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SOLUBLE GROUPS OF FINITE RANK

John C. Lennox and Derek J. S. Robinson

in The Theory of Infinite Soluble Groups

This chapter details the ranks of an abelian group. It also talks about the structure of soluble groups with finite rank, minimax groups, and residual finiteness.

THE THEORY OF FINITELY GENERATED SOLUBLE GROUPS II

John C. Lennox and Derek J. S. Robinson

in The Theory of Infinite Soluble Groups

This chapter discusses simple modules over polycyclic groups, the Jategaonkar-Roseblade theorem, the Artin-Rees property, and residual finiteness. It also talks about the Frattini subgroup.

Hilbert on Theology and Its Discontents: The Origin Myth of Modern Mathematics

Colin McLarty

Apostolos Doxiadis and Barry Mazur (eds)

in Circles Disturbed: The Interplay of Mathematics and Narrative

This chapter examines the myth surrounding Paul Gordan's response to David Hilbert's finiteness theorems. A proof introduced by Hilbert in 1888 became the paradigm of modern axiomatic mathematics. In ... More

This chapter examines the myth surrounding Paul Gordan's response to David Hilbert's finiteness theorems. A proof introduced by Hilbert in 1888 became the paradigm of modern axiomatic mathematics. In the myth, Gordan denounced Hilbert's proof, and his anathema rebounded against himself when he said, “This is not Mathematics, it is Theology!” After providing the background to the various interpretations that Gordan's comment has generated, the chapter considers the so-called “Gordan's problem”—to find finite complete systems of invariants for forms. It then discusses Hilbert's theorem and Gordan's reaction to Hilbert's fuller version of the invariant theorem, as well as Gordan's mythic quotation. It also explores the role played by Gordan's one and only doctoral student, Emmy Noether, in the Gordan–Hilbert controversy and concludes by emphasizing Gordan's story as an example of the deliberate use of narrative in mathematics.Less

Truth and Meaning

Donald Davidson

in Inquiries into Truth and Interpretation

Develops the criterion of learnability proposed in Essay 1 into a constraint for a satisfactory theory of meaning: such a theory must show how, on the basis of a finite stock of semantic primitives, ... More

Develops the criterion of learnability proposed in Essay 1 into a constraint for a satisfactory theory of meaning: such a theory must show how, on the basis of a finite stock of semantic primitives, speakers come to understand a potentially infinite number of sentences; more precisely, how the meaning of each and every sentence in the language is owed to the sentence's structural composition. Davidson notes that introducing distinct entities as meanings for each expression (‘intensions’, ‘senses’) does not illuminate these structural workings of the language (this complaint, he says, operates quite independently of ontological reservations about intensions being ‘abstract’ or lacking clear individuation conditions). If, on the other hand, we dispense with intensions, we must define ‘meaning’ by mere appeal to reference; but this clearly won’t work since, if we concede that sentences ‘refer’ to their truth‐values, all sentences alike in truth value would be alike in meaning. Davidson therefore individuates a sentence's meaning by its ‘logical place’ in the language to which it belongs, i.e. by its entailment relations to other sentences in the language; the meaning of a subsentential expression is likewise individuated holistically, namely in terms of the systematic effect the expression has on the entailment relations of the sentence in which it occurs. Since a Tarskian truth theory captures these entailment relations for each sentence (in a specified language) by associating it with its ‘truth‐condition’, the truth theory can be said to give the ‘meaning’ thus construed for each sentence (a fortiori for their subsentential expressions), while meeting both the constraint for finiteness and the restriction to purely referential means. Davidson discusses various objections to this proposal (since called ‘Davidson's programme’), e.g. that it cannot accomodate sentences containing indexicals or contexts in which extensionality breaks down; and he shows its merit in its varied applicability to questions of logical form, the success of which is evaluated by whether the aforementioned entailments are preserved and explained for a variety of sentences (see Essays 6–8, and Essays 6–10 of the companion volume Essays on Actions and Events).Less

Semantics for Natural Languages

Donald Davidson

in Inquiries into Truth and Interpretation

Davidson rehearses the demands for an acceptable theory of meaning for a natural language—that it give the meaning for each (actual and potential) utterance in the language, that it do so by finite ... More

