Disclaimer: I do not know a whole terrible lot about the intricacies of either chaos theory or quantum mechanics, let alone the combination of the two, this is more a philosophical thing than a scientific one, I know I get a lot of things wrong (on both sides)

Further disclaimer (thank to ariels for the information): The 'snapshot' mentioned above is a well defined object in dynamics (its mathematical form containing firm proofs and a specific ontology). However, I think the point below still stands. Though the 'snapshot' as defined mathematically may be vastly different than the 'snapshot' the lay person is familiar with, I think Feyerabend would still argue that the very choosing of the term 'snapshot' is a metaphorical/rhetorical one, that cannot be encompassed by an easy rationality...

Applying Philosophy of Science

Feyerabendian and Lakatosian analyses of Quantum Chaos

I will be discussing an article entitled “Chaos on the Quantum Scale” by Mason A. Porter and Richard L. Liboff from the November-December 2001 issue of American Scientist. The article discusses recent advances in recent attempts to model systems that behave chaotically on the quantum (sub-atomic) scale. It will be helpful to briefly summarize the main points of the article:

The first few introductory paragraphs relate quantum mechanics and chaos theory by placing emphasis on their respective uses of uncertainty. From this common point of uncertainty, the authors state that because scientists seem to ‘find’ chaotic phenomena at all scales, they cannot rule out the possibility of chaos at the sub-atomic level. The next section of the article is a brief history of chaos theory that describes the early work of Henri Poincaré and mentions the later work in the 1960’s by meteorologist Edward Lorenz. They then explain that chaos has been found in so many disparate disciplines of science, and once again reiterate that they cannot rule it out at the quantum level. Here they also mention possible applications of such quantum level chaos in nanotechnology.

From here they move into the largest section of the article, the billiard-themed thought experiment/model. They move from a simple two dimensional billiard table to increasingly more chaotic and quantum-like billiard tables. There is a two-dimensional table with a circular rail, a spherical ‘table’, a spherical table with wave-particles as ‘balls’, and finally, a spherical table with an oscillating boundary and with wave-particles of different frequencies. Within this section, they also explain the more technical aspects of their attempt to model quantum chaos. They explain their plotting methods (the Poincaré section) as well as their mathematical methods as well (the Schrödinger equation and Hamiltonians). With the final few examples they show us that they cannot as yet model true quantum chaos, but only semi-quantum chaos (which requires mathematics from the realm of classical physics as well as quantum mechanics). After this admission, they go on to describe in detail future applications that successful quantum chaotic modeling will have in nanotechnology, from superconducting quantum-interference devices (SQUIDs) to carbon nanotubes. The final sentence of the article sums up the general attitude of the authors: “As we have shown… this theory possesses beautiful mathematical structure and the potential to aid progress in several areas of physics both in theory and in practice” (Porter 537).

I shall now attempt to analyze the article in light of two very different ‘theories’ (though one can certainly not firmly be called a ‘theory’): namely, those of Paul Feyerabend and Imre Lakatos. I will begin my discussion with Feyerabend’s thought, and then move on to Lakatos. After these analyses, I will engage both authors with each other, and attempt to bring out certain problems in each of their ‘theories’ that I see myself.


Paul Feyerabend introduces the Chinese Edition to his book Against Method by stating his thesis that:

the events, procedures and results that constitute the sciences have no common structure; there are no elements that occur in every scientific investigation but are missing elsewhere. Concrete developments… have distinct features and we can often explain why and how these features led to success. But not ever discovery can be accounted for in the same manner, and procedures that paid off in the past may create havoc when imposed on the future. Successful research does not obey general standards; it relies now on one trick, now on another…(AM 1).

So, we can (and do) explain why certain scientific developments/revolutions do occur, but we should not expect these explanations to bud into theories, and we should definitely not expect that our explanations should apply in all cases. This inability for universally applicable theories to be universally applied, is not a result of our inability to hit upon the correct theory, but is a result of the non-uniform character of what we call ‘science’. Science is not a homogenous enterprise. It comprises everything from sociology to quantum mechanics. Before we can expect to have an absolute theory (which Feyerabend thinks is neither possible nor desirable) we would have to have an absolute definition of what ‘science’ is. (Here we can see the influence of Wittgenstein’s idea of language games on Feyerabend’s thought). Perhaps science isn’t something we can have a theory about.

