In 1931 eminent chemist Fritz Paneth maintained that the modern notion of “element” is closely related to (and as “metaphysical” as) the concept of element used by the ancients (e.g., Aristotle). On that basis, the element chlorine (properly so-called) is not the elementary substance dichlorine, but rather chlorine as it is in carbon tetrachloride. The fact that pure chemicals are called “substances” in English (and closely related words are so used in other European languages) derives from philosophical compromises made by (...) grammarians in the late Roman Empire (particularly Priscian [fl. ~520 CE]). When the main features of the constitution of isotopes became clear in the first half of the twentieth century, the formal (IUPAC) definition of a “chemical element” was changed. The features that are “essential” to being an element had previously been “transcendental” (“beyond the sphere of consciousness”) but, by the mid-twentieth century the defining characteristics of elements, as such, had come to be understood in detail. This amounts to a shift in a “horizon of invisibility” brought about by progress in chemistry and related sciences. Similarly, chemical insight is relevant to currently-open philosophical problems, such as the status of “the bundle theory” of the coherence of properties in concrete individuals. (shrink)

During a long and distinguished career, Belgian physical chemist Ilya Prigogine (1917–2003) pursued a coherent research program in thermodynamics, statistical mechanics, and related scientific areas. The main goal of this effort was establishing the origin of thermodynamic irreversibility (the ‘‘arrow of time’’) as local (residing in the details of the interaction of interest), rather than as global (being solely a consequence of properties of the initial singularity – the ‘‘Big Bang’’). In many publications for general audiences, he stated the opinion (...) that this scientific research had great philosophical importance. Prigogine and his colleagues considered that the most recent stages of this research program have been successful, so that the local origins of the arrow of time are now established. There is no scientific consensus as to whether or not this claim is valid. Similarly, there is no consensus on whether the competing global (initial singularity) explanation has been proven. (shrink)

The main scientific problems of chemical bonding were solved half a century ago, but adequate philosophical understanding of chemical combination is yet to be achieved. Chemists routinely use important terms ("element," "atom," "molecule," "substance") with more than one meaning. This can lead to misunderstandings. Eliminativists claim that what seems to be a baseball breaking a window is merely the action of "atoms, acting in concert." They argue that statues, baseballs, and similar macroscopic things "do not exist." When macroscopic objects like (...) baseballs move, exceedingly large numbers (1025) of microscopic components coordinate their activities. Understanding how this happens requires attention to the interactions that link parts into larger units. Eliminativists say that everything that truly exists has causal relationships in addition to those of its components—"nonredundant causality." This paper holds that if a number of entities interact in such a way that the effect of that collection on test objects is different than it would have been in the absence of the interaction, then identification of that collection as a single composite agent is warranted, for purposes to which that difference is relevant. Ordinary "chemical substances" (both elementary materials such as dihydrogen and compounds such as water) fulfill this version of the requirement of nonredundant causality. Other sorts of chemical coherences, including chemical dissipative structures (e.g., flames), also fulfill that criterion. All these types of coherences qualify as "substances" (as that term is used in philosophy) even though they are not all "chemical substances.". (shrink)

Are the conclusions reached by mature sciences merely “likely stories” or are they “really true”? Questions of this sort have been live issues from Étienne Tempier’s March 1277 condemnation of theses of the Radical Aristotelians of Paris to the May 2005 Kansas State Board of Education deliberations on Darwinism. One major difficulty with the view that scientific findings are “really true” is that, from the historical record, all such statements are recognized as being revisable, even replaceable. Several attempts to formalize (...) a notion of “approximate truth” have addressed this difficulty—without notable success. This book proposes a new solution to the problem—a resolution that stands in the tradition of the American Pragmatists: Peirce, James, and Dewey. (shrink)

The author approves “a decidedly nonclassical concept, suggested by Heisenberg, that actuality and potentiality constitute two fundamental species of reality”. “Potentialities of definiteness,” designated “potentia,” which are closely related to Whitehead’s “eternal objects,” are central to this discussion.

