Philosophical Foundations

However, for other problems, notably those of statistical mechanics, entropy describes a force of nature. In those cases, constraints such as averages over a row of probabilities are not only numbers that are known to an observer, they are also physical constraints that are imposed upon a physical system from outside (see Chapter 10). In this case, the Principle of Fair Apportionment is a description of nature s symmetries. When there is an underlying symmetry in a problem (such as t outcomes that are equivalent), then Fair Apportionment says that external constraints are shared equally between all the possible states that the system can occupy (grid cells, in our earlier examples). In this case, entropy is more than a description about an observ-er. It describes a tendency of nature that is independent of an observer. 

Historically, statistical mechanics has been framed in terms of the frequency interpretation. JW Gibbs (1839-1903), American mathematical physicist and a founder of chemical thermodynamics and statistical mechanics, framed statistical mechanics as a counting problem. He envisioned an imaginary collection of all possible outcomes, called an ensemble, which was countable and could be taken to the limit of a large number of imaginary repetitions. To be specific, for die rolls, if you want to know the probability of each of the six outcomes on a roll, the ensemble approach would be to imagine that you had rolled the die N times. You would then compute the number of sequences that you expect to observe. This number will depend on N. On the other hand, if the outcomes can instead be described in terms of probabilities, the quantity N never appears. For example, the grid of cells described in Table 6.2 describes probabilities that bear no trace of the information that N = 1000 die roUs w-ere used to obtain Table 6.1. 

These issues are largely philosophical fine points that have had little implication for the practice of statistical mechanics. We will sometimes prefer the counting strategy, and the use of S/k = InW, but S/k = - Ptb Pi wdl be more convenient in other cases. 

The entropy Sipii. pij. is a function of a set of probabilities. The distribution of p,j s that cause 5 to be maximal is the distribution that most fairly apportions the constrained scores between the individual outcomes. That is, the probability distribution is flat if there are no constraints, and follows the multiplication rule of probability theory if there are independent constraints. If there is a constraint, such as the average score on die rolls, and if it is not equal to the value expected from a uniform distribution, then maximum entropy predicts an exponential distribution of the probabilities. In Chapter 10, this exponential function will define the Boltzmann distribution law. With this law you can predict thermodynamic and physical properties of atoms and molecules, and their averages and fluctuations. How-ever, first we need the machinery of thermodynamics, the subject of the next three chapters. 

Calculating the entropy of mixing. Consider a lattice with N sites and n green particles. Consider another lattice, adjacent to the first, with M sites and m red particles. Assume that the green and red particles cannot switch lattices. This is state A. 

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We use the term philosophical substructure of a curriculum (Van Berkel, 2005, pp. 33, 52), whereas Schwab uses the term syntactical or also methodological substructure. The philosophical substructure of a curriculum contains besides methodological assumptions a number of, often implicit, philosophical foundations of a curricnlnm. 

Carnap, R. Philosophical foundations of physics. New York Basic Books, Inc. 1966. 

It is interesting to conjecture that if chemists have spent less effort than physicists reflecting on the philosophical foundations of their subject, the reason may be illuminated by Bas Van Frassen s claim that the closer contemporary physicists are to experimental work, the less interested they are in fundamental questions. Chemists have always been closer to experimental work, more thoroughly involved in the laboratory than natural philosophers, and less interested in idealizations of phenomena. 

We have seen how Daltons idea of quantized units of weight was quickly accepted and its virtues recognized. But serious reservations about Daltons philosophical foundation were expressed by nearly everyone. Few had doubts that matter was made of atoms, but it was widely believed with Lavoisier that we know nothing at all about the atoms themselves. All this was neatly summed up by Berthollet when he wrote We know bodies only by the effects which they produce by their action, but nothing in this action can inform us of the distinct properties of their ultimate atoms. Flenrys... 

The Institute is named for Cato s Letters, libertarian pamphlets that were widely read in the American Colonies in the early 18th century and played a major role in laying the philosophical foundation for the American Revolution. 

Beginnings of cosmochemistry (and geochemistry) Philosophical foundations Meteorites and microscopy Spectroscopy and the compositions of stars Solar system element abundances Isotopes and nuclear physics Space exploration and samples from other worlds New sources of extraterrestrial materials Organic matter and extraterrestrial life ... 

Microscopic approaches have scored many notable successes, including the entire worlds of chemistry, nuclear power, and solid state electronics. To those who are very much concerned with the logical and philosophical foundations of things, the logical untidiness of micro approaches has been a bit of an embarrassment. It is indeed a brilliant accomplishment to deduce the second law in the style of Carnot, but the accomplishments in electronics in developing, say, a theory of amorphous semiconductors are also impressive even if the theory seems less firmly grounded. 

M.A. Grodin (ed.) Meta Medical Ethics The Philosophical Foundations of Bioethics. 1995... 

Szasz, T. 1988c. The Theology of Medicine The Political-Philosophical Foundations of Medical Ethics. Syracuse, NY Syracuse University Press [Baton Rouge Louisiana State University Press, 1977]. 

Shrader-Frechette, K. S. (1991). Risk and Rationality. Philosophical Foundations for Populist Reforms. University of California Press, Berkeley. 

Moehalov I.I. (1971) Scientific aitd Philosophical Foundations of V.I. Vernadsky s Worldview. 

Shelby, Tommie. 2005. We Who Are Dark The Philosophical Foundations of Black Solidarity. Cambridge, Mass. Harvard University Press. 

The study of statistics is, for the most part, a study of various tests and their calculations. However, the application of statistics depends very strongly on a philosophical foundation. In order to avoid bias in the results, the appropriate statistical test must be decided upon before the data are seen. Once the data are known, even partly, it is tempting to apply a particular statistical procedure that seems to fit patterns in the data. Although there is no difference in the outcome of the calculation when done this way, the interpretation of the result can be erroneous. Studying the data before a specific statistical test is applied violates the assumption of random error. 

His book reflected his various interests, and it was truly transdisci-plinary, providing a methodological foundation for science and engineering for centuries to come. In part two of the book, he described the scientific method and its four governing rules, which can be considered as a philosophical foundation of morphological analysis. These rules are as follows (Descartes 1960) ... 

The reason for including a discussion of the philosophical foundations of mathematics and science is to demonstrate that traditional two-valued logic is but one way of setting up a deductive system for communicating scientific ideas there are alternative logics, and fuzzy logic as presented in Chapter 6 is one of them. 

In this chapter, I have tried to lay, as far as possible, a,philosophical foundation for the rest of this book. Therefore, it is important that the basic ideas presented are clearly appreciated and I will attempt now to summarise them. 

Bennett MR, Hacker PMS (2003) Philosophical foundations of neuroscience. Blackwell, Oxford Binnig G, Rohrer H (1986) Scanning tunneling microscopy. IBM J Res Dev 30 4... 

Since the publication over a quarter of a century ago of my original paper Philosophical Foundations of Classical Evolutionary Classification, I have become increasingly interested in systems of explanation in science and especially in evolutionary biology (see Bock, 1978, 1981, 1988, 1991, 1992, 2000 Szalay and Bock, 1991). This was prompted by concerns in 1974 when I introduced into systematics Popper s ideas that the nature of explanation and the methods of testing theoretical statements in classification and phylogeny do not fit easily or at all into the Popperian mold. 

Bock, W.J., Philosophical foundations of classical evolutionary classification, Syst. Zoo/., 22, 375-392, 1974. 

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