Science has become, as all nonspecialists know to their cost, increasingly mathematical; science textbooks and research papers, even popularising articles in Scientific American, are littered with graphs, numbers, mathematical symbols and equations. This has prompted the question “What exactly is the function of mathematics in science?” For example, could one understand a theory such as Einstein’s theory of special relativity without having knowledge of any sophisticated mathematics?

One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also (...) varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics. (shrink)

This dissertation makes two primary contributions. The first three chapters develop an interpretation of Carnap's Meta-Philosophical Program which places stress upon his methodological analysis of the sciences over and above the Principle of Tolerance. Most importantly, I suggest, is that Carnap sees philosophy as contiguous with science—as a part of the scientific enterprise—so utilizing the very same methods and subject to the same limitations. I argue that the methodological reforms he suggests for philosophy amount to philosophy as the explication of (...) the concepts of science through the construction and use of suitably robust meta-logical languages. My primary interpretive claim is that Carnap's understanding of logic and mathematics as a set of formal auxiliaries is premised upon this prior analysis of the character of logico-mathematical knowledge, his understanding of its role in the language of science, and the methods used by practicing mathematicians. Thus the Principle of Tolerance, and so Carnap's logical pluralism, is licensed and justified by these methodological insights. This interpretation of Carnap's program contrasts with the popular Deflationary reading as proposed in Goldfarb & Ricketts. The leading idea they attribute to Carnap is a Logocentrism: That philosophical assertions are always made relative to some particular language, and that our choice of syntactical rules for a language are constitutive of its inferential structure and methods of possible justification. Consequently Tolerance is considered the foundation of Carnap's entire program. My third chapter argues that this reading makes Carnap's program philosophically inert, and I present significant evidence that such a reading is misguided. The final chapter attempts to extend the methodological ideals of Carnap's program to the analysis of the ongoing debate between category- and set-theoretic foundations for mathematics. Recent criticism of category theory as a foundation charges that it is neither autonomous from set theory, nor offers a suitable ontological grounding for mathematics. I argue that an analysis of concepts can be foundationally informative without requiring the construction of those concepts from first principles, and that ontological worries can be seen as methodologically unfruitful. (shrink)

The paper is the final, fifth part of a series of studies introducing the new conceptions of “Hilbert mathematics” and “ontomathematics”. The specific subject of the present investigation is the proper philosophical sense of both, including philosophy of mathematics and philosophy of physics not less than the traditional “first philosophy” (as far as ontomathematics is a conservative generalization of ontology as well as of Heidegger’s “fundamental ontology” though in a sense) and history of philosophy (deepening Heidegger’s destruction of it from (...) the pre-Socratics to the Pythagoreans). Husserl’s phenomenology and Heidegger’s derivative “fundamental ontology” as well as his later doctrine after the “turn” are the starting point of the research as established and well known approaches relative to the newly introduced conception of ontomathematics, even more so that Husserl himself started criticizing his “Philosophy of arithmetic” as too naturalistic and psychological turning to “Logical investigations” and the foundations of phenomenology. Heidegger’s “Aletheia” is also interpreted ontomathematically: as a relation of locality and nonlocality, respectively as a motion from nonlocality to locality if both are physically considered. Aristotle’s ontological revision of Plato’s doctrine is “destructed” further from the pre-Socratics' “Logos” or Heideger’s “Language” (after the “turn”) to the Pythagoreans “Numbers” or “Arithmetics” as an inherent and fundamental philosophical doctrine. Then, a leap to contemporary physics elucidates the essence of ontomathematics overcoming the Cartesian abyss inherited from Plato’s opposition of “ideas” versus “things”, and now unifying physics and mathematics, particularly allowing for the “creation from nothing” instead of the quasi-scientific myth of the “Big Bang”. Furthermore, ontomathematics needs another interpretation of arithmetic, propositional logic and set theory in the foundations of mathematics, where the latter two ones are both identified with Boolean algebra, and the former is considered to be a “half of Boolean algebra” in the exact meaning to be equated to it after doubling by a dual anti-isometric counterpart of Peano arithmetic. That unified algebraic realization of the foundations of mathematics is related to Hilbert mathematics in both “narrow and wide senses” where the latter is isomorphic to the qubit Hilbert space, thus underlying all the physical world by the newly introduced substance of quantum information being physically dimensionless and generalizing classical information measured in bits. The substance of information, whether classical or quantum, visualizes the way of the unification of physics and mathematics by merging their foundations in Hilbert arithmetic and Hilbert mathematics: thus how ontomathetics is a “first philosophy”. The relation of ontomathematics to the Socratic “human problematics”, furthermore being fundamental for Western philosophy in Modernity, is discussed. Ontomathematics implies its “substitution” by abstract information (or by “subjectless choice” relevant to it), thus “obliterating the human outline on the ocean beach sand” (by Michel Foucault’s metaphor). A reflection back from the viewpoint of mathematic to Western philosophy as the philosophy of locality ends the study. (shrink)

The economist Paul Samuelson acknowledged that he was a disciple of Edwin Bidwell Wilson (1879–1964), an American polymath who was a protégé of Josiah Willard Gibbs. Wilson’s influence on the development of sciences in America has been relatively neglected, as he mostly acted behind the scenes of academia at the organizational and pedagogical fronts. At the basis of his activism were original ideas about the foundations of mathematics and science. This essay reconstructs Wilson’s career and foundational discussions, which evolved as (...) he reacted to David Hilbert’s mathematics, Bertrand Russell’s logic, Henri Poincaré’s conventionalism, Karl Pearson’s statistics, Charles Sanders Peirce’s inference theory, and Lawrence Henderson’s work in social sciences. In brief, after having started his professional life as a mathematician at Yale University (1902–1907), Wilson marginalized himself from mathematics and joined other academic communities and fields, working in mathematical physics at MIT (1907–1922) and in statistics, social science, and economics at Harvard University (1922–retirement). He defined mathematics as a language, meaning by this that mathematics and science were reciprocally indispensable, with the former providing structure and the latter meaning. In an American exceptionalist spirit, Wilson also meant to suggest that homegrown mathematical and scientific traditions should prevail over European ones. (shrink)