Davidson rehearses the demands for an acceptable theory of meaning for a natural language—that it give the meaning for each (actual and potential) utterance in the language, that it do so by finite means and shed light on the structural workings of the language (see Essay 2)—and argues that a theory of truth for that language provides ‘a precise, profound and testable answer’ to these demands. In addition, he demands that the stock of concepts used by metalanguage in which the truth theory is stated (its ‘ideology’) extends minimally beyond, and ideally includes, that of the object language (cf Essay 14); he looks at how alternative semantic programs fare under this constraint (cf Essay 5). He then addresses the more general question of how truth‐theoretic semantics copes with recalcitrant, i.e. non‐extensional, locutions current in natural languages, and addresses Tarski's pessimism of formal frameworks to ‘tame’ them. He concludes by stressing, as he did in Essay 2, that recognizing the role played by logical form (as analysed in truth theories) for the analysis of sentential meaning is all‐important; and questions whether this form or ‘deep structure’ of the sentence can be identified, as Chomsky suggested, with an underlying psychological mechanism in the speaker.Less

Theories of Meaning and Learnable Languages

Donald Davidson

in Inquiries into Truth and Interpretation

Davidson asks what properties a language must have to be learnable. He criticizes a (then) popular response that models the order of language acquisition on the epistemological priority of the types ... More

Davidson asks what properties a language must have to be learnable. He criticizes a (then) popular response that models the order of language acquisition on the epistemological priority of the types of expressions learnt; he labels this position the ‘building‐block theory’ (see further Essay 16). He discusses Strawson's critique of Quine's elimination of singular terms and shows how it is likewise premissed on the questionable derivation of claims about language learning from purely a priori considerations. On the positive side, Davidson proposes that a language is learnable by a creature with finite means if the language's number of semantic primitives or undefinables is finite. Using this criterion, he demonstrates that various theories in the philosophy of language introduce an infinite number of semantic primitives into the language and thus make it unlearnable; theories he alleges of this error (1) model quotations on names of expressions (Tarski, Quine; cf Essay 6), (2) analyse belief attributions in terms of linguistic marks (Scheffler, Carnap) or distinct one‐place predicates for each attributed belief (Quine; cf Essay 7), or (3) postulate intensional entities into their overall semantic framework (Frege, Church).Less

Some Thoughts on Radical Indeterminacy

Hartry Field

in Truth and the Absence of Fact

Discusses some issues about indeterminacy of reference and truth, from two points of view about reference and truth: that of a correspondence theory and that of a disquotational theory. It is argued ... More

Discusses some issues about indeterminacy of reference and truth, from two points of view about reference and truth: that of a correspondence theory and that of a disquotational theory. It is argued that a correspondence theorist can continue to accept the usual disquotation schemas for reference and truth, despite the indeterminacy. And it is argued that the disquotationalist can accept indeterminacy even in his own conceptual scheme. Together, these claims mean that the two views on truth are much closer in their treatments of indeterminacy than one might have thought. Also discusses indeterminacy in our logical and mathematical vocabulary.Less

Mathematical Objectivity and Mathematical Objects

Hartry Field

in Truth and the Absence of Fact

Focuses on an issue about the objectivity of mathematics—the extent to which undecidable sentences have determinate truth‐value—and argues that this issue is more important than the issue of the ... More

Focuses on an issue about the objectivity of mathematics—the extent to which undecidable sentences have determinate truth‐value—and argues that this issue is more important than the issue of the existence of mathematical objects. It argues that certain familiar problems for those who postulate mathematical objects, such as Benacerraf's access argument, are serious for those with highly ‘objectivist’ pictures of mathematics, but dissolve for those who allow for sufficient indeterminacy about undecidable sentences. The nominalist view that does without mathematical entities is simply one among several ways of accomplishing the important task of doing without excess objectivity. There is also a discussion arguing for one kind of structuralism but against another.Less

Which Undecidable Mathematical Sentences Have Determinate Truth Values?

Hartry Field

in Truth and the Absence of Fact

Argues that typical undecidable sentences of set theory (e.g. about the size of the continuum) can have no determinate truth‐value, since nothing in our practice can determine which ‘universe of ... More

Argues that typical undecidable sentences of set theory (e.g. about the size of the continuum) can have no determinate truth‐value, since nothing in our practice can determine which ‘universe of sets’ we are talking about. But it gives an account of how typical undecidable number‐theoretic sentences can have determinate truth‐value. The account has it though, that, whether these undecidable number‐theoretic sentences have determinate truth‐value, turns on assumptions about the physical world that could fail, and it explores the consequence of their failure. It concludes with a critique of an argument that at least the Gödel sentence of our fullest mathematical theory must be determinately true: the critique is that the argument depends on the same sort of reasoning that leads to the semantic paradoxes.Less

