The Dialogues of Saint Gregory the Great, book II, ch. 4

How do your cure an attention deficit in the sixth century? By magic, it seems.

In his biography of Saint Benedict, Pope Gregory I relates a story of how he "cured a monk that had an idle and wandering mind." Here it is:

In one of the monasteries which he had built in those parts, a monk there was, which could not continue at prayers; for when the other monks knelt down to serve God, his manner was to go forth, and there with wandering mind to busy himself about some earthly and transitory things. And when he had been often by his Abbot admonished of this fault without any amendment, at length he was sent to the man of God, who did likewise very much rebuke him for his folly; yet notwithstanding, returning back again, he did scarce two days follow the holy man's admonition; for, upon the third day, he fell again to his old custom, and would not abide within at the time of prayer: word whereof being once more sent to the man of God, by the father of the Abbey whom he had there appointed, he returned him answer that he would come himself, and reform what was amiss, which he did accordingly: and it fell so out that when the singing of psalms was ended, and the hour come in which the monks betook themselves to prayer, the holy man perceived that the monk, which used at that time to go forth, was by a little black boy drawn out by the skirt of his garment; upon which sight, he spake secretly to Pompeianus, father of the Abbey, and also Maurus, saying : "Do you not see who it is, that draweth this monk from his prayers?" and they answered him, that they did not. "Then let us pray," quoth he, "unto God, that you also may behold whom this monk doth follow": and after two days Maurus did see him, but Pompeianus could not. Upon another day, when the man of God had ended his devotions, he went out of the oratory, where he found the foresaid monk standing idle, whom for the blindness of his heart he strake with a little wand, and from that day forward he was so freed from all allurement of the little black boy, that he remained quietly at his prayers, as other of the monks did : for the old enemy was so terrified, that he durst not any more suggest such cogitations : as though that blow, not the monk, but himself had been strooken. (pp. 61-62)

This is remarkably un-psychological: It seems like the only way these people were able to explain lack of distraction was by externalizing the problem in a devil or spirit, or as here, in the "allurement of the little black boy."

And of course, a problem so concrete should be treated with a good dose of Harry Potter-style spell-casting. Unless, that is, the wand is a club, and the spell is a beating.

Edwards: Likelihood (1972)

Edwards (from his Cambridge site)

The geneticist A. F. W. Edwards is a (now retired) professor of biometry who was massively influenced by Ronald Fisher in his scientific writings. His books Likelihood argues that the likelihood concept is the only sound basis for scientific inference, but it reads at times almost like one long rant against Bayesian statistics (particularly ch. 4) and Neyman-Pearson theory (particularly ch. 9).

Don't Do Probs

As an alternative to these approaches to statistics, Edwards proposes that we limit ourselves to makes assertions only in terms of likelihood, "support" (log-likelihood, p.12), and likelihood ratios. In the brief epilogue of the book, he states that this

… allows us to do most of the things which we want to do, whilst restraining us from doing some things which, perhaps, we should not do. (p. 212)

In particular, this approach emphatically prohibits the comparison of hypotheses in probabilitistic terms. The kind of uncertainty we have about scientific theories is simply not, Edwards states, of a nature that can be quantified in terms of probabilities: "The beliefs are of a different kind," and they are "not commensurate" (p. 53)

The Difference Between Bad and Worse

He briefly mentions Ramsey and his Dutch book-style argument for the calculus of probability, and then goes on to speculate that, had not died so young,

… perhaps he would have argued that his demonstration that absolute degrees of belief in propositions must, for consistency's sake, obey the law of probability, did not compel anyone to apply such a theory to scientific hypotheses. Should they decline to do so (as I do), then they might consider a theory of relative degrees of belief, such as likelihood supplies. (p. 28)

In other words, it might be true that you cannot assign numbers to propositions in any other way than according to the calculus of probabilities, but you can always reject to have a quantitative opinion in the first place (or not make a bet).

Nulls Only

Consistently with Fisher's approach to statistics, Edwards finds it important to distinguish between null and not-null hypotheses: That is, in opposition to Neyman-Pearson theory, he refuses to explicitly formulate the alternative hypothesis against which a chance hypothesis is tested.

Here as elsewhere, this is a serious limitation with quite profound consequences:

It should be noted that the class of hypotheses we call 'statistical' is not necessarily closed with respect to the logical operations of alternation ('or') and negation ('not'). For a hypothesis resulting from either of these operations is likely to be composite, and composite hypotheses do not have well-defined statistical consequences, because the probabilities of occurrence of the component simple hypotheses are undefined. For example, if $p$ is the parameter of a binomial model, about which inferences are to be made from some particular binomial results, '$p=\frac{1}{2}$' is a statistical hypothesis because its consequences are well-defined in probability terms, but its negation, '$p\neq\frac{1}{2}$', is not a statistical hypothesis, its consequences being ill-defined. Similarly, '$p=\frac{1}{4}$ or $p=\frac{1}{2}$' is not a statistical hypothesis, except in the trivial case of each simple hypothesis having identical consequences. (p. 5)

This should also be contrasted with Jeffreys' approach, in which the alternative hypothesis has a free parameter and thus is allowed to 'learn', while the null has the parameter fixed at a certain value.

Scientists With Attitude

At several points, in the book, Edwards uses the concerns of the working scientist as an argument in favor of a likelihood-based reasoning calculus. He thus faults Bayesian statistics for "fail[ing] to answer questions of the type many scientists ask" (p. 54).

This question, I presume, is "What does the data tell my about my hypotheses?" This is distinct from "What should I do?" or "Which of these hypotheses is correct?" in that it only supplies the objective, quantitative measure of support, not the conclusion:

The scientist must be the judge of his own hypotheses, not the statistician. The perpetual sniping which statisticians suffer at the hands of practising scientists is largely due to their collective arrogance in presuming to direct the scientists in his consideration of hypotheses; the best contribution they can make is to provide some measure of 'support', and the failure of all but a few to admit the weaknesses of the conventional approaches has not improved the scientists' opinion. (p. 34)

In brief form, this leads to the following tirade against Bayesian statistics:

Inverse probability, in its various forms, is considered and rejected on the grounds of logic (concerning the representation of ignorance), utility (it does not allow answers in the form desired), oversimplicity (in problems involving the treatment of frequency probabilities) and inconsistency (in the allocation of prior probability distributions). (p. 67–68)

Fisher: The Design of Experiments (4th ed., 1947), Chapter I

Fisher; from Judea Pearl's website.

Now, here's a revealing turn of phrase:

In the foregoing paragraphs the subject-matter of this book has been regarded from the point of view of an experimenter, who wishes to carry out his work competently, and having done so wishes to safeguard his results, so far as they are validly established, from ignorant criticism by different sorts of superior persons. (p. 3)

You could hardly spell out more explicitly the philosophy that lies behind Fisher's concept of statistics: It's a strategic ritual, not designed to ensure a result, but to protect against criticism.

Perfectly Rigorous and Unequivocal

Such protection only goes as far as the mathematical consensus on the validity of the logic. But Fisher goes on to state that "rigorous deductive argument" is possible even in the context of random events, citing gambling as a proof of concept:

The mere fact that inductive inferences are uncertain cannot, therefore, by accepted as precluding perfectly rigorous and unequivocal inference. (p. 4)

This seems to confuse the issues of probability and statistics, unless his argument here really only amounts to saying that distributions are non-stochastic entities.

Useless for Scientific Purposes

This leads him to a discussion of "inverse probability," which he gives three reasons for rejecting: First,

… advocates of inverse probability seem forced to regard mathematical probability, not as an objective quantity measured by observed frequencies, but as measuring merely psychological tendencies, theorems respecting which are useless for scientific purposes. (p. 6–7)

Second, Bayes' axiom (about the flat prior for a coin flip) is not self-evident, that is, the choice of prior is not unequivocal (p. 7).

Ever Since the Dawn of Man…

And third,

… inverse probability has been only very rarely used in the justification of conclusions from experimental facts, although the theory has been widely taught, and is widespread in the literature of probability. Whatever the reasons are which could give experimenters confidence that they can draw valid conclusions from their results, they seem to act just as powerfully whether the experimenter has heard of the theory of inverse probability or not. (p. 7)

That's a funny sociological proof, given that he has just rejected Bayesian statistics for its psychologism. But he himself sometimes seems to think that his statistics is a kind of theory of learning, whatever that means:

Men have always been capable of some mental processes of the kind we call "learning by experience." … Experimental observations are only experience carefully planned in advance, and designed to form a secure basis of new knowledge; (p. 8)

Chernoff: "A career in statistics" (2014)

Chernoff makes some interesting remarks about the philosophy of statistics in his recent autobiographical essay.

