MA Thesis of Timothy Hilgenberg
(c)
MM Timothy Hilgenberg
Problems with Reduction
9. Issues
The previous sections have clearly marked out several central issues that are causing problems for any reductive account. From Nagel’s attempt to salvage reduction as a form of correspondence to Feyerabend’s defiant stance essentially denying its practical application. Attempts to introduce a multiplicity from Nickles as a way forward as well as a view that once beyond a certain complexity threshold, it becomes virtually impossible to reduce, suggested by Kitcher, all seem to leave a sentiment of dissatisfaction with the general situation.
What are the issues then that seem to make reduction so unpalatable? The correspondence approach from Nagel has some merit in that it seems to connect with our intuitive understanding of the reductive approach. While we would agree that Newton’s and Einstein’s understanding of the term mass are not the same, we do see how they appear to be connected. We can follow the development from one to the other.
On the other hand we would have to agree with Feyerabend that it is neither here nor there how we are able to understand the development historically. What matters is whether one theory really does have a logically deductive link from another and on the face of it there seem to be few examples that could be counted on as providing some evidence to contradict this interpretation.
Splitting reduction into two or more concepts, as is suggested by Nickles seems a neat way out of the situation. While it allows us to use reduction in its fully fledged glory of logical deduction for those few examples where it may be agreed to work, the other type of reduction could be used in a transitive explanatory kind of way, where we would be able to recount a story as to how two theories are connected.
Finally Kitcher seeks to persuade us that while reduction might work on simple, linear causally connected systems, it appears to be inadequate when it has to deal with multi-layer and self-organising ones. He does not seem to hold up much hope for a solution there, seemingly saying that a tiny part may be reductively explained, but on the whole it is an approach not likely to be worth the effort. (top)
10. Homogenous vers Inhomogenous
A point Nagel raises and raises for good reason is the distinction between homogeneous and inhomogeneous reduction. While reducing one theory to another within the same domain is apparently less fraught with difficulty than reducing across different domains, there are still some difficulties. Although Feyerabend would still have none of it, a case for this type of reduction could be argued along the lines of empirical research and domain probing analysis explaining why the findings are such as they are. Feyerabend does make an important point when objecting, his claim being, that research will be done within the domain as we perceive it, from the framework of the theory that defines it, that is it would indeed appear that what is found is such that it will fit somehow within the explanatory framework. To illustrate the point it might be worth looking at the move from the geo-centric to the helio-centric world view. Although complicated, knowing the intricate formulae for the planetary movement, astronomers were well capable of calculating planetary positions correctly. Not until 150 years after the abandonment of Ptolemy’s view in favour of the Copernican idea was observational support found. The geo-centric system worked well and each new discovery matched the pattern, albeit with the occasional need for some minor adjustments.
There are two ways of dealing with this kind of problem, first we can agree with Feyerabend and say that a paradigm shift took place and this is a clear example of theory replacement. There is another way, however, of looking at this kind of problem. The underlying description of the theory in such matters, as with much of physical science, is mathematics. When we compare Ptolemy and Copernicus, it is clear that Copernicus’ algorithms are vastly simpler than Ptolemy’s! And for a simple reason; Copernicus is using the "natural" centre, that means there is no offset that needs to be taken into account. If we were travelling on the train at 70 mph, facing each other and you looking back, and I were to throw you a book at 5 mph the book would travel at 75 mph(28). Although to you and me it would be travelling at 5 mph we have to make a "co-ordinate transformation" for the observer at rest, that is we have to take the speed of the train, its direction and add that to the speed of the book and its direction. With both directions aligned it is a simple addition of the two speeds. In a similar way, though much more complicated formulae are needed, we can connect a geo-centric to a helio-centric system by adding those elements to the formulae that describe the movement of the earth in respect of the "natural" centre.
If we now take this, here two formulae describing essentially the same event, we can break down them down deductively into a helio-centric system, plus a co-ordinate transformation matrix to describe the geo-centric offset, mathematically we can therefore reduce the geo-centric to a helio-centric system by discovering that the co-ordinate transfer matrix is applicable through the whole system and can therefore omit it as an unnecessary element in the calculations, in the same way as we can say one-half instead of two-quarters.
So, in fact, even though it is usually seen as a "replacement" of the older theory by the newer one, there is a logically deductive path that can lead from one to the other. If this can be seen as dealing with Feyerabend’s objection, then it seems Nagel’s correspondence approach, at least for homogeneous reduction stands a much better chance. Scientific theories are often best expressed using the "neutral" language of mathematics and using such a technique we can see which theories can been connected and under which circumstances.