This article aims to clarify how aspects of current chemical understanding relate to some important contemporary problems of philosophy. The first section points out that the long-running philosophical debates concerning how properties stay together in substances have neglected the important topic of structure-determining closure. The second part describes several chemically-important types of closure and the third part shows how such closures ground the properties of chemical substances. The fourth section introduces current discussions of structural realism (SR) and contextual emergence: the (...) final sections reconsider the coherence of the properties of substances and concludes that recognition that structures qualify as determinants of specific outcomes—as ‘causes’ as that designation is used in Standard English—clarifies how properties stay together in chemical entities, and by analogy, how characteristics cohere in ordinary items. (shrink)

This volume addresses relations between macroscopic and microscopic description; essential roles of visualization and representation in chemical understanding; historical questions involving chemical concepts; the impacts of chemical ideas on wider cultural concerns; and relationships between contemporary chemistry and other sciences. The authors demonstrate, assert, or tacitly assume that chemical explanation is functionally autonomous. This volume should he of interest not only to professional chemists and philosophers, but also to workers in medicine, psychology, and other fields in which relationships between explanations (...) based on diverse levels of description and investigation are important. (Listed on Google books). (shrink)

The main scientific problems of chemical bonding were solved half a century ago, but adequate philosophical understanding of chemical combination is yet to be achieved. Chemists routinely use important terms with more than one meaning. This can lead to misunderstandings. Eliminativists claim that what seems to be a baseball breaking a window is merely the action of “atoms, acting in concert.” They argue that statues, baseballs, and similar macroscopic things “do not exist.” When macroscopic objects like baseballs move, exceedingly large (...) numbers of microscopic components coordinate their activities. Understanding how this happens requires attention to the interactions that link parts into larger units. Eliminativists say that everything that truly exists has causal relationships in addition to those of its components—“nonredundant causality.” This paper holds that if a number of entities interact in such a way that the effect of that collection on test objects is different than it would have been in the absence of the interaction, then identification of that collection as a single composite agent is warranted, for purposes to which that difference is relevant. Ordinary “chemical substances” fulfill this version of the requirement of nonredundant causality. Other sorts of chemical coherences, including chemical dissipative structures, also fulfill that criterion. All these types of coherences qualify as “substances” even though they are not all “chemical substances.”. (shrink)

What, precisely, is `salt'? It is a certainwhite, solid, crystalline, material, alsocalled sodium chloride. Does any of that solidwhite stuff exist in the sea? – Clearly not.One can make salt from sea water easily enough,but that fact does not establish thatsalt, as such, is present in brine. (Paper andink can be made into a novel – but no novelactually exists in a stack of blank paper witha vial of ink close by.) When salt dissolves inwater, what is present is no (...) longer `salt' butrather a collection of hydrated sodium cationsand chloride anions, neither of which isprecisely salt, nor is the collection. Theaqueous material in brine is also significantlydifferent from pure water. Salt may beconsidered to be present in seawater, but onlyin a more or less vague `potential' way.Actually, there is no salt in the sea. In bothancient and modern treatments of otherimportant chemical concepts, including thenotions of `element', related complication,especially polysemy (terms with multiplemeanings), also occurs.In a recent paper, Paul Needham discussed the(predicable) properties of chemical substances,phases, and solutions. He provided a valuablecharacterization of cases in which severalquantities occupy the same space. He alsoconcluded that solution properties are not`intensive', because solvent and solute do nothave parts in common. He tacitly assumed thatingredients are not altered by their inclusionin a solution. This may be the case in somespecial cases (deutero-benzene dissolved inbenzene, say) but is not true in general – andcertainly does not apply to the case of brine,which Needham used as an example – since theions that exist in the solution, and also theaqueous material there, are quite differentfrom the pure ingredients used in making thesolution. An adequate theory of wholes andparts (mereology) must take into account thatwhen individuals enter combinations ofinteresting sorts they no longer are the verysame individuals that existed prior to thecomposition. It appears that no such formaltheory now actually exists. (shrink)

