that induction cannot be justified. (The problem of Hume’s problem of induction)

Gorham says Kant provided a synthesis of empiricist and rationalist conceptions. For Kant, empiricism is misguided into thinking the world could be experienced as it is “in itself” – free from any involvement of our intellectual faculties. (pp. 72-73)

On the contrary, he maintained, the world we attempt to understand in science – the “phenomenal world” is perceived through a set of hard-wired intellectual categories such as space, time, causality etc.

The world in itself, the noumenal world, is beyond our grasp (cf. § 11), hence the synthesis of both approaches.

How would this play out in reality? Hahnemann was a contemporary of “our Kant” as he referred to the philosopher in a letter to von Villers, the French translator of Kant.1 Hahnemann often uses the word “phenomena” and he does so to explain that we cannot access the cause of disease as disease is invisible; disease presents itself in a totality of symptoms which is what homeopaths prescribe on rather than looking for an a priori cause:

Two thousand years were wasted by physicians in endeavouring to discover the invisible internal changes that take place in the organism in diseases, and in searching for their proximate causes and a priori nature; because they imagined that they could not cure before they had attained to this impossible knowledge. (“The Medicine of Experience”, Lesser Writings, p. 40)

He further explains in the first footnote to the first paragraph of his Organon that we cannot theorise about disease:

His mission is not, however, to construct so-called systems, by interweaving empty speculations and hypotheses concerning the internal essential nature of the vital processes and the mode in which diseases originate in the interior of the organism, (whereon so many physicians have hitherto ambitiously wasted their talents and their time); nor is it to attempt to give countless explanations regarding the phenomena in diseases and their proximate cause (which must ever remain concealed), wrapped in unintelligible words and an inflated abstract mode of expression, which should sound very learned in order to astonish the ignorant – whilst sick humanity sighs in vain for aid. Of such learned reveries (to which the name of theoretic medicine is given, and for which special professorships are instituted) we have had quite enough, and it is now high time that all who call themselves physicians should at length cease to deceive suffering mankind with mere talk, and begin now, instead, for once to act, that is, really to help and to cure.

Mill and Whewell tried to refine the argument in favour of induction further, Mill being a particular reference for homeopaths. Mill and Bacon made a similar point, that logicians are more interested in logical consistency, in ensuring conclusions follow their premises. Philosophers scarcely considered induction as it was undervalued by Aristotelians, Bacon had said. (Cf. Copleston, p. 300ff)

Dr Stuart Close explains further in his The Genius of Homeopathy (p. 246): “Formal logic takes no account of the matter of our reasoning, of the things reasoned about.” He claims:

He does not know, nor wish to know what some of us have learned and forgotten – the Inductive Logic, the logic of Bacon, Mill and Hahnemann, has a higher function than the logic of Aristotle, which exists and is used largely for the purpose of mere argumentation.

Inductive Logic does concern itself with facts, with reality. Its primary purpose is the discovery and use of Truth.

The first requirement of Inductive Logic is that the premises must be true, the result of true and valid observation of facts, based, if need be, upon pure experimentation.

Others, more recently, have seen fit to defend induction. In Shifting the Paradigm: alternative perspectives on induction, the editors, Biondi and Groarke put together a number of essays on induction. One essay, by Hugo Meynell, explores Bernard Lonergan’s solution to the problem of induction. Meynell says Hume’s “problem” is an artefact of classical empiricism. The correct epistemology, he explains, is found in Aristotle and which Lonergan fully worked out in his “generalised empirical method” expounded in his Insight: A Study in Human Understanding. Once this is grasped, Meynell believes, induction will no longer be a problem.

Coming to know, for Lonergan, is a threefold process. Knowledge involves judgements but some judgements are self-defeating or self-destructive without being self-contradictory (p. 416). Good judgments for good reason and for them to be well founded require attentiveness, intelligence and reasonableness.

Attentiveness, the first quality, was largely forgotten by the classical empiricists who acknowledged our sensations, feeling and imaginings but who forgot the operation of our minds upon the inner realities of our sensations, feelings and resultant imaginings by questioning, hypothesising and marshalling evidence, judging, deciding and so on.

The second condition, intelligence, is a matter of envisaging possibilities, forming hypotheses – what intelligent people do all the time and which stupid people are in capable of.

Thirdly, reasonableness – the capacity to judge that some hypotheses are actually or possibly accurate, in the light of evidence to which one has attended.

This is the threefold process of coming to know. For Lonergan, the role of questions and their answers are to be noted, Meynell says (p. 418). There are those questions whose answers are acts of intelligence and call for a hypothesis, a possibility. The second type of question presupposes an answer to the first type of question and the answers to which are acts of reasonableness or reflection: is the hypothesis true or not? Is the possibility the case or not?

Deductive logic is a matter of inferring strictly from judgements and is necessary if those judgments are to be tested adequately. But even after this has been done, the deductions still have to be matched with observations.