Inhomogeneous reduction, though, is a different kettle of fish. As Nickles puts it, this is domain combining reduction. Kuhn’s objection here looks much more powerful with different sets of terms valid and defined within each domain, it seems obvious that they are incommensurable, at least on the face of it. Feyerabend, too with his meaning invariance objection, has an easy target here, any mapping from one domain onto another, say with Nagel’s bridge laws, would entail somehow explaining that those two domains have sufficiently enough in common as to allow for such a link. At the start this may seem like trying to compare bananas and steak, there is no direct connection between the two, but overall they both are edibles, part of a human’s diet, so in a way there is an over arching framework into which they both fit. This kind of approach is taken when reduction is attempted in a domain combining way. An oft cited example is the subsumption of optics within the branch of electromagnetic radiation. While most of us are able to see light in different colours, we are not able to electromagnetic radiation beyond infrared or ultraviolet, so it seems it is a hopeless task to try and combine them, and one without much call for either when the geometric laws of optics are found to be working well. However closer inspection again of mathematical principles underlying wave mechanics shows that the same descriptions hold across the whole band of electromagnetic radiation and visible light being only a small fraction in the middle of it. We have found the overarching framework in which describing these phenomena is reduced to a single set of formulae.
Moving once again via mathematics we are able to circumvent Feyerabend’s objections as the terms within the new description are mapped to terms within a calculus and these in themselves are meaning neutral. In the appropriate frame conditions we can further derive those theories that we know to be well established, but at the same time we can also test for consequences that our work hitherto would have failed to pick up on. A case from chemistry makes this point. While noble gases with a full set of valence electrons(29) are inert, that is they do not combine with other elements or even themselves, research through the literature showed that there were energy levels lower than those of Xenon’s valence electrons, therefore it seemed Xenon should enter into a chemical reaction(30). Here is thus an example that shows Feyerabend to have a point: no-one would normally have experimented with noble gases to see whether they did or did not react. The theory that reactivity relied on an attempt to somehow have a full set valence electrons clearly rules out reactivity for noble gases. However by moving away from a descriptive form of the theory and taking into account other laws of chemistry, such as reactions are always initiated where the result would be a release of energy, it was discovered that Xenon’s levels could be reduced by reacting with certain other elements.
So at the level of physical chemistry it was possible to theoretically predict something that had been utterly discounted previously and was yet empirically verifiable. Inhomogeneous reduction is thus possible if we can find the appropriate reference frame which fundamentally is the basis of the old theory and at the same time extends it to cover empirical observations previously either not expected or inexplicable.
Suggestions, such as the introduction of different kinds of reduction, by Nickles can appear to help overcome problems. Where it is not easily appreciated that apparently different domain concepts actually are describing one and the same thing or how two accounts that look differently are historically linked. However simply by saying there are different kinds of reduction it does little to advance the issue. While we are grappling, and I would suggest not wholly unsuccessfully, with the logically deductive kind, introducing a secondary type, one able to deal with "non-logically deductive" reduction, seems only to move the problem sideways, facing the same objection to the one Feyerabend uses on analogy as explanation. While Nickles does try to frame his Reduction2 in such a way as to prevent an "anything goes" approach, he says little as to how he such a type would be defined other than suggesting a somehow linguistically founded explication could be used to make the account be made plausible by using limits and approximations that are commonly used in science, which however is not exactly the same as empirically verified.
