CONTENTS

1    THE PROBLEM

2    A CONJECTURAL SOLUTION

3    TESTING THE CONJECTURE

4     CONCLUSIONS - A THEORY OF THEORY

3   TESTING THE CONJECTURE

The results of science have been produced by the joint labours of experimentalists and theoreticians. From a realist-empiricist perspective it is easy to see what input is provided by experiment. This essay develops a conjecture about what additional input is required from theory, based on the notion of articulation, itself derived by making more precise the ideas of Fregean analysis. In this section we look at some brief sketches of advances in science to test this conjecture, to see if the picture of articulation developed above fits the historical processes.

Some of this job has already been started, in the subsection immediately above looking at failures of conceptual frameworks and scientific revolutions. In the course of this examination a distinction began to emerge between concept explication and the change of concepts under the impact of new facts. In the former case the changes can be described as sense-driven, that is, driven by a discontent of theoreticians with the clarity of the current modes of expression. By contrast the second sort of changes are reference-driven. (Of course, many actual changes will be characterised by a mixture of and interaction between these two types of driver.)

This distinction can be further refined by considering whether the senses are already grasped or are newly discovered, and whether the referents are already known or newly discovered. The cases can be presented on a two-by-two matrix.

We now look for advances in science under each of these headings.

3.1    Refining Existing Senses

As a paradigm of this case, let us return to one of our original mysteries of understanding: how to deal with instantaneous velocities, and changes in velocities. Here the basic phenomena of motion were directly accessible, and a reasonable vocabulary was already in place to describe them. However the new science of dynamics required a greater precision of expression, which was supplied by the discovery of the differential calculus by Newton and Leibniz. This new, more powerful vocabulary had great utility, but it could be used only at the price of ignoring serious logical flaws in the foundations of the subject. The advent of the differential calculus made profound problems almost trivial, effectively overnight, but it put in its place another set of profound problems.

It would be another two hundred years before mathematicians developed the subject called mathematical analysis to the point where differentiation could be articulated precisely, in terms of a deeper concept, the limit. At the end of this long conceptual struggle, we arrive at the following fully articulated definition. If  q is the function mapping time instants t onto the successive positions q(t) of an object, then at each instant the instantaneous velocity of the object is given by:

v(t)  =  ü y "e [e > 0  Þ  $d [d > 0  Ù  "x [ | x | < d  Þ  | (q(t-x) – q(t))/x  -  y | < e ]]]

This “epsilon-delta” form of the definition is displayed here to show two things. The first is that it involves multiple quantification, and so could not be properly articulated until Frege developed his Begriffsschrift. In this example the end-point is very clearly a Fregean analysis. The second point is that the full articulation exposes just what conceptual complexities lurked inside the superficially harmless notion of instantaneous velocities. The velocity is formed by linking the definite description operator deep into the heart of complex predicate involving three layers of quantification.

None of these heroic struggles with the notion of instantaneous velocity, and more generally with the notion of the derivative of a function, required or was influenced by additional empirical input. They were entirely sense-driven. From the anachronistic perspective of our current division of subjects, we would say that they took place within pure mathematics. This fits in with the picture of pure mathematics as being a systematic study of sense.

From the point of view of theoretical physics, the level of articulation displayed above can be regarded as the end of the story. It is probably already going further than is needed. But for pure mathematics it was only the start of new adventures, uncovering deeper concepts and deeper problems. Beyond the idea of limits on the real line lies the question: what structure must a set possess if limits are to be defined on it? This takes us into topology, and links up with a whole new conceptual vocabulary. It also uncovers a deeper set of problems to do with axiomatic set theory, such as the status of the axiom of choice.

3.2     Case II:   New Senses for Old

In this case the changes are still sense-driven, but instead of a progressive evolutionary refinement of a sense we have the replacement of old senses with new ones (the distinction already explored in Subsection 2.3.7 above). The example given there was the replacement of the Copernican view of the solar system with the Newtonian view. The same planets, and the same apparent motions, were being spoken of, but in a completely different way. (This is of course an oversimplification; the role of new empirical facts about the planets in bringing about the revolution is discussed in Subsection 3.4 below.) The explanation of the motions in terms of dynamical laws became embedded in our understanding of what the planets are: heavenly bodies were replaced by physical objects.

As another example, let us consider the revolution in chemistry that took place either side of the year 1800, focussing to begin with on one aspect, namely the overthrow of the phlogiston theory of fire and its replacement by the oxygen theory. The phlogiston theory is more natural in terms of the direct appearances. Wood sits in the hearth. It is set alight and something flows out of it, as evidenced by the flames. In the end what is left behind is the ash. In the oxygen theory, something, namely oxygen flows in and combines with the fuel as the essence of the process of fire.

