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Logical
consequences are the scarecrows of fools and the beacons of wise men.
— T. H. Huxley, Science and Culture, ix, On the Hypothesis that Animals are Automata.
The usual reason given for teaching proving, the importance of proving as the method of verification is mathematics, does not agree with either the role of proving in mathematics or with students’ needs to prove. This observation led me to raise the question “Why teach proving?” at the end of the previous chapter. It could have as easily led to the question “Why are the curriculum designers asserting that proving has a role that it does not have either in mathematics or for students?” The answer to this question lies in the role of Rationalism in our society. The problems that have resulted from the application of proving in inappropriate contexts and weaknesses in the basic assumptions of Rationalism suggest that the role of Rationalism in society needs to be reconsidered. In fact, I would assert that if education involves preparing students to play “a role in constantly making and remaking the culture” (Bruner, 1986, p. 123), then Rationalism is a part of our culture that needs remaking.
In this chapter I describe Rationalism and some problems it has given rise to. I then analyze some of its weaknesses and limits. These limits suggest both a need for other modes of thinking and a need for a remaking of Rationalism. In the next chapter I describe one possible remaking of Rationalism, and its implications for teaching and research. My critique cannot pretend to be exhaustive. A thorough description and critique of Rationalism would fill many volumes. I will be ignoring the critical perspectives of feminism, post-modernism, and phenomenology, among others. These perspectives are certainly valuable, but my purpose here is simply to suggest that a strong critique of Rationalism exists from a Rationalist perspective, and so I will be limiting myself to that perspective. I will also be providing a simplified description of Rationalism, which contains what I believe are its central points, but which necessarily neglects subtleties which would be included in a more thorough history.
Rationalism is based on two beliefs: that deductive reasoning can determine absolute truths, and that deduction is applicable to all situations. The implication of these two beliefs is that in any situation in which we want to know something, the best way to reason is deductively. The basic ideas of rationalism can be traced to Descartes. Descartes published his Discourse on the Method for Rightly Conducting One’s Reason and for Seeking Truth in the Sciences in 1637. The “Method for Rightly Conducting One’s Reason” he wrote of is deductive, rational thought.
As an aside, I should note that my tracing of Rationalism to a few words of Descartes could be seen as a misrepresentation of his work. Descartes’ ideas
occurred in a context and have been reinterpreted many times in many other contexts. My use of his name and words here is largely a rhetorical and heuristic device. It allows me to describe in a simple way a belief or attitude held by many people at many times. Modern Rationalism can be, as I have done here, traced to Descartes, but just as easily traced to Russell, or Leibniz, or Plato.
Descartes’ method was inspired by the proofs of Euclid. His innovation was to imagine that such deductions might illuminate other areas:
Those long chains of reasoning, each of them simple and easy, that geometricians commonly use to attain their most difficult demonstrations, have given me an occasion for imagining that all the things that can fall within human knowledge follow one another in the same way and that, provided only that one abstain from accepting anything as true that is not true, and that one always maintains the order to be followed in deducing the one from the other, there is nothing so far distant that one cannot finally reach nor so hidden that one cannot discover. (Descartes, 1637/1993, p. 11)
In fact, Descartes believed that the simple and easy reasoning that geometricians use was the only way of reasoning which could succeed in revealing truth:
Of all those who have already searched for truth in the sciences, only the mathematicians were able to find demonstrations, that is, certain and evident reasons. (p. 11)
The influence of Rationalism has extended beyond its origins in mathematics, science, and philosophy. In the eighteenth century, the Age of Reason, the successes of Rationalist science became known to all educated Europeans and had effects on their vision of the world:
Science was for them ... living growing evidence that human beings, using their “natural” reasoning powers in a fairly obvious and teachable way, could not only understand the way things really are in the universe; they could understand what human beings are really like, and by combining this knowledge of nature and human nature, learn how to live better and happier lives. (Brinton, 1967, p. 519)
This Enlightenment vision has continued into present day rhetoric, curriculum documents, textbooks, and teaching. Rationalism also continues to play an important role in research, both in the definition of reasoning and as the basis of methodology. Lakoff (1987) describes the dominant understanding of what “reasoning” means in this way:
In this century reason has been understood by many philosophers, psychologists, and others as roughly fitting the model of formal deductive logic:
Reason is the mechanical manipulation of abstract symbols which are meaningless in themselves, but can give meaning by virtue of their capacity to refer to things either in the actual world or in possible states of the world. (p. 7)
The two beliefs that form the basis of Rationalism, that deductive reasoning can determine absolute truths and that deduction is applicable to all situations, turn out to be problematic. The weaknesses in these fundamental beliefs permeate all of Rationalism (because Rationalism has a deductive structure). In this section I describe the weaknesses of Rationalism, and also suggest that Rationalism is not only flawed, but dangerous when applied to many situations.