First, an anecdote about the three classical decision criteria considered in decision theory:

I had always been interested in the philosophical issues in statistics, and Jimmie Savage claimed to have resolved one. Wald had proposed the minimax criterion for deciding how to select one among the many “admissible” strategies. Some students at Columbia had wondered why Wald was so tentative in proposing this criterion. The criterion made a good deal of sense in dealing with two-person zero-sum games, but the rationalization seemed weak for games against nature. In fact, a naive use of this criterion would suggest suicide if there was a possibility of a horrible death otherwise. Savage pointed out that in all the examples Wald used, his loss was not an absolute loss, but a regret for not doing the best possible under the actual state of nature. He proposed that minimax regret would resolve the problem. At first I bought his claim, but later discovered a simple example where minimax regret had a similar problem to that of minimax expected loss. For another example the criterion led to selecting the strategy A, but if B was forbidden, it led to C and not A. This was one of the characteristics forbidden in Arrow’s thesis.

Savage tried to defend his method, but soon gave in with the remark that perhaps we should examine the work of de Finetti on the Bayesian approach to inference. He later became a sort of high priest in the ensuing controversy between the Bayesians and the misnamed frequentists. (pp. 32–33)

He immediately moves on to one of his own more dismal conclusions about the issue:

I posed a list of properties that an objective scientist should require of a criterion for decision theory problems. There was no criterion satisfying that list in a problem with a finite number of states of nature, unless we canceled one of the requirements. In that case the only criterion was one of all states being equally likely. To me that meant that there could be no objective way of doing science. I held back publishing those results for a few years hoping that time would resolve the issue (Chernoff, 1954). (p. 33)

I haven't read the paper he is referring to here, but it seems like the text has jumbled up the conclusions: I think what he meant to say is that there is no single good decision function when we have infinitely many states, since the criteria essentially require us to use a uniform distribution. But I would have to check the details.

Finally, he moves on to a more on-record explication of his position:

In the controversy, I remained a frequentist. My main objection to Bayesian philosophy and practice was based on the choice of the prior probability. In principle, it should come from the initial belief. Does that come from birth? If we use instead a non-informative prior, the choice of one may carry hidden assumptions in complicated problems. Besides, the necessary calculation was very forbidding at that time. The fact that randomized strategies are not needed for Bayes procedures is disconcerting, considering the important role of random sampling. On the other hand, frequentist criteria lead to the contradiction of the reasonable criteria of rationality demanded by the derivation of Bayesian theory, and thus statisticians have to be very careful about the use of frequentist methods. 

In recent years, my reasoning has been that one does not understand a problem unless it can be stated in terms of a Bayesian decision problem. If one does not understand the problem, the attempts to solve it are like shooting in the dark. If one understands the problem, it is not necessary to attack it using Bayesian analysis. My thoughts on inference have not grown much since then in spite of my initial attraction to statistics that came from the philosophical impact of Neyman–Pearson and decision theory. (p. 33)

Duda and Hart: Pattern Classification and Scene Analysis (1973)

A lot of references seem to indicate that this book played an important role in the popularization of Bayesian methods in machine learning. It also provides an interesting missing link between the statistics and decision theory of the 1950s and the field of machine learning in the form it now has.

Interestingly, their rejection of minimax approaches to decision theory is rather casual, relative to how toxic the debate actually was:

In fact, the Bayesian approach is avoided by many statisticians, partly because there are problems for which a decision is made only once (so that average loss is not meaningful), and partly because there may be no reasonable way to determine the a priori probabilities. Neither of these difficulties seems to present a serious problem in typical pattern recognition applications, and for simplicity we have taken a strictly Bayesian approach. (p. 36)

Compare this to David Blackwell's compact statement of intent in Basic Statistics (1969):

This book indicates the content of a lower-division basic statistics course I have taught several times at Berkeley. […] The approach is intuitive, informal, concrete, decision-theoretic, and Bayesian. (p. v)

Duda and Hart also provide a number of quite interesting references:

The text by Nilsson (1965) provides an exceptionally clear treatment of classification procedures. (p. 8)

There are many interesting subject areas that are related to this book but beyond its scope. […] Those interested in philosophical issues will find the books by Watanabe (1969) and Bongard (1970) thought provoking. (p. 8)

We are also fond of the the text by Ferguson (1967), who presents many topics in statistics from a decision theoretic viewpoint. (p. 36)

Chow (1957) was one of the first to apply Bayesian decision theory to pattern recognition. His analysis include a provision for rejection, and he later estiablished a funcamental relation between error and reject rates (Chow 1970). (p. 36)

Blackwell and Girshick: Theory of Games and Statistical Decisions (1954), Ch. 4

There's an interesting representation theorem in Blackwell and Girshick's textbook in statistics: It provides a set of sufficient conditions for a preference ordering over lotteries to be expressible as a prior probability distribution (Th. 4.3.1, p. 118).

I assume the theorem comes from either Wald, Savage, or de Finetti, but no reference is given.

Well-Behaved Preferences

A lottery can here be defined as a function from the sample space to the real numbers. The conditions in the theorem are then the following:

• The ordering of two lotteries $f$ and $g$ cannot depend on the availability of other lotteries.
• If a lottery $f$ provides a higher payoff than a lottery $g$ at all points in the sample space $\Omega$, then $f$ must be preferred to $g$.
• If $f$ is preferred to $g$, then $f+h$ must be preferred to $g+h$.

An inspection of the proof also shows that they should have included a continuity condition:

• If $f_1, f_2, \ldots$ is a series of lotteries converging to a limit $f$, and if $f_i$ is preferred to $g$ for all $i$, then $f$ must also be preferred to $g$.

When these conditions are met, the preference ordering over the lotteries can be expressed as a distribution over the sample space, unless it categorizes all lotteries as equally good.

The Large and the Good

The proof of the theorem uses the fact that two convex, disjoint, open sets can be separated by a hyperplane. Here's a sketch:

If we let $e$ be the lottery that pays zero in all situations $\omega \in \Omega,$ then we can define the following sets:
\begin{eqnarray}
F[>] &=& \{f\ |\ \forall \omega \in \Omega: f(\omega) > 0\},
\\
F[\gtrsim] &=& \{f\ |\ f\gtrsim e\},
\end{eqnarray}
and we can further define, in the usual way, $F[>] + F[\gtrsim]$ to be the sums of lotteries from those two sets.

Now, $F[>]$ is an open set, and hence $F[>] + F[\gtrsim]$ is, too. Further, $F[>]$ and $F[\gtrsim]$ are both closed under addition (due to the third assumption), and hence their sum is, too. They are also both closed under multiplication with a scalar, and again, so is their sum — but this latter argument requires a bit more spelling out.

Rational and Real Convexity

Suppose a lottery $f$ is preferred to the zero lottery, that is, $f \in F[\gtrsim]$. The third assumption of the theorem then tells us that
$$
e \;\lesssim\; f \;\lesssim\; f + f \;\lesssim\; f + f + f \;\lesssim\; \ldots \;\lesssim\; nf.
$$
By further adding multiple copies of the zero lottery to both sides of this preference ineqality, we can see that
$$
me \;\lesssim\; (m-1)e + nf \;=\; nf,
$$
where we have selectively used the fact that $e$ is the zero lottery. Putting these facts together, we then have the preference inequality
$$
e \;\lesssim\; \left(\frac{n}{m}\right)f.
$$
By using a positive sequence of rational approximations $(n/m) \rightarrow \lambda$, we can use this fact along with the continuity assumption to conclude that $F[\gtrsim]$ is closed under multiplication with any positive, real scalar $\lambda$.

I don't think there's a way around this last technicality. It is, incidentally, the same proof technique used to prove that the logarithmic functions are the only continuous functions that turn products into sums.