In 1931 eminent chemist Fritz Paneth maintained that the modern notion of “element” is closely related to (and as “metaphysical” as) the concept of element used by the ancients (e.g., Aristotle). On that basis, the element chlorine (properly so-called) is not the elementary substance dichlorine, but rather chlorine as it is in carbon tetrachloride. The fact that pure chemicals are called “substances” in English (and closely related words are so used in other European languages) derives from philosophical compromises made by (...) grammarians in the late Roman Empire (particularly Priscian [fl. ~520 CE]). When the main features of the constitution of isotopes became clear in the first half of the twentieth century, the formal (IUPAC) definition of a “chemical element” was changed. The features that are “essential” to being an element had previously been “transcendental” (“beyond the sphere of consciousness”) but, by the mid-twentieth century the defining characteristics of elements, as such, had come to be understood in detail. This amounts to a shift in a “horizon of invisibility” brought about by progress in chemistry and related sciences. Similarly, chemical insight is relevant to currently-open philosophical problems, such as the status of “the bundle theory” of the coherence of properties in concrete individuals. (shrink)

Intra-molecular connectivity (that is, chemical structure) does not emerge from computations based on fundamental quantum-mechanical principles. In order to compute molecular electronic energies (of C 3 H 4 hydrocarbons, for instance) quantum chemists must insert intra-molecular connectivity “by hand.” Some take this as an indication that chemistry cannot be reduced to physics: others consider it as evidence that quantum chemistry needs new logical foundations. Such discussions are generally synchronic rather than diachronic —that is, they neglect ‘historical’ aspects. However, systems of (...) interest to chemists generally are metastable . In many cases chemical systems of a given elemental composition may exist in any one of several different metastable states depending on the history of the system. Molecular structure generally depends on contingent historical circumstances of synthesis and separation, rather than solely or mainly on relative energies of alternative stable states, those energies in turn determined by relationships among components. Chemical structure is usually ‘kinetically-determined’ rather than ‘thermodynamically-determined.’ For instance, cyclical hydrocarbon ring-systems (as in cyclopropene) are produced only in special circumstances. Adequate theoretical treatments must take account of the persistent effects of such contingent historical events whenever they are relevant—as they generally are in chemistry. (shrink)

One of the main functions that introductory chemistry courses have fulfilled during the past century has been to provide evidence for the general validity of 'the atomic hypothesis.' A second function has been to demonstrate that an analytical approach has wide applicability in rationalizing many kinds of phenomena. Following R.G. Collingwood, these two functions can be recognized as related to a philosophical 'cosmology' (worldview, weltanshauung) that became dominant in the later Renaissance. Recent developments in many areas of science, and in (...) chemistry, have emphasized the central importance of understanding synthetic, developmental, and evolutionary aspects of nature. This paper argues that these scientific developments, and changes in other aspects of culture, amount to a widespread shift to an alternative cosmology, a quite different general worldview. To the extent that this is the case, introductory chemistry courses ought to be changed in fundamental ways. Rather that having a main focus on analysis to microscopic components, introductory chemistry instruction should emphasize current scientific understanding of the (synthetic) evolutionary origins of the present world. This altered approach would provide good preparation for future professional work, while also making better contact with the perceived concerns of students. (shrink)

By the 1960s many, perhaps most, philosophers had adopted 'physicalism' – the view that physical causes fully account for mental activities. However, controversy persists about what counts as 'physical causes'. 'Reductive' physicalists recognize only microphysical (elementary-particle-level) causality. Many, perhaps most, physicalists are 'non-reductive' – they hold that entities considered by other 'special' sciences have causal powers. Philosophy of chemistry can help resolve main issues in philosophy of mind in three ways: developing an extended mereology applicable to chemical combination; testing whether (...) 'singularities' prevent reduction of chemistry to microphysics; and demonstrating 'downward causation' in complex networks of chemical reactions. (shrink)

Combinations of molecules, of biological individuals, or of chemical processes can produce effects that are not simply attributable to the constituents. Such non-redundant causality warrants recognition of those coherences as ontologically significant whenever that efficacy is relevant. With respect to such interaction, the effective coherence is more real than are the components. This ontological view is a variety of structural realism and is also a kind of process philosophy. The designation ‘process structural realism’ (PSR) seems appropriate.