The aim of Reverse Mathematics (RM for short) is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five (...) systems together are called the ‘Big Five’. The aim of this paper is to greatly extend the Big Five phenomenon as follows: there are two supposedly fundamentally different approaches to RM where the main difference is whether the language is restricted to second-order objects or if one allows third-order objects. In this paper, we unite these two strands of RM by establishing numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalizations and variations of the aforementioned third-order theorems fall far outside of the Big Five. (shrink)

As idealized descriptions of mathematical language, there is a sense in which formal systems specify too little, and there is a sense in which they specify too much. On the one hand, formal languages fail to account for a number of features of informal mathematical language that are essential to the communicative and inferential goals of the subject. On the other hand, many of these features are independent of the choice of a formal foundation, so grounding (...) their analysis on a particular choice of a formal system introduces unnecessary specificity. This chapter begins to map out the design features of mathematical language without descending to the level of formal implementation, drawing on examples from the mathematical literature and insights from the design of computational proof assistants and their libraries. (shrink)

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes. Yanofsky describes (...) simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known. Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there. (shrink)

During the second half of the 20th century, attempts were made to operationally redefine various social activities, including those related to science, the military, administration and industry. These attempts were aided by scientific and technical innovations developed in the Second World War, and subsequently by the increase in use of automation in various domains. This Ph.D. thesis addresses these attempts from a sociohistorical perspective, focusing on the specific case of archaeology. During this period, the domain of archaeology underwent a process (...) of disciplinarisation and professionalisation. The same occurred in applied mathematics and then computer science: this thesis focuses on the relationships between these three domains. In France, during the 1950's and 1960's, there were significant methodological and conceptual innovations. Their subsequent scientific recognition, was, however, relatively minor. In archaeology, innovations related to applied mathematics and automatics did not lead to the emergence of an archaeological speciality based on computation. This situation was in striking contrast to what happened in other scientific domains and in archaeology in other countries, where new theoretical and methodological Anglophone definitions in ‘New Archaeology’ were spreading worldwide. This thesis explores three collective attempts to redefine the conceptual and methodological basis of archaeology, led by Georges Laplace, Jean-Claude Gardin and Jean Lesage, across France, Spain and Italy. These cases are completed by other people who had significant careers in both engineering and archaeology. In general, this thesis studies a scientific activity by investigating the cognitive and social aspects of peoples’ methodological contributions. Three models of the relationships between experts in a scientific domain and experts in an applied science (here mathematics and computing) are empirically identified and described. The effects of introducing mathematical and automation procedures on the division of labour and the distribution of recognition are analysed. The success or failure of the methodological propositions are discussed with reference to several factors and models of scientific innovation. This thesis generates new information on the development of rescue and preventive archaeology and on the use of digital technologies in human sciences. The analysis draws on 82 interviews, 23 archives and several bibliometric datasets (extracted from pre-existing databases or constructed for the purpose of this research). Mirroring the archaeological propositions under study, this research also intends to illustrate the possible use of computing and formalised procedures in social sciences. The documentation and demonstrative principles underlying this work, implemented by using Wiki, the methods of literate programming and reproducible research, are themselves analysed. (shrink)

Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and to (...) computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives. Predicates are constructed from two atomic binary predicates (which denote the relations of identity and set-theoretic membership) by repeated applications of four operators that are analogues of the well-known operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics. The book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; first-order logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and decision problems in equational logic. (shrink)

Language, through the discrete nature of linguistic names and strictly determined grammatical rules, creates absolute, “quantized”, sharply separated “facts” within the external world that is continuous, “fuzzy” and relational in its essence. Therefore, it is similar, in some important sense, to magic, which attributes causal and creative power to magical words and formulas. On the one hand, language increases greatly the effectiveness of the processes of thinking and interpersonal communication, yet, on the other hand, it determines and distorts (...) to a large extent the picture of the world created within the mind. The relatively smallest (but still significant) magical admixture is present in science, because of its relatively reliable methodology, while the largest is found in religion and a large part of philosophy. The magical nature of language also manifests itself in logic and mathematics that refer to ill determined, fuzzy objects, sets and relations in the real world. The meaning of linguistic names is based on the conceptual network—an epiphenomenon (continuous in its essence) of the neural network—where interactions between particular concepts are based on the relation of connotation. The names and formulas of language correspond to these concepts which are best separated and determined. A direct relation of denotation between the elements of language and “facts” of the world is an illusion. While we cannot dispense with language because of its immense usefulness, we must remember about its “fact-creating” nature and influence on our thought and cognitive processes. The picture of the reality created as the result of them is to a large extent formed and deformed by the nature of language, and not by the “immanent” properties of the world in itself. (shrink)