Cutting the Cake

At any rate, $F[>] + F[\gtrsim]$ is an open and convex set separated from the singleton set $\{e\}$. We can therefore conclude that there is a lottery (or vector) $p$ which defines the hyperplane $\{f\ |\ f\cdot p=0\}$ separating $F[>] + F[\gtrsim]$ from $\{e\}$. The set $F[>] + F[\gtrsim]$ is thus a subset of the half-space $\{f\ |\ f\cdot p \geq 0\}$.

This vector $p$ must have nonnegative coordinates, since the set $F[>]$ is unbounded in all positive directions. If $p$ had a negative coordinate, $p(\omega) \leq 0$, we could choose a lottery for which the corresponding coordinate, $f(\omega)$, was so large that $f\cdot p < 0$. This would violate the definition of $p$, and $p$ hence has to be a nonnegative vector which can be interpreted as a probability distribution.

It would also have the property that $f\cdot p \geq g\cdot p$ if and only if $f \gtrsim g$. This follows from the fact that $f\cdot p \geq g\cdot p$ if and only if $(f - g) \in F[>] + F[\gtrsim]$, which holds if and only if $f$ can be expressed as the sum of $g$ and some lottery preferable to the zero lottery.

David Publishing Co. is (also) a scam

Yet another unsolicited email from a predatory publishing company:

… we invite you to submit this paper and related research papers to Psychology Research.

The new journal of Psychology Research, an award-winning peer-reviewed, multidisciplinary periodicals published by David Publishing Company, New York, NY 10034, USA., since July, 2011, welcomes the submission of original manuscripts reporting innovations or investigations in the Psychology area. Successful general submission manuscripts may report interdisciplinary efforts or be of a sufficiently broad nature to be of interest to those centered in related disciplines. Manusripts reporting innovations or collaborations leading to enhancements in Psychology are of particular interest to Psychology Research.

If you have the idea of making our journals as vehicles for your research interests, please send your WORD format manuscripts (papers or books) through e-mails/submission system (for more details refer journal Web page). We appreciate your support.

We also seek researchers who have deep research in and outstanding contribution to Psychology area to be our reviewers/editors. Good review board has insightful understanding in Psychology field, and can provide professional suggestions to authors. Anyone who is interested in our journals can send us CV. We are looking forward to your contribution!

Sincerely yours,

Lily, R.

Editor OfficePsychology Research, ISSN 2159-5542

David Publishing Company 240 Nagle Avenue #15C, New York, NY 10034, USA.

Italics, boldface, and typos in the original.

Some comments by other people:

• Daniel Sage: "Young researchers beware: potential scams, paying for articles and David Publishing" (2012)
• Jeffrey Beall: "David Publishing: Flipping Its Model" (2012)
• Brian Leiter: "On David Publishing, Once Again--and the Philosopher's Index" (2013)
• Paige Alfonzo: "Beware of predatory journals" (2013)

Lambert Academic publishers is a scam

I've just received an email from company "Lambert Academic Publishing," a daughter company of the VDM group.

After making a reference to a conference abstract of mine which appears on a university website, the email continues:

As an international publisher whose aim is to disseminate research to a global audience, we would be especially interested in publishing your dissertation (or a recent scientific monograph) in paperback form.

I would greatly appreciate if you could confirm your interest in receiving a brochure with details about our services.

I am looking forward to hearing from you.

--
Kind regards/Freundliche Grüße

Ilie Tsilea
Acquisition Editor

LAP LAMBERT Academic Publishing is a trademark of:
OmniScriptum GmbH & Co. KG

Heinrich-Böcking-Str. 6-8,
66121, Saarbrücken, Germany

This email and the company behind it is a scam — not exactly in the sense that the company doesn't exist, but in the sense that it applies aggressive and misleading sales tactics to feed off the insecurity and publication pressure of the academic world. If you receive an email from them (probably one that will be eerily reminiscent of the one quoted above), I recommend that you don't reply.

A number of other people have already written about the company, so I'll let them explain:

• Jeffrey Beall: Lambert Academic Publishing: A Must To Avoid;
• Christopher Collins: Lambert Academic Publishing Continues To Spam;
• Hristina Petrova: Reply to 'Is it worth publishing with Lambert Academic publishers?';
• Joseph Stromberg: I Sold My Undergraduate Thesis to a Print Content Farm.

I'm sure there are more cautionary tales out there, but these should do.

Urban: Language and Reality (1938)

The American philosopher Wilbur Urban seems to have been one of the most vocal advocates for the philosophy of Ernst Cassirer in the English-speaking world. Most references I've seen to his work point to the 1938 book Language and Reality.

I haven't familiarized myself very much with this massive book, but it seems that its general thesis is that a phenomenological metaphysics can be reconstructed on the basis of a careful analysis of language. This involves, among other things, an inspection of what he calls "radical metaphor."

Wilbur M. Urban; uncredited photo snatched from this website.

Urban writes:

It has repeatedly been pointed out that metaphor, in the sense here understood, must be clearly distinguished from metaphor in the conscious reflective activity of the poet. It is rather the unconscious activity that is creative in language itself. (p. 176)

The essence of radical metaphor is that it is intuitive and involves the intuitive meaning which we analysed out and described in the preceding chapter. This is seen especially in the Erlebnis-Wert of the adjective. The transfer of the coldness of ice to the coldness of a reception, the height of a tree or mountain to nobility, the dirt of the streets to the dirt of the Yellow Press—in all such cases we recognize a certain natural affinity or likeness between the objects which makes the transferences natural and even inevitable. (p. 177)

It is true that such transfers may often be conceptually justified, but such a "verification" is always ex post facto. The transfer of the cold of the ice to the cold of a reception … may be thus justified. It consists in arguing that when A is to B as C is to D, the name A can be used to indicate C. But such validation is significantly different from the self-authenticating process by which the transfer takes place. (p. 178)

A recurring theme in these analyses is that language "moves upwards" in the sense of transferring meaning from concrete, physical concepts to more abstract or "spiritual" ones. Thus:

The general upward movement of language "from the physical to the spiritual," the transfer of names, analogical predication with the accompanying development of ontological predicates, creates an entire region or universe or discourse which is post-logical or metalogical in this sense. (p. 331)

… this "figurative" use of language is not so much a fanciful expression for something otherwise known, as the means of apprehending and fixating new aspects and meanings. The natural movement of language, the upward movement as we described it, is from the physical to the spiritual. All words are physical in their origin and have a physical reference. It is through metaphorical transfer that they acquire their new references, their second intentions, but they acquire these new references because they becomes the vehicle for the intuition and description or expression of new entities. (p. 345)

But these very figurative expressions are expressions precisely because they are representative, or rather constitutive, of the intuitions themselves. Here intuition and expression are one, for the reason that the figurative expression itself is part of the apprehension or knowing. (p. 345)

Although Urban muddles the point a bit, I think his take on the relation between metaphorical language and thought is that the language influences the thought. At least, this is what he appears to be focus on most of the time, perhaps because it is the most controversial direction of the claim.

His remark about "expressions" does seem to indicate, however, that he has some kind of phenomenological authenticity criterion in mind too. This could mean that he also considers figurative language to be an expression of certain cognitive "intuitions," although these intuitions may also, in a circular fashion, have been put there by the language shared by a given culture.

I. A. Richards: Interpretation in Teaching (1938)

Now, here's a thought:

Thinking is radically metaphoric. … To think of anything is to take it as of a sort … and that 'as' brings in (openly or in disguise) the analogy, the parallel, the metaphoric grapple or ground or grasp or draw by which alone the mind takes hold. (pp. 48–49)

My point is not that language is full of metaphors … It is that thought is itself metaphoric – not merely that it expresses itself in linguistic metaphors. The metaphor that a thought is using need not correspond to the metaphor that its language displays, though it usually does, and the thought will often adopt the verbal metaphor when it is noticed. But equally often we discount and disown the metaphors in our speech, treat them as dead, or kill them as we go. (p. 49)

Gardner: "Metaphors and Modalities" (1974)

This paper (by the later very famous developmental psychologist Howard Gardner) reports on an experiment in which college students and children of different ages were asked to apply adjectives like hot/cold to stimuli like uncommon stimuli like faces, color samples, or small objects. The young children were quite good at this, with some caveats.

The experiment is notable for employing materials that are not entirely obvious clichés, like the connection between big/small size and high/low tones.

