MA Thesis of Timothy Hilgenberg
(c)
MM Timothy Hilgenberg
Views on Reduction
5. Ernst Nagel
For Nagel reduction is a form of explanation, a relation between the complex and the elements that is used as an aid to understanding. He sees reduction as a certain relation between those elements of a given subject and those explained by them, albeit realising that there are questions as to what the logical structure of such a reduction would be.
In "Issues in the Logic of Reductive Explanations(7)", Nagel says:
…[For] strictly speaking, it is not phenomena which are deduced from other phenomena, but rather statements about phenomena from other statements. This is obvious if we remind ourselves that a given phenomenon can be subsumed under a variety of distinct descriptions, and that phenomena make no assertions or claims(8).
It is the systematicity of the connection between two sets of statements about phenomena that is at issue here, reduction is to somehow link these different sets.
He distinguishes two major kinds of reduction: Homogeneous and Inhomogeneous(9). Similar to deduction, in homogeneous reduction all descriptive and fundamental elements eventually identified are present in, or directly derivable from, the premises. That is, it is a "closed system" reduction. On the other hand, with inhomogeneous reduction there is at least one descriptive term in the conclusion [Nagel’s words] which is not found in the premises or derivable from them.
For Nagel it seems homogeneous reduction is a form of deduction where a succeeding theory can explain an earlier one and where there are discrepancies or shortcomings, explain why they are there. For an example he uses Kepler’s first and second laws(10). Within a two body system, they agree completely with Newton’s theory of motion. However he points out that nearly all of today’s physical theories that are said to be mathematically deduced are so only with a number of approximations and idealisations, such as the idealisation of a body’s mass being a point mass, that is it is concentrated in one single one point or restricting the scope of an observation for which those rules are applicable(11).
For him these approximations and limitations do not mean however that what is actually happening, as some other philosophers claim, is that an old theory is being replaced by a radically new one. Even where an issue arises from the introduction of new concepts, Nagel defends the deductive approach over a replacement view. He cites the example of Kepler’s third law, which says that that the periods of two planets squared are to each other as the cubes of their mean distances from the sun. In fact this ratio varies with their masses, a concept only introduced by Newton. However as the masses of the planets relative to that of the sun are very small, this is a good approximation. He argues, that although they cannot be equated, they are not radically different. In fact he points out that Newton has added to the theory by identifying a further causal factor, one unknown to Kepler.
While homogeneous reduction has thrown up some discord, it is, says Nagel, the inhomogeneous kind that causes the most controversy. One example where he says a fairly complete inhomogeneous reduction has been possible is the explanation of thermal laws by the kinetic theory of matter. One where it is far from complete is the reduction of biological laws to those of physicochemical theory. The problem is that in addition to any deductive process there is a need for concept mapping, i.e. analogues have to be found for concepts in the reduced theory to those in the reducing theory.
Nagel distinguishes three separate approaches to theory reduction: instrumentalist, correspondence and replacement. These are based on views others have put forward. In reviewing them, he points out shortcomings with all three. The problem with the instrumentalist position comes from limiting statements to comparing the ranges of observable phenomena over which the reducing theory and the reduced theory are applicable. The shortcoming of this view, in Nagel’s opinion, is that it depends on the plausibility of interpreting statements in science as rules of inference. While instrumentalists neither accept nor deny the existence of things unobservable, it makes it difficult to explicate macro-processes through the action of unobservable micro-processes postulated by any theory. The approach suffers further from the fact that is has nothing to say about how theoretical terms from laws may connect with matters of observation. This also leaves the question of how the concepts of the reducing and the reduced theories are connected essentially unanswered.
With the correspondence approach, connections between the reducing and the reduced theory are made using bridge laws or rules of correspondence that match distinctive terms in the original theory to terms or combinations of terms in the reduced theory. However there too are problems. Several assumptions have been made. One such is that the terms used are "theory-impregnated", they are not neutral terms for sense data, but are already defined on the basis of some theoretical commitments. A further assumption is that notwithstanding idealisations and approximations as would be made in homogeneous reduction, the same schemata are taken to apply within inhomogeneous cases. The most problematic aspect of the correspondence approach is the abuse of bridge laws. Nagel distinguishes two kinds of connections, one which essentially lays down the conditions of equivalence, such as for example that gas temperature can be equated with the mean kinetic energy of its molecules in the kinetic theory of gases. The other shows the identity of entities described by two logically non-equivalent expressions, Nagel uses the example of optics being reduced to electromagnetic theory by the fact that light is an electromagnetic wave within a certain range.