Descartes modeled his method on the proofs of Euclid, and saw them as the ultimate example of thinking which produced certainty. For this reason it seems to me that the relationship of absolute truth to proving in mathematics is a sensible place to explore this aspect of Rationalism in general. In the past two centuries the relationship of absolute truth in mathematics has shifted radically. In the late eighteenth century mathematics, and especially geometry, was seen as the most absolute of truths. By the late nineteenth century, it was acknowledged that what was true depended on the assumptions, the axioms and postulates, which form the basis of a mathematical system. There could be two equally valid systems, based on different assumptions, but within each system proving could reveal all truth and engender no contradiction. By the mid-twentieth century, even this hope was lost since it was shown that all mathematical systems are necessarily incomplete; there are truths that can be known but not proven. What Kline (1980) called the “loss of certainty” in mathematics has implications for Rationalism as a whole although they are barely beginning to be felt.
In the Age of Enlightenment Euclid’s geometry was often held to be the epitome of certainty. Descartes based Rationalism of Euclid’s proofs, and Kant (1781/1927) used the certainty of geometry to support the necessity of space being a priori.
On this necessity of an a priori representation of space rests the apodictic certainty of all geometric principles, and the possibility of their construction a priori. For if the intuition of space were a concept gained a posteriori, borrowed from general external experience, the first principles of mathematical definition would be nothing but perceptions. They would be exposed to all the accidents of perception, and there being one straight line between two points would not be a necessity, but only something taught in each case by experience. Whatever is derived from experience possesses a relative generality only, based on induction. We should therefore not be able to say more than that, so far as hitherto observed, no space has been found to have more than three dimensions. (p. 19)
The “apodictic certainty of all geometric principles”* was undermined by the discovery, in early nineteenth century, of non-Euclidean geometries. These geometries begin with different assumptions than Euclid’s, but rather than collapsing in a mess of contradictions, as Kant might have predicted, they turn out to be just as consistent as Euclidean geometry.
* “Apodictic” means “established on incontrovertible evidence. (By Kant applied to a proposition enouncing a necessary and hence absolute truth.)” (Oxford English Dictionary, 1971)
The first of these non-Euclidean geometries, hyperbolic geometry, was independently discovered by Lobachevsky (published in 1829), Bolyai (published in 1832), and the renowned mathematician Gauss (never published, but he claimed he thought of it first). Hyperbolic geometry replaces Euclid’s “parallel postulate” (There is exactly one line through a given point parallel to a given line) with the postulate “There is more than one line through a given point parallel to a given line.” This postulate leads to many surprising results, like the sum of the angles in a triangle is less than 180˚, but it does not lead to contradictions.
Strictly speaking, what was shown was not that the non-Euclidean geometries are consistent, but rather that they are consistent if Euclidean geometry is consistent. This raised for the first time the question “How do we know Euclidean geometry is consistent?” The old answer, that it is the absolutely true geometry of space and so must be consistent, was no longer acceptable. Instead, a mathematician named Hilbert answered this question by proving that Euclidean geometry is consistent if basic arithmetic is consistent. Now the problem was to show that arithmetic is consistent.
Hilbert presented this problem, along with about twenty others, at the Second International Congress of Mathematicians held in Paris in 1900. At that time great progress had been made in making mathematical reasoning more formal, which made gaps in logic easier to spot and fix. The mathematical community had great confidence that the formal structures they were developing would, for mathematics at least, achieve what Leibniz had dreamed of in the eighteenth century, “an exhaustive collection of logical forms of reasoning—a calculus ratiocinator—which would permit any possible deductions from initial principles” (Kline, 1980, p. 183).
It quickly became apparent that the problem of verifying the consistency of arithmetic was not going to be a simple one. And the problem was not just showing consistency. By selecting a very small number of initial assumptions or axioms, it was easy to produce a system that was consistent. But a small number of axioms was not enough to allow the derivation of all the statements one might make about arithmetic. In this case the system would be a consistent but incomplete arithmetic.