In one task, for instance, the subjects to felt sandpaper of various degrees of coarseness, then deciding which of the two types of sandpaper was the more light, and which was the more dark. Other stimuli included as ping-pong ball vs. a jack (presumably meaning the edgy die used in playing the jacks), or the sound of a triangle vs. the sound of a recorder.

Table of materials used in the experiment; p. 86.

In general, Gardner found that even preschool children agreed more or less with the judgments college students made on these tasks. A notable exception was the application of color terms to new stimuli, something that seems to indicate a heavily conventional component in color metaphors, such as a blue tap for cold water (p. 88).

The smaller children also gave a number of bizarre, funny, and "incorrect" explanations for their choices. For instance,

The 7-year-old[s] equated "getting lots of presents on your birthday" with dark and "getting no presents" with light, because, as several subjects explained, "lots of presents are heavy to carry." Here the contrast to light/dark was apparently assimilated to light/heavy. (p. 88)

Also, the children equated

… a tactile-perceived Ping-Pong call with loud and a tactile-perceived jack with quiet; in this case, subjects reported that the Ping-Pong ball would make more noise if it fell, Here an action in which the elements might be involved overwhelmed the more conventional association of pointed compactness with noise and smooth emptiness with silence. (p. 88)

In both of these cases, the older children and the college students tended to arrive at the opposite conclusions.

The 11-year-olds in the study, by the way, also tended to answer that

[an] angry face is cold "because it makes you feel cold" (p. 88)

a finding that sits somewhat uneasily with Zoltán Kövecses' insistence on the universality of ANGER IS HEAT metaphors.

Wigod: The Matter of Metaphor and Its Importance for Linguistics (1972)

Chapter 10 of this MA thesis contains some useful references to state of metaphor theory in the early 1970s. I don't know anything about the author, but a bit of googling suggests that she now works in Canada as a journalist.

Notables

Wigod's own take on the cognitive underpinnings of metaphorical speech is that "metaphor is man's cognizing tool par excellence, and quite possibly his only such" (p. 84).

She also quotes John Middleton Murry as saying that "metaphor appears as the instinctive and necessary act of the mind exploring and ordering experience," and that the lexis of a language is "the record of past cognitive exploration." Both of these quotes come from his essay in the anthology Essays on Metaphor edited by Warren Shibles (1972; pp. 28 and 93, respectively).

A number of interesting quotes are also given from William Hilton Leatherdale's The Role of Analogy, Model, and Metaphor in Science (1974). Leatherdale claims that metaphor is "indispensable for positing hypotheses" in science (p. 133), and it leads to "a multi-dimensional, gestalt-like insight into new ways of looking at phenomena" (p. 22).

Wigod also explains that

… Leatherdale cites another two-fold classification of analogies. The first type is the observation of a simple resemblance, which is basically transparent common-sense cognizing. The other type is more prodigious, and compares one relation (in the mathematical sense) to another. (p. 87)

This sounds amazingly like the distinction between metaphor and analogy later emphasized by Dedre Gentner.

A Digression on Margaret Mead

Wigod also quotes Terence Hawkes as quoting Margaret Mead as saying that

… the metaphor we may embody in the statement 'Love will find a way' may simply not exit in some countries, or may have an utterly different role (and so call forth appropriately different responses in others. (p. 96 in Wigod's text)

But this is almost certainly a spurious quote. The source seems to be a passage from Mead's 1949 book Male and Female:

To get some sense of the experience an anthropologist brings to the consideration of a human problem, let us take the simple statement "Love will find a way," a well-worn and well-loved adage of our own tradition. To a young American, this phrase will conjure up images of difficult transportation, a determined young man thumbing his way across the United States, or driving thirty-six hours stopping only for hot dogs, to get there in time to see his girl—before she sails or decides to marry someone else. Or it may mean the way in which a girl plans, and saves, and even sews, to divise the dress that she then wears to the dance where she knows her estranged lover will see her and may choose her again. Through one's head will pass a variety of plots and incidents: motor cars, jobs, shortages of cash, failures of plane connections, occasionally even recalcitrant parents if the lovers are young enough or the parents rich enough for their views to matter. Mixed up with images from one's own experience will be snatches of scenes from movies, from novels, from radio serials, occasional images of Tom Mix riding across the plains, or Ingrid Bergman in some over-intense role, perhaps a line or so from Romeo and Juliet, or a couplet from an old valentine.

I don't have the rest of the quote, but I can see that it occurs in chapter 2 of the book ("How an Anthropologist Writes"), on page 35 of the 1949 edition, if I'm not mistaken.

The Works

In addition to these texts, Wigod cites the following useful references:

• Christine Brooke-Rose: A Grammar of Metaphor (1958)
• Weller Embler: Metaphor and Meaning (1966)
• Martin Foss: Symbol and Metaphor in Human Experience (1949)
• Owen Thomas: Metaphor, and Related Subjects (1969)
• Paul Henle (ed.): Language, Thought, and Culture (1968)

Bloomfield: Language (1933), chs. 9 and 24

Leonard Bloomfield was a die-hard behaviorist in his approach to linguistics, and he aggressively against any attempt to explain metaphors or any other linguistic phenomena in "mental" terms (§9.4, pp. 142–144).

He was, however, well aware of the many metaphorical etymologies change in English and other languages, and he did put forward a few speculations about their forms and causes. The relevant parts of his book are in particular Chapter 9 ("Meaning"), particular pp. 149–50, and Chapter 24 ("Semantic Change").

The Ubiquity of Metaphor

My first claim—that he recognizes how widespread everyday metaphor is—can supported by the following quote:

Very many linguistic forms are used for more than one typical situation. In English, we speak of the head of an army, of a procession, of a household, or of a river, and of a head of cabbage, of the mouth of a bottle, cannon, or river; of the eye of a needle, and of hooks and eyes on a dress; of the teeth of a saw; of the tongue of a shoe or of a wagon; of the neck of a bottle and of a neck of the woods; of the arms, legs, and back of a chair; of the foot of a mountain; of hearts of celery. A man may be a fox, an ass, or a dirty dog; a woman, a peach, lemon, cat, or goose; people are sharp and keen or dull, or else bright or foggy, as to their wits; warm or cold in temperament; crooked or straight in conduct; a person may be up in the air, at sea, off the handle, off his base, or even beside himself, without actually moving from the spot. The reader will be able to add examples practically without limit; there is no greater bore than the enumeration and classification of these "metaphors." (p. 149)

He proposes what now appears like a kind of pragmatic repair theory of the comprehension of such phrases:

The remarkable thing about these variant meanings is our assurance and our agreement in viewing one of the meanings as normal (or central) and the others as marginal (metaphoric or transferred). The central meaning is favored in the sense that we understand a form (that is, respond to it) in the central meaning unless some feature of the practical situation forces us to look to a transferred meaning. … He married a lemon forces us to the transferred meaning only because we know that men do not go through a marriage ceremony with a piece of fruit. (p. 149)

He adds a cautionary note on ignoring the difference between languages in this respect:

We are likely to make the mistake of thinking that the transferred meanings of our language are natural and even inevitable in human speech — the more so, as they appear also in other European languages. This last, however, is merely a result of our common cultural traditions; while transferred meanings occur in all languages, the particular ones in any given language are by no means to be taken for granted. Neither in French nor in German can one speak of the eye of a needle or of an ear of grain. To speak of the foot of a mountain seems natural to any European, but it would be nonsense in Menomini and doubtless in many other languages. (p. 150)

Although he doesn't use the concept of "dead metaphors," it is also clear from his discussion of the "isolation" of meaning (Ch. 24, p. 432) that he thinks that many metaphors have no imagistic content for contemporary English-speakers.

The Dynamics of Meaning Change

Bloomfield; photo from the Yale website.