Both, the instrumental and the correspondence view, have been criticised and Nagel takes up these objections introducing a third approach which he terms the "replacement view." This view put forward essentially by Paul Feyerabend, is not one that Nagel agrees with, favouring as he does the correspondence approach. He agrees with Feyerabend’s main assumption, that the meaning of terms used is bound up within the theoretical context, at least to a certain degree. For Nagel, Feyerabend’s main argument is that when a theory changes all its terms change too, the primitives as well as those defined within the theory. As a result, it is not possible in Feyerabend’s view to connect the old theory with the new one. The terms are as Kuhn(12) put it, incommensurable. For Nagel, however, the conclusion does not follow. The main problem is that according to Nagel’s take on Feyerabend, there are two theories T and T* which tested over a common range meet the observations for each theory well, but because they share no common terms, it is impossible to give any interpretation of how statements from T connect to T*. The threat of relativism is circumnavigated for Feyerabend, by introducing what he calls the "pragmatic theory of observation" however, Nagel questions this by asking why only such observation statements would remain meaning neutral. Another difficulty for Feyerabend, so Nagel, is that theories are not always quite the monolithic structure, Feyerabend takes them to be. The terms’ dependence on the theoretical framework varies, some may be more profoundly connected than others.
Showing that replacement fails to deal fully with correspondence Nagel says:
… [This] difficulty of the replacement view in explaining how the "wider" theory, which allegedly replaces a "narrower" one, may nevertheless have a domain of common application, does not arise in the correspondence account of reduction(13).
Something that is much more akin to what is found in scientific investigations. (top)
6. Paul Feyerabend
Eleven years before Ernest Nagel wrote his essay on Reductionism, Paul Feyerabend set the discussion going with his paper called "How to Be a Good Empiricist - A Plea for Tolerance in Matters Epistemological(14)" in which he deals with the issue of theory reductionism. Right at the start of the paper he already points out one potential problem that may belie the empirical approach, namely the assumption that only a well founded observational procedure can progress knowledge, when he says:
… [This] predilection for empiricism is due to the assumption that only a thoroughly observational procedure can exclude fanciful speculation and empty metaphysics as well as to the hope that an empiristic [his word ed.] attitude is most liable to prevent stagnation and to further the progress of knowledge(15).
He sets out two conditions for theory reduction, one he calls the "consistency condition" the other the "condition of meaning invariance". The consistency condition is met only when the new theory either contains older and established theories or is at least consistent with them across the domain in which they apply. The condition of meaning invariance requires any new theory to be explanatory in such away that its terms are not dependent on the theory. These two conditions, Feyerabend holds, are restrictive and in their nature will have an influence on the "growth of knowledge".
His main criticism in this paper is aimed not so much at reductionism per se, but at the way it is employed. He lists a number of examples, one Quantum Mechanics à la Copenhagen Interpretation(16) which he says was "constructed" with the two conditions in mind and other theories, such as Einstein’s General Relativity Theory, which he claims contradicts both conditions in several places - at least logically as some of the predictions are too small to be measured empirically. His approach is that today, those conditions are used more to hinder the advancement of knowledge rather than assist it. His claim is that overbearing adherence to these principles has prevented alternative theories being developed which, so Feyerabend, would permit improved testing of "established" theories compared to merely taking into account observable results.
A further assumption he takes issue with is that of the "relative autonomy of facts" that is, that a set of observation statements is available independently from any theory. Thus an alternative theory can be tested against the very same facts as have been used to show the validity of the current theory. However Feyerabend asserts that this cannot hold by contending that in many cases the set of statements of observation are very much linked to the theory they are used to confirm. He argues:
… [All] these investigations use a model in which a single [Feyerabend’s italics] theory is compared with a class of facts (or observation statements) which are assumed to be ‘given’ somehow.