We propose a model of physics that blends Rovelli’s relational quantum mechanics (RQM) interpretation with the language of finite information quantities (FIQs), defined by Gisin and Del Santo in the spirit of intuitionistic mathematics. We discuss deficiencies of using real numbers to model physical systems in general, and particularly under the RQM interpretation. With this motivation for an alternative mathematical language, we propose the use of FIQs to model the world under the RQM interpretation, wherein we view (...) the propensities that make up a FIQ as quantifications of potential interaction. Under this model, the stable facts, relative facts, and shifting perspectives that make up the relational interpretation correspond to shifting digits and propensities of the FIQs. The model’s predictions agree with those of both classical and quantum physics, and it is indeterministic. We also propose explanations, with examples, for how the propensities of a FIQ are distributed, and how its digits become actualized. This is equivalent to the notion of the measurement problem, and the question of what causes wave function collapse. In short, by stepping through the “new door” opened by the language of FIQs, we attempt to describe the world under the relational interpretation. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviews 91 THE ROOTS OF MODERN LOGIC ALASDAIR URQUHART Philosophy/ U. ofToronto Toronro, ON, Canada M5S IAI [email protected] I. Grattan-Guinness. The Searchfor Mathematical Roots,r870--r940: logics, Set Theoriesand the Foundations of Mathematicsfrom Cantor through Russellto Godel Princeron: Princeton U. P.,2000. Pp. xiv,690. us$45.oo. Grattan-Guinness's new hisrory of logic is a welcome addition to the literature. The title does not quite do justice ro the book, since it begins with (...) the prehistory of English work in algebraic logic, including the work of the French analycicalschool, and extends to just after Godel's great incompleteness paper of 1931.The core of the book, though, is the philosophical and mathematical developmems leading up to and immediately following from chework ofWhicehead and Russell.Russell is ac chehearcof che book, and as a whole che hisrory forms an imporranr conrribucion ro Russell scudies. Commemarors on Russell have often confined themselves to a rather narrow hisrorical perspeccivein which Russell is seen as the (problemacic) heir of Gottlob Frege, and few ocher hisrorical figures (ocher than Peano) emer imo che picture. Graccan-Guinness corrects chis hisrorica.limbalance by placing Russell in che mucl1 wider context of the development of machemacics on che continent, and in parricu.laremphasizing strongly che key influence of Camor on 92 Reviews Russell's work. Cantor has usually received short shrift from philosophers, as unlike Frege,he appears rarher naive from rhe philosophical point of view. The story proper begins in Chaprer 2 wirh L-igrange'sversion of analysis in which rhe basic concepts were to be defined in rerms of algebraic manipularion of power series. This lead to rhe founding of rhe Analyrical Socierywhere Babbage, Herschel and Peacock were active. It is in chis English tradicion of algebraic analysis char rhe pioneering work of De Morgan and Boole found irs roors. Grarran-Guinness givesa derailed account of the work of both logicians, although his discussion of Boole's merhods does not seem entirely adequate. The question is:what are we to make of Boole'spuzzling insistence chat expressions like x + y are "uninterpretable", while he nevertheless manipulated them freely in his mathematical derivations? It is not correct to say that the addition sign can only link disjoint class symbols, since ir is belied by Boole's formal pracrice. A possible solution has been suggested by Hailperin, in his book on Boole's logic, in which he proposes interpreting the "uninterpretable" expressions as denoting signed mulrisets. Oddly, Grattan-Guinness refers to Hailperin 's work,' bur elsewhere (p. 42) adopts rhe view that Boole's addition sign could only link disjoint classes.The chapter concludes with brief accounts of the work of Cauchy, Weierstrass and Bolz.'lno. The next chapter is a detailed account of rhe work of Cantor and his creation of Mengenlehre. The origins of set theory in the theory of trigonometrical series, and rhe ensuing discoveryof rransfinire ordinal and cardinal numbers, are described clearly and succinctly. In addition, the chapter contains an account of Dedekind's philosophy of arithmetic and Cantor's philosophy of mathematics. Cantor's philosophy is an uneasy blend of formalism, platonism and idealism, and has tmderstandably aroused little enthusiasm among philosophers of matl1ematics, alrhough Michael Hallerr has recently smdied it in detail. GrammGuinness emphasizes, rhough, Cantor's magnificent marhematica.l achievements, in defining and clarifying basic conceprs such as measure, dimension and cardinality of sets. Chaprer 4 is a rather miscellaneous chapter, in which six partly independent, partly interrwined stories are told. It begins with developments in sectheory in Germany and France up to the turn of rhe century, goes on to discuss American logic in rhe work of C. S. Peirce and his students, and continues the rheme of algebraic logic with Schroder and his logic of relatives. The remainder of the chapter is given over to Frege, Husserl and Hilbert. The section on Frege is one of rhe more idiosyncratic parts of the book. Grattan-Guinness distinguishes berween Frege, "a mathematician who wrore in ' T Hailperin, Book's logic and Probability, 2nd ed. (Amsterdam: North-Holland, 1986). Reviews 93 German, in a markedly Platonic spirit", and Frege, "a philosopher of language and founder of... (shrink)

We introduce a notion of relative efficiency for axiom systems. Given an axiom system Aβ for a theory T consistent with S12, we show that the problem of deciding whether an axiom system Aα for the same theory is more efficient than Aβ is II2-hard. Several possibilities of speed-up of proofs are examined in relation to pairs of axiom systems Aα, Aβ, with Aα ⊇ Aβ, both in the case of Aα, Aβ having the same language, and in the (...) case of the language of Aα extending that of Aβ: in the latter case, letting Prα, Prβ denote the theories axiomatized by Aα, Aβ, respectively, and assuming Prα to be a conservative extension of Prβ, we show that if Aα — Aβ contains no nonlogical axioms, then Aα can only be a linear speed-up of Aβ; otherwise, given any recursive function g and any Aβ, there exists a finite extension Aα of Aβ such that Aα is a speed-up of Aβ with respect to g. (shrink)

Taken at face value, a programming language is defined by a formal grammar. But, clearly, there is more to it. By themselves, the naked strings of the language do not determine when a program is correct relative to some specification. For this, the constructs of the language must be given some semantic content. Moreover, to be employed to generate physical computations, a programming language must have a physical implementation. How are we to conceptualize this complex package? (...) Ontologically, what kind of thing is it? In this paper, we shall argue that an appropriate conceptualization is furnished by the notion of a technical artifact. (shrink)

The current technical report of the research project investigating how the mathematical dimension of gambling is reflected in the communication and texts associated with the gambling industry raises the problem of the adequacy of sampling and proposes a new approach in this respect. The qualitative analysis of the reviewed websites is extended to a deeper analysis of language and also to the organization and structure of websites’ content. Although not stated as a goal of the initial project, the (...) research will also try to identify relationships between the specificity of content relative to the mathematical dimension of gambling, and the management and marketing of the website. An overview of the results of the statistical analysis of the current sample is also presented. (shrink)