Whitehead’s cosmology centers on the self-creation of actual occasions that perish as they come to be, but somehow do combine to constitute societies that are persistent agents and/or patients. “Instance Ontology” developed by D.W. Mertz concerns unification of relata into facts of relatedness by specific intensions. These two conceptual systems are similar in that they both avoid the substance-property distinction: they differ in their understanding of how basic units combine to constitute complex unities. “Process Structural Realism” (PSR) draws from both (...) of these approaches in developing an account of how combinations of processes may produce ontologically significant coherences. When a group of processes achieves such closure that a set of states recurs continually, the effects of that coherence differ from what would occur in the absence of that closure. Such altered effectiveness is an attribute of the system as a whole, and would have consequences. This indicates that the network of processes, as a unit, has ontological significance. The closed network of processes, together with the conditions that prevail, constitute the form of definiteness of the coherence. That form continues to obtain as long as the coherence persists. Constituents contribute to, rather than share, that characteristic. Aspects of some recent research in systems biology, microeconomics, and social psychology illustrate the application of PSR. (shrink)

In the characterization of the ArCl2 'van der Waals complex', a recognizable pattern of well-defined peaks is observed in the microwave absorption spectrum. In the control of chaos in a chemical oscillatory reaction the power spectrum progressively becomes simpler, at length yielding a single peak. Since both of these cases generate coherences that are centers of agency, they should be considered to produce new chemical entities. Applicability of this ontological approach to coherences of wider societal interest is suggested.

Nature abounds in compound individuals. Discrete, functioning entities are made up of components which are, in some sense, also individuals. Scientists sometimes need to be concerned with whether aggregates (e.g.. species of plants) or components (e.g., quarks) exist. but such questions are not generally regarded as having great importance for science. It has often happened, however, that scientific developments have had major significance for subsequent philosophical discussion of problems of the one and the many. Recently, there has been considerable increase (...) in scientific understanding of spontaneous development of spatial and temporal organization (structure) in physical. chemical, and biological systems. In an earlier note (PS 11:35), I suggested that this progress in science raises points that may be helpful in dealing with a question of current importance for process philosophy. This paper provides support for that suggestion. The first section introduces the philosophical problem. The middle sections provide brief non-technical introduction to scientific concepts. The final section combines both topics. (shrink)

When a group of processes achieves such closure that a set of states of affairs recurs continually, then the effect of that coherence on the world differs from what would occur in the absence of that closure. Such altered effectiveness is an attribute of the system as a whole, and would have consequences. This indicates that the network of processes, as a unit, has ontological significance. Whenever a network of processes generates continual return to a limited set of states of (...) affairs, the system may function as a “whole”— with respect to appropriate interaction partners. The balance achieved by the processes provides the form of definiteness of a unified agent. The causal powers of such coherent aggregates are indeed just the powers of the “constituents acting in concert”. However, the components act in concert in the specific way they do only because of their inclusion in the closed set of interactions that defines the coherence. This renders the causal powers of the coherence defined by that closure non-redundant, and hence the coherence, as a unit, is ontologically significant. The form of definiteness that provides internal coherence also grounds external efficacy of the societal aggregation. The closure is a structural feature of the coherence — possibly, but not necessarily, apparent in spatial structuring. This approach can provide a unified account that includes quantum microphysics, systems biology, and the philosophy of organism ─ without reducing any of these to another. (shrink)

Philosophers have long debated ‘substrate’ and ‘bundle’ theories as to how properties hold together in objects ― but have neglected to consider that every chemical entity is defined by closure of relationships among components ― here designated ‘Closure Louis de Broglie.’ That type of closure underlies the coherence of spectroscopic and chemical properties of chemical substances, and is importantly implicated in the stability and definition of entities of many other types, including those usually involved in philosophic discourse ― such as (...) roses, statues, and tennis balls. Characteristics of composites are often presumed to ‘supervene on’ properties of components. This assumption does not apply when cooperative interactions among components are significant. Once correlations dominate, then adequate descriptions must involve different entities and relationships than those that are involved in ‘fundamental-level’ description of similar but uncorrelated systems. That is to say, descriptions must involve different semantics than would be appropriate if cooperative interactions were insignificant. This is termed ‘Closure Henri Poincaré. Networks of chemical reactions that have certain types of closure of processes display properties that make other more-complex coherences possible. This is termed ‘Closure Jacques Cauvin.’ Each of these three modes of closure provides a sufficient basis for warranted recognition of causal interaction, thus each of them has epistemological significance. Other modes of epistemologically-important closure probably exist. It is important to recognize that causal efficacy generally depends on closure of relationships of constituents. (shrink)