And:
… [Facts] and theories are much more intimately connected than is admitted by the autonomy principle(17).
that contrary to practice, the only way to seek out the full set of statements is through the use of mutually inconsistent alternative theories. As an example he uses the Brownian Particle, which on the face of it would seem to contradict the Second Law of Thermodynamics(18). This states that entropy cannot decrease in an isolated system, yet it would seem that Brownian molecular motion, could be directly converted into work. Without developing the kinetic theory, Feyerabend suggests, this finding would have been regarded as an oddity. In fact kinetic theory shows that the statistical chance of Brownian particles to line up in a way as to be able to convert molecular motion directly into work, is extremely low and it therefore taken that Brownian motion cannot be used to produce useable work… hence no real threat to the Second Law of Thermodynamics.
In his essay, Feyerabend, takes the position where instead of letting empiricism rule, as he feels it does elsewhere, he wants philosophy to reconsider the position of a metaphysical approach. Contrary to what he holds as widespread practise, he wants to see metaphysical systems as alternatives to further testing of well-confirmed theories. He believes this approach alone allows us to examine the already observed phenomena to be properly assessed and reassessed.
For Feyerabend the application of reductive processes as means to explain an older theory within the context of the newer one, is misguided. For him too much of the terminology is theory context laden in such a way as confirming facts are not really validated equally by the two theories at hand:
… [Moreover], if our methodology demands the use of mutually inconsistent, partly overlapping, and empirically adequate theories, then it thereby also demands the use of conceptual systems which are mutually irreducible [Feyerabend’s italics] (their primitives cannot be connected by bridge laws which are meaningful and factually correct) and it demands meanings of terms be left elastic and that no binding commitment be made to a certain set of concepts(19).
As a result he views reductive endeavours in modern times as wholly misguided and liable to retard the acquisition of new knowledge because of the deductive element which binds the scientist and philosopher to a certain path; possibly aiding the discovery of more facts confirming the theory, but also preventing them from finding those elements that may well have posed serious problems for the theory. Feyerabend draws on a point that is easily missed, reductionism in science is often viewed somewhat haphazardly. That is, although a new theory is able to explain an older one, it is not an exact logical reduction that has led to it. In fact the examples cited above show time and again, that some paradigm shift had to take place and then in the new light, the older theories are re-assessed and a reduction of ideas takes place, allowing then a pseudo-logical deduction. (top)
7. Thomas Nickles
A year after Nagel’s critique on Feyerabend, Thomas Nickles published his paper "Two Concepts of Intertheoretic Reduction(20)". In it taking issue with the common view that theory reduction involves only the relation of a theory to its special case. He suggests there are at least two types of reduction, differing both in function and purpose. Reduction1 as he terms it follows Nagel’s approach by being a type of derivational reduction, i.e. an explanation of one theory by another. This type is most useful where it is possible to combine domains. Reduction2 in Nickles view involves other relations rather than logical or mathematical. The role of this second type of reduction is that of justification in science.
… [But] rather than agree with Nagel’s critics that we find no reduction here, I prefer to recognize certain of these of these important nonderivational intertheoretic relationships as a distinct type of reduction(21).
He also takes issue with the question which way round reduction actually takes place, using the example similar to one used earlier, he suggests that actually Einstein’s special theory of relativity (SRT) reduces to Newton’s classical mechanics (CM) when we make certain assumptions. For example at very low speeds we can say that the momentum calculated using Einstein’s SRT and that calculated using CM will be the same (c.f. footnote 5). Looking at the historical record, Nickles finds that what he has termed reduction1, namely the kind of elimination, trimming down or consolidation [his words] does not neatly fit with what has actually been happening. The fact the that classical mechanics can be used in place of the special theory of relativity within the right limits just confirms that for Nickles.
A crucial difference for the distinction is that Reduction2 would potentially permit a reduction into several different non-competing ways depending on what limits are taken. This being possible because, as Nickles sees it, there is no logical compatibility issue with this type. On the other hand Reduction1 links reduced and reducing theories logically and is therefore far more constrained.