Most attempts to develop a formal structure for arithmetic used axioms about sets as their basis. But the theory of sets, which was chosen for its simplicity and obviousness, turned out to produce paradoxes. The central problem involves sets that contain themselves. The set of all apples does not contain itself because a set is not an apple. The set of all mathematical objects does contain itself because a set is a mathematical object. We can distinguish between sets that do contain themselves and sets that do not. But what of the set of all sets which do not contain themselves? Does it contain itself?
This paradox is called the Barber Paradox, which was first noticed by Bertrand Russell in 1902. The name comes from the following story, which expresses the same paradox in different terms.
In a village there is a barber, who claims that he shaves every man who does not shave himself. Of course, he does not shave those who do shave themselves. Who shaves the barber?
Implicit in the paradox is a fashion statement: all men are shaved. Either Russell’s barber shaves himself, or he doesn’t. If he does, then he belongs to the class of men who shave themselves, and he does not shave those men so he cannot shave himself. If he doesn’t shave himself, then he belongs to the class of men who do not shave themselves, and he shaves all such men. Either way, a contradiction arises.
This amusing story caused great consternation in the small, but famous, circle of mathematicians working on the problem of showing that arithmetic is consistent. Their proofs became more and more formal. Solutions to the paradoxes were proposed, but many of these solutions had undesirable side effects since they barred methods of proving that had been used with much success in the past. It began to look like showing arithmetic to be consistent might require redoing most of mathematics or even rejecting parts of it.
Hilbert, Russell, Whitehead, Peano, Frege, Zermelo, Brouwer, Weyl, and many others worked on the problem of consistency. They divided into various schools, employing different bases and limitations on logic in developing arithmetic. The conflict between these schools raised another issue. Critics from other schools raised the point that it might be possible that such and such a school’s position might guarantee consistency but only at the cost of completeness. There might be important parts of mathematics that would left out. The baby might go with the bath water.
Russell and Whitehead decided to approach the problem by basing arithmetic on logic itself. As logic would be used in any proof of consistency, using logic as a basis added no new potential source of contradiction. Russell and Whitehead’s efforts resulted in their Principia Mathematica, first published in 1913. Although no one felt that they had completely solved the problem of consistency, Russell and Whitehead had, through careful use of formal proving, clarified the problem further. Progress continued, and things looked hopeful.
In 1931 the situation changed radically. Kurt Gödel published a paper entitled “On Formally Undecidable Propositions of Principia Mathematica and Related Systems.” In this paper Gödel delivered a powerful double whammy. The first blow related specifically to “Principia Mathematica and Related Systems.” Gödel proved that the task which had occupied the greatest mathematical minds of the first three decades of the century could not be done. The consistency of arithmetic cannot be established using the logical principles of Russell and Whitehead. The second blow was even worse. Gödel proved that any system which does manage to show that arithmetic is consistent, must be incomplete. That is, we can use formal deductive logic to know that part of arithmetic is free of contradictions, but we can never know that all of arithmetic is free of contradictions.
Gödel’s proof depends on producing a true statement, which he then shows cannot be proved without resulting in a contradiction. He does this by encoding the familiar Epimenides paradox into formal mathematics. The simplest formulation of this paradox is the sentence “This sentence is false.” If this sentence is true, then it is false. If it is false then it is true. Gödel produced a formal sentence (encoded as a number), which asserted that it could not be proved. If the statement is assumed to be true, then there is a true statement that cannot be proved and mathematics is incomplete. If it is false, then there is a false statement that can be proved and mathematics is inconsistent. (A readable description of Gödel’s Theorem is
Hofstadter, 1980. Readers with backgrounds in computer science may prefer the description in Penrose, 1994, based on computability. The empty set {readers who are mathematicians and read German, but are unacquainted with Gödel’s Theorem} will find a reference to the original paper in the list of References.)
Gödel’s Theorem established that deductive reasoning has limits even in mathematics, the original model of Rationalism. Its significance lies in establishing that there are some questions that deductive reasoning is powerless to answer.
Gödel’s Theorem undermines the foundations of Rationalism by invalidating Descartes’ original assumption that mathematics is complete. This is enough to cast doubts on any sort of Rationalist description of the world. Gödel’s Theorem and analogous arguments can also be applied directly to Rationalist world views. As examples, consider Penrose’s critique of Artificial Intelligence research, and Putnam’s critique of Objectivist semantics.
Penrose (1989, 1994) argues that Gödel’s Theorem implies that Artificial Intelligence (AI), as it is usually understood, is impossible, and that a scientific understanding of the mind will require major revisions to current theories in physics.