So, it being understood that Feyerabend believes that ‘science’ is not homogenous, and that we can only explain individual cases with individual criteria, what processes would he think applicable in the article at hand? Obviously this is a difficult question to answer. I think a fruitful way of approaching the task is through a very un-Feyerabendian process. By seeing what he has done in the past (e.g. in his previous analyses of scientific ‘developments’) we may be able to surmise what he would be likely to note in our particular example. In Feyerabend’s analysis of Galileo (specifically in chapter 7 of Against Method) he emphasizes the role of rhetoric, and ‘propaganda’ in scientific change. He states that:>/p>

Galileo replaces one natural interpretation by a very different and as yet (1630) at least partly unnatural interpretation. How does he proceed? How does he manage to introduce absurd and counterinductive assertions, such as the assertion that the earth moves, and yet get them a just and attentive hearing? One anticipates that arguments will not suffice - an interesting and highly important limitation of rationalism – and Galileo’s utterances are indeed arguments in appearance only. For Galileo uses propaganda (AM 67).

So, as it seems that an analysis of non-argumentative (rhetorical) uses of language aided Feyerabend in his discussion of Galileo. Thus, one possibly fruitful method of analysis may be to search out similar uses of language in our article. Which is precisely what I will do. Here is a good example of the use of non-rational, non-argumentative means of convincing someone of your point:

The trail of evidence towards a commingling of quantum mechanics and chaos started late in the 19th century, when … Henri Poincaré started working on equations to predict the positions of the planets as they rotated around the sun (Porter 532).

Here we are led to believe by Porter/Liboff that Poincaré’s work is part of a ‘trail of evidence’ that provides support for their work (‘the commingling of quantum mechanics and chaos’). By the appeal to an accepted authority (it is generally accepted in the chaos community that Poincaré is the ‘father of chaos theory’) we are supposed to lend further credence to their own work (though, as we are told in the last portion of the article, this work has not provided a true connection between the two theories). But, is there, in Poincaré’s work any evidence of this commingling of chaos and quantum mechanics? Hardly. The ‘evidence’ they refer to is simply the birth of chaos theory. If we accept their claim, one might analogously state that my birth contains ‘evidence’ for whom I will marry in the future. (Putting aside genetic predisposition toward certain possible mates, this is absurd.) We cannot (rationally) justify the claim that the birth of chaos theory provides evidence for the future ‘commingling’ of that theory with quantum mechanics. It does, however, provide a nice segue for the authors into a historical summary of the birth of chaos theory. Rather than an argument, it is a literary device (like exaggeration, alliteration, etc.) that aids both the achievement of the authors’ goal (describing quantum chaos) and making the text itself more fluid.

Staunch rationalists would argue (Feyerabend might say) that this example mistakes a literary device for a scientific argument, and that if we simply separated the two, the problem would dissolve. Feyerabend’s position however, is that we are unable to separate the two. He states in Against Method

That interests, forces, propaganda and brainwashing techniques play a much greater role than is commonly believed in …the growth of science, can also be seen from an analysis of the relation between idea and action. It is often taken for granted that a clear and distinct understanding of new ideas precedes, and should precede, their formulation and institutional expression. (An investigation starts with a problem, says Popper.) First, we have an idea, or a problem, then we act, i.e. either speak, or build, or destroy. Yet this is certainly not the way in which small children develop. They use words … they play with them, until they grasp a meaning that has so far been beyond their reach… There is no reason why this mechanism should cease to function in the adult. We must expect, for example, that the idea of liberty could be made clear only by means of the very same actions, which were supposed to create liberty (AM 17).

Putting aside the theory of language acquisition proposed here, we see that Feyerabend believes that the form of our investigation is just as important as the content or result of it. Thus, we cannot understand an argument separately from the language it is phrased in, language that often contains suggestive (propagandistic) phrases. In other words what you say is often inseparable from how you say it Analogies to real world objects are also used by Porter/Liboff. For example: “A buckyball has a soccer-ball shape…” (Porter 536); “Nanotubes can also vibrate like a plucked guitar string…” (Porter 537); and, “Such a plot represents a series of snapshots of the system under investigation” (Porter 534). These analogies appear to be used simply to enhance the more abstract qualities of the quantum-chaotic world the authors are describing, and make them more understandable. But, it seems there is more going on here. If we view the article in the Feyerabendian sense that I have been developing above, the choice of metaphor can also affect the readers’ conception of the ‘ideas’ that the authors are attempting to put across.