Homogeneous and inhomogeneous reduction are further distinguished by Nagel’s distinction of on the one hand a set of theories where the descriptive terms of one are a subset of the other and the case where phenomena of different domains are explicable in one combined theory, that is terms used for each domain are not easily shown to be logically derivable. Nickles says that Nagel attempts to deal with this by introducing a set of correlative laws connecting the different terms from one theory to the other.
Agreeing with Nagel’s detractors, that theories such as classical mechanics and the special theory of relativity are not sufficiently logically connected as to fulfil his requirements for Reduction1 they are nonetheless connected and this is where he feels his Reduction2 approach can do the work. He underlines this by arguing that models used for reduction are often approximations or idealisations to an extent that they are not themselves intended to do the reductive work.
Reduction2 is not meant to be some kind of harebrained scheme where anything goes. Instead Nickles’ concept is still one of derivation, but one that takes account of those approximations and limit processes common in mathematics.
A set of examples, one that Nagel himself uses, the kinetic theory of gases and another advanced by Lawrence Sklar(22), involving the Wiedemann-Franz law describing the connection between electrical and thermal conductivity show, so Nickles that the reduced and reducing theories could be used interchangeably to derive the theories the "other way round", that is classical mechanics could be extracted from the kinetic theory of gases. The problem lies, so Nickles, with the "bridge laws" which on the one hand are too weak, allowing a two way relation without directionality of the reduction. On the other though these bridge laws are too strong, requiring very precise links. He points to the problem:
… [This] problem becomes especially thorny when one attempts to reduce non-statistical theories like CM or phenomenological thermodynamics to statistical theories like QM or statistical mechanics(23).
However he does not take these objections to make Nagel’s account appear fundamentally flawed.
Another approach to the problem for reduction, thrown up by Feyerabend’s meaning change objection, is that of Schaffner, who does not use the historical theory, but a corrected historical theory. Instead to trying to show a reduction between the old and new theories, he attempts to explain the old theory adjusted in view of modern findings. To do this his he has to give reasons for the adjustments, such as missed variables and why they might have been missed. The problem with this attempt, though, so Nickles, is that it extends reduction to analogy, which in effect only moves the problem from one to another explanatory device.
Like Nagel, Schaffner and other philosophers seem to regard Reduction2 as an "imperfect" form of Reduction1, that is as a form of logical deduction, says Nickles, missing entirely the point of Reduction2. For him it is false that any reduction that does not fit logical deduction exactly has to be an approximate logical reduction, this is exactly where Reduction2 comes in:
Reduction2 can sometimes help us analyze approximative reduction1, as indicated, but it can do so only be being a separate kind of reduction(24).
8. Philip Kitcher
In his paper "1953 and All That: A Tale of Two Sciences(25)" Kitcher deals with reduction using the example of genetics, picking up the discussion nine years on from Nickles. Problems with reduction in physics are magnified when it comes to biological entities Kitcher claims. One of the first problems, for Kitcher is, that there does not seem to be an exact way to map genes onto segments of DNA.
To show a simple mapping from classical genetics to molecular biology is not established he sets out what it would take for such a reductive process to be successful. The three claims he devises are a result of an "adjusted" classical theory to fit the reduction schemata found in examples from physics.
The claims are that the laws detailing gene transmission from classical genetics can be instantiated in the reduced theory. That some kind of mapping exists which can match the distinctive terms from genetics onto those of molecular biology and lastly that molecular biology would be able to give an explanation for the efficacy of the laws of genetics. Stepping through each of these requirements, detailing the implications opposing them to what is actually the case, he demonstrates that none of them stand the test.
The problem for Kitcher is that any attempt of a simple mapping is ill conceived when it comes to biological reduction. By looking at the systematic elements in each theory, he approaches the problem from a different angle:
As we have seen, the main difficulty in trying to axiomatize classical genetics is to decide what body of statements one is attempting to axiomatize(26).
Although he finds is that molecular biology has been able to solve some of the problems classical genetics had, those examples are far to specific to make a reduction in the normal sense on the basis of that.
An attempt at reduction is made again with his example of sickle-cell anaemia. This works well as the mutation that causes the illness is simple. However this is an atypical case as mutations are only rarely this simple, leading to a much more complex and widespread chemical interaction than in this simple case. It may be that altered chemical reactions will have effects on other intercellular reactions, leading to changes to cell geometry, which in turn feed back into other chemical reactions.