While it is relatively clear what the subject matter of empirical sciences is, puzzles persist about the proper subject matter of mathematics. The logicists took mathematics to be concerned solely with deductive arguments. Their programme attempted to combine three incompatible claims: that mathematics is a body of truths, that it is non‐empirical, and that it employs proofs obeying the rules of classical logic. By giving up the third contention, it becomes possible to salvage the logicist programme and to explain better (...) mathematical practice, including the fact of its application in empirical disciplines. (shrink)

The paper discusses a relatively underexamined element of Wittgenstein’s philosophy of mathematics associated with his critique of set theory. I outline Wittgenstein’s objections to the theories of Dedekind and Cantor, including the confounding of extension and intension, the faulty definition of the infinite set as infinite extension and the critique of Cantor’s diagonal proof. One of Wittgenstein’s major objections to set theory was that the concept of the size of infinite sets, which Cantor expressed by means of symbols אₒ and (...) c, had no application, i.e. that there was no grammatical technique that could show how such expressions were to be used. Notions of set theory are, so to speak, exterior – they find themselves outdoors, outside of what we usually do. They form a discourse that takes us beyond the horizon of everydayness and commonality. They are like an engine idling of the language of mathematics. (shrink)

The paper introduces mathematical encoding for subjective experience and meaning in natural cognition. The code is based on a quantum-theoretic qubit structure supplementing classical bit with circular dimension, functioning as a process-causal template for representation of contexts relative to the basis decision. The qubit state space is demarcated in categories of emotional experience of animals and humans. Features of the resulting spherical map align with major theoreties in cognitive and emotion science, modeling of natural language, and semiotics, suggesting (...) several generalizations and improvements. The developed model bridges psychological, quantum-theoretic, and semiotic perspectives, allowing for an integrative account of subjectivity, agency, and meaning in living Nature.Graphical abstract. (shrink)

The aim of this paper is to elucidate the question of whether Newtonian mechanics can be derived from relativity theory. Physicists agree that classical mechanics constitutes a limiting case of relativity theory. By contrast, philosophers of science like Kuhn and Feyerabend affirm that classical mechanics cannot be deduced from relativity theory because of the incommensurability between both theories; thus what we obtain when we take the limit c in relativistic mechanics cannot be Newtonian mechanics sensu stricto. In (...) this paper I focus on the alleged change of reference of the term mass in the transition from one theory to the other. Contradicting Kuhn and Feyerabend, special relativity theory supports the view that the mass of an object is a characteristic property of the object, that it has the same value in whatever frame of reference it is measured, and that it does not depend on whether the object is in motion or at rest. Thus mass preserves the reference through the change of theory, and the existence of a Newtonian limit of relativity theory provides a good example of the rationality of theory change in mathematical physics. (shrink)

We answer three questions: 1. Can we give a wholly mathematical explanation of a physical phenomenon? 2. Can we give a wholly mathematical explanation for a whole physical theory? 3. What is gained or lost in giving a wholly, or partially, mathematical explanation of a phenomenon or a scientific theory? To answer these questions we look at a project developed by Hajnal Andréka, Judit Madarász, István Németi and Gergely Székely. They, together with collaborators, present special relativity (...) theory in a three-sorted first-order formal language. (shrink)

For most of the 1980s, Jon Barwise focused much of his research in the area of natural language semantics. This article surveys his research publications in that area.Most, but not all, of those publications were in the area of situation semantics, a new approach to natural language semantics Barwise developed jointly with his colleague John Perry in the first half of the 1980s. That work was both blessed, and cursed, by becoming closely identified in academic circles with the (...) award of a $23 million gift to Stanford from the System Development Foundation for the establishment in 1983 of its Center for the Study of Language and Information. The development of situation semantics, and in due course its underlying mathematical foundation Situation Theory, carried out for the most part within the framework of CSLI's STASS research group, was actually a relatively small part of the overall research program at CSLI. But because Barwise and Perry were leading figures at CSLI, were centrally involved in securing the SDF gift, and were respectively the first and second directors of CSLI, a general impression sprung up that the two of them had been awarded millions of dollars to develop situation semantics.A major consequence of this false impression was that from the theory's early days, a great deal of interest was shown in the project. With so many accomplished scholars pouring over the work while the ink was still wet on the manuscript pages, Barwise and Perry were able to take advantage of a broad range of helpful criticism. This meant that they were able to develop the new theory much more rapidly than would otherwise have been the case. (shrink)

In the World Library of Educationalists series, international experts themselves compile career-long collections of what they judge to be their finest pieces--extracts from books, key articles, salient research findings, major theoretical and/practical contributions--so the work can read them in a single manageable volume. Readers will be able to follow the themes and strands of their work and see their contribution to the development of a field. A developmental psychologist by training, Howard Gardner has spent the last 30 years researching, thinking (...) and writing about the development and education of the mind. He has contributed over 30 years researching, thinking and writing about the development and education of the mind. He has contributed over 30 books and 700 articles to the field. He is best known for his critique of the notion that intelligence is one single human intelligence that can be assessed through psychometric tests. Instead Gardner developed the theory of "multiple intelligence" which states that an individual has eight relatively autonomous intelligence: · Language · Music · Emotional · Logical-mathematical · Spatial · Kinesthetic · Creative · Interpersonal (understanding oneself) This theory has proved popular, particularly with those who see the IQ testing a relatively narrow set of abilities. In this book, he brings together over 20 of his key writings in one place. The book begins with a specially written Introduction, which gives an overview of Howard's career and contextualizes his selection in this book. Through his selection we can see the development of his thinking as well as the development of the field. This is the only book that offers this insight into this great scholar's work. (shrink)