His argument against AI is essentially that a computer based AI is a formal system, and so by Gödel’s Theorem there are statements that are true, but which the AI cannot know because they cannot be proven within the formal system defined by the AI. Penrose asserts that an intelligent being could understand Gödel’s Theorem, but that an AI could not, at least as far as Gödel’s Theorem applies to the AI itself. If it could, then it would know some statement was true but be unable to prove it. Knowing the statement is true, however, implies that it is proven within the formal system of the AI.
Penrose’s argument for the need to revise the theories of physics is based on his AI argument. According to current physical theories, the human brain operates according to physical laws which, in theory, could be represented by an incredibly complicated formal system. If this were the case, then the same argument he used to show an AI cannot exist would show that human intelligence cannot exist. Given that human intelligence does seem to exist, there must be some physical property of brains that makes them essentially unlike a formal system. If such a property exists, however, current theories of physics must undergo a radical modification.
Lest you be tempted to assume that Penrose is a crackpot, which would be reassuring given the sweeping nature of his conclusions, I should assure you that he is a well respected mathematical physicist. He has had many critics, which is a measure of the significance of what he has to say. People may not agree on what significance Gödel’s theorem has, but at the very least there is no doubt that its significance is not restricted to mathematics.
Semantics is the part of linguistics that shows how abstract symbols are related to the world and that characterizes ‘meaning’. In what Lakoff (1987) calls Objectivist semantics, these two processes are one and the same. Recall the description of reason according to Objectivism:
Reason is the mechanical manipulation of abstract symbols which are meaningless in themselves, but can give meaning by virtue of their capacity to refer to things either in the actual world or in possible states of the world. (Lakoff, 1987, p. 7)
A symbol has meaning only because of the way it is related to the world. This relation is based on the idea of truth value. A sentence has meaning if it has a well defined truth value, which in turn is supposed to be based on the way the terms in it are related to the world. Putnam (1981) argues against this position by showing that a sentence might have two interpretations; that is, its terms might relate to the world in two different ways while the truth value, and hence meaning, of the sentence remain the same. If this is the case, then meaning cannot be based on the relation between symbols and the world.
Putnam’s argument is analogous to Gödel’s Theorem in at least two ways. It plays a similar role in limiting Rationalism, and the arguments are similar in structure. Gödel’s Theorem does not show that all of mathematics is inconsistent, or even that deduction cannot be used to safely determine truth in mathematics. What Gödel showed is that there are limits to the power of deduction to reveal truth. Similarly Putnam showed not that it is impossible for symbols to be related to the world in a way that gives them meaning, but instead that there are limits to this process of making relations to the world to give meaning. According to Gödel there is a class of true statements that cannot be accounted for by deduction. According to Putnam there is a class of meaningful statements that cannot be accounted for by reference to the world.
Recall that Gödel’s proof involved producing a sentence that asserted that it could not be proved. Putnam produced a sentence that could be given two interpretations. In both interpretations the sentence is true, so it has a truth value and is meaningful. But the two interpretations use very different references for the terms involved, making its meaning under the two interpretations different. In other words, Putnam produced a well-defined, meaningful sentence (according to the Objectivist idea of ‘meaning’), with a completely ambiguous meaning (according to common sense). Thus there must be something more to ascribing meaning to a sentence than the criteria employed by Objectivism.
Putnam illustrates his proof with an example, and I am not able to provide a clearer synopsis of his proof, so I will quote his example in full. Further details can be found either in Putnam’s work, or in Lakoff (1987).
Consider the sentence
(1) A cat is on a mat. (Here and in the sequel ‘is on’ is tenseless, i.e. it means ‘is, was, or will be on’.)
Under the standard interpretation this is true in those possible worlds in which there is at least one cat on at least one mat at some time, past, present, or future. Moreover, ‘cat’ refers to cats and ‘mat’ refers to mats. I shall show that sentence (1) can be reinterpreted so that in the actual world ‘cat’ refers to cherries and ‘mat’ refers to trees without effecting the truth-value of sentence (1) in any possible world. (‘Is on’ will keep its original interpretation.)
The idea is that sentence (1) will receive a new
interpretation in which what it will come to mean is:
(a) A cat* is on a mat*.
The definition of the property of being a cat* (respectively, a mat*) is given by cases, the three cases being:
(a) Some cat is on some mat, and some cherry is on some tree.
(b) Some cat is on some mat, and no cherry is on any tree.