In particular, the ‘snapshot’ analogy seems suggestive to me. What the authors describe as ‘snapshots’ are Poincaré sections taken from higher-than-three dimensional systems. In effect, two-dimensional plots that are, by a mathematical process, abstracted from ‘multi-dimensional masses.’ These are possibly some of the most theoretical objects ever created yet the authors describe them as ‘snapshots’. Obviously there are qualities of the Poincaré section that lend it to the comparison: both a snapshot and a Poincaré section are thought to be reports of a particular time and space. But, other aspects of the comparison may (hopefully, for the Porter/Liboff) lead the reader into accepting highly theoretical concepts as real objects, more so than they would have without the analogy. Obviously the creation of a photographic snapshot is itself based on theory, but it is one that we use (and accept) in everyday life, one that we accept without reservations. Not only that, but the real-life snapshot (as opposed to the Poincaré section snapshot) represents things which we already accept as existing in the real world. In comparing the Poincaré section to a snapshot, the authors attempt to further solidify the reality of the objects that the section represents. Rather than seeing the n-dimensional objects of the Poincaré section as abstract objects, we are now more suggested to picture them as objects like our vacation slides, or wedding photos.


Imre Lakatos’ great contribution to the history and philosophy of science (and the historiography of science) is the concept of the research programme. As a general illustration of the role of a research programme, the following quote may be helpful:

the great scientific achievements are research programmes which can be evaluated in terms of progressive and degenerating problemshifts; and scientific revolutions consist of one research programme superseding (overtaking in progress) another (Lakatos 115).

How can we apply such a methodology to the emergence of quantum-chaos? Well, to start with, we might ask just what research programme, or programmes we are working with. Are quantum mechanics, chaos theory and quantum-chaos all individual research programmes, and, if so, how do we explain the emergence of quantum-chaos (a theory that contains elements of both quantum mechanics and chaos theory) in relation to the other two? I shall attempt to answer these two questions in order.

To answer the first, we should define more firmly what Lakatos means by the term ‘research programme’. He states that:

The basic unit of appraisal must be not an isolated theory or conjunction of theories but rather a ‘research programme’, with a conventionally accepted (and thus by provisional decision ‘irrefutable’) ‘hard core’ and with a ‘positive heuristic’ which defines problems, outlines the construction of a belt of auxiliary hypotheses, foresees anomalies and turns them victoriously into examples, all according to a preconceived plan. The scientist lists anomalies, but as long as his research programme sustains its momentum, he may freely put them aside. It is primarily the positive heuristic of his programme, not the anomalies, which dictate the choice of his problems (Lakatos 116).

So, in order to determine whether or not our three ‘categories’ can be aptly described as research programmes they must have a ‘hard core’ (which I take to mean principles or examples that one has to accept in order to work within the research programme), and also a ‘positive heuristic’ that determines what problems will be addressed (and how to address them). For brevity’s sake I shall limit my discussion to the ‘hard core’ and the problem-determining function of the positive heuristic while ignoring the role of anomalies in negative determination of problems (a role that Lakatos, unlike Popper, believes is secondary to that of the positive heuristic).

Quantum mechanics definitely seems to have a ‘hard core’ that its adherents agree is irrefutable, and essential to its elaboration. Historical examples of such an irrefutable core can be found in papers (from the late 19th century to the first quarter of the 20th) by Planck, Bohr, Einstein and others. These papers contain principles that form the unshakeable core of quantum mechanics even now. Here is just one example, which should suffice to illustrate the point:

Today we know that no approach which is founded on classical mechanics and electrodynamics can yield a useful radiation formula. … Planck in his fundamental investigation based his radiation formula…on the assumption of discrete portions of energy quanta from which quantum theory developed rapidly (Einstein 63).

So, quantum mechanical theory develops directly from Planck’s assumption of quanta. Although this is an oversimplification, it does illustrate that there are basic assumptions which quantum theorists are unwilling to sacrifice. We have our ‘hard core’, now the question is: does quantum mechanics have its own ‘positive heuristic’? I think the easiest way to answer this is to rephrase the question slightly: has quantum mechanics generally determined its own problems positively (i.e. set out to solve them) before they are negatively determined by emergent anomalies? Obviously searching out the ‘general’ answer to this question is well beyond the scope of this essay, but finding a few examples can at least allow us to provisionally classify quantum mechanics as a research programme. One example is the full, and accurate, derivation of Planck’s law. Planck proposed the idea of quanta (discrete units of energy) in 1900 and the perfection of a law describing this idea was worked on until 1926. The idea of quanta was proposed as a basic tenet of quantum mechanics (it was ‘anomalous’ only for the then degenerating research programme of classical mechanics), though it could not be perfectly derived. So, setting it up as a problem, quantum mechanics attempted to ‘solve’ it (and eventually did). The problem of splitting the atom, though it may have been motivated by outside political factors, was internally posed to quantum mechanics as well, and consequently solved as ‘predicted’ by theory.

