JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Journal of Philosophy, Inc. is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Philosophy.http://www.jstor.org FOUR-COLOR THEOREM 803Such a slippery slope covered with absurd science fiction examples does not constitute much of an argument, perhaps, for the conclusion that mathematics is not an essentially human activity. But I think it shows that an argument is needed for the positive claim, and that one sensibly remains skeptical until someone provides some pretty strong reasons. In other places in this note I have similarly relied, rather cavalierly, on imperfectly drawn distinctions and controversial claims. But I think that enough has been said to put the present points at issue beyond much doubt. The fact that mathematicians have extended their powers of surveying proofs by relying on computers, or might get computers to produce proofs by acknowledged mathematical means of proof, does not show that a new concept of proof is at hand, nor that knowledge in mathematics is more like knowledge in the natural sciences than might be thought on other grounds. PAUL TELLER University of Illinois at Chicago Circle THE FOUR-COLOR THEOREM AND MATHEMATICAL PROOF .I N a recent paper,* Thomas Tymoczko has attributed fundamental significance and novelty to the lately published proof, by Appel, Haken, and Koch, of the famous Four-color Conjecture. Among the theses that Tymoczko puts forth is one in which it is claimed that the Appel-Haken-Koch proof of the Four-color Theorem (hereafter, 4CT) utilized empirical evidence in a manner theretofore unheard of in mathematics. Hence, the fundamental novelty of the Appel-Haken-Koch proof. The need for the appeal to empirical evidence is brought about, in Tymoczko's view, by the fact that the calculation performed by an IBM 370-160A in order to determine the reducibility I of certain configurations is too long to be *"The Four-Color Problem and Its Philosophical Significance," this JOURNAL, LXXVI, 2 (February 1979): 57-83; parenthetical page references in the text will be to this article, unless otherwise noted. 1 We shall not, for the most part, define technical terms (e.g. 'reducibility') that occur in discussing the proof of the 4CT. The interested reader may refer to-Tymoczko's paper or to some of the papers (e.g., those by Bernhart and Haken in the Journal of Graph Theory) cited by Tymoczko in his paper. 0022-362X/80/7712/0803$01.70 X) 1980 The Journal of Philosophy, Inc.

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Abstract. Control sets on grammars are extended to depth-first derivations. It is proved that a context-free language is generated by the depth-first derivations of an arbitrary context-free grammar controlled by an arbitrary regular set. This result is sharpened to obtain a new characterization of the family of derivation-bounded languages: a language L is derivation bounded if and only if it is generated by the depth-first derivations of a context-free grammar G controlled by a regular subset R of the Szilard language of G. The left-derivation-bounded languages are characterized analogously using leftmost derivations. It is proved that a grammar G is nonterminal bounded if and only if the Szilard language defined using only the depth-first derivations of G is regular. Finally, it is proved that if a family of languages C is a trio, a semi-AFL, an AFL, or an AFL closed under k-free substitution, then the family of languages generated using arbitrary context-free grammars controlled by members of C is full, is closed under reversal, and has the closure properties assumed of C.

Abstract.A generalization of leftmost derivation called depth-first derivation is defined. The main result, that the depth-first derivations of an arbitrary phrase-structure grammar generate a context-free language, is proved using a new technique in which families of equivalent depth-first derivations of one grammar are represented by single productions in a new grammar. This result is related to several others, including an analogous result for leftmost derivations, through the theorem of B. Baker [1] that every terminal-bounded grammar generates a context-free language.

I he National Science Foundation played I a pivotal role in the development of the Internet through a program of gisants that provided startup funding for a national backbone net^vork, a system of regional networks, and initial connections for college campuses. By 1995 many of these elements were connected and had begun to rely on the network to support core activities in research and education.The Internet has now grown far beyond its original applications and customers, has hundreds of commercial service providers, and offers a variety of new services. Accordingly, NSF is turning its attention beyond basic connectivity to die next step-high-performance networking-to better sei"ve the increased demands and new opportunities in networking for die research and education community.NSF's new grant program, 96-64, Connections to the Internet, pro\'ides startup funding-this time for upgrades to higher bandwidth (including expanded access to the vBNS, NSF's very high-speed backbone network for research). Its main focus, however, is on the implementation of emerging methods to conu^ol network congestion and guai^antee appropriate levels of sei-vice for different t)'pes of applications.