The results established in this paper are in connection with the Relative Internal Set Theory (R.I.S.T.). The main result is the general principle of choice: Let α be a level and let Φ(x, y) be anαexternalαbounded formula of the language of R.I.S.T.. Suppose that to each elementx, dominated by α, corresponds an elementy x such that Φ(x, y x ) holds, then there exists a function of choice ψ such that, which is a very general principle of choice, for (...) everyx dominated by α, Φ(x, ψ(x)) holds. More than that, we establish that if all the elementsy x are uniformly dominated by a level β then we can prescribe that the function of choice is also dominated by β. (shrink)

Traditional logic as a part of philosophy is one of the oldest scientific disciplines. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, Russell and others to create a logistic foundation for mathematics. It steadily developed during the 20th century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy. While there are already several well-known textbooks on mathematical logic, this book is unique in that (...) it is much more concise than most others, and the material is treated in a streamlined fashion which allows the professor to cover many important topics in a one semester course. Although the book is intended for use as a graduate text, the first three chapters could be understood by undergraduates interested in mathematical logic. These initial chapters cover just the material for an introductory course on mathematical logic combined with the necessary material from set theory. This material is of a descriptive nature, providing a view towards decision problems, automated theorem proving, non-standard models and other subjects. The remaining chapters contain material on logic programming for computer scientists, model theory, recursion theory, Godel’s Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. The author has provided exercises for each chapter, as well as hints to selected exercises. About the German edition: …The book can be useful to the student and lecturer who prepares a mathematical logic course at the university. What a pity that the book is not written in a universal scientific language which mankind has not yet created. - A.Nabebin, Zentralblatt. (shrink)

ArgumentThis article discusses a number of heterogeneous Yiddish monographs on Einstein's theory of relativity. It presents background information on Einstein's relationship with the Yiddish language, with the cultural movement, Yiddishism, and with its leading institution, YIVO. Although Einstein avoided taking sides in the conflict between Yiddishism and its rival Zionism, his Zionist friends were successful in establishing at least a “primacy of palestinocentric Zionism” in his thinking. Of special interest are two books by philosophic writers and one by (...) an author trained in mathematics and physics, Tuvia Schalit, an acquaintance of Einstein's assistant Jacob Grommer. This article describes the underlying motivations and the philosophical horizon of these authors, in particular with respect to their view on the relationship between Kantianism and the theory of relativity. In contrast to many German academic philosophers, the Yiddish authors thought it possible to reconcile Kantianism with the new physics. (shrink)

Despite the fact that natural language has always been one of the most important resources for the study of ontology, many authors continue to regard it as a deceptive guide to the inquiry into what there is. The notion of natural language as a trap is carried over into contemporary metaontological studies, which typically reject natural language as ontologically committing. From a deflationary perspective, this paper aims to argue that ontological commitment occurs in natural languages, with implications (...) for the linguistic relativity hypothesis. To this end, a view based on naturalized epistemology and other aspects of Quine’s philosophy is presented. The perspective of Natural Language Ontology proposed by Moltmann is also introduced, with the goal of offering a new approach that allows a specific analysis of the ontological commitments of natural languages. While Moltmann herself indicates some motivations for this, its potential attractiveness for the study of linguistic relativity will be emphasized here. Finally, it will be suggested that there may be a linguistic bias around the proposed criteria of ontological commitment. (shrink)

A set theory model of reality, representation and language based on the relation of completeness and incompleteness is explored. The problem of completeness of mathematics is linked to its counterpart in quantum mechanics. That model includes two Peano arithmetics or Turing machines independent of each other. The complex Hilbert space underlying quantum mechanics as the base of its mathematical formalism is interpreted as a generalization of Peano arithmetic: It is a doubled infinite set of doubled Peano arithmetics having (...) a remarkable symmetry to the axiom of choice. The quantity of information is interpreted as the number of elementary choices (bits). Quantum information is seen as the generalization of information to infinite sets or series. The equivalence of that model to a quantum computer is demonstrated. The condition for the Turing machines to be independent of each other is reduced to the state of Nash equilibrium between them. Two relative models of language as game in the sense of game theory and as ontology of metaphors (all mappings, which are not one-to-one, i.e. not representations of reality in a formal sense) are deduced. (shrink)

Journal of Mathematical Logic, Ahead of Print. We initiate the study of a generalization of Kim-independence, Conant-independence, based on the notion of strong Kim-dividing of Kaplan, Ramsey and Shelah. A version of Conant-independence was originally introduced to prove that all [math] theories are [math]. We introduce an axiom on stationary independence relations, essentially generalizing the “freedom” axiom in some of the free amalgamation theories of Conant, and show that this axiom provides the correct setting for carrying out arguments of (...) Chernikov, Kaplan and Ramsey on [math] theories relative to a stationary independence relation. Generalizing Conant’s results on free amalgamation to the limits of our knowledge of the [math] hierarchy, we show using methods from Conant as well as our previous work that any theory where the equivalent conditions of this local variant of [math] holds is either [math] or [math] and is either simple or [math]. We observe that these theories give an interesting class of examples of theories where Conant-independence is symmetric. This includes all of Conant’s examples, the small cycle-free random graphs of Shelah, and the (finite-language) [math]-categorical Hrushovski constructions of Evans and Wong. We then answer Conant’s question on the existence of non-modular free amalgamation theories. We show that the generic functional structures of Kruckman and Ramsey are examples of non-modular free amalgamation theories. We also show that any free amalgamation theory is [math] or [math], while an [math] free amalgamation theory is simple if and only if it is modular. Finally, we show that every theory where Conant-independence is symmetric is [math]. Therefore, symmetry for Conant-independence gives the next known neostability-theoretic dividing line on the [math] hierarchy beyond [math]. We explain the connection to some established open questions. (shrink)

