Notes to The Problem of Induction - or: How we learn from experience
Note 1 : By now, there is more literature on induction than anyone can read in a life. Also, there are quite a few treatments of book length some of which are listed in the references. The interested reader is referred to Broad and Von Wright for older treatments, written before Goodman's sharpening of the problem of induction, and to Skyrms for a more recent treatment that discusses Goodman's problem. Incidentally, the term "scandal of philosophy" (in the present sense) seems to have been first used by C.D. Broad in 1919. Back to text of Note 1
Note 2: Hume's position is stated in the Treatise on Human Nature, and - in a considerably clearer form - in the Enquiries. I shall quote Hume in the body of the text, and show he was mistaken (thereby reserving for myself a place in hell, according to another of Russell's sayings). Back to text of Note 2
Note 3: This is e.g. Max Black's position, and rather well-known through the Encyclopedy of Philosophy, Ed. P. Edwards. The only merit of this position has is that it may have prevented some of adding to the existing nonsense on induction. Back to text of Note 3
Note 4: It seems rather likely to me that both Berkeley and Hume came to their philosophies through Newton's Principia (and through Locke's "Essay on Human Understanding", which was also motivated by Newton). It should also be added - in fairness to Berkeley - that Newton added his Rules of Reasoning in 1714, while Berkeley published his "Principles of Human Knowledge" in 1704. And it is as well to remark here that the present essay also shows why this Berkeleyan reasoning does not hold, and that to my knowledge no one has remarked on its formal similarity to Goodman's new riddle of induction. Back to text of note 4
Note 5: Sir Karl Popper obtained fame and a knighthood by insisting on the virtues of falsifiability. When I was a teenager I was more impressed by this, until I bought the Logic of Scientific Discovery, which starts with the problem of induction, and promises that his theory of falsifiability will dissolve the problem of induction. Then, on p. 33, I read this concerning falsification:
"But if the decision is negative, or in other words, if the conclusions have been falsified, then their falsification also falsifies the theory from which it cannot be logically deduced."
Here is my - rather disgusted and scandalised - marginal note of December 1970:
"The key-question here, of course, is: why should we accept a falsification as final, if, as it does, this depends on other deductive theories - for no one proves, deductively, that each falsification is not an outcome of "abnormal" conditions. You are, in other words, as far as Hume's causation."
Popper got quite popular in the 70-ies and 80-ies in academic circles, which was an object-lesson to me of how well average academics reason. Back to text of note 5.
Note 6: The many discussions I have read of Goodman's sharpening of the problem of induction are all vitiated by the introduction of artificial terms like "grue" and "bleen". For example, Brian Skyrms introduces snarfs ("A snarf is something presented to you in a box named "Excelsior!" and is either an insect, a ball of wax, a feather, or a mask") and murkles (murkles come out as green insects or yellow balls of wax or purple feathers or anything red).
But such artificial terms - that may be helpful in Goodman's original presentation, to make the point in a rhetorically satisfying manner - are neither the essence of the problem nor essential to the problem, and are consequently as misleading as any red herring.
The general point of the sharpened version of the problem of induction is that anything whatsoever may declared relevant to a prediction or a theory, while all the existing evidence is compatible both with its being relevant and its being irrelevant, whence either assumption may be true.
Hence a much better example - to choose one from many - is Galton's investigation of the efficacy of prayer: Galton noted that if prayer is relevant to health (as many people in his time believed), those who pray should have longer lives than those who do not pray. An investigation of the death-registers, that gave the ages of death and creeds of people buried, convinced him that prayer is not relevant to health. So here relevance was mistakenly assumed.
And a medical example illustrating a mistaken assumption of irrelevance is the Semmelweiss story, as given by Hempel: Semmelweiss - a 19th Century doctor - believed, contrary to the great majority of his fellow doctors, that better hygiene could prevent many deaths due to purpureal fever (which was unwittingly given to pregnant women in the hospital Semmelweiss worked in, because it was then a habit for students of medicine to investigate corpses manually before investigating pregnant women manually, without any washing or disinfecting of the hands in between). So Semmelweiss insisted, correctly as it turned out, that a factor hitherto regarded as irrelevant to purpureal fever in fact was relevant.
Two other good classical examples are Pliny the Elder on astrology
"If a man's destiny is caused by the star under which he is born, then all men born under the same star should have the same fortune."
and Francis Bacon on prayer
"And therefore it was a good answer that was made by one who, when they showed him hanging in a temple a picture of those who had paid their vows and then escaped by shipwreck, and would have him say whether he did not now acknowledge the power of the gods - `Aye', asked he, "but where are they painted that were drowned after their vows?" And such is the way of all superstition, whether in astrology, dreams, omens, divine judgements, or the like; wherein men, having a delight in such vanities mark the events where they are fulfilled, but where they fail, though this happens much oftener, neglect and pass them by.") Back to text of note 6
Note 7: The interpretations of probability will be considered in the next chapter. Meanwhile, what matters is that the reader can make both mathematical and intuitive sense of the principles to be proved in this chapter. Incidentally, in this chapter I follow Polya's excellent "Plausible Reasoning". Back to text of Note 7.
Note 8: Those who know something about the more mathematical foundations of probability, involving sigma-algebra's and Lebesgue-integrals should realise that I am not working on that level of abstraction, and also that even so, a sigma-algebra is very well presented as a Boolean Algebra over normalised areas (which to the non-mathematical reader is very well explained by saying that it enables us to sum and multiply the proportions of areas in a unit area). Back to text of Note 8.
