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Frank Quinn

Publications and source records attributed to Frank Quinn.

At least 19 recordsLinked to original sources
arXiv2024

On foundations for deductive mathematics

This article was motivated by the discovery of a potential new foundation for mainstream mathematics. The goals are to clarify the relationships between primitives, foundations, and deductive practice; to understand how to determine what is, or isn't, a foundation; and get clues as to how a foundation can be optimized for effective human use. For this we turn to history and professional practice of the subject. We have no asperations to Philosophy. The first section gives a short abstract discussion, focusing on the significance of consistency. The next briefly describes foundations, explicit and implicit, at a few key periods in mathematical history. We see, for example, that at the primitive level human intuitions are essential, but can be problematic. We also see that traditional axiomatic set theories, Zermillo-Fraenkel-Choice (ZFC) in particular, are not quite consistent with mainstream practice. The final section sketches the proposed new foundation and gives the basic argument that it is uniquely qualified to be considered {the} foundation of mainstream deductive mathematics. The ``coherent limit axiom'' characterizes the new theory among ZFC-like theories. This axiom plays a role in recursion, but is implicitly assumed in mainstream work so does not provide new leverage there. In principle it should settle set-theory questions such as the continuum hypothesis.

math.HO↗
arXiv2023

Object generators, categories, and everyday set theory

In "Object generators, relaxed sets, and a foundation for mathematics", we introduced ``object generators'', a logical environment much more general than set theory. Inside this we found a `relaxed' version of set theory. That paper is focused on construction of the universal Zermillo-Fraenkel-Choice theory, and the argument that it alone is consistent with mainstream mathematical practice. This paper is oriented toward potential users. The first topic is that if the general context is not needed then there is a simpler description of the set theory. In particular this uses only familiar binary logic, and the resut is almost the same as na\"ıve set theory. The second topic collects facts about the smallest object that is not a set (the traditional Ordinal numbers, or ``class of all sets''). Quite a bit is known, but it heavily involves non-binary assertion logic. For instance the powerset of this object is the disjoint union of the bounded subobjects, and the cofinal subobjects. However there is no function to {yes, no} that detects this decomposition. The third topic illustrates how the general object-generator context enables natural and full-precision work with categories.

math.LO↗
arXiv2023

A construction of set theory

We begin with a context more general than set theory. The basic ingredients are essentially the object and functor primitives of category theory, and the logic is weak, requiring neither the Law of Excluded Middle nor quantification. Inside this we find "relaxed" set theory, which is much easier to use with full precision than traditional axiomatic theories. There is also an implementation of the Zermillo-Fraenkel-Choice axioms that is maximal in the sense that any other implementation uniquely embeds in it.

math.LO↗
arXiv2021

A foundation for deductive mathematics

Set theory is widely believed to provide a secure foundation for deductive mathematics, but current set theories do not quite do this. The mainstream essentially uses na\"\i ve set theory. After Russell's paradox showed this to be inconsistent, the patch ``don't say `set of all sets' '' was added. The resulting methodology has been extremely successful, but still lacks a consistent foundation. The set theory community extracted properties of na\"\i ve set theory to use as axioms, culminating in the Zermillo-Fraenkel-Choice (ZFC) axioms. Unfortunately they missed an axiom, and ZFC as it stands is not consistent with standard methodology. This paper addresses these issues. The first dozen pages (Sections 1--5) gives primitives, defines sets in this context, and verifies that these have the properties used in standard practice. Sections 6--7 relates this to traditional axiomatic set theory. We show the sets here correspond to the sets in a maximal model for the ZFC axioms. Section 8 gives the ``coherent limit axiom'', considered obviously true in mainstream practice, and shows it holds in the maximal model and fails in all others. There are several qualitative conclusions. First, standard mainstream practice implicitly takes place in the set theory described here. This also shows there are no ``hidden axioms'': we already have the full toolkit. Second, most of the axiomatic set theory of the last hundred years is irrelevant to standard mathematical practice. The ZFC models produced by forcing, for example, are essentially never maximal, and therefore do not constrain or inform standard practice.

math.LO↗
arXiv2013

The triangulation of manifolds

A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the questions.

math.GT↗
arXiv2013

Fractions in elementary education

This paper is one of a series in which elementary-education practice is analyzed by comparison with the history of mathematics, mathematical structure, modern practice, and (occasionally) cognitive neuroscience. The primary concerns are: Why do so many children find elementary mathematics difficult? And, why are the ones who succeed still so poorly prepared for college material needed for technical careers? The answer provided by conventional wisdom is essentially that mathematics is difficult. Third-graders are not developmentally ready for the subtlety of fractions, for instance, and even high-performing students cannot be expected to develop the skills of experienced users. However we will see that this is far from the whole story and is probably wrong: elementary-education fractions are genuinely harder and less effective than the version employed by experienced users. Experts discard at least 90% of what is taught in schools. Our educational system is actually counterproductive for skill development, and the reasons for this are an important secondary concern.