The difference for Kitcher between a life science such as biology and sciences such as physics and chemistry is that the dispute about reductionism has an added dimension. On the one hand anti-reductionists face the charge of a woolly "vitalism" and on the other, reductionists are under attack for failing to take account of the complexity of nature. Although it seems that today virtually all biologists agree in at least a minimal physicalist account, the question is the level at which these processes take place and have influence. Kitcher sees essentially two options, one of molecular biology providing an explanatory extension and another, stronger one, that any biological explanation can be reformulated through means of molecular biology.
Demonstrating earlier, that an exact mapping of molecular biology onto classical genetics was unsuccessful, he draws the conclusion that the strong form is not likely to be successful. Even the weaker position, however suffers from problems. He argues that where genetic mutation impacts on say the cell shape for example, the consequences can affect the whole organism. In his example he continues, the cell’s abnormality prevents it from connecting to another group of cells and this failure in turn means another important chemical reaction fails to start It is not simply a molecular biological event, so Kitcher, that here leads to the development, if anything it would have to be the combination of molecular biology with something else. In a quote from Oster and Alberch he makes plain the problem, it is not a linearly organised causal process that is at work here, but a complex one, where any number of consequences can and do feed back into the system triggering or preventing yet other developments.
The main issue, and the one that in the end must lead to the failure of reduction for Kitcher is his analysis of classical genetics; he describes it as:
… [a] family of related patterns of reasoning for solving the pedigree problem(27).
Effectively what is being attempted with reduction, therefore, is a mapping from a domain where some sort formalistic representation is possible, molecular biology, to one that is defined by recognised patterns of causal conjunctions. (top)
Footnotes (top)
(7) Reprinted in Curd, M. & Cover, J.A., Philosophy of Science, The central Issues (1998) pages: 905-921, originally published in Teleology Revisited, 1974, Columbia University Press. (back)
(8) Curd & Cover (1998), page 907 (back)
(9) Some philosophers prefer to refer to inhomogeneous reduction as heterogeneous reduction (back)
(10) The area velocity of planets circling the sun in ellipses is constant, that is closer to the sun a planet travels faster than farther away, but the areas marked by the radii and the arc travelled for the same time are equal. (back)
(11) The period of a pendulum being proportional to the square root of its length is an example, this is directly derivable from Newton’s Law of Gravitation, but only when a number of assumption are made, including the oscillation angle being very small and gravity being constant. (back)
(12) In Kuhn, T. S. (1977), The Essential Tension: Selected Studies in Scientific Tradition and Change, University of Chicago Press. (back)
(13) Curd & Cover (1998), page 920. (back)
(14) Reprinted in Curd, M. & Cover, J.A., Philosophy of Science, The central Issues (1998) pages: 922-949, originally from Bernard Baumrin, ed., Philosophy of Science, The Delaware Seminar, vol. 2 (1963), Interscience Publishers. (back)
(15) Curd & Clover (1998), page 922. (back)
(16) To this day this interpretation is taken as the standard understanding of QM, advocated by Niels Bohr, Werner Heisenberg, Max Born and others. It is named after the town where Niels Bohr worked. (back)
(17) Curd & Cover (1998), page 934. (back)
(18) Initially Second Law of Thermodynamics was taken to be a general law, but it has been realised to be a statistical one instead. (back)
(19) Curd & Cover (1998) page 939. (back)
(20) Reprinted in Curd, M. & Cover, J.A., Philosophy of Science, The central Issues (1998) pages: 950-970, originally from Journal of Philosophy 70 (1975) (back)
(21) Curd & Cover (1998), page 956. (back)
(22) Types of Intertheoretic Reduction, British Journal of the Philosophy of Science xviii, 2 (August 1967 (back)
(23) Curd & Cover (1998) page 959 (back)
(24) Curd & Cover (1998) page 962 (back)
(25) Reprinted in Curd, M. & Cover, J.A., Philosophy of Science, The central Issues (1998) pages: 971-1003, originally from Philosophical Review 93 (1984) (back)
(26) Curd & Cover (1998), page 980 (back)
(27) Curd & Cover (1998), page 984 (back)