This book argues that our current best theories of fundamental physics are best interpreted as positing spacetime as non-fundamental. It is written in accessible language and largely avoids mathematical technicalities by instead focusing on the key metaphysical and foundational lessons for the fundamentality of spacetime. -/- According to orthodoxy, spacetime and spatiotemporal properties are regarded as fundamental structures of our world. Spacetime fundamentalism, however, faces challenges from speculative theories of quantum gravity – roughly speaking, the project of applying (...) the lessons of quantum mechanics to gravitation and spacetime. This book demonstrates that the non-fundamentality of spacetime does not rely on speculative physics alone. Rather, one can give an interpretation of general relativity that supports some form of spacetime non-fundamentalism. The author makes the case for spacetime non-fundamentalism in three steps. First, he confronts the standard geometrical interpretation of general relativity with Brown and Pooley’s dynamical approach to relativity theory. Second, he considers an alternative derivation of the Einstein field equations, namely the classical spin-2 approach, and argues that it paves the way for a refined dynamical approach to general relativity. Finally, he argues that particle physics can serve as a continuity condition for the metaphysics of spacetime. -/- The Non-Fundamentality of Spacetime will be of interest to scholars and advanced students working in philosophy of physics, philosophy of science, and metaphysics. (shrink)

An insight, Central to platonism, That the objects of pure mathematics exist "in some sense" is probably essential to any adequate account of mathematical truth, Mathematical language, And the objectivity of the mathematical enterprise. Yet a platonistic ontology makes how we can come to know anything about mathematical objects and how we use them a dark mystery. In this paper I propose a framework for reconciling a representation-Relative provability theory of mathematical truth with platonism's (...) valid insights. Besides helping to clarify the ontology of pure mathematics, I think this approach suggests a novel philosophical interpretation of some central results of modern mathematics, Including godel's incompleteness theorems and the independence theorems for the continuum hypothesis. (shrink)

Taking Wittgenstein's love of music as my impetus, I approach aporetic problems of epistemic relativity through a round of three overlapping (canonical) inquiries delivered in contrapuntal (higher and lower) registers. I first take up the question of scepticism surrounding ‘groundless knowledge’ and contending paradigms in On Certainty (physics versus oracular divination, or realism versus idealism) with attention given to the role of ‘bedrock’ certainties in providing stability amidst the Heraclitean flux. I then look into the formation of sedimented bedrock (...) knowledge, or practices of knowing, by comparing Wittgenstein's remarks on animal habituation and initiate training into human forms of life. In the latter case, mastery of techniques—our common education—secures agreement in judgment. Finally, I entertain Wittgenstein's obscure references to Einstein's Relativity in Zettel, showing initiate training as a way of ‘setting the clocks’ with variable degrees of certainty, relative to the language‐games played. Together, these three approaches help us to stop the ‘endless circling’ when philosophers try to address knowledge questions through the logic of object and designation, or verification of correspondence between propositions and things. Instead, attention moves to the way we educate our children and how we employ agreements and bedrock certainties in practices. (shrink)

The title essay was originally presented as two lectures inaugurating the John Dewey lectures at Columbia. It is an important essay for understanding Quine's work for it brings together many themes at the center of his thinking since Word and Object. Quine quotes with approval Dewey's statement "meaning is primarily a property of behavior" and then goes on to consider a thesis which, according to Quine, is a consequence of such a behavioral theory of meaning, i.e., the thesis of the (...) indeterminacy of meaning and translation. Quine relates this indeterminacy thesis, which he has been defending for some time, to language learning, the foundations of mathematics, and to a general view of ontological relativity. Other essays in the volume concern natural kinds and the various paradoxes of confirmation, propositional objects, quantification and existence and the empirical basis of science. All the essays are post-1965 except the introductory essay which was Quine's Presidential Address to the Eastern Division of the American Philosophical Association in 1956. This address was something of an introduction to the ideas to appear in Word and Object and is placed at the beginning of this collection to emphasize that all the essays collected here expand on and defend some of the positions of Word and Object. Quine's fluid style is everywhere in evidence.--R. H. K. (shrink)

This essay is a contribution to the historical phenomenology of science, taking as its point of departure Husserl’s later philosophy of science and Jacob Klein’s seminal work on the emergence of the symbolic conception of number in European mathematics during the late sixteenth and seventeenth centuries. Sinceneither Husserl nor Klein applied their ideas to actual theories of modern mathematical physics, this essay attempts to do so through a case study of the conceptof “spacetime.” In §1, I sketch Klein’s account (...) of the emergence of the symbolic conception of number, beginning with Vieta in the late sixteenth century. In §2,through a series of historical illustrations, I show how the principal impediment to assimilating the new symbolic algebra to mathematical physics, namely, thedimensionless character of symbolic number, is overcome via the translation of the traditional language of ratio and proportion into the symbolic language of equations. In §§3–4, I critically examine the concept of “Minkowski spacetime,” specifically, the purported analogy between the Pythagorean distance formula and the Minkowski “spacetime interval.” Finally, in §5, I address the question of whether the concept of Minkowski spacetime is, as generally assumed, indispensable to Einstein’s general theory of relativity. (shrink)

The work of George Smith has illuminated how Newton’s scientific method, and its use in constructing the theory of universal gravitation, introduced an entirely new sense of what it means for a theory to be supported by evidence. This new sense goes far beyond Newton’s well known dissatisfaction with hypothetico-deductive confirmation, and his preference for conclusions that are derived from empirical premises by means of mathematical laws of motion. It was a sense of empirical success that George was especially (...) well placed to identify and to understand, through his experience as an engineer specializing in failure analysis. For Newton, to understand how well his theory was supported by evidence, he had to anticipate, as far as possible, all the ways in which it might be wrong. This paper explores how Newton's empirical method shaped his thinking about space, time, and the relativity of motion. (shrink)