(c) Neither of the foregoing.
Here is the definition of the two properties:
DEFINITION OF ‘CAT*’
x is a cat* if and only if case (a) holds and x is a cherry; or case (b) holds and x is a cat; or case (c) holds and x is a cherry.
DEFINITION OF ‘MAT*’
x is a mat* if and only if case (a) holds and x is a tree; or case (b) holds and x is a mat; or case (c) holds and x is a quark.
Now, in possible worlds falling under case (a), ‘A cat is on a mat’ is true, and ‘A cat* is on a mat*’ is also true (because a cherry is on a tree, and all cherries are cats* and all trees are mats* in worlds of this kind). Since in the actual world some cherry is on some tree, the actual world is a world of this kind, and in the actual world ‘cat*’ refers to cherries and ‘mat*’ refers to trees.
In possible worlds falling under case (b), ‘A cat is on a mat’ is true, and ‘A cat* is on a mat*’ is also true (because in worlds falling under case (b) ‘cat’ and ‘cat*’ are coextensive terms and so are ‘mat’ and ‘mat*’). (Note that although cats are cats* in some worlds — the ones falling under case (b) — they are not cats* in the actual world.)
In possible worlds falling under case (c), ‘A cat is on a mat’ is false and ‘A cat* is on an mat*’ is also false (because a cherry can’t be on a quark).
Summarizing, we see that in every possible world a cat is on a mat if and only if a cat* is on a mat*. Thus, reinterpreting the word ‘cat’ by assigning to it the intension we just assigned to ‘cat*’ and simultaneously reinterpreting the word ‘mat’ by assigning to it the intension we just assigned to ‘mat*’ would only have the effect of making ‘A cat is on a mat’ mean what ‘A cat* is on a mat*’ was defined to mean; and this would be perfectly compatible with the way truth-values are assigned in every possible world. (Putnam, 1981, pp. 33-35, emphasis in original)*
* The asterisk (*) in Putnam’s example is used to distinguish the word ‘cat’ from the word ‘cat*’, and does not indicate a reference to a footnote.
Examples of cases in which Rationalism seemed not to be applicable have occurred many times in the past three hundred years. Mere examples, however, are not enough to shake the Rationalist belief that proving is universally applicable. In some cases it was suggested that human weakness had introduced some bias into the course of the deduction, which made the problem people rather than Rationalism. In other cases, it was acknowledged that Rationalism could say nothing about a phenomenon, but that was taken to indicate that the phenomenon did not really exist. This approach has been taken by some Artificial Intelligence researchers to deal with the phenomenon of consciousness (Searle, 1992, pp. 6-7).
In order to show that Rationalism is not universally applicable, it is necessary to prove it, just as Gödel proved that Rationalism cannot establish all truths. In order to do so, we need to consider the relationship between logical implication and causation. The application of Rationalism to events in the world requires that physical causes must be almost as certain as logical implications. Otherwise Rationality has no predictive power, which is its whole point. Not even the most die hard Rationalist would assert that causes can be used to predict effects exactly, but there is an underlying assumption that causes predict effects approximately. The great successes of Rationality in the physical sciences bear this out. The orbits of the planets are almost exactly what Newton’s laws predicts, and Einstein’s theories improve the accuracy even further. In the Rationalist world all phenomena can be predicted, and things like the weather, for which prediction is currently very approximate, will be predicted with more and more accuracy as science progresses.
I bring up the planets and the weather as examples because they are two examples of dynamical systems, which are the topic of chaos and complexity theory. The simplest description of chaos theory is that it is the study of how chaos can emerge from order. Complexity theory considers how order can arise out of chaos. To take the planets as an example, physics provides precise laws which govern the motion of the planets. However, the interacting gravities of several bodies in motion result in a system for which questions like “Will the moon fall out of the sky one day?” cannot be answered. Not only is it impossible to predict the path of the moon exactly, it is not even possible to do so approximately over long periods of time. In weather systems this phenomenon is more obvious. In fact it was in attempts to simulate weather systems that the emergence of chaos from order was first observed (Gleick, 1987, p. 16). Approximate prediction is not possible because very tiny changes to initial conditions result in radical changes in final conditions. One might expect that knowing approximately what the current state of things is would be sufficient to predict approximately future events. This would allow refinements in our knowledge of the present to improve predictions of the future. In dynamical systems, however, the sensitivity to initial conditions is such that prediction is simply impossible.