In the chapter on semantic change, Bloomfield first reviews the many metonymic and metaphoric paths by which a word can change its meaning over time, noting:

The surface study of semantic change indicates that refined and abstract meanings largely grow out of more concrete meanings. (p. 429)

He then moves on to the more complex question of what the cause of these changes are. As one might expect, he dismisses

… so-called psychological explanations, such as Wundt's, which merely paraphrase the outcome of the change. Wundt defines the central meaning as the dominant element of meaning, and shows how the dominant element may shift when a form occurs in new typical contexts. … This statement leaves the matter exactly where it was. (p. 435)

As an internative to these tautological theories, Bloomfield suggests that the engine of semantic change has to be the context in which the language is embedded:

The shift into a new meaning is intelligible when it merely reproduces a shift in the practical world. A form like ship or hat or hose designates a shifting series of objects because of changes in the practical world. If cattle were used as a medium of exchange, the word fee 'cattle' would naturally be used in the meaning 'money,' and if one wrote with a goose-feather, the word for 'feather' would naturally be used of this writing-implement. (p. 436)

Linguist Hermann Paul, from Spiegel.de.

The two latter examples are references to examples he had discussed previously (pp. 428 and 435): The English fee comes from Old English feoh (live-stock, cattle, property, money), and the English pen comes from the Latin penne (feather).


While this contextual theory in itself is clear enough, it still leaves open the question of why people stop using an old word or, in some cases, start using a new one when alternatives exists.

In the last couple of pages of the chapter, Bloomfield sketches a couple of possibilities, mostly of stylistic nature. One of the more interesting ones is his perceptive description of how "the salesman" might not want to sell a house as a house, since the availability of the alternative home will make house call up an image of "an empty shell that has never been inhabited" (p. 442).

Johnson and Malgady: "Some Cognitive Aspects of Figurative Language" (1979)

The purpose of this paper is to map the correlations between a number of variables related to metaphor comprehension, such as the difficulty of understanding the metaphor and the similarity of the concepts it is constructed out of. The experiment and its results are them selves utterly forgettable, but the paper is a good source of references as well as a probe of the mood of the cognitive science of metaphor around 1979.

First some references:

• Bolinger: "The Atomization of Meaning" (1965)
• Sheehan: The Function and Nature of Imagery (1972)
• Thomas: Metaphor and Related Subjects (1969)
• Verbrugge and McCarrell: "The Role of Inference in the Comprehension of Metaphor" (1973; probably lost in the seas of time, but possibly also similar to this 1977 paper)
• Weinrich: "Explorations in Semantic Theory" (in Current Trends in Linguistics 3, which was either published in 1966 as the paper says, or in 1970, as the publisher says)

About the last reference: Note that the the Polish-American linguist Uriel Weinrich (1926–1967) is distinct from the German classist, philogist, and literary scholar Harald Weinrich (born 1927). The latter also wrote a number of texts on metaphor ("Semantik der Metapher" and "Wieder die Bildstürmer"/"Against the iconoclasts"). I was myself confused about this for a while.

Now for some quotes.

In the introduction of the paper, the authors approvingly quote Bolinger as saying that metaphor should be covered by a good semantic theory, and they continue:

Others … have also suggested that metaphor and related figurative language use ought to be thought of as an intrinsic (perhaps central) part of language, and not something that can be dismissed by simply shunting it off to the poet's corner as a deviant curiosity—albeit an aesthetically satisfying one. (p. 250)

In the concluding discussion section:

The ubiquity of both metaphor and association (alluded to in the introduction and elsewhere) gives the impression that both must play a central role in everyday cognition—and that they both may be simply "symptoms" of a single underlying process. It is no more correct, therefore, to say that metaphor is simply similarity or simply association than it would be to say that association or similarity judgments are simply examples of metaphor. (p. 263)

And finally:

It is the flexibility of the relationship between words and categories, augmented by the linguistic device labeled "metaphor," that allows productivity in thinking. Perhaps the chief function of metaphor is to provide—by setting the stage for the perception of similarity between dissimilar words—a way of forming new categories. (p. 264)

Osgood, Suci, and Tannenbaum: The Measurement of Meaning (1957)

Charles Osgood was a student of the psychologist Theodore Karwoski, who did a number of studies of synaesthesia between the 1930s and the 1950s. His own work partly continued this line, but also partly shifted the focus more explicitly to the expression of synaesthetic thought in language.

The Dimensions of Meaning

In this book, Osgood and his two coauthors thus use factor analysis to show that the actual dimensionality of semantic space is much smaller than the nominal, since there are strong correlations between oppositions such as up/down, good/bad, etc.

A test item used on one of the experiments reported in the book (p. 81).

In 1980, Osgood would summarize the findings as follows:

Happy is UP, COLORFUL, LIGHT, and CLEAR, but _sad_ is DOWN, COLORLESS, DARK and HAZY; heavy is DOWN, THICK, DARK, and LARGE, but light (weight having been specified) is UP, THIN, LIGHT and SMALL; excitement is VERTICAL, COLORFUL, CROOKED and SHARP, but _calm_ is HORIZONTAL, COLORLESS, STRAIGHT and BLUNT; woman is COLORFUL, THIN, (except for Mexicans), LIGHT, BLUNT and ROUNDED (except for Navajos), but man is (VERTICAL (woman tending to be HORIZONTAL), COLORLESS, THICK, DARK, SHARP and ANGULAR. These trends for four cultures suggest certain "universal" tendencies. (Quoted from the 1981 reprint, p. 59)

In the 1957 book, the authors explain:

 … it was found that, as used by our subjects in making their judgments, the semantic scales fell into highly intercorrelated clusters. For example, fair-unfair, high-low, kind-cruel, valuable-worthless, Christian-antiChristian, and honest-dishonest were all found to correlate together .90 or better. Such a cluster represents the operation of a single, general factor in social judgments, obviously here an evaluative factor. Scales like strong-weak, realistic-unrealistic, and happy-sad were independent of this evaluative group and pointed to the existence of other dimensions of the semantic framework. (pp. 24--25)

Funny how the metaphor "ANTICHRISTIAN IS DOWN" has escaped from the current literature on cognitive metaphor theory.

Watch the Music

Front cover of the book.

For his undergraduate thesis, Osgood had also studied a number of anthropological reports of "primitive man," coming to similar conclusions. He reports in the book that,

for example, good gods, places, social positions, etc., were almost always up and light (white), whereas bad things were down and dark (black). A prevalent myth tells how the gods helped the original man to struggle from the dark, cold, wet, sad world below the ground up to the light, warm, dry, happy world on the surface. (p. 23)

Given Osgood's background, all of these things of course had to be connected to synaesthesia:

It seems clear from these studies that the imagery found in synesthesia is intimately tied up with language metaphor, and that both represent semantic relations. (p. 23)

Or, more elaborately:

Whereas fast, exciting music might be pictured by the synesthete as sharply etched, bright red forms, his less imaginative brethren would merely agree that words like "red-hot," "bright," and "fiery," as verbal metaphors, adequately describe the music; a slow melancholic selection might be visualized as heavy, slow-moving "blobs" of somber hue and be described verbally as "heavy," "blue," and "dark." The relation of this phenomenon to ordinary metaphor is evident: A happy man is said to feel "high," a said man "low"; the pianist travels "up" and "down" the scale from treble to bass; souls travel "up" to the good place and "down" to the bad place; hope is "white" and despair is "black." (p. 21)

Metaphorical Grounding

They also proposed a genealogical explanation for these phenomena, essentially the same as the one given today by cognitive metaphor theorists once you strip away the behaviorist language:

Take the case of parallelism between auditory pitch and visual size (synesthetes typically represent high tones as small and low tones as large): it is characteristic for the physical world that large-sized resonators produce low frequency tones and small-sized resonators, high frequency tones (think of a series of organ pipes, bells, or even hollow logs and sticks, and of the voices of men vs. boys, large dogs vs. little dogs, or lions vs. mice). This means that repeatedly the visual stimulus of large objects will be paired with the auditory stimulus of low-pitched tones, and so on consistently throughout the continuum. Any representational processes associated with one (e.g., danger significance of threatening big dog vs. play significance of little dog) will tend to be associated with the other as well (e.g., sounds produced). This will a hierarchy of equivalent signs come to be associated with a common mediation process. Any encoding responses associated with this mediator, such as "large" drawing movements and saying the word "large," will tend to transfer to any sign which elicits this mediator — thus "synesthesia" when a deep tone produces "large" drawing movements and "metaphor" when the word "deep" is associated with the word "large." (p. 24)

Michel Bréal: Semantics (1900)

Bréal; from Wikimedia Commons.