Ernst Cassirer (1874–1945) occupies a unique place in 20th-century philosophy. His view that human beings are not rational but symbolic animals and his famous dispute with Martin Heidegger at Davos in 1929 are compelling alternatives to the deadlock between 'analytic' and 'continental' approaches to philosophy. An astonishing polymath, Cassirer's work pays equal attention to mathematics and natural science but also art, language, myth, religion, technology, and history. However, until now the importance of his work has largely been overlooked. -/- (...) In this outstanding introduction Samantha Matherne examines and assesses the full span of Cassirer’s work. Beginning with an overview of his life and works she covers the following important topics: Cassirer’s neo-Kantian background; philosophy of mathematics and natural science, including Cassirer’s first systematic work, Substance and Function, and subsequent works, like Einstein’s Theory of Relativity; the problem of culture and the ground-breaking The Philosophy of Symbolic Forms; Cassirer’s ethical and political thought and his diagnosis of fascism in The Myth of the State; and Cassirer’s influence and legacy. -/- Including chapter summaries, suggestions for further reading, and a glossary of terms, this is an ideal introduction to Cassirer’s thought for anyone coming to his work for the first time. It is essential reading for students in philosophy as well as related disciplines such as intellectual history, art history, politics, and literature. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviews 105 NEW RUSSIAN WORK ON RUSSELL IRVING H. ANELLIS Modern Logic Publishing I Box 1036, Welch Ave. Station Ames, JA 5°010-1036, USA A. S. Kolesnikov. cI»HJIOCOcPHJl BepTPaHa PacceJIa [Filosofija Bertrana Rassela]. Leningrad: Izdatel'srvo Leningradskogo Universiteta, 1991. Pp. 232. 3 rub. 30 kop.. Anatolii Sergeevich Kolesnikov is a relatively new name in Russell studies,.r1.a1though his book shows a deep knowledge of the material available on Russell in Russian and (...) a wide acquaintance with Russell's publications in English and in Russian translation.1 In this work, which translates as The Philosophy ofBertrand Russel~ Kolesnikov traces the evolution of Russell's "world-view", while presenting a traditional Soviet interpretation of Russell's place in "bourgeois" philosophy. This monograph presents for the first time in Russian a thorough analysis of the evolution of Russell's philosophy as the outstanding representation of contemporary bourgeois philosophy, and is the first major study on Russell's philosophy in Russian since the appearance in'1962 ofSoviet philosopher 1. S. Narskii's The Philosophy ofBertrand Russell2 Russell himself is viewed by Kolesnikov as the best representative of the bourgeois humanist, philosopher, and mathematician. The author seeks a critical understanding of the historical and philosophical sources of Russell's ideas and conceptions and of the influence which these exercised and continue to exercise on contemporary Western philosophy and science. The author's aim is to "uncover" the neo-realist empiricist direction of Russell's philosophy as it manifested itself as a condition of his scientific and epistemological thinking. As had been usual for Soviet studies of Western "bourgeois" philosophers and their philosophies, Lenin and his empiriocriticism serves as a foil for the elucidation of Russell's thought and its development. Probably the most famous example of the dialectical attack on anaI Kolesnikov is also the author of The Freethought ofBertrand Russell [Svobodomyslie Bertrana Rassela] (Moscow: Mysl', 1978). 2 The Philosophy ofBertrand Russell: Lectures for Students in the University Philosophy Faculty [Filosofija Bertrana Rassela: lekcija dlja studentovfilosofikih fakul'tov universitetov] (Moscow: 1962). In the first footnote (on p. 60) to his translation of the article on "Bertrand Russell in the Great Soviet Encyclopedia: Translation from Russian", Russel~ nos. 23-4 (1976): 60-2, Charles Haynes wrote that "Narskiy... appears to be a leading Soviet writet on Russell's philosophy." In fact Narskii wrote extensively on philosophy of logic, for which work he is bcst known. 106 Reviews lytic philosophy revolved around the rather rough treatment accorded to A. J. Ayer when he lectured at Moscow State University in 1962. This methodology for criticizing "bourgeois idealism" has declined in recent years as a consequence of perestroika; from as early as 1987 Soviet philosophers have managed to refrain from employing this tactic in their writings (as one may readily see, e.g., from Zinaida Sokuler's recent paper on "Wirrgenstein on the Contradictions in Logic and in the Foundations of Mathematics"3). Kolesnikov's discussion of political-ideological, social and moral issues is limited to the Preface, which also presents a brief sketch of Russell's life, especially his education and the earliest of the philosophical influences at Cambridge, of course Russell's visit in 1920 to Soviet Russia and the writings that derived from that trip, especially his book The Practice and Theory of Bolshevism, and of his travels in China. Here Kolesnikov notes Russell's ties to the Fabian socialists and names in particular the Webbs, H. G. Wells and "other members representative of the bourgeois intelligentsia" (p. 5). Mention is also made here of his activism for nuclear disarmament and against the American war in Indochina, and of the essay "Why I Am Not a Christian". We are told at the very outset (p. 3) that "the name of this philosopher is widely known in our country." The remainder of the book is concerned with Russell's technical philosophy, i.e. with his work in philosophy of mathematics, logic, philosophy of language, metaphysics and epistemology. Kolesnikov divides Russell's philosophical evolution into three stages (p. 22). The "early" period (1894-1910) is the developmental stage, characterized by the influence ofneo-Hegelianism and neo-Kantianism and by the development of the conception of... (shrink)