The distinction of homogeneous and inhomogeneous reduction is essentially only an apparent one. Looking at what constitutes a domain it is immediately plain that the description is initially based on empirical observations that are grouped together to form a human concept. It is essentially our human understanding of what it is that is out there. Taking Frege’s Hesperus and Phosphorus it was quite clear that they were distinct - at least until it was discovered that actually they were not, but instead both are instantiations of the planet Venus. In the same way it could be argued that the domain distinctions we apply are so arbitrarily and when we come to have a better, more profound look, we discover that what seemed to be two separate domains in fact turn out to be merely two different descriptions or facets of essentially the same domain. Taking this approach, in cases where reduction would be successful, not only does inhomogeneous reduction reduce itself, like the Cheshire cat, and vanish, but it leaves us "solely" with the problems of homogeneous reduction. (top)
11. Complex Issues
Kitcher brings up reduction at a point that is far from the clean cut cases of essentially linearly causal physics, as he interprets it, where we have been fairly successful at setting frame conditions within which certain phenomenological observations are predicted and then verified. With his example from the world of life sciences Kitcher brings out a whole set of problems for reduction that seem intractable. On the one hand we have a theory, classical genetics, which works well and on the other he introduces molecular biology, which more fundamental is not able to make the same successful predictions that have become commonplace in genetics. What has gone wrong? In the section discussing homogeneity versus heterogeneity we found that we are dealing with human concepts and their limits. When looking more closely we discovered what had appeared as two distinct domains was in fact fundamentally only one, housing the very same phenomena. Could this be a similar situation here? Kitcher makes a good point when he uses the example with sickle-cell anaemia although this case shows how reduction can work even in a complex system such as biology, it is an atypical example. In the normal case by contrast, it seems any attempt to construct bridge laws between genetics and molecular biology would be ill founded because of causal factors coming into play which cannot be predicted from within molecular biology alone. The example he cities is one where a genetic mutation leads to an abnormally formed cell which in turn then fails to connect to other cells and this then prevents the start of yet another process necessary to complete the organism’s development.
While this is undoubtedly true that we would be hard pressed to explicate this from within molecular biology alone, it seems we are not comparing like with like. Successful reduction in this kind of inhomogeneous situation would mean having essentially to show two things, one, there is a domain into which the domain distinction used so far collapses and two, there is a law like connection. However what we have here is not so much a case of reductionism but instead causality. What Kitcher is describing is a causal chain, one that connects a certain genetic mutation to the development of the whole organism. But causal chains have their own inherent problems. One of the problems with causal chains is that there is some dispute as to the type of connection between cause and event. Of course a case could well be constructed where a plausible causal chain can seem to show how one thing impacts on another, but this is not really the same as a logically deductive reduction. We have moved from a strict law-like connection to one that is at best plausible, at worst fuzzy and vague.
Reduction is the attempt to use a minimal set to explicate a phenomenon, causality on the other hands attempts to show a connection between two events where without one the other might not have happened.
There is an issue though, that warrants investigation. Fundamental sciences such as physics and chemistry are looking at basic elements and how they are connected. All phenomena that are dealt with are the sum of quite clearly describable processes and elements used. A good example is particle physics, where entities are postulated as the result of theory and then the hunt for them begins. Unlike that though life sciences appear to develop "emergent" properties, properties that are elusive to empirical verification. As an example, mixing together in the right proportions the chemical elements found in a human, warming it to a temperature of roughly 310 K will produce a tepid watery mush, but no Frankenstein. It seems we are not able to reduce these kind of emergent properties to mere physical or material elements. However this is not quite accurate. While it is true that we are not able to produce life in a laboratory (yet), we are able to synthesise the amino acids that are the building blocks for what are described as self organising systems. Today it is even possible to give a mathematical description of how chlorophyll uses light energy and water to convert carbon dioxide into glucose, starch and oxygen. But this still does not provide a mechanism for reduction where complicated feed-back mechanisms are at work. (top)
12. Hidden Assumptions
The last issue I want to deal with is that of hidden assumptions. As we have seen in the previous sections that to an extent we are prisoners of our own concepts. This is a point Feyerabend alludes too with his meaning invariance objection. The example of domains is one that can highlight the problem well. As shown earlier, domains are artificially defined regions within which empirical investigation are undertaken. It is not until we somehow discover that the domain is actually part of a larger, more fundamental one, that we realise our limitations. Feyerabend is right to say that there is a fundamental shift in meaning from one theory to its successor. It is important to note that once this domain combining feature has been discovered we cannot easily retreat - What has been though cannot be "unthought", as Friedrich Dürrenmatt, the Swiss playwright, says towards the end of the play "The Physicists" - highlighting how it makes it difficult for us to try and understand concepts that are prior to our own. As Feyerabend suggests there is a greater term-theory entanglement than we often realise. Within this we often are subject to assumptions that seem wholly natural to us, so much so that we do not always realise that these assumptions are in fact only that. The Newton-Einstein comparison is a good one, since we do not regularly travel at speeds at which relativistic effects start to play a role, it makes little difference to our everyday life if we leave out that element from our calculations. This is not true for everyone though, Neil Armstrong and his space travelling colleagues have actively experienced what is called time dilatation, that is time for them has gone slower than for us, Armstrong’s cells are not as "old" as they would be had he never gone to the moon. A difficult thing to imagine, but one that shows how we are "trapped" within our own concepts.