Undoubtedly, then, Lakatos would define quantum mechanics as a research programme, and not merely a theory contained within a larger research programme. Can the same be said of chaos theory? Well, chaos theory seems to have its own ‘hard core’. This much we can see from the Porter/Liboff. The theory’s basic assumption is that

some phenomena… depend intimately on a system’s initial conditions, so that an imperceptible change in the beginning value of a variable can make the outcome of a process impossible to predict (Porter 532).

All applications of chaos theory work outward from this core principle, which is also historically situated (in the article) through the work of Poincaré:

Poincaré started working on equations to predict the positions of the planets as they rotated around the sun… Note the starting positions and velocities, feed them into a set of equations based on Newton’s laws of motion, and the results should predict future positions. But the outcome turned Poincaré’s expectations upside down. With only two planets under consideration, he found that even tiny differences in the initial conditions… elicited substantial changes in future positions (Porter 532).

So, like quantum mechanics, the hard core of chaos is situated historically in a few irrefutable examples and principles. For quantum mechanics some examples of the core principles are the Heisenberg uncertainty principle and Planck’s assumption of discrete quanta. The individuals most often recognized historically as exemplars of quantum mechanical theory are Einstein, Bohr, Born, Ehrenfest, to name a few. These examples are constantly cited and referred to both pedagogically, and in scientists’ description of the birth of their field. Chaos theory’s core principle is that we cannot accurately predict the future state of a dynamical (i.e. chaotic) system. This principle is exemplified in the early work of Poincaré (which is generally seen as proto-chaotic) and the later meteorological studies of Lorenz (who is also mentioned by Porter/Liboff).

Now we move on to the question of whether or not chaos theory has a positive heuristic, which determines the problems to be solved. It seems, at least prima facie (which is as far as such a limited study can go) that, unlike quantum mechanics (whose scope is internally limited to the ‘quantum realm’) that chaos theory has the potential to be applied to any system. In this respect, can it be considered a research programme? If it has historically been applied only within other research programmes (meteorology, electrodynamics, planetary motion, to name only a few mentioned in the article itself) it does not seem plausible that it can define its own problems and attempt to solve them in seclusion from other research programmes. Rather than a research programme, I propose that chaos theory is a self-contained theory (a modeling or mathematical tool) that functions within a variety of established and independent research programmes.

On this view, it would appear that quantum-chaos, far from being an independent research programme, is the result of a development that is internal to the progressive research programme of quantum mechanics. Quantum-chaos is not an entirely new system of ideas, but a growth of new ideas within the boundaries of the quantum realm. That is, without quantum mechanics, there would be no realm in which to create quantum-chaos, and no ‘rules’ with which to describe it.

Critique of Feyerabend and Lakatos

Now that we have seen a few of the ideas of Feyerabend and Lakatos in application (albeit forcefully) I shall move on to a critical engagement of the two, playing off their views (as well as my own) against one another. I will start with Lakatos.

It seems that though the research programme is a valuable historiographical lens with which to view scientific history, it has obvious limitations. Although it enables the historian of science to encompass more examples than something like (what Lakatos calls) a ‘conventionalist’ historiography, it is by no means all encompassing. The main problem that I see with his methodology is one that Lakatos states himself.

The methodology of research programmes – like any other theory of scientific rationality – must be supplemented by empirical-external history. No rationality theory will ever solve the problems like why Mendelian genetics disappeared in Soviet Russia in the 1950’s, or why certain schools of research into genetic racial differences or into the economics of foreign aid came into disrepute in the Anglo-Saxon countries in the 1960’s… (Lakatos 119).

So, like most other rationalist reconstructions of the history of science, his attempt must be supplemented by psychological, sociological and other explanations. The difference between a falsificationist like Popper and someone like Lakatos is that Lakatos at least admits that there are other factors in the history of science than rational ones. But, for a rationalist project, whose aim is to explain all scientific change, this fundamental problem simply cannot be overcome. The problem is that the human agents in science (who, despite any talk of a ‘third world’ are key agents in scientific change) are never fully, or exclusively, rational. If we are bound by a purely rational reconstruction of the history of science, then the irrational in science (which Lakatos admits exists) will always elude our methodological understanding. Lakatos denies that any theory of scientific rationality can succeed in this task.