Interrogatives

John M. Anderson

in The Substance of Language Volume I: The Domain of Syntax

Interrogative is a mood, a secondary category of the functional category of finiteness. Different kinds of interrogative structure are described in terms of the framework of Chapter 3. These include ... More

Interrogative is a mood, a secondary category of the functional category of finiteness. Different kinds of interrogative structure are described in terms of the framework of Chapter 3. These include alternative questions, argument questions (wh- in English), re-opened questions, which question an answer or a question, proxy questions, allowing predicators to be questioned, and so-called ‘indirect questions’. The form of these is determined by the meaning of interrogative, so that they involve either a structure whose truth value is open or an argument or predicator that is open. A localist account of truth is offered and its place in relation to finiteness is described. ‘Indirect questions’, whose construction does not appear in main clauses, are regarded as demoted to non-finite. The role of phonology in the expression of interrogatives is described and emphasized. Notionally and lexically based interrogative representations involving argument sharing need not appeal to structural mutation or empty categories.Less

Finiteness and mood

John M. Anderson

in The Substance of Language Volume I: The Domain of Syntax

Finiteness is interpreted here as the category licensing potential independence of a predication, and mood is a secondary category that grammaticalizes speech‐act type. As such it is limited to ... More

Finiteness is interpreted here as the category licensing potential independence of a predication, and mood is a secondary category that grammaticalizes speech‐act type. As such it is limited to main‐clause finites. Declarative is the prototypical mood, and its prototypical interpretation is as a speech act of assertion; and the indicative construction is the prototypical expression of declarative. The indicative varies in its properties from language to language, but it is likely to attract secondary categories associated with the circumstances of utterance, such as tense and person‐number. Most like indicatives are typically expressions of negative and insistent assertion. The expression of non‐declaratives differs more, though interrogatives, in questioning a potential assertion, are less deviant than non‐propositional moods such as imperatives, hortatives, optatives, etc. The structures of a range of types of mood expression are described. This and the two following chapters illustrate the workings of a notional grammar in this fundamental area of the syntax.Less

Unpacking finiteness

Irina Nikolaeva

in Canonical Morphology and Syntax

Since no implicational relations suggested for finiteness parameters so far have been without exceptions, it is argued that finiteness is a clausal notion which is best characterized by independent ... More

Since no implicational relations suggested for finiteness parameters so far have been without exceptions, it is argued that finiteness is a clausal notion which is best characterized by independent criteria belonging to different linguistic components: morphology, syntax, and semantics. There is no uniform dimension called ‘finiteness’. Instead, the relevant category (if it is a category) is sensitive to different kinds of cross-cutting information and each of them can vary independently.Less

Homotopy Groups and CW Complexes

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built ... More

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.Less

Free Groups and Folding

Matt Clay

in Office Hours with a Geometric Group Theorist

This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show ... More

This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show the rank of the free group H and whether every finitely generated nontrivial normal subgroup of a free group has finite index. The edge paths and the fundamental group of a graph are discussed, along with subgroups via graphs. The chapter also considers five applications of folding: the Nielsen–Schreier Subgroup theorem, the membership problem, index, normality, and residual finiteness. A group G is residually finite if for every nontrivial element g of G there is a normal subgroup N of finite index in G so that g is not in N. Exercises and research projects are included.Less

Entropy for hyperbolic Riemann surface laminations I

Tien-Cuong Dinh, Viet-Anh Nguyen, and Nessim Sibony

Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)

in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of ... More

This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of the entropy. The chapter first focuses on compact laminations, which are transversally smooth, before turning to the case of singular foliations, showing how the Poincaré metric on leaves is transversally Hölder continuous. In addition, the chapter considers the problem in the proof that the entropy is finite for singular foliations is quite delicate and requires a careful analysis of the dynamics around the singularities. Finally, the chapter discusses a notion of metric entropy for harmonic probability measures and gives some open questions.Less

Entropy for hyperbolic Riemann surface laminations II

Tien-Cuong Dinh, Viet-Anh Nguyen, and Nessim Sibony

Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)

in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

This chapter studies Riemann surface foliations with tame singular points. It shows that the hyperbolic entropy of a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated ... More

This chapter studies Riemann surface foliations with tame singular points. It shows that the hyperbolic entropy of a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated singularities on a compact complex surface is finite. The chapter then proves the finiteness of the entropy in the local setting near a singular point in any dimension, using a division of a neighborhood of a singular point into adapted cells. Next, the chapter estimates the modulus of continuity for the Poincaré metric along the leaves of the foliation, using notion of conformally (R,δ‎)-close maps. The estimate holds for foliations on manifolds of higher dimension.Less