LA PROPUESTA PEDAGÓGICA EN EL DISEÑO DE LOS MATERIALES

According to conventional wisdom, theorists use a very simple strat- egy for coming up with their theories: they make guesses and then they test them.18

This approach is summarized in figure I.8 (‘The Hypothetico-Deductive Method’).19

In fact we can subdivide ‘The Hypothetico-Deductive Method’ (or H-D) into four basic steps:

First, observe some distinct phenomenon in a well-defined test sample (figure I.8a).

Second, guess some laws that seem to explain these observations (figure I.8b).

Third, deduce some predictable consequences that are implied if these laws are correct (figure I.8c).

Fourth, see if these predictions are confirmed by further obser- vations (figure I.8d).

If the predictions are indeed confirmed, then theorists carry on using their laws; if they aren’t, then they must either modify them, or they must replace them with new laws and start the procedure all over again. This final step is crucial to the entire process; ideally, it implies not only that the predictions are testable, but also that they are testable by someone else and under different conditions.

Now, if we want to make a theory to explain the behavior of tonal voice leading and harmony, then we might adopt the follow- ing plan. We might begin by using certain familiar concepts to study a specific corpus of pieces that a given community regards as

quintessentially tonal. Chances are we would pick pieces by such composers as Handel, J. S. Bach, Scarlatti, C. P. E. Bach, Haydn, Mozart, Beethoven, Schubert, Schumann, Mendelssohn, Chopin, and Brahms.20_{Next, we might develop general laws that cover the}

behavior of these concepts. For example, we might generalize about how lines and chords behave in specific contexts. We could then predict how the lines and chords might behave in other contexts, preferably those that are slightly different from the ones used to make the original theory. If our predictions are confirmed, then we will keep on using our theory to explain the behavior of similar works from the same corpus; if, however, they are disconfirmed, then we must either modify the theory or find an alternative one that does work.

So much for conventional wisdom; we find it perpetuated in many introductions to science. But when we look at how theorists actually work, we soon find that the process of building and testing theories is a good deal more complex than figure I.8 suggests. To begin with, it is very unlikely that any music theorists would actually build a new theory of tonality from scratch. Instead, they are more likely to start by taking some preexisting model and seeing if indi- vidual laws stand up to close scrutiny. Once they encounter a prob- lem, then they will propose new covering laws. But disconfirming existing laws and confirming new laws is no easy task; on the con- trary, these activities are riddled with problems and inconsistencies.21

These difficulties stem from the fact that, as David Hume famously remarked, “all inferences from experience, therefore, are effects of

a. Observe phenomenon in some well-defined test sample.

b. Guess laws to explain these observations.

c. Deduce some consequences that are implied if the laws are correct.

d. See if these predictions are confirmed by further observations. If they are, then keep using the new laws, if they aren’t, then modify them or replace with some new laws and start procedure over.

custom, not of [logical] reasoning.”22

Hume’s claim does not mean that inferences from experience are necessarily untrue; rather it sug- gests that they always fall short of certainty. As a result, our theories will always be fallible, though we may not always know where they will fail.

Besides recognizing the general problems noted by Hume, philosophers have discussed several other difficulties. One of the most famous is ‘The Raven Paradox.’23

This paradox arises because law statements of the form “for all x, if x is F, then x is G” are log- ically equivalent to those of the form “for all x, if x is not G, then x is not F.” If x stands for piece of music, F stands for Beethoven and G stands for tonal, then the first law-statement “all pieces of music by Beethoven are tonal” is equivalent to the observation that a particular non-tonal piece, say Babbitt’s Philomel, is indeed not by Beethoven. What is paradoxical is that an atonal piece by Babbitt should count as evidence that confirms the generalization that all pieces by Beethoven are tonal; after all, Babbitt’s music appears to have little in common with Beethoven’s and it hardly seems relevant to any claims about whether the latter is tonal or not.

Relevance also features prominently in another paradox known as ‘The Grue Paradox.’ This paradox was first discussed by Nelson Goodman.24

According to him, the issue of whether a generalization is supported by its instances depends on the nature of the properties that appear in that generalization. Paraphrasing Goodman, let us imagine a new property, ‘gronality’ which we define as follows: a piece is ‘gronal’ if it is classified as tonal before the year 2010 and atonal after that point. Now consider the following generalizations: 1) all works by Burt Bacharach are tonal and 2) all pieces by Burt Bacharach are ‘gronal.’ All works examined before the year 2010 will support not only the first generalization, but also the second one. This result is problematic because we want to use our general- izations to predict what will happen at some later date; as it stands, we have no basis for knowing whether the piece will be tonal or ‘gronal.’ To resolve this paradox, Goodman proposed that the law-like status of a generalization is a matter of entrenchment and projectibility. According to him, a predicate is entrenched if it is true as a matter of historical fact and has been used to formulate

true predictions.25

He suggests that this property is the only one that allows us to project what will happen in the future. Tonality is just such a predicate; it is a trait that we naturally project from past observation to future expectation. ‘Gronality’ is not, however, because we have no reason to suppose that Burt Bacharach wrote music that can been classified as tonal at one point in time and atonal at some later date.26

‘The Grue Paradox’ leads to a more general problem in confir- mation; even if we agree on the same body of evidence, there is no reason to suppose that this data can be explained by only one theory; as we have seen, we can always invent new predicates, such as ‘grue-ness’ or ‘gronality,’ that capture some aspect of the piece. This means that, in principle at least, the evidence always underde- termines theories; there are always a variety of theories that will accommodate any given set of data. Pierre Duhem and W. V. Quine have gone even further to claim that, taken on its own, a particular piece of experimental evidence is seldom used to falsify an entire theory, because each element of the theory is somehow related to another element in the theory. As Quine puts it, “our statements about the external world face the tribunal of sense experience not individually but only as a corporate body.”27

In other words, “any seemingly disconfirming observational evidence can always be accommodated to any theory.”28

This claim is usually known as ‘The Duhem-Quine Thesis.’