Seen at its simplest, oxygen looks like nothing more than anti-phlogiston. The shift could be seen merely as a change in sign convention. This however ignores the bigger picture. This was part of a bigger conceptual shift, as, under the pressure of a new body of empirical evidence, the old sense of the word “element” was replaced by the modern one. The new chemical knowledge, particularly of the properties of different gases, could not be fitted into the alchemical framework of the four elements. It made increasingly less sense to treat the new gases as different forms of air. Chlorine for example, had quite different properties to air, and could not be transformed into it. A new concept of chemical substance had to be developed, based upon sameness of chemical reactions. Within this, a special class of chemical substances, the chemical elements, began to be perceived, based upon the inability to decompose these substances, on their own, into other substances. The elements could have different physical forms, such as the allotropes of sulphur, but they were all chemically sulphur. The sulphur dioxide formed by burning the different allotropes in oxygen were indistinguishable.

Within this new framework, oxygen was not simply a hypothetical essence introduced to explain fire, something which is like phlogiston but flows in the opposite direction. Instead, it was an element, in the new sense; something that could be bottled, examined and characterised independently of its property of being an essential ingredient of fire. The oxygen theory of fire then becomes a contingent conjecture to be tested, and not the definition of oxygen. As part of the same conceptual shift, fire was no longer a substance, one of the four elements, but a type of chemical process, namely exothermic oxidation. Indeed there was no longer a sense which was worth separating out and expressing with the word “fire” in an exact, scientific manner. It remains in fuzzy, everyday use, but is not a part of rigorous chemistry.

As a third example, consider the Darwinian revolution in biology. This started life with the conjecture within biology, namely that species are not immutable and can evolve into new ones. Darwin’s first job was the collection of evidence to support this conjecture. The next step was the emergence of a conjecture about the mechanism of speciation, and this in turn required more evidence in support. In the following century a whole new body of information, about genetics, far from demolishing the Darwinian framework, greatly strengthened it. This reference-driven evolution of neo-Darwinism was accompanied by a deeper conceptual shift, which changed the way in which we look at life. Darwinism became constitutive of the sense of the word “life”, defined now largely as that which is driven by the Darwinian dynamic, seen as a necessary feature of the combination of the transmission of information and the competition for resources.

This shift moved Darwinism from a conjecture within biology to the driving principle for the whole of biology. It turned biology from a descriptive, historical science (“Natural History”) into a modal science with a dynamics of its own. The ideological enemies, who want to rip the pages on Darwinism out of the biology textbooks, have completely failed to notice this deep paradigm shift.

3.3     Case III:    New Reference for Existing Senses

At first it may seem questionable that case III is possible. Unless some phenomena were known initially, how could we have developed, either in an intuitive or articulated form, the senses which refer to them? What has made case III possible is the development of modern pure mathematics, seen as a systematic exploration and development of senses, freed from any constraints of reference. In this picture, pure mathematicians are seen as going out hunting new senses without any drive from reference whatsoever.

The classic, and maybe unique, example of sense getting ahead of reference in this way in theoretical physics is provided by the work of Einstein. By contrast, in his work on the photoelectric effect and on the heat capacity of solids, Einstein worked in the conventional direction, taking unexplained phenomena and providing for them a theory. With special relativity, and even more so, with general relativity, he went ahead of experiment, being guided instead by a form of theoretical aesthetic. In this he found that the senses he needed had already been put in place by pure mathematicians. The results of his aesthetic were subsequently validated by experiment. In his third attempt, with unified field theory, his luck ran out. The world stubbornly refused to conform to Einstein’s aesthetic vision.

3.4    Case IV:   Empirical Revolution

The paradigm of case IV, were new facts make necessary the development of new conceptual structures, is the advent of quantum mechanics. This has been discussed already above. As an additional example let us return to the Copernican-Newtonian revolution, and examine the extent to which this was driven by new facts. In the earlier discussions the sense-driven aspects were emphasised: how a full understanding of the logical power of dynamical laws, especially developed by Newton, led to the shift from heavenly geometry to universal dynamics. This is however only part of the story. The new facts which also helped drive the shift can be represented by those provided by Tycho and by Galileo.

The new, more accurate measurements of planetary motions provided by Tycho drove Kepler to reform radically the heavenly geometry left behind by Copernicus. Out when the circles, to be replaced by ellipses. In addition, Kepler introduced simple but non-trivial laws about the speed at which the planets moved around the elliptical paths, which in turn paved the way for a dynamical explanation.

Tycho’s measurements were however the old type of facts obtained more accurately. Galileo’s telescopic discoveries, reported in Siderius Nuncius, were of a new sort of fact. These went beyond points of light and their apparent motion. The heavenly bodies were revealed to have internal features similar to earthly bodies. The markings on the moon were found to be elevations and depressions. The sun was found to have spots. The “fixed stars” were still fixed in terms of motion, but their number was found to be anything but fixed. The number one could see appeared to depend only on the power of one’s telescope. This torrent of discoveries blew apart the old world-view based on the concepts “x is heavenly” and “x is earthly”, making it possible instead to think in terms of universal law, in which the causes of motion of an apple in a garden in Woolsthorpe could be thought of as “extending to the orb of the moon”.

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