An homage to the men and women of science, and an exposition of Einstein's place in scientific history In this fascinating collection of articles and speeches, Albert Einstein reflects not only on the scientific method at work in his own theoretical discoveries, but also eloquently expresses a great appreciation for his scientific contemporaries and forefathers, including Johannes Kepler, Isaac Newton, James Clerk Maxwell, Max Planck, and Niels Bohr. While Einstein is renowned as one of the foremost innovators of modern science, (...) his discoveries uniquely his own, through his own words it becomes clear that he viewed himself as only the most recent in a long line of scientists driven to create new ways of understanding the world and to prove their scientific theories. Einstein's thoughtful examinations explain the "how" of scientific innovations both in his own theoretical work and in the scientific method established by those who came before him. This authorized Philosophical Library book features a new introduction by Neil Berger, PhD, and an illustrated biography of Albert Einstein, which includes rare photos and never-before-seen documents from the Albert Einstein Archives at the Hebrew University of Jerusalem. "What place does the theoretical physicist's picture of the world occupy among all these possible pictures? It demands the highest possible standard of rigorous precision in the description of relations, such as only the use of mathematical language can give" -Albert Einstein, "Principles of Research" Albert Einstein (1879-1955) was born in Germany and became an American citizen in 1934. A world-famous theoretical physicist, he was awarded the 1921 Nobel Prize for Physics and is renowned for his Theory of Relativity. In addition to his scientific work, Einstein was an influential humanist who spoke widely about politics, ethics, and social causes. After leaving Europe, Einstein taught at Princeton University. His theories were instrumental in shaping the atomic age. Neil Berger, an associate professor emeritus of mathematics, taught at the University of Illinois at Chicago in the Mathematics, Statistics, and Computer Science department from 1968 until his retirement in 2001. He was the recipient of the first Monroe H. Martin Prize (1975), which is now awarded by the University of Maryland every five years for a singly authored outstanding applied mathematics research paper. He has published numerous papers and reviews in his fields of expertise, which include elasticity, tensor analysis, scattering theory, and fluid mechanics. (shrink)

The energy-momentum tensor for a particular matter component summarises its local energy-momentum distribution in terms of densities and current densities. We re-investigate under what conditions these local distributions can be integrated to meaningful global quantities. This leads us directly to a classic theorem by Max von Laue concerning integrals of components of the energy-momentum tensor, whose statement and proof we recall. In the first half of this paper we do this within the realm of Special Relativity and in the (...) traditional mathematical language using components with respect to affine charts, thereby focusing on the intended physical content and interpretation. In the second half we show how to do all this in a proper differential-geometric fashion and on arbitrary space-time manifolds, this time focusing on the group-theoretic and geometric hypotheses underlying these results. Based on this we give a proper geometric statement and proof of Laue's theorem, which is shown to generalise from Minkowski space to space-times with significantly less symmetries. This result, which seems to be new, not only generalises but also clarifies the geometric content and hypotheses of Laue's theorem. A series of three appendices lists our conventions and notation and summarises some of the conceptual and mathematical background needed in the main text. (shrink)

1. The early philosophy--language as picture -- 2. Logic and ontology -- 3. "My world and its value" -- 4. The later philosophy--views and reviews -- 5. Method and essense -- 6. Meaning -- 7. Criteria -- 8. Knowing, naming, certainty, and idealism -- 9. The private language argument -- 10. Logical necessity and rules -- 11. Philosophy of mathematics -- 12. Persons -- 13. Psychology and conceptual relativity -- 14. Aesthetics, ethics, and religion -- 15. Elective (...) affinities. (shrink)

In classical logic, a contradiction allows one to derive every other sentence of the underlying language; paraconsistent logics came relatively recently to subvert this explosive principle, by allowing for the subsistence of contradictory yet non-trivial theories. Therefore our surprise to find Wittgenstein, already at the 1930s, in comments and lectures delivered on the foundations of mathematics, as well as in other writings, counseling a certain tolerance on what concerns the presence of contradictions in a mathematical system. ‘Contradiction. Why (...) just this spectre? This is really very suspicious.’ ( Philosophical Remarks III–56) In the last decades, several authors (e.g. Arrington, Hintikka, Van Heijenoort, Wright, Wrigley) have been digging into Wittgenstein’s rather non-standard standpoint on what concerns the interpretation and import of contradiction in logic and mathematics, and many other authors (e.g. da Costa, Goldstein, Granger, Marconi) have been investigating the possibility of taking Wittgenstein seriously as one of the early forerunners of paraconsistency. While many advances have been made on the first front, the second set of investigations has led almost exclusively to negative results: no, no operational proposal about the construction of a logic in which (some) contradictions are made inoffensive can be read from Wittgenstein’s philosophical work; in fact, it appears that the most one can find there is the exhortation for mathematicians to alter their attitude with respect to contradictions and to consistency proofs. The play is done, and one looks for a resume of the opera. This paper fills that blank, as a thorough investigation of the possible relations between Wittgenstein and paraconsistency. DOI:10.5007/1808-1711.2010v14n1p135. (shrink)

There are three elements in this paper. One is what we shall call ‘the Hungarian project’. This is the collected work of Andréka, Madarász, Németi, Székely and others. The second is Molinini’s philosophical work on the nature of mathematical explanations in science. The third is my pluralist approach to mathematics. The theses of this paper are that the Hungarian project gives genuine mathematical explanations for physical phenomena. A pluralist account of mathematical explanation can help us with appreciating (...) the significance of the Hungarian project. The significance consists in the fruitfulness and spread of the project. The spread is wide because the explanations are written in the very familiar language of first-order logic with identity. For this reason, the explanation is understandable to many mathematicians. Because of the methodology adopted in the Hungarian project, the explanations are fruitful in another sense. In the Hungarian project certain questions are asked that would not be asked with a more usual methodology. The Hungarians are distinguished from other scientists in asking logical and mathematical questions, and these both deepen our understanding of the physical theories and induce further spread to mathematics and philosophy. (shrink)

After the publication of Begriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions of lingua characterica and calculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a mere calculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but did (...) not explain Frege’s and Schröder’s conceptions of the fulfilment of Leibniz’s scientific ideal. -/- In this paper I explain the reasons for Frege’s and Schröder’s mutual accusations of having created a mere calculus ratiocinator. On the one hand, Schröder’s construction of the algebra of relatives fits with a project for the reduction of any mathematical concept to the notion of relative. From this stance I argue that he deemed the formal system of Begriffsschrift incapable of such a reduction. On the other hand, first I argue that Frege took Boolean logic to be an abstract logical theory inadequate for the rendering of specific content; then I claim that the language of Begriffsschrift did not constitute a complete lingua characterica by itself, more being seen by Frege as a tool that could be applied to scientific disciplines. Accordingly, I argue that Frege’s project of constructing a lingua characterica was not tied to his later logicist programme. (shrink)