Although extremely controversial, ‘The Duhem-Quine Thesis’ is significant because it threatens to undermine the most famous alternative to H-D. Given the many paradoxes of confirmation, Karl Popper and others have suggested that, instead of defending their theories by finding more and more supporting evidence, scientists should actually spend their time trying to show that some hypotheses are false.29

In this sense, the guiding principle of test- ability is not confirmation but falsification. The rationale behind Popper’s thinking is simple enough and is apparent from the argu- ments given in figure I.9 (The logic of falsification). According to Popper, H-D seems to follow the plan given in figure I.9a. Let us assume that, if a particular explanation E is valid, then it will make a given prediction P. When researchers test this prediction and find that it is indeed accurate, they regard this as confirmation of their

explanation. But, according to Popper, this line of reasoning is invalid. Since explanation E was used to come up with prediction P, prediction P cannot then be used to confirm explanation E. That would commit the fallacy of affirming the antecedent (see figure I.9b). To avoid this problem, Popper insists that H-D should be used, not to confirm, but to falsify a theory. This means that, given explanation E and prediction P, knowing that P is false allows us to deduce that E is false as well (see figure I.9c). Such an argument follows the principle of modus tollens given in figure I.9d.

Popper used the notion that H-D can be used to falsify an explanation to reach two important conclusions. First, he proposed that even our best knowledge is fallible or conjectural. To quote him, “we cannot reach certainty . . . all we can do is to criticize [our theories], and to test them, as severely as our ingenuity per- mits.”30

Second, Popper decided that “the criterion of the scientific status of a theory is its falsifiability, or refutability, or testability.”31

He even claimed that falsifiability serves as a criterion for making demarcations between science and non-science; whereas scientific theories must always have discrete boundaries and can be falsified, non-science need not have such boundaries and cannot be falsified. Critics, however, have responded that strict falsification is hard to uphold in light of ‘The Duhem-Quine Thesis.’ If, as Duhem and Quine insist, “any seemingly disconfirming observational evidence can always be accommodated to any theory,” then it is hard to see how we can decisively falsify any theory. Each time we come up with a counter example, we can simply adjust our theory to make it

a. If Explanation E is valid, b. If X→ Y then Prediction P is true.

Prediction P is true Y

∴ Explanation E is valid ∴ X (invalid) c. If Explanation E is valid, d. modus tollens

then Prediction P is true. If X→ Y Prediction P is false ⫺Y

∴ Explanation E is false ∴ ⫺X (valid)

fit. If we cannot use specific observations to falsify a given theory, then we cannot pick one theory over another on purely evidential grounds. Popper’s claim that falsifiability provides us with a defini- tive means of discriminating science from non-science seems, therefore, too strong.32

As it happens, Popper’s views have also been challenged by recent findings in cognitive science. Research by Tweney, Doherty, Mynatt, and others has suggested that scientists do not usually set out simply to falsify existing theories; on the contrary, they nor- mally start out by seeking confirmatory data; only when this data has been obtained does it make sense to engage in rigorous falsifi- cation.33

Thus, while it might be true that successful theories are initially conjectural, the accumulation of supporting evidence will eventually move them beyond that status.34

Most people do, in fact, believe that theories become more strongly confirmed the more supporting evidence has been amassed. This point suggests that our understanding of what makes a successful music theory must eventually take account of the ways in which music theorists actu- ally work, rather than simply relying on their logical or empirical content.

All in all, just as ‘The Covering Law Model’ provides an idealized picture of explanation, so ‘The Hypothetico-Deductive Method’ presents an idealized account of how music theorists confirm or refute a particular theory. While the latter conveys many aspects of how music theorists work, the process of building and testing theories involves a far more complex interplay between confirmation and falsification. As we have seen, this process is always open ended; music theorists do not begin with a blank slate, they do not have foolproof methods, and they do not reach definitive solutions. Instead, they plunge in medias res. They start working within the context of an existing music theory, even if they know some portions of that theory are surely wrong. They then try to overcome certain specific problems, using the rest of the theory to support their work. To borrow an image from Neu- rath and Quine, this situation is like that facing sailors at sea on a leaking boat.35

Unable to rebuild their vessel from the keel up in a dry dock, the crew is forced to fix the leaks while adrift on the open water. As they work on leaks in one area of the boat, the

sailors rely on the remaining timbers to keep the craft afloat. But as one leak is patched so another appears; bit-by-bit the boat becomes transformed into something new. In fixing the leaks, music theorists typically try to balance what Quine has described as “the drive for evidence and the drive for system.”36 _According

to him, the former demands that “theoretical terms should be sub- ject to observable criteria, the more the better, the more directly the better, other things being equal” while the latter insists that these terms “should lend themselves to systematic laws, the sim- pler the better, other things being equal.” Quine adds, “If either of these drives were unchecked by the other, it would issue in some- thing unworthy of the name scientific theory: in the one case a mere record of observations, and on the other a myth without foundation.”37