This paper recommends that chemistry educators shift to a different ‘idea of nature’, an alternative ‘worldview.’ Much of contemporary science and technology deals in one way or another with dynamic coherences that display novel and important properties. The notion of how the world works that such studies and practices generate (and require) is quite different from the earlier concepts that are now integrated into science education. Eventual success in meeting contemporary technological and social challenges requires general diffusion of an overall (...) outlook that focuses on creative generation of novel and useful coherences, replacing a worldview that concentrates on analysis of pre-existing items to minimum constituents. Such a shift in emphasis would amount to general adoption of a new basic model of how nature functions. Chemistry educators can and should provide leadership for this urgently-needed development. (shrink)

During a long and distinguished career, Belgian physical chemist Ilya Prigogine (1917–2003) pursued a coherent research program in thermodynamics, statistical mechanics, and related scientific areas. The main goal of this effort was establishing the origin of thermodynamic irreversibility (the ‘‘arrow of time’’) as local (residing in the details of the interaction of interest), rather than as global (being solely a consequence of properties of the initial singularity – the ‘‘Big Bang’’). In many publications for general audiences, he stated the opinion (...) that this scientific research had great philosophical importance. Prigogine and his colleagues considered that the most recent stages of this research program have been successful, so that the local origins of the arrow of time are now established. There is no scientific consensus as to whether or not this claim is valid. Similarly, there is no consensus on whether the competing global (initial singularity) explanation has been proven. (shrink)

Origin of a dissipative structure in a chemical dynamic system: occurs under the following constraints: 1) Affinity must be high. (The system must be far from equilibrium.); 2) There must be an auto-catalytic process; 3) A process that reduces the concentration of the auto-catalyst must operate; 4) The relevant parameters (rate constants, etc.) must lie in a range corresponding to a limit cycle trajectory. That is, there must be closure of the network of reaction such that a state sufficiently close (...) to the prior condition is achieved at the corresponding part of each oscillation. If these constraints continue to be met interactions of the system with the rest of the world are quite different in the presence of the dissipative structure than they would be in the absence of that self-organized coherence. Closure of a network of relationships which gives rise to a dissipative structure is a 'unit-determining' feature. The effects of the structure as a whole are the resultants of the effects of the components, but the concentrations of the components at any instant are effects of the closure of the limit cycle. This is an influence on components arising from what those components constitute --- downward causation. (shrink)

A report on an international meeting held at Georgetown University on the Philosophy of Chemistry.

Among the many recent volumes dealing with tension between "creation" and "evolution," this one is highly unusual. It does not presuppose that these two ideas are intrinsically at odds. Rather, the general thrust of the work is to explore the possibility that "the two doctrines are not incompatible, that they may indeed be taken to complement one another in important ways.".

This book aims to "develop the idea that evolution is an axiomatic consequence of organismic information and cohesion systems obeying the second law of thermodynamics in a manner analogous to, but not identical with, the second law's usual application in physical and chemical systems." The authors "adhere to a particular methodological approach called phylogenetic systematics." They have "devoted most of their primary research efforts to discovering historical effects in developmental patterns." Finding that "such historical effects seem ubiquitous," they "began to (...) believe that these historical effects were constraints on the evolutionary process, suggesting the influence of a natural law of history.". (shrink)

At the end of this impressive work, Michael Lockwood observes: "Philosophers, especially British philosophers tend, in my experience, to combine a rather complacent ignorance of science with an excessive respect for it". The author himself seems to be a definite exception to this generalization, since he reports that he has spent more than twenty years "thinking about the mind body problem and the interpretation of quantum mechanics" and displays a critical attitude toward statements based on scientific research. This book outlines (...) a common interpretive framework that may be able to resolve tensions that have long characterized understandings of human mental functioning and of microphysics. The work is addressed to both philosophers and scientists. (shrink)

Magnani, Lorenzo (2001), Abduction, Reason, and Science: Processes of Discovery and Explanation. New York: Kluwer Academic/ Plenum Publishers. Magnani. Lorenzo, and Nancy Nersessian (eds.) (2002), Model-Based Reasoning: Technology, Science, Values. New York: Kluwer Academic/ Plenum Publishers. Joseph E. Earley, Sr. (ed.), Chemical Explanation: Characteristics, Development, Autonomy, Annals of the New York Academy of Sciences, vol. 988. New York Academy of Sciences (2003), 370 pp., $130.00 (cloth).