Note 9: The reason I consider it likely Newton was aware of most of the reasoning in this essay is that he did add his Rules of Reasoning to the second edition of the Principia; and that the most important of these Rules, Rule III and IV, follow from the Inductive Postulate. But it is also clear that he was not aware of the possibility of deriving his rules from probability theory, for if he had been he would have given the derivation (and 250 years of bad philosophy, including Kant's, would have been prevented). Back to the text of Note 9.
Note 10: It should also be pointed out here that Hume's postulate that we reason inductively and in terms of causes by habits is itself an inductive causal explanation that ought to be regarded as invalid on his own principles: it is an inductive generalisation. (And since an inconsistency is less probable than any consistent assumption, the assumption that in fact there are regularities in nature, that can be found by experiment, is less improbable on Hume's own principles than his assumption of "habits"). Back to the text of Note 10.
Note 11: Since this essay is not conducive to the reputation of some philosophers who acquired fame by publishing on induction and its related topics (such as Berkeley, Hume, Kant and Popper), I like to submit that after 30 years of reading in philosophy, science and mathematics, the truly great minds in Western philosophy seem to me to be Plato, Aristotle, Leibniz and Peirce, and that apart from these the minds that set up science and mathematics (say, Euclid, Archimedes, Galileo, Newton, Euler, Gauss) seem to me generally of considerable better quality than those of the more common run of philosophers, especially modern academic ones. Back to the text of Note 11.
This is a personal note that is merely incidental to the rest of this essay.
Since I believe this essay to solve a fundamental philosophical problem, I want to make a personal dedication and remark. I dedicate this essay to the memory of my parents, and to my G.P. if only because these people were or are a living proof that human beings may be intelligent and good.
My personal remark is this: I arrived in 1977 in Holland from Norway, were I had lived several years, to study in my native country, in the municipal university of the place of my birth, which is Amsterdam. I did not know when arriving that a revolution had taken place in the universities, and especially in the one were I was to study, since it had been taken over in the early 70-ies by leftist students, and was in fact directed by a number of socialist bureaucrats and student-members of the communist party. Not everyone was happy with this state of affairs, but the great majority of the Dutch academics were resigned to the situation, since they were and are well paid, and are almost all far less interested in science, philosophy or ethics than they are interested in their incomes, status and careers. I was not happy of receiving as academic education the opinions than current in the communist party, delivered by people I knew to be members of that party, because I had been raised in a communist family, and myself had refuted Marx and left the party in 1970, when I was 20 (and hordes of my studying contemporaries, who also did not have communist parents, joined the party). Since I am a very effective debater, I was soon asked by a few of the academic minority that still opposed the so-called democratisation of the university to set up a student party to do something against the influence of Marxist students and Socialist bureaucrats. I did so, for the same sort of reasons as moved Lakatos some years before to protest against the same kind of events in England - with this difference that Lakatos was at the time a respected academic with a job, and I was a beginning student of philosophy, psychology and mathematics. The result was that I was rather well known and much hated, and often told by my student opponents that I was a fascist, and by my academic teachers that "you will never get a job in the university", though everyone agreed I was very intelligent. The first of January 1979 I fell ill, and 10 days later so did my common law wife. We are still ill, and it has meanwhile, in 1990, transpired that we have M.E. (Chronic Fatigue and Immune Deficiency Syndrome). This nearly wrecked our lives, the more so as the Board of Directors of the university started court-proceedings against me (on a pretext, but in fact because I was the only effective opponent against the then current Marxism, Feminism and Gay Studies, which the proponents - members of the communist and socialist parties, feminists, and homosexuals, and very often all three combined - tried to make the dominant force, both in teaching and in administration, in the UvA). The Board of Directors lost the court-case (which took three years) but succeeded for the time being putting an end to my studies. In 1987 I enrolled again, still ill, but determined to graduate, and with some verbal assurances from a few members in the faculty of philosophy that they would help me get a degree. In 1988 I was an invited speaker to philosophy graduates, which, when the academic staff lost their discussions with me, ended in screams that I am a fascist and a terrorist (because I am a strong proponent of objective science!), after which I left - one important reason being that my whole family was in the Dutch Resistance in the 2nd W.W., and both my father and grandfather were sent to a German concentration-camp in 1941, where my grandfather was murdered, and whence my father only was released at the end of the war. Being members of the resistance, they were - by an SS-court - convicted of "terrorism". At the time I did not publicise my background, but it is no less real. Also in 1988 the Board of Directors - presided by the same men and the same secretary as at present - moved again against me, and sent me a letter in which they said, after acknowledging that I was chronically ill and could not leave Holland or Amsterdam:
"Your stated opinions about the quality of the administration, education and research in this university, and about the (in)competence of its directors and staff, in our opinion should make you leave this university and try to get a degree in some other university in this country or abroad."
At the same time, the faculty of philosophy - with a majority, at the time, of self-confessed Marxists and Feminists, the latter mostly homosexual: at the time that was the desired qualification to become an academic in the UvA - absolutely refused to do give me a degree or do anything for me. I had to give up my studies again. Since I also had a B.A. in psychology, and since I was (and am) still ill, I finally moved to the faculty of psychology, where I did get an M.A. in 1993 (again after a lot of trouble due to my outspoken opinions), with the highest possible marks.
My apology for the length of this footnote, but it may interest the reader that its writer has been ill for the last 20 years and has done all the research and reached all the conclusions of this paper between 1980 and 1983, while he certainly is the only student in a European or American university whose human rights, career, and academic opportunities, have been wantonly destroyed by the board of directors of his university because of his "outspoken opinions". Also, though this essay has not been published, I have lectured on it in the University of Amsterdam in November 1989, in the faculty of psychology, where I was an invited speaker. The reason I have not attempted to publish this before is ill health and my personal history. Back to the text of Note 0.