Complexity theory examines dynamical systems in order to describe how order emerges from the chaos produced by the interactions within them. To return to the example of weather, one might expect that a system which changes radically in response to slight variations in initial conditions would be essentially random. However, when we examine satellite photographs, for instance, we see patterns in this chaos. Not all chaotic systems give rise to patterns, but some of the ones that do are very significant in our lives. They are those systems whose internal interactions are such that they are self-sustaining. This characteristic means that
within such systems the second law of thermodynamics, entropy, does not apply. Some examples are living things, species, herds, and societies.
The emergence of such orderly systems out of the chaos of dynamical systems implies that they are, for all practical purposes, unpredictable. This sharply limits the use of Rationalism in understanding them. At the same time the basis of dynamical systems is interactions, which are causal when taken individually. This means that Rationalism has a role to play in the study of such systems. But it is a role that involves some fundamental changes in the use of proving. It is a role that replaces proving to verify with proving to explain and explore. It is proving that the students who participated in my research might relate to. An example of such a transformation of Rationalism is Enactivism, which is the topic of the next chapter.
I would like to briefly mention some of the effects attempts to treat Rationalism as if it can be applied in all domains have had. I do this in order to make it plain that applying Rationalism to “all the things that can fall within human knowledge,” as Descartes suggested, is not only a logical error, but dangerous to individuals and societies.
Descartes spent a number of years as a gentleman soldier of fortune, and so it is perhaps appropriate to begin by describing the Rationalism of war. War has become increasingly “scientific,” especially in the past hundred years. When Gilbert and Sullivan wrote The Pirates of Penzance in 1880 mathematics was as important to the training of a “modern Major General” as statecraft or strategy. By the First World War, the planning of the generals was so rationally perfect that the Austrian declaration of war on Serbia led to the British declaration of war on Germany eight days later with all the inevitability of one of Euclid’s proofs (Taylor, 1974, pp. 25-28).
Modern technology provides further scope for Rationalist military planning. Consider the age old problem of ground troops becoming frightened or disturbed at the carnage of war. The U.S. military is developing remote control, “telerobotic,” tanks and planes, which can be operated from sufficient distances to eliminate the risk to the driver or pilot operating them (Rheingold, 1991, pp. 357-358), and simulators, like SIMNET which is capable of connecting 200 four person tank crews, each in their own simulated M-1 tank, into a virtual tank battle. The combination of these two technologies could solve the problem of troop morale. If the simulators used to train soldiers are equipped to operate the robot tanks and planes, then the combatants need never know that the images on their simulator screens are real people, and they will feel no remorse at their deaths. One more human element will have been eliminated from the planning of war on Rationalist principles.
Many people have realized that, “if there are ‘objective’ criteria on which to base a decision, then one cannot be blamed for being biased, and consequentially one cannot be criticized, demoted, fired, or sued” (Lakoff, 1987, p. 184). The association of mathematics with objectivity, which is an integral part of Rationalism, has lead to the use of mathematics as an “objective” way of determining which people are accepted to some desirable position. For example, in universities many academic programs include a mathematics course as part of their requirements. These “service” courses have high failure rates. In the case of the
introductory calculus course at one major Canadian university about a third of the students who register either fail or drop the course. This limits the number of students enrolled in programs with mathematics requirements. Mathematics serves as a social filter, imbued with the appearance of rationality and objectivity.
Davis and Hersh (1986) describe this situation in the case of the calculus requirement of the typical business school:
There seems to be no necessity to make math a requirement. There is a practical necessity to make a selection among the students who want to go to the business school. The business professors decide to use math for that purpose. Is that OK? How should we (math teachers) feel about it? First of all, there is nothing inevitable about the choice of math as a filter. Some other filters that could be used, or that have been used are: family connections, political connections, income, ability in sports, personal charm, brutality and aggressiveness, trickiness and sneakiness, devotion to public welfare, etc. The first five have been relevant criteria in admission of students to U.S. institutions of higher learning; the last three are suggested, somewhat in jest, for particular relevance to a school of business. (pp. 101-102)
Davis and Hersh go on to consider the effects on mathematicians that result from spending a great deal of effort teaching students who have no desire to learn mathematics, and who see it mainly as an impediment to their success in some unrelated academic program. In this case, it is not only the students who are denied access to their chosen career on the basis of a Rationalist criteria who suffer. The mathematicians who participate in this process also find it demoralizing and try to avoid teaching such courses.