There are many things we regard as unpredictable, the lottery numbers drawn each Saturday for example, but there are other things which to us appear random and yet it is possible to describe them mathematically. "Chaos mathematics" is the popular name for this branch of mathematics. Against our common understanding, it is possible to show that there is order in chaos for certain things. One example is a double jointed pendulum, something that would seem to move clearly randomly, but yet it follows mathematically describable curves. Graphic representations of this branch of mathematics are popular and colourful instantiations of Mandelbrot sets are amongst the most famous of these fractal sets.
Conditioning, too, plays a role in the way we perceive things. No-one would expect to open a working watch to find it devoid of all mechanical parts(31). Some underlying mechanism, whatever that might be, just has to be there. Were we to find such a watch, it would leave us bewildered. Our assumption, that there is some explanation as to how a mechanical device works leads us, late 20th Century Western observers to refuse point blank to believe in such a possibility. A child, faced with the same situation would be much more at ease with this and would probably be happy with the explanation that it is an invisible fairy that moves the watch’s hands.
This mechanistic world view and its apparent failure to map across to the life sciences is what leaves us with a kind of tension. While on the one hand we are happy to build the universe from exotic nuclear particles, that fit nicely into a super-symmetric theory we stop in our tracks when it comes to biological entities, unwilling to contemplate that all that this is, is an instantiation of these exotic nuclear particles arranged in a highly complex way. It seems that once beyond a complexity threshold we are unable or unwilling to follow our mechanistic world instincts. Living matter we assume is not simple nor mechanistically explicable. We find ourselves smiling at Shelley’s notion of Frankenstein: some body parts stitched together and the right bolt of lightning is all it takes to make a living being.
A further problem for reductive accounts is their abuse. Devised for mapping one domain onto another, the very same technique is then applied on completely different domains, ones that might not even match the criteria. As pointed out at the beginning, humans develop a set of solution attempts to problems they encounter and they use these as a first try when they come up against a so far unknown problem.
However this could lead to problems. It is clear that using Euclidean geometry in non-Euclidean space will lead to errors. However if we fail to appreciate that we are dealing with non-Euclidean space we could run into trouble. Looking out at sea there is no reason for me to think the world is not a flat discoid, rather then the geodes that it really is. As a result it would seem only natural for me to use lessons learned in geometry to work out my position. So if I were to travel from London to the North Pole, turn 90 degrees to my left, headed south as far as Pickle Crow in Ontario, Canada and from there heading due east (a 90 degree turn from the north-south direction) I would eventually arrive back in London, perpendicular to the route I had set out on… my journey will have described a triangle, but one with 270 degrees and that’s 90 degrees more than any Euclidean triangle can have! So we see that perfectly obvious assumptions can lead to errors.
A final assumption, but one that is of major concern is that how terms used in one theory are connected to the same linguistic expression, but used in another theory. Although to many it may seem that mass is just mass, when looking at it more precisely it is clear that Newton’s concept differed quite a bit from that of Einstein. This point has already received a fair amount of discussion and is principally what Kuhn is attacking when he describes those terms as being incommensurable. It is along the same lines as Feyerabend’s criticism that meanings of terms are entangled with the theories in which they are used. Taking his suggestion at one step removed we must realise that what statements we derive from the observation of phenomena through the application of theories must necessarily be couched within our concepts, that is we are not really able to see thing other than within the contemporary theoretical framework. This is a very difficult position to disentangle oneself from, in a way it mirrors the kind of line drawings that looked at one way show an old lady another way a dancing ballerina. However we try we can only ever see one or the other.
This is a very powerful argument and one that certainly carries much weight when it comes to "radically" different theories, but that is not necessarily what we deal with most of the time. The problem is one of degree, where does a new theory start to depart "radically" from the old one and up to which point are changes merely subtle alterations with little overall impact. (top)
Footnotes (top)
(28) Note at these speeds no relativistic effects are perceptible - the example was chosen specifically. (back)
(29) Valence electrons are said to be the active elements in chemical reactions. (back)
(30) The first Xenon compounds were discovered by N. Bartlett using dioxygenyl platinum hexaflouride O2+PtF6- to form Xe+PtF6-, a white powder. (back)
(31) I am referring to a watch with an hour and minute hand, not a digital watch. (back)