MA Thesis of Timothy Hilgenberg

Conclusion

13. A Take on Reduction

The problem with reduction appears essentially twofold, on the one hand we have the question as to what it actually is: Is Nickles right and we should split it different ways or is it only really one thing?On the other hand we have the problem of the entanglement of terms and theory: Does Feyerabend have a point that replacement is the only transition between old and new theory?

Throughout I have sought to highlight what objections to reduction amounted to. In the case of Nagel’s distinction between homogeneous and heterogeneous reduction, it seems that this was but an apparent distinction, should the reduction be successful, it is clear that there were no two different domains to start with, but the distinction was wholly artificial, imposed by our human concepts.

Nickles’ approach of having two types of reduction, one fulfilling the standard logically deductive expectations and another some kind of linguistic justification clears the path a little for one, but muddies the waters for the other. While he does not want to admit to the second type being a "loose fitting" cousin of the logically deductive kind, he still employs mathematical elements that would essentially describe some kind of logically deductive type within these.

Feyerabend’s meaning invariance objection is another one of those points where the logically deductive account seems to founder on our linguistic definitions. Simply because we might not be able to exactly describe a possible meaning shift between two linguistically identical terms which belong to different theories it seems we cannot make the reduction.

And finally we have Kitcher who seems to want to show the limits of reduction by demonstrating a causal chain in a complex system. However if any reduction in complex systems is to be achieved we need to be sure the system under investigation is one that is describable with law-like rules and where possibly necessary approximations are mathematically justifiable. Where a phenomenon, however, is best regarded as the consequence of a causal chain, it would seem reduction is the wrong tool to employ.

It seems many objections to reduction are based on linguistic and human concepts being in turn mapped on to a logically deductive domain. Alone the fact that mathematics is a formal language, unlike natural language should alert us to some of the potential problems. Any mathematical description will always be embedded within natural language and it is a matter of interpretation as to what the various terms of any mathematical expression may refer too. As is sometimes said by physicists, mathematicians are people who do not know what they are talking about. Alluding to the way mathematicians may deal with the same formulae as used in physics, but without making the link between the terms and the world out there.

Einstein said in his lecture at the Prussian Academy in 1921 "How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality? " Is this then the answer? It seems one approach for a reductive argument could be the attempt to formulate it in a mathematical or formalistic language. This would prevent problems arising such as implied meaning and meaning change over time. Once such a transformation has been achieved we would be able to logically compare it to any other under investigation. Reductive arguments not following this schemata will continue to face the challenges outlined in this dissertation with little by way of recourse.

In a way, mathematics could provide the bridge laws that Nagel was looking for, logically connecting two theories and at the same time avoiding Feyerabend’s meaning invariance objection, by permitting perhaps linguistic value variations while the mathematical instantiations would remain "meaning neutral".

This approach would also deal with Nickles two types of reduction, on the one hand we would maintain the strict rigours for logically deductive systems while on the other hand a story could be told using the appropriate approximations and limits, even within a mathematical description.

Complex systems too could be approached this way. If it could be described mathematically then it can be "reduced", that is, there is a description that uses the minimal set of elements to explicate the complex. That this may not be a very simple description is neither here nor there. Reduction as we have taken it, was the attempt to describe empirically observable phenomena with the minimal set, it says nothing about our ability to comprehend the result.