Online research in philosophy

Unlike physics and chemistry, the behavioral sciences are historical sciences that explain the fuzzy complexity of social life through historical narratives. Unifying the behavioral sciences through evolutionary game theory would require a nested hierarchy of three kinds of historical narratives: natural history, cultural history, and biographical history. (Published Online April 27 2007).

Relational complexity provides a metric for measuring task demands, and in this respect has much in common with the cognitive complexity and control theory. However, relational complexity does not account for the relative difficulty of different relational types, and appears to underestimate the importance of changes in children's ability to act on the basis of their understanding.

Mathematics, and mechanics conceived as a formal science, have their own proper subject matters, their own proper unities, which ground the characteristic way of constituting problems and solutions in each domain, the discoveries that expand and integrate domains with each other, and so in particular allow them, in the end, to be connected in a partial way with empirical fact. Criticizing both empiricist and structuralist accounts of mathematics, I argue that only an account of the formal sciences which attributes to them objects as well as structure, proper semantics as well as syntax, can do justice to their intelligibility, heuristic force and explanatory power.

A dialogue between the exact, the physical and the natural sciences along with the social and human sciences is both possible and necessary. The place where such a dialogue is truly prosperous is the one of the new sciences of complexity. However, with the study of complex systems, the very traditional status of the sciences changesdramatically.

The views of some historians and philosophers of history as to the possibility of fruitful historical generalization seem at odds with the underlying methodology of the other social sciences. A formal model of the world historical process is here presented within which this apparent contradiction is seen to be resolvable in terms of modern theories of probability and stochastic processes. This is done by giving rigorous form to procedures and statements in the social sciences. A formal treatment of the dependence of an investigation in one discipline on previous studies both in that area and in other social and natural sciences then follows naturally.

Among the aims of the author in this wide-ranging article is to draw attention to the numerous formal sciences which so far have received little scrutiny, if at all, on the part of philosophers of mathematics and of science in general. By the formal sciences the author understands such mathematical disciplines as operations research, control theory, signal processing, cluster analysis, game theory, and so on. First, the author presents a long list of such formal sciences with a detailed discussion of their subject matter and with extensive references to the pertinent literature. Turning to the nature of the formal sciences, the author states that “the formal sciences, though they arose in most cases out of engineering requirements, are sciences and can be pursued without reference to applications”. It is argued, through a wealth of examples, that in a great number of cases the formal sciences permit the attainment of provable certainty about actual parts of the world. As Franklin puts it, “knowledge in the formal sciences, with its proofs about network flows, proofs of computer program correctness, and the like, gives every appearance of having achieved the philosophers’ stone; a method of transmuting opinion about the base and contingent beings of this world into the necessary knowledge of pure reason.” Franklin clearly distinguishes between certainty and necessity: “ g¤g¤g what the mathematician in offering is not, in the first instance, absolute certainty in principle, but necessity. This is how his assertion differs from one made by a physicist. A proof offers a necessary connection between premises and conclusion. One may extract practical certainty from this g¤g¤g but this is a separate step.” Though Franklin explicitly states that there is a gap between necessity and certainty as one passes from mathematical reasoning to applications, the main thrust of the article consists in arguing that the gap is considerably smaller than generally claimed..

The last fifty years have seen the creation of a number of new "formal" or "mathematical" sciences, or "sciences of complexity". Examples are operations research, theoretical computer science, information theory, descriptive statistics, mathematical ecology and control theory. Theorists of science have almost ignored them, despite the remarkable fact that (from the way the practitioners speak) they seem to have come upon the "philosophers' stone": a way of converting knowledge about the real world into certainty, merely by thinking.

Any discussion of the concept of “formal science” must acknowledge that the term is used in different ways, for different purposes, by different people. For some, the formal sciences are defined by the exclusive use of deductive methods for discovering, or reasoning about, the properties of formal, abstract systems. On this view, the formal sciences are synonymous with mathematics, formal logic, and certain branches of linguistics and computer science that emphasize the study of formal languages. For others, “formal science” means something like “exact science”, or “formalized science”. On this view, any scientific discipline that places heavy emphasis on mathematical or logical formalization of key theoretical concepts and theories, could be described as a formal science. This latter conception of formal science is much more liberal than the former, and would include all of physics, much of chemistry, and some parts of biology, ecology, psychology and economics, as well as newer computationoriented disciplines like artificial life and artificial intelligence that do not fit easily within the traditional classification of the sciences.

Professor Franklin is correct to say that there are significant areas of agreement between his account of formal science (Franklin, 1994) and my critique of his account. We both agree that the domain-independence exhibited by the formal sciences is ontologically and epistemically interesting, and that the concept of ‘structure’ must be central in any analysis of domain-independence. We also agree that knowledge of the structural, relational properties of physical systems should count as empirical knowledge, and that it makes sense to talk about an empirical ‘science’ of structure. Where we disagree is over the frequency of occasions where ‘practical certainty’ is actually attained: Franklin argues that practical certainty is not uncommon in the formal sciences, while I argue that there are barriers to practical certainty that Franklin fails to appreciate. In Franklin’s response, he briefly presents and offers criticisms of my two main arguments against his central thesis. Franklin asserts that the flaw in my first argument is that it does not appreciate that modern mathematical models exhibit structural stability, and thus that many of their predictions are robust under small variations in input or parameter values (and hence, insensitive to the inevitable gaps between estimates and actual values). Nevertheless, what I had in mind in using the term ‘realistic modelling situations’ were models of fairly complex natural systems, such as the dripping faucet, spruce budworm and ecosystem ecology examples. My objection is not that mathematical models of such systems cannot legitimately and accurately describe structural properties that are genuinely predicable of real-world systems, but simply that for..

James Franklin has argued that the formal, mathematical sciences of complexity — network theory, information theory, game theory, control theory, etc. — have a methodology that is different from the methodology of the natural sciences, and which can result in a knowledge of physical systems that has the epistemic character of deductive mathematical knowledge. I evaluate Franklin’s arguments in light of realistic examples of mathematical modelling and conclude that, in general, the formal sciences are no more able to guarantee certainty than the natural sciences. Yet the formal sciences are characterized by a ‘domain-independence’ that is philosophically interesting, and I argue that it is this property that Franklin actually employs to distinguish the formal from the natural sciences. I use Einstein’s ‘principle’/‘constructive’ theory distinction to contrast the domain-independence of physical theories with the domain-independence of formal mathematical theories, and show how both kinds of domain-independence function to generate the domain-independence that is observed in the complex systems sciences. © 1999 Elsevier Science Ltd. All rights reserved.