Apparently, this book by Michel Bréal is considered by some to be a foundational document in semantics. However, it's not mentioned in Saeed's textbook, and I wouldn't have heard of it if it weren't for a footnote in a paper by Harald Weinrich — so think it's fair to say it's fallen pretty much into obscurity by now.

I looked at because I was interested in seeing what Bréal had to say about metaphor. It turns out that his observations are that the phenomenon is (1) widespread (2) mostly dead (3) often synaesthetic.

The Psychological Basis of Metaphor

As for the widespread nature of metaphor:

Our business is not to claim admiration for these images, which indeed have ceased to be images, but to show that language is full of them. (p. 123)

His take on the emergence of metaphors seem to be that they are invented for communitative purposes, not that that it reflects deep psychological structure. Most of his attention is devoted to the connection between cultural variables (e.g., Roman land ownership institutions) and the choice of metaphors.

However, he says that they demonstrate "the universal intelligence, which does not vary much from one nation to another" (p. 123). In his discussion of grammatical analogy, he also claims that opposites like night/day and dead/alive tend to take on similar forms, and that "Language here reveals to us a fact of psychology;" providing a hand-wavy sketch of the psychology of this phenomenon (p. 68).

The Death of a Metaphor

There is not doubt that Bréal thinks that metaphors wither and die over time:

But the metaphor remains such at its outset only; soon the mind becomes accustomed to the image; its very success causes it to pale; it fades into a representation of the idea scarcely more coloured than the proper word. (p. 122)

But for the child who learns to speak them the complication [of faded historical meanings] does not exist: the last meaning, the meaning farthest removed from it original, is often the first learnt. (p. 133)

More indirectly, this is also evidenced by his interest in the difficulty of etymological problems. He notes, for instance, that it is not immediately obvious which of the meanings of the Latin gemma are the older, "pearl" or "bud." In fact, Cicero seems to have gotten the historical order wrong (p. 125).

Synaesthetic Metaphors

Bréal also notes the importance of cross-sensory transfers of meaning:

A special kind of Metaphor, extremely frequent in all languages, comes from the communication between our organs of sense, which permit us to transport the sensations of sight into the domain of hearing, or the ideas of touch into the domain of taste. We speak of a "a warm reception," "a broad style," "a bitter reproach," "a black grief," with the certainty of being understood by everybody. … A deep sound, a high note were originally images. (pp. 129-30)

Again, however, he warns the reader not to pull etymologies out of a hat:

Sometimes it is difficult to tell exactly from what organ of the body these expressions came: for example, it was long considered doubtful whether the adjective clarus came from sigh or from hearing. Without the words acies, acus, acutus, acer, we should not know that acid (the French aigre) did not always belond to the sense of taste. (p. 130)

 So much for introspective psychology, then.

Asch: "The Metaphor: A Psychological Inquiry" (1958)

The social psychologist Solomon Asch was interested in how we perceive, understand, and describe other people, and this let him (among other things) to conduct this small, cross-linguistic study on the polysemy of some common sensory adjectives.

He wanted to investigate whether the cross-modal ambiguity of English words like sharp was a language-specific coincidence, so he selected a set of promising terms and asked some informants what other meanings the corresponding word in their language could have.

Cutting the Cake

Asch presents his materials as follows:

Included among the terms were the following: warm, cold, hot; right, left; dull, bright, pale, shining; straight, twisted, crooked; sweet, bitter; colorful, colorless, white, black (and some other color terms); rought, smooth, slippery; dry, wet; clear, cloudy; deep, shallow, high, low; broad, rounded, sharp; hard, soft. (p. 88)

The languages he compared were ancient Hebrew, ancient Greek, "Chinese" (not further specified), Thai, Malayalam, Hausa, Burmese (p. 88).

The conclusion he came to was that "… all the languages examined here contain terms that simultaneously describe both physical and psychological qualities" (p. 89). However, he also warned about overextrapolating these findings:

From the linguistic evidence alone, even if it were more complete, we could not, of course conclude about the responsible operations. For this purpose we need a psychological analysis. (p. 91)

Solomon Asch with his wife Florence (from the Asch Center website).

Cognitive Operations

After having presented his results, Asch also speculates about the "conceptual basis" for this dual function of sensory words:

Dual terms may be the consequence of stable associative connections established between dissimilar physical and psychological  conditions that regularly share some stimulus properties. (p. 92)

The concepts in question have little in common with abstract logical operations. They are not generalizations of what is common to an array of different instances. Rather they are concrete cognitive operations in terms of which we naïvely comprehend events and similarities between them. (p. 93)

We see natural events as conductors of the same fundamental forces that we find in the human sphere. Therefore we speak spontaneously of seeing a point, of shedding light on or illuminating a problem, of penetrating to the heart of a matter. (p. 93)

The Schema of Interaction

Thus:

What now is the sense of hard when it refers to a person? It describes an interaction that is formally similar. We see a man refusing the appeal of another. This interaction we experience as a force proceeding from one person, having as its aim the production of a change in the other, which, however, fails to move him, or which produces resistance. The hardness of a table and of a person concerns events radically different in content and complexity, but the schema of interaction is experienced as dynamically similar, having to do with the application of force and of resulting actions in line with or contrary to it. (p. 93)

Williams: "Synaesthetic Adjectives" (1976)

Based on an inspection of English and Japanese adjectives, this paper claims that the etymological development of meaning by metaphorical extension always follows a particular pattern, shown in the figure below.

Figure 1 from Williams' paper (p. 463).

For instance, sound can be conceptualized as touch (soft music), but the opposite is not the case (loud touch). Williams emphasizes, though, that the law is quantitative, not logical:

Even in English, the generalization is not exceptionless. But its regularity varies between 83% and 99%, depending on how we compute what counts as an observation of it. (p. 463)

These numbers refer to the number of dictionary entries inspected (not, for instance, token frequencies in running text).

Towards the end of the paper, he wonders whether "these sequences [of consecutive word meanings] might be reflected in any physical basis of sensation" (p. 472) like its evolutionary history, or the developmental history of the individual. In the conclusion, he writes:

Obviously it is presumptuous, to say the least, to seek a biological foundation for a phenomenon that may not universally exist, in an aspect of human cognition about which very little is known. But the parallels that do exist … indicate … a point of interaction between mind an brain. (p. 473)

However, it is important to keep in mind that he explicitly presents his hypothesis as one about "rule-governed semantic change" (p. 473), not about cognition.

Hull: Touching the Rock (1990)

John M. Hull is a former (now retired) professor of religious studies who struggled with poor heath and receding eyesight for decades, finally going completely blind at age 48. In this sort-of-autobiography, based on a series of tape-recorded diary entries, he recounts the slow and painful process of adjusting to his new condition.

He writes about the everyday challenges and frustrations of his new life: How to lecture without notes, how he struggles to find his way around the snow-covered streets of Birmingham, how he copes with the difficulties of meeting new people at conferences and in bars. He is honest about his moments of panic and depression, but you also sense how he gradually, over the span of about five years, learns to live with his blindness.

I just finished reading the book now, and I'm completely in love with it. It's frankness and its small nuggets of wisdom and lived experience give an extraordinary sense of the difficulty of living with blindness, but also of the new and different world that comes with it. I really recommend reading it; but since we're all busy people, I'll also give a few long quotes below.

'See You'

I originally picked up the book because of its testimony of the strange linguistic situation blindness causes: Not being able to see yet having to use the visual metaphors of sighted language.

Hull explains:

'Well, I'll see you around.'

'Nice to see you again.'

'I see what you mean.'

When I use expressions like these, some of my sighted friends are surprised. They laugh, perhaps teasing me, and say, 'You don't really mean that, do you John?' I explain that, when I say I am pleased to see you, what I mean is that I am pleased to meet you, pleased to be with you, glad to be in your presence. I explain that this is surely what anybody, blind or sighted, would mean by that expression. (p. 21)

Not just words like see that are affected by these changes. So does the role of people's eyes, faces, and expressions, and the language surrounding them:

Another strange feature of not knowing what people look like is the effect this has upon reported speech. When I am describing an encounter with someone, I may want to say, 'He looked blankly at me'. I feel a little sensitive about this, because I cannot help thinking that the sighted person to whom I am talking would know that I could not possibly know how my friend looked at me. To say, 'He responded in a blank manner' is absurd and pedantic. … To say, 'He paused before replying and seemed to be at a loss' would be perfectly accurate, but to use the brief, concrete idiom of the sighted exchange is so natural and vivid. What am I to do? (p. 16–17)

This is interesting in itself, but it also makes me wonder to what extent such descriptions ever really report visual observations. Do I, as a sighted person, describe my social or my visual experience when I say a cold stare, give me the eye, and sparkling eyes?