Gould (1981) described the unfortunate effects of Rationalist attempts to quantify mental attributes have had on individuals and groups. The connection between mathematics and Rationalism leads to situations such as Gould described, in which the use of statistical methods in psychometrics gave the field an air of objectivity. This Rationalist claim has been the basis for the acceptance of psychometrics as the fair way to determine employment, immigration, and scholastic opportunity since the development of statistical techniques in the late nineteenth century.
Gould’s most disturbing example describes the effects of the IQ testing of 1 750 000 U.S. Army recruits during the First World War on social policy. One important “discovery” that came out of the interpretation of this data was the mental inferiority of immigrants, especially immigrants from Mediterranean and Slavic backgrounds, including Jews. This led to the U.S. Immigration Restriction Act of 1924, which sharply limited immigration in general, especially from southern and eastern Europe. This Act prevented the immigration to the U.S. of millions of people, including Jews attempting to leave Hitler’s Germany. As Gould (1981) says:
We know what happened to many who wished to leave but had nowhere to go. The paths to destruction are often indirect, but ideas can be agents as sure as guns and bombs. (p. 233)
That particular misapplication of Rationality is still a serious problem is indicated by the periodic appearance of books that make use of data from psychometric testing to argue for the inferiority of various groups within our society, or globally (e.g., Jensen, 1979; Herrnstein & Murray, 1994).
Our model of thinking defines what we can think. Not surprisingly, Rationalism most strongly determines how scientists and mathematicians can think. This can limit the possible explanations scientist can provide for phenomena, perhaps resulting in false conclusions. This is illustrated by the existence of something known in psychology as the ‘base-rate fallacy’ which is usually illustrated by studies done by Kahneman & Tversky (see, for example, Bruner 1986, p. 89 or Holland, Holyoak, Nisbett & Thagard, 1986, pp. 217-222). In a typical study subjects are shown psychological profiles drawn from a sample of 70 engineers and 30 lawyers. They are then asked to guess whether the profile is that of an engineer or a lawyer. When the subjects ignore the information that 70% of the profiles are of engineers, even when the individuating information is completely useless for making a decision, the researchers label this irrational behavior the “base-rate fallacy.” The subjects are not thought to be thinking differently, but incorrectly. The interpretation given by the scientists involved is an illustration of the powerful hold Rationalism has on the paths that their thoughts can take and cannot take. As Wittgenstein points out: “So much is clear: when someone says: ‘If you follow the rule, it must be like this’, he has not any clear concept of what experience would correspond to the opposite” (1956, §III-29, p. 121, emphasis in original).
The previous sections could be very discouraging if we believe that thinking means thinking deductively. And we would not be alone in believing that:
In this century reason has been understood by many philosophers, psychologists, and others as roughly fitting the model of formal deductive logic:
Reason is the mechanical manipulation of abstract symbols which are meaningless in themselves, but can give meaning by virtue of their capacity to refer to things either in the actual world or in possible states of the world. (Lakoff, 1987, p. 7)
Any mode of thought defines how one sees the world and acts in the world, which in turn defines what one is in the world. Rationalism is no different. It acts as a filter and a lens for perception, eliminating some objects and relationships from view, distorting others, and bringing some into clear focus. It “enable[s] us to keep an enormous amount in mind while paying attention to a minimum of detail” (Bruner, 1986, p. 48) in much the same way that a wide angle lens provides an enormous view, but distorts details.
As Rationalism has developed, it has become more and more difficult to see the world in other ways. This is a general feature of “ideas of mind.”
Perhaps once a culture has become gripped by an idea of mind, its uses, and their consequences, it is impossible to shed the idea, even when one has lost faith in it.
For the impact of ideas about mind does not stem from their truth, but seemingly from the power they exert as possibilities embodied in the practices of culture. (Bruner, 1986, p. 138)
The adoption of Rationalism gives it power to affect the way that society develops. As Rationalism is more generally considered to be the model of thinking, we construct our society in such a way that it must be our model of thinking.
Isn’t it like this: so long as one thinks it can’t be otherwise one draws logical conclusions. This presumably means: so long as such-and-such is not brought into question at all.