The problem of irrationality in science is one that I believe Feyerabend can overcome more easily. To him it seems that if a completely rational reconstruction (based on the rigorous application of a specific ‘system’) is bound to fail, then should we not look at the possibility of an irrational, even non-systematic explanation of the history of science? Obviously such an explanation could not be termed a ‘methodology’ but through something like it, we could attempt to explain any historical stage of science. Such an irrational, anti-methodological approach is precisely what Paul Feyerabend calls for. Feyerabend’s explanations do not rely on the constancy of a specific method or concept, but fluctuate based on the particular situation they are attempting to ‘explain’. When talking about a series of lectures he had given at the London School of Economics, Feyerabend sketches out for us his intent:

My aim in the lectures was to show that some very simple and plausible rules and standards which both philosophers and scientists regarded as essential parts of rationality were violated in the course of episodes (Copernican Revolution; triumph of the kinetic theory; rise of quantum theory; and so on) they regarded as equally essential. More specifically I tried to show (a) that the rules (standards) were actually violated and that the more perceptive scientists were aware of the violations; and (b) that they had to be violated. Insistence on the rules would not have improved matters, it would have arrested progress (SFS 13).

Feyerabend suggests here that not only are rules not always fruitful in science, but that strict adherence to those rules sometimes hinders its progress. The same can be said about historiography of science. If we insist on strict adherence to specific rules in all cases then not only are going to get it ‘wrong’, but we may make it harder to get it ‘right’ (i.e. more useful, less problematic historical descriptions).

So, we have discussed a specific problem with Lakatos’ methodology of research programmes and ended up at the seeming inadequacy of all methodologies. But neither I, nor Feyerabend, believe that there are never times when rules can be applied fruitfully to historical analyses. Indeed, Lakatos’ concept of the research programme seems to provide criteria that are more widely applicable than many others proposed before it. It does not fall prey to the rash assumption that science is strictly rational, thought it admits science’s rationality is all that it can explain. This is precisely what Feyerabend wants the rationalists (and particularly the other LSE rationalists) to admit: that we cannot always fit history into the box of rationality (regardless of whether the box is that of falsificationism or the methodology of research programmes). So, on the one hand, Lakatosian research programmes explain more than any other rationalist reconstruction can, but on the other hand, Lakatos admits that (unlike Feyerabend) he cannot explain irrationality in science.

How can I criticize Feyerabend? If I accused him of incoherence, or self-contradiction, he would take it as a complement. If one can accept any standard at any time, depending upon the circumstances, then of course one can seem to be contradictory, he would say. I tend to agree with Feyerabend that no rules can be applied absolutely, for all time. But, one might criticize him in his specific historical analyses. For instance, his emphasis on the rhetorical (non-rational) use of language and irrational ‘methods’ of Galileo and Copernicus may ignore some of the important rational features in their work. Though this problem may be inherent to an attack on rationalist reconstructions of science, I think that Feyerabend often ignores salient features of history simply because they are instances of rationality. That being said, I believe that Feyerabend’s philosophy of science provides us with the mindset to build a number of very unique perspectives on the history of science. He tells us that no method can work absolutely, but some methods can work sometimes. Our task is to think for ourselves and create our own interpretations of science, and not to rely on the grandiose systems of our predecessors.


Bennett, Jesse. The Cosmic Perspective (1st edition), Addison Wesley Longman, 1999, New York.

“Chaos on the Quantum Scale”, Mason A. Porter and Richard L. Liboff, pp.532-537 in American Scientist Volume 89, No. 6 November-December 2001.

“Chaos Theory and Fractals”, Jonathan Mendelson and Elana Blumenthal 2000-2001, URL: http://www.mathjmendl.org/chaos/index.html

“Early Quantum Mechanics”, J J O'Connor and E F Robertson 1996, URL: http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/The_Quantum_age_begins.html

Einstein, Albert. “On the Quantum Theory of Radiation” pp63-77 in Sources of Quantum Mechanics. Ed. B.L. Van der Waerden, 1968, Dover Publications, New York.

Feyerabend, Paul. Against Method, Verso, 1988 1975 New York. (Referred to in the text as AM)

Feyerabend, Paul. Science in a Free Society, New Left Books, 1978, London (referred to in the text as SFS).

Lakatos, Imre. “History of Science and its Rational Reconstructions” pp.107-127 in Scientific Revolutions, ed. Ian Hacking, 1981, Oxford University Press, New York.

Wittgenstein, Ludwig. Philosophical Investigations. Translated by G.E.M. Anscombe (No publishing information provided).

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