But how do we get from our subjective judgements to objective knowledge of the world which supposedly exists independently of our judgements? Meynell asks. Importantly, one is objective to the degree one is rational but “it looks as if ‘genuine objectivity’ were nothing more than the ‘fruit of authentic subjectivity’ as unrestrictedly attentive, intelligent and reasonable. Such ‘authentic subjectivity’ is transcultural; cultures may be transculturally evaluated as more or less frustrating or encouraging it.” (p. 426)

So what is objective reality that we can know? “Real things”, Meynell explains, “are nothing other than what are to be known by intelligent inquiry followed by reasonable reflection.” (p. 427)

On the problem of induction, Meynell tell us we can distinguish between the “primary” and “secondary” problems. The primary is the logical inference we make from the observation of ninety-nine black swans, all of which are black, to the generalisation that “all ravens are black.” The latter involves the kind of judgement we make from evidence to a statement of what is supposed to explain such evidence when no deductively valid inference can be made from the evidence to the explanation.

He concludes by summarising as follows:
“1. To deny we ever make true or well-founded judgements is self-destructive.
2. Well-founded judgements can be wrong but head toward the truth.
3. The world, reality, is nothing other than what true judgements are about.
4. Judgements are well-founded if based on three principles: paying attention to the relevant experience (in a broad sense), using intelligence in envisaging a sufficiently wide range of possibilities or hypotheses that might account for that experience, and thirdly, being reasonable in affirming to be so in each case the judgement which does best account for it.
5. “Induction” is a misleading term for the somewhat complex process that I have just summarised: of inference from experience to sound judgement based on that experience. It is only a problem if one approaches it from the point of view of classical empiricism rather than by way of the “generalised empirical method” which is not only attentive to our experience in a narrow sense but to our awareness of the operations of our minds upon such experience (pp. 435-436).”

William Briggs is a statistician who has also come to the defence of induction. He confidently asserts in his Uncertainty (ch. 3):

There is no knowledge more certain than that provided by induction. Without induction, no argument could, as they say, get off the ground floor; this is because induction provides that ground floor. No argument could even be phrased if it were not for induction, because phrasing requires language and language requires induction. When we say apple, we know it applies to all apples via a form of induction.

All arguments must trace eventually back to some foundation. This foundational knowledge is first present in the senses. Through noesis or intellection, i.e. induction, first principles, universals, and essences are discovered. Induction is what accounts for our being certain, after observing only a finite number of instances or even one and sometimes even none, that all flames are hot, that all men are mortal, that white is always white, that for all natural numbers x and y, if x = y, then y = x, and for the content and characteristics of all other universals and axioms. Because we know these indubitable propositions more surely than any other, induction produces greater certainty than deduction.

Mistakes in induction occur, as they do in every area of intellectual activity. When a man sees several white swans and reasons, “All swans are white”, he is proved wrong when a black swan in sighted (as in Australia). It would be a gross mistake to condemn induction because of this error. It would be like throwing out algebra because middle schoolers miscalculate…

… One reason induction is widely misunderstood, even considered a “problem” in the academic sense is because it is analogical. Here, I largely follow Louis Groarke’s wonderful An Aristotelian Account of Induction, which is must reading. There is no way to adequately summarize the entire work, which is long and deep. Only a few highlights sufficient to dispel the sense that induction is “problematic” are given here.2

Elsewhere, Briggs refers to David Stove who has defended induction in his The Rationality of Induction. Briggs maintains Stove succeeds in defending induction as rational and, further:

He [Stove] also shows that the common belief that ordinary logic is formal is a myth. Knowledge of the validness of certain arguments must come from intuition, as Carnap argued, and Stove proves. He shows that certain forms of logical arguments do not always give valid conclusions, and that all arguments must be judged individually. In his words, “Cases Rule”.3

Then, using Groarke’s defence of induction, Briggs explains the purpose and the mechanism of induction:

“The goal of induction,” Groarke tells us, “is not simply to prove that something is the case but to provoke an understanding of the general case.” Induction moves from the particularities collected by the senses, and moves to unobservable, unsensible generalities or universals, such as knowledge of a thing’s essence. Induction starts with the finite and progresses to the infinite; so although we can never entirely grasp the infinite, we can and even must know part of it.

According to Groarke’s view, induction is “the cognitive/psychological mechanism that produces the leap of insight” necessary for all understanding. He gives five flavours, aspects, or facets of induction. These are (in my modified terms more useful for probability) (1) induction-intellection, (2) induction-intuition, (3) induction-argument, (4) induction-analogy, and (5) the most familiar induction-probability. The order is from that which provides the most to the least certainty.4

Briggs proceeds to summarise each of the above aspects of induction.

In an earlier blogpost entitled “Occam’s Razor and Induction” Briggs addresses the problem of inductive conclusions not necessarily following from the premises. He says:

Inductions are just one-off deductions, so to speak. Like a deductive one, an inductive conclusion is always seen to follow from its premises, but unlike the deductive conclusion which must always follow, the inductive conclusion is not logically necessary.