Face to Face

Hull continues:

Another result of all this is that the face no longer has the central place for me which it has in normal human relationships. The face is merely the place from which the voice comes. I look towards the face with conscious effort, for there is no real reason why I should do so. … I no longer have any natural sense of needing to be face to face. (p. 17)

The marine biologist Geerart Vermij (who had his eyes removed at age three) makes a similar point in his discussion of "sentimentalists" who lament the loss of sight as a tragedy for social competence:

What these sentimentalists forget is that the face is only part of the whole person. The voice—its quality, its intonation, the use of language—is unique and every bit as informative as the face. I can detect surprise, disgust, pleasure, boredom, dishonesty, thoughtfulness, and a hundred other states of mind from the voice. (p. 17)

On Pretty Women

Hull mostly remains quite vague and abstract about sex, but he does make some very interesting remarks about his continued desire to associate with pretty women:

Sometimes I ask one of my sighted friends to give me a quick impression of what somebody else looks like. I am often interested in a sort of thumb-nail sketch of a new acquaintance. This is particularly true if my new acquaintance is a woman. What colour is her hair? What is she wearing? Is she pretty? Sometimes I long to know. I remain, after all, a man, reared in a certain sighted culture, conditioned to certain male expectations. Perhaps I should change, and be less influenced in my judgment of women by my male conditioning, but it is painful to have this change forced upon me by mere blindness.

It makes a difference to the way I feel about a new female acquaintance if a colleague, having caught sight of her, remarks on her beauty or plainness. There is a double irrationality in this. In the first place, by feelings should not be so dependent upon a woman's appearance. I know that, and I apologize. But I still feel it. The second thing is that it is surely a deplorable lack of independence on my part to be so affected by a criterion which can be of no significance to me.

What can it matter to me what sighted men think of women, when I, a blind man, must judge women by quite different means. Yet I do care what sighted men think, and I do not seem able to throw off this prejudice. (p. 17)

Again, this says as much about sighted relationships as it does about blind. Do men pursue pretty women for pleasure or prestige? To have sex or to assert their power over other men?

The thought does not come up again, perhaps because its importance to Hull gradually wanes. However, he does later note (with, perhaps, an undertone of despair) that the loss of vision has cut him off from one of the most intense sources of sexual excitement:

So it is possible, I think, for a heterosexual blind man to be bored by women and yet to be conscious of sexual hunger. The trace of a perfume and the nuance of of a voice are so insubstantial when compared with the full-bodied impact upon a sighted man of the appearance of an attractive woman. It must take a long time for a man who loses sight in adult life to transfer the cues of sexual arousal from the visual to other senses. There must be many men in that position who wonder whether they will ever again be capable of genuine sexual excitement. (p. 38)

This is interesting because it negates the popular conception that the sense of smell somehow bypasses our "rational minds" and penetrates directly into our subconsciousness. (Billy Jean may have looked "like a beauty queen from a movie scene," but the intensity only goes up as Michael Jackson can "smell the sweet perfume.") Whatever the physiological facts are, Hull does not seem to find the effect very impressive.

What Is It Like to Be a Bat?

A few months after Hull finally lost his sight completely, he started to noticing that he was picking up a new sensitivity to the space around him, a kind of intuitive sonar:

I gradually realized that I was developing some strange kind of perception. … As the months so past, sensitivity seems to be increasing. I find that I am now quite often aware of approaching lampposts … (p. 19)

He elaborates:

Not only have I become sensitive to thinner objects, but the range seems to have increased. When walking home, I used only to be able to detect parked cars by making contact with my cane. These days I almost never make contact with a parked car unexpectedly. Nearly always, I realize there is an obstacle in my path before my stick strikes against it. This is in spite of the fact that I am now using the very long cane. I think the range for detecting parked cars must be approximately six to eight feet. Another feature of this experience is that it seems to be giving me a sort of generalized sense of the environment. There is one part of my route where I must step aside to avoid an upward flight of steps. I am expecting these, of course, since I come this way every day. Nevertheless, I am now aware of their approach, and not merely of the lower, closer steps, but of the whole massive object, looming up and somehow away from me. The phenomenon seems to be partly dependent on attention, since at home I can easily walk into the edge of doors, having had no warning of their proximity. Possibly in a house where sound is muffled by carpets and curtains, echoes would be less easily perceived?

The experience itself is quite extraordinary, and I cannot compare it with anything else I have ever known. It is like a sense of physical pressure. One wants to put up a hand to protect oneself, so intense is the awareness. One shrinks from whatever it is. (p. 20)

Space and Time

These and other changes to his perception naturally changes the way he relates to his environment. In particular, he reports that the difference between front and back becomes less important:

Here is another feature of the acoustic world: it stays the same whichever way I turn my head. This is not true of the perceptible world. It changes as I turn my head. New things come into view. The view that way is different from the view looking this way. It is not like that with sound. New noises do not come to my attention as I turn my head around. I may allow my head to hand limply down upon my chest; I may lean right back and face the sky. It makes little difference. (p. 63)

This also sounds a lot like a remark a blind informant made to the (sighted) anthropologist Gili Hammer during one of her interviews: "Hearing operates in 360 degrees; however, you can't see what's behind you."

Hull also notes that his relationship to time changes, although in a different way and for quite different reasons:

When I had sight, I would have worked with a feverish haste, correcting forty footnotes in a single morning. Now, I am happy if, with the help of a sighted reader, by the end of the morning I have corrected ten. I do not think to myself, 'Oh damn. I've only done ten'. I think, 'Good. That's ten done. Only another three mornings like this and the job will be finished.' I am so glad that I am able to do it at all. (p. 60)

He seems to find some hope or comfort in the indefatigable spirit others bring to the fight:

I think of my friend Chris with his multiple sclerosis. … It takes him 45 minutes to tie his shoelaces in the morning. It doesn't matter. He does not get impatient. He just does it. That is how long it takes to tie shoelaces. (p. 60)

As he notes towards the end of the book, his solution cannot exactly be Stoicism, not exactly acceptance of the situation, but perhaps some sense of reaching the other side of the ocean of despair.

Cardano: The Book on Games of Chance (1961/1663)

Cardano; from Wikimedia.

My library has refused to buy the new translation of Jacob Bernoulli's Ars Conjectandi (1713), possibly because the price tag is on the order of $3000. Consequently, I don't have an English translation of the complete book, arguably the most influential one in the history of probability (perhaps with the exception of Kolmogorov's little pamphlet).

Meanwhile, they hold two copies of Cardano's weird, chatty, and uneven book on gambling. I checked out both of them, only two find out that one (the 1961 version) was a reprint of the other, the translation by Øystein Ore appended to his own 1953 commentary on the book (which, incidentally, also included an interpretation of Cardano's erroneous calculus of probabilities).

I'll deal with Ore's book later. Now a few comments on quotes on Cardano.

Contents

Cardano's book can roughly be divided up into sections as follows. If it looks jumbled up, it's because it is.

• Chapters 1–10: Preliminaries, survey of games, moralistic preaching, etc.
• Chapters 9–11: Combinatorics of dice games.
• Chapters 16–19: On card games (mostly non-mathematical discussions of cheating, moral issues, and the like).
• Chapters 20–21: On luck.
• Chapters 22–25: More game taxonomies and definitions.
• Chapter 26: On theoretical and practical knowledge.
• Chapter 27: More on luck
• Chapter 28: Recommendations on playing styles in backgammon.
• Chapter 29: On why you shouldn't play against hotheads.
• Chapters 30–31: Games in ancient Greece.
• Chapter 32: Computing expectations for a die.

From the perspective of the history of mathematics proper, the most directly relevant parts are those on combinatorics and those on luck.

"If Fortune Be Equal"

One of the more remarkable features of Cardano's book is the prominent place it gives to "luck." As Lorraine Daston has noted, this concept seemed to fill out any gap his calculus didn't account for, including the gap between frequencies and probabilities, or between expectations and actual outcomes.