Introduksiyon -/- 50-50. Fifty-fifty. Sa bigkas ng mga Pinoy, “pipti-pipti.” Sa kontekstong Pilipino, pantukoy ito sa kalagayan ng taong nasa kritikal na kalagayan. Limampung porsyento ng tsansang makaligtas, limampung porsyento ng tsansang masawi. Kilala ito sa iba pang katawagan sa Pilipinas bilang “naghihingalo,” “nasa bingit ng kamatayan” o “nag-aagawbuhay.” Ganito mailalarawan ang kalagayan ng lipunang Pilipino sa ilalim ng Batas Militar ni Ferdinand Marcos, Sr. Para sa mga pabor sa Batas Militar at mga loyalista ni Marcos, 50-50 dahil sa banta (...) raw ng komunismo at tangkang pagpapabagsak sa Republika. Para sa mga kritiko’t lumaban sa diktadura, 50- 50 dahil sa pang-aabuso’t paniniil ng rehimen laban sa taumbayan. 50-50. Ito ang sa tingin nami’y nababagay na pamagat ng aklat na ito. Una, ginugunita ng sambayanang Pilipino ngayong 2022 ang ika-50 taong anibersaryo ng Batas Militar. Idineklara ni Marcos noong Setyembre 21, 1972 ang Proclamation 1081 na naglagay sa buong bansa sa ilalim ng kamay na bakal. Pangalawa, itinatampok ng aklat ang 50 personalidad na may mga direkta at aktibong papel para o laban sa Batas Militar. Dalawampu’t lima (25) ang nagmula sa bakuran ng rehimeng Marcos. Bukod sa diktador at kanyang pamilya, sila ang mga pangunahing tagapagbalangkas at tagapagpatupad ng Batas Militar. Kasabwat sila sa kabuuang operasyon ng mga mekanismo ng awtoritaryanismong ito – mapapulitikal, ekonomiko o sosyokultural man. Sila rin ang mga nakinabang, kung hindi man nagpakasasa, sa lahat ng mga ganansyang dulot ng kolaborasyon at paglilingkod sa diktadura. Ang nalalabing 25 ay binubuo naman ng mga humamon at nanindigan laban sa kamay na bakal. Magkakaiba ang kanilang pinagmulan, mga adhikain at dulog sa pagtindig laban sa Batas Militar – armadong pakikibaka, pagkilos sa mga lansangan, paglaban mula sa pulpito, at iba pa. (shrink)

Theories of discourse bring to realism new ideas about how knowledge develops and how representations of reality are influenced. We gain an understanding of the conceptual aspect of social life and the processes by which meaning is produced. This collection reflects the growing interest realist critics have shown towards forms of discourse theory and deconstruction. The diverse range of contributions address such issues as the work of Derrida and deconstruction, discourse theory, Eurocentrism and poststructuralism. What unites all of the contributions (...) is a sense that it is essential to provide a realist alternative to the hitherto dominance of social constructionism, hermeneutics and postmodernism, over many of the issues discussed. By developing a realist perspective the different authors attempt to embed discourse within the structured nature of the reality of the world. Realism can situate language, discourse and ideology within context specific, or 'causally efficacious' circumstances. Realism can help to uncover issues of power, representation, and subjectivity and how discursive and other social practices produce real effects. This can help us understand the manner in which social structures are reproduced through various forms of ideology and discourse. And by knowing this, we can start to address questions concerning human emancipation and how the world is to be transformed. (shrink)

Topics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory. An accessible, precise, and comprehensive approach is adopted in the treatment of each topic. The ability (...) of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills. All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers. As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book. (shrink)