The steps which are not brought into question are logical inferences. But the reason why they are not brought into question is not that they ‘certainly correspond with the truth’—or something of the sort,—no, it is just this that is called ‘thinking’, ‘speaking’, ‘inferring’, ‘arguing’. There is not any question at all here of some correspondence between what is said and reality; rather is logic antecedent to any such correspondence; in the same sense, that is, as that in which the establishment of a method of measurement is antecedent to the correctness or incorrectness of a statement of length. (Wittgenstein, 1956, §I-155, p. 45, emphasis in original)
If we accept that all thinking is deductive, and combine that idea with the knowledge that systems of deductive logic are essentially incomplete, we might be tempted to believe that there are things we cannot think about at all. Rather than do that I would take up Bruner’s (1986) suggestion that thinking can occur in a number of modes.
Bruner identified two main modes of thinking in our society, paradigmatic (which is Rationalism), and narrative.
The ‘reality’ of most of us is constituted roughly into two spheres: that of nature and that of human affairs, the former more likely to be structured in the paradigmatic mode of logic and science, the latter in the mode of story and narrative. The latter is centered around the drama of human intentions and their vicissitudes; the first around the equally compelling, equally natural idea of causation. (p. 88)
It should be noted that these modes of thought are complementary. While it seems that some individuals have developed one of these modes of thought to a higher degree than the other, all humans possess the ability to think in these ways. In this modes of thought seem to correspond to what Lakoff (1987) calls “conceptual schemes”.
The paradigmatic mode of thought is closely allied to Rationalism.
[It] attempts to fulfill the ideal of a formal, mathematical system of description and explanation. It employs categorization or conceptualization and the operations by which categories are established, instantiated, idealized, and related one to the other to form a system. Its armamentarium of connectives includes on the formal side such ideas as conjunction and disjunction, hyperonymy and hyponymy, strict implication, and the devices by which general
propositions are extracted from statements in their particular contexts. At a gross level, the logico-scientific mode ... deals in general causes, and in their establishment, and makes use of procedures to assure verifiable reference and to test for empirical truth. Its language is regulated by requirements of consistency and noncontradiction. Its domain is defined not only by observables to which its basic statements relate, but also by the set of possible worlds that can be logically generated and tested against observables—that is, it is driven by principled hypotheses. (Bruner, 1986, pp. 12-13)
The narrative mode, on the other hand, “deals in human or human-like intention and action and the vicissitudes and consequences that mark their course. It strives to put its timeless miracles into the particulars of experience, and to locate the experience in time and place” (Bruner, 1986, p. 13).
Bruner briefly touches on one other mode of thinking, faith, and notes its power in the Middle Ages. This was: “an unmediated knowing of eternal truths revealed by God (or ... by virtue of man’s endowment with an intuition of pure knowledge). It was revelation” (p. 108). Barrow (1992) goes into more detail on the subject of faith, or theological thinking:
Abstract ideas and concrete realities were once interwoven and interdependent to such an extent that no significant wedge could be driven between them. For the ancients and the medievals symbolic meanings of things assumed a natural significance that rests upon associations of ideas that we no longer possess. ... In this way numbers came to possess one aspect that was within the reach of human computation, whilst always possessing others which could be fathomed only by divine revelation. ... Every user of numbers adds their own subjective ingredient to the question of their true meaning and its link to the meanings of other aspects of reality. (p. 106-107, emphasis in original)
Barrow’s description of faith in the Middle Ages resembles what Gödel called “the theological worldview” which is:
the idea, that the world and everything in it has meaning and reason, and in particular a good and indubitable meaning. It follows immediately that our worldly existence, since it is in itself at most a very dubious meaning, can only be the means to the end of another existence. The idea that everything in the world has a meaning is an exact analogue of the principle that everything has a cause, on which rests all of science. (quoted in Barrow, 1992, p. 124)
Note that just as paradigmatic thought is based on causality, and narrative thought is based on intention, theological thought has its basis, which Gödel called meaning.
Narrative, paradigmatic and theological thinking do not necessarily exhaust the possible modes of thinking. Davis’ (1993) statement “mathematics also displaced religion, history, and narrative to become the primary model of reason for the modern era” (p. 3) suggests that history could also be seen as a mode of thinking. In an historical mode of thought, truth would be derived from the truths
of the past, from traditions and customs. The basis of the historical mode thought could be called ‘conservation of truth over time’. If scientists best typify paradigmatic thinking (as Bruner asserts, 1986, p. 15), then perhaps members of conservative political parties and movements best typify the historical mode of thinking.
The existence of different modes of thought, each suited to thinking about the world in different ways, provides one important answer to the question “How can we know what Rationalism cannot tell us?” We can know in many ways. At the same time it is important to ask, “If Rationalism cannot tell us everything, what can it tell us?” The answer to this question is the topic of the next chapter.