math.HO↗
arXiv2011

Algebraic K-theory over the infinite dihedral group: a controlled topology approach

We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture can be reduced to only those virtually cyclic groups which admit a surjection with finite kernel onto a cyclic group. The second is that the Waldhausen Nil groups for a group which maps epimorphically onto the infinite dihedral group can be computed in terms of the Farrell-Bass Nil groups of the index two subgroup which maps surjectively to the infinite cyclic group.

math.KT↗
arXiv2009

Problems on homology manifolds

A list of problems prepared for the proceedings of the Workshop on Exotic Homology Manifolds, Oberwolfach June 29-July 5 2003.

math.GT↗
arXiv2006

Hyperelementary assembly for K-theory of virtually abelian groups

Controlled $K$-theory is used to show that algebraic $K$-theory of virtually abelian groups is described by an assembly map defined using possibly-infinite hyperelementary subgroups. The Farrell-Jones summand (coming from infinite subgroups) is parameterized by the rational projective space of the group, and a reduced version is torsion. Includes general material on assembly and universal spaces.

math.KT↗
arXiv2005

Controlled K-theory I: Basic theory

This paper provides a full controlled version of algebraic $K$-theory. This includes a rich array of assembly maps; the controlled assembly isomorphism theorem identifying the controlled group with homology; and the stability theorem describing the behavior of the inverse limit as the control parameter goes to 0. There is a careful treatment of spectral cosheaf homology and related tools, including an ``iterated homology identity'' giving a spectrum-level version of the Leray-Serre spectral sequence.

math.KT↗
arXiv2005

Homology manifolds and 4-dimensional surgery

Withdrawn May 2005. There is an error in the even-dimensional case of the proof in the April 2005 version. The hoped-for 4-dimensional applications are unlikely to survive the repairs.

math.GT↗
arXiv2004

Cores of s-cobordisms of 4-manifolds

The main result is that an s-cobordism (topological or smooth) of 4-manifolds has a product structure outside a ``core'' sub s-cobordism. These cores are arranged to have quite a bit of structure, for example they are smooth and abstractly (forgetting boundary structure) diffeomorphic to a standard neighborhood of a 1-complex. The decomposition is highly nonunique so cannot be used to define an invariant, but it shows the topological s-cobordism question reduces to the core case. The simply-connected version of the decomposition (with 1-complex a point) is due to Curtis, Freedman, Hsiang and Stong. Controlled surgery is used to reduce topological triviality of core s-cobordisms to a question about controlled homotopy equivalence of 4-manifolds. There are speculations about further reductions.

math.GT↗
arXiv2004

Dual decompositions of 4-manifolds II: linear invariants

This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index $\leq2$. Part I gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental group, and use this to define algebraic-topological invariants. These reveal a basic asymmetry in these decompositions: subtle changes on one side can force algebraic-topologically detectable changes on the other. A solvable iteration of the basic invariant gives an ``obstruction theory'' using lower commutator quotients. By thinking of a 2-handlebody as essentially determined by the links used as attaching maps for its 2-handles this theory can be thought of a giving ``ambient'' link invariants. The moves used are related to the grope cobordism of links developed by Conant-Teichner, and the Cochran-Orr-Teichner filtration of the link concordance groups. The invariants give algebraically sophisticated ``finite type'' invariants in the sense of Vassilaev.

math.GT↗
arXiv2002

Lectures on controlled topology: mapping cylinder neighborhoods

The existence theorem for mapping cylinder neighborhoods is discussed as a prototypical example of controlled topology and its applications. The first of a projected series developed from lectures at the Summer School on High-Dimensional Topology, Trieste Italy 2001

math.GT↗
arXiv2001

Controlled surgery with trivial local fundamental groups

We provide a proof of the controlled surgery sequence, including stability, in the special case that the local fundamental groups are trivial. Stability is a key ingredient in the construction of exotic homology manifolds by Bryant, Ferry, Mio and Weinberger, but no proof has been available. The development given here is based on work of M. Yamasaki.

math.GT↗
arXiv2000

Relation between quantum invariants of 3-manifolds and 2-dimensional CW-complexes

We show that the Reshetikhin-Turaev-Walker invariant of 3-manifolds can be normalized to obtain an invariant of 4-dimensional thickenings of 2-complexes. Moreover when the underlying semisimple tortile category comes from the representations of a quantum group at a primitive prime root of unity, the 0-term in the Ohtsuki expansion of this invariant depends only on the spine and is the mod p invariant of 2-complexes defined previously from the second author. As a consequence it is shown that when the Euler characteristic is greater or equal to 1, the 2-complex invariant depends only on homology. The last statement doesn't hold for the negative Euler characteristic case.

math.GT↗
arXiv2000

Subexponential groups in 4-manifold topology

We present a new, more elementary proof of the Freedman-Teichner result that the geometric classification techniques (surgery, s-cobordism, and pseudoisotopy) hold for topological 4-manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reformulate the growth estimates in terms of coarse geometry.

math.GT↗