Mean playing dice; illustration from a 1531 print of Cicero's On Duties.

The first time the word occurs is during Cardano's dicussion of fair bets. Apparently, the most common dice game at the time was a bet on whether a specific throw (e.g., double six) would show up or not show up within n throws of a number of dice.

Cardano was therefore interested in computing the number n for which this bet would be fair — or in his words, for which "there is equality." He explains in Chapter 11:

This gives eighteen casts for equality for a throw (1,1); for in that number of casts this throw can appear and not appear with equal probability; and similarly for the throws (2,2) and (3,3).

But the throw (1,2) can turn up in two ways, so that for it there is equality in nine casts; and if it turns out up more frequently or more rarely, that is a matter of luck. (p. 11; my emphasis)

Later, in Chapter 14:

If, therefore, someone should say, "I want an ace, a deuce, or a trey," you know that there are 27 favorable throws, and since the [size of the] circuit [= sample space] is 36, the rest of the throws in which these points will not turn up will be 9; the odds will therefore 3 to 1.

• Therefore in 4 throws, if fortune be equal, an ace, deuce, or trey will turn up 3 times and only one throw will be without any of them;
• if therefore, the player who wants an ace, deuce, or trey were to wager three ducats and the other player one, then the former would win three times and would gain three ducats; and the other once and would win three ducats;
• therefore in the circuit [= observation] of 4 throws they would always be equal. (p. 16; my epmhasis; my itemization)

His language use here suggests that, on some level, he believes that the expected values must somehow be realized — they are what the game should pay off if it weren't for all these kinks and imperfections in the universe.

"The Length of Time … shows Forth All Forms"

Apparently, he is completely serious about this. In a later chapter on skill (ch. 27), he lists two "methods" by which fortunes can change, the second being the more occult one:

But of the other method there is also some secret principle. To these matters belong amulets, witchcraft, and the like, and just as in each case (as they say) the sword fits its own sheath and the foot its own show, so the hour, the day, the year, and the place must fit; so also in this question, what will make one man happy will make another wretched. (p. 44)

Austrian 16th century woodcut of two soldiers playing dice.

Cardano does seem to think, however, that time tends to cancel out luck. In the chapter on sequential successes, he computes the probability of observing an unbroken string of 20 repetitions of a probability 1/2 event, wrongly getting the answer to be 1/8000. He comments:

Yet it can scarcely be believed that in 7,999 throws a player can cast an even number twenty times in a row. But we have demonstrated this; because, although in 8,000 performed only once, this reasoning can deceive us, still in an infinite number of throws, it is almost necessary for it to happen; for the magnitude of the circuit is the length of time which shows forth all forms. (pp. 19–20).

Here, "luck" almost seems to be synonymous with "noise."

Squares and Cubes

Since I'm talking about this error anyway, and since this is essentially the sole remaining mathematical component of the book, let me just quickly summarize what Cardano seems to be doing in the rest of Chapters 14 and 15.

At the end of Chapter 14, he claims that if the odds in favor of a single success is p : 1 – p, then odds in favor of k successes in a row are p^k : (1 – p)^k. This is not correct, since (1 – p)^k is not in general equal to 1 – p^k. He recognizes this in the opening of Chapter 15, noting that if it were really true, any run of consecutive probability 1/2 events would also have probability 1/2:

But this reasoning seems to be false, even in the case of equality, as, for example, the chance of getting one of any three chosen faces in one cast of one die is equal to the chance of getting one of the other three, but according to this reasoning there would be an even chance of getting a chosen face each time in two casts, and thus in three, and four, which is most absurd. For if a player with two dice can with equal chances throw an even and an odd number, it does not follow that he can with equal fortune throw an even number in each of three successive casts. (p. 19)

 Cardano 1 

 Cardano 2  

Correct 

1 : 1

1 : 1

1 : 1

1 : 1

3 : 1

3 : 1

1 : 1

8 : 1

7 : 1

1 : 1

15 : 1

15 : 1

1 : 1

24 : 1

31 : 1

1 : 1

35 : 1

63 : 1

1 : 1

48 : 1

123 : 1

So now Cardano owes us a different argument. He therefore goes on to claim that the correct answer for p = 1/2 in fact is k² – 1 : 1. This coincides with the correct answer for a couple of small values, but then diverges exponentially from it. This leads him to make the "infinity" remark quoted above.

Parenthetically, I'm not sure why he would cube rather than square the number 20 in that example. Perhaps Ore has something intelligent to say about this.

Late 15th century book illustration showing a dice game.

How To Gamble If You Must

In Chapter 20, Cardano tells a little autobiographical anecdote as an illustration of his points about fortune, luck, etc. This story is not strictly relevant to my topic here, but it is simply to bizarre not to quote. Hence, Ladies and Gentlemen, Cardano without filter:

Yet I have decided to submit to the judgment of my readers what happened to me in the year 1526 in the company of Thomas Lezius, the patrician of Venice, leaving it to each reader to form his own opinion. I had just duly resigned from the office of rector of the scholars in the University of Padua on the third of August, and now I was journeying with Hieronymus Rivola, a scholar from Bergamo, on a certain night of the same month toward Venice. We were playing a game (called Bassette) and I won all the money he had. Then he asked me to play with him on credit, if I am not mistaken up to two or three aurei, and I won again. Then, finally, he wanted to carry it on endlessly, but I refused. He promised to pay what he owed me within three ways; he did not come.

Then he chanced to meet me and said that he would come to pay the money on Saturday (which was the day of the Nativity of the Virgin) and promised to take me to a beautiful prostitute. At that time I was just completing my twenty-fifth year, but I was impotent. Nevertheless, I accepted the condition; there was not a word about the game. He came on the day agreed; and in that year the festival of the Blessed Virgin was on Saturday. He took me to the home of Thomas Lezius; there was no Thais there, but a bearded man with a young servant. No money was paid but we played with marked cards. I lost to him all the money which he owed me, and he reckoned it as part of his debts just as though he had given it to me. I list about twenty-five aurei or even a few more which I had, and played on, giving my clothes and my rings as security.

I returned home in sadness (as was natural), especially since there was no hope of getting money from home because uprisings and plots were raging at Milan. And so (and now I tell the truth, there being no reason why I should lie), I contrived for myself a certain art; I do not now remember what it was, since thirty-eight years have passed, but I think it took its rise in geomancy, by which I kept in mind on up to twenty-four plays all the numbers whereby I should win and all those whereby I should lose; by chance the former were far more numerous than the latter, even in the proportion (if I am not mistaken) of seven to one; and I do not recall now in what order these were against me.

But when I saw that I could not safely hold more numbers in my memory, I admonished my young servant, whose name was Jacob, that when he saw I had won back my clothes and my money he was to call me. I threatened that if he did not do it I would beat him severely. He promised and we went. As the game went on I won and lost in all the plays just as I had foreseen and after the third play I realized that there was no trickery or deceit about it. They laid down money freely and I accepted the wagers, but he was delighted be the example of the previous day and also on account of the marked cards (as I have said).

Thus his thoughts were inflamed by his youthful ardor; but the result was otherwise, for, on those plays in which I saw (as it were, without any real foreknowledge) that I would win, I did not rehect any amount of money and made large bets of my own, and in the other cases, where I knew he would win, I refused if he was the first to wager, and wagered very meagerly myself: thus the result was that within twenty plays I regained my clothes, my rings, and money and also what he had added besides. As for the clothes, the rings, and a collar for the boy, I sent them home piecemeal. Out of the total number there remained four deals; I played and won, and also came out victor in a few deals which were not contained in the number.

He was already perturbed and full of admiration, since he saw that in all the plays in which we played for high stakes I came out the victor, and in those in which he won I myself wagered little and when he wished to wager a great deal I refused. So (he said) I believe some demon is advising you, or that you know by some enchantment what is going to happen. What happened after that I remember that I have narrated elsewhere. (pp. 32–34)

He also gives a bit of extra detail in Chapter 17, the chapter on fraud in card games:

There are also some who smear the cards with soap so that they may slide easily and slip past one another. This was the trick practiced upon me by the well-known Thomas Lezius of Venice, patrician, when in my youth I was addicted to gambling. (p. 27)

Extraordinary, isn't it?