And both of these differ from non-deductive conclusions, which only sometimes, or rarely, or usually—anything but never or always—follow from their premises. These are crucial distinctions, easily misunderstood and the cause of much confusion.5

Conclusion

In conclusion, it is evident philosophers don’t easily side with Hume and his followers in believing induction to be a big problem for science. Criticisms of induction have not been as damaging as many presume. For example, the Cambridge mathematician and philosopher Jacob Bronowski in his fascinating series for the BBC in the 1970s, The Ascent of Man and its accompanying book, stated:

And Mendeleev’s guesses [when he discovered the Periodic Law and created the Periodic Table] showed induction is a far more subtle process in the hands of a scientist than Bacon and other philosophers supposed. In science we do not simply march along a linear progression of known instances to unknown ones. Rather, we work as in a crossword puzzle, scanning two separate progressions for the point at which they intersect: that is where the unknown instances should be in hiding. Mendeleev scanned the progression of atomic weights in the columns, and the family likenesses in the rows, to pinpoint the missing elements at their intersections. By doing so, he made practical predictions, and he also made manifest (what is still poorly misunderstood) how scientists actually carry out the process of induction.6 (Emphasis mine)

It’s as if the Humean philosophers don’t appreciate that the scientist operates in a three-dimensional world as opposed to their two-dimensional world, as a quote from the Oxford Dictionary explains: “The difference between the 3D and 2D is like in music, the difference between a musical note and the actual music.”7

Hahnemann observed cured cases (inductively) throughout history, concluding they were cured because the medication obeyed a law (deductively) whereby symptoms from which patients suffered were removed by drugs which caused similar symptoms (as Hippocrates said they could and which “may be a very effectual method,” Mill claimed). With further experimentation, observation and therapeutic application by countless others, even inadvertently in conventional medicine, (most drugs in medicine are in fact homeopathically prescribed; for examples see “Ssh Most Drugs are Homeopathic”), Hahnemann discovered the law of cure – unequalled in medicine.

Homeopaths can say – to Chalmer’s and Mayo’s satisfaction – all possible alternatives to homeopathy, the law of cure discovered by Samuel Hahnemann, were proven unable to cure or even to do harm; only like can cure like.

Notes

1 Haehl, Richard, Samuel Hahnemann: His Life and Work Vol. 2, (p. 387) available at: Archive.org

A letter to Mr. von Villers Torgau, January 30th, 1811.

I had known for some time that you had made our Kant available in France, but had not considered what an enormous effort it must have cost you even to understand his “Critique of Pure Reason”, as so many German-born scientists cannot fathom or understand Kant, let alone translate him into a language which is hardly capable of reproducing his modes of expression. This has been done for the good of mankind. Your health, however, has suffered through this and other similar work, and you must now try to regain it completely. This the world asks of you through me.

I admire Kant very much, particularly because he draws the line of philosophy, and of all human knowledge, where experience ends. If the remaining part of what he has thought and written had only unfolded itself a little more clearly before his inner vision, I think that he would not have enveloped himself in a cloud of such obscure sentences. His whole accepted philosophy ought, I think, to have been easily understood at least by all educated people, and to have been so comprehensible that no misunderstanding could arise. It is, however, only my humble self who thinks this, and perhaps I am wrong. It is for this reason that I only value Plato when he is quite comprehensible and speaks clearly.

If the so-called philosophers who followed Kant had not written even more mystically and allowed their imagination so much play, if in one word they had kept, as Kant wanted them to, within the boundaries of experience. My fight to-day with the reform in medical science would have been an easier one. . .

2 The full excerpt is available at http://wmbriggs.com/post/22364/
3 “The Rationality of Induction: David Stove”, by William Briggs, http://wmbriggs.com/?s=rationality+of+induction
4 Ibid.
5 William Briggs, “Occam’s Razor and Induction”, http://wmbriggs.com/post/4323/
6 Bronowski, Jacob, The Ascent of Man, BBC books, p. 249.
7 en.oxforddictionaries.com, “definition of 2D in English”, in “more examples”, https://en.oxforddictionaries.com/definition/2d

Addendum
Of note; a new book of philosophical essays, Retrieving Realism, the authors Hubert Dreyfus and Charles Taylor question the Cartesian understanding of knowledge. The book’s description reads: “This ‘mediational’ epistemology―internal ideas mediating external reality―continues to exert a grip on Western thought, and even philosophers such as Quine, Rorty, and Davidson who have claimed to refute Descartes remain imprisoned within its regime. As Hubert Dreyfus and Charles Taylor show, knowledge consists of much more than the explicit representations we formulate. We gain knowledge of the world through bodily engagement with it―by handling things, moving among them, responding to them―and these forms of knowing cannot be understood in mediational terms. Dreyfus and Taylor also contest Descartes’s privileging of the individual mind, arguing that much of our understanding of the world is necessarily shared.”

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