This is a book that opens up an area of contemporary experience that rarely sees the light of day. I believe readers from all walks of life and different educational backgrounds will be as excited to read about these experiences as my subjects were delighted to talk about them. One measure of the public's interest in relevant oral history is the current popularity of Brandon Stanton's Humans of New York, Stories, found in museums and bookstores throughout the city. And the (...) fact that the recent Nobel went to Svetlana Alexievich, a writer whose work is grounded in oral history, will expand the market for such books, and is a sign that more and more readers are interested in what ordinary people, whose voices are usually muted, have to say about crucial aspects of modern life. The book's interdisciplinary appeal will attract faculty and students in cultural studies, English, sociology, and other departments and programs interested in aesthetics. The book will have considerable crossover appeal to a wide general audience that especially includes women (most of my subjects are women) and women of color (a third of my subjects are members of a minority). The book is, in fact, eminently "Oprah-ble" because of its undoubted appeal to her vast audience. The challenge for an academic press, of course, will be to gain this audience's attention, a task to which I will be more than happy to devote substantial time and effort. (shrink)

PREFACE. We present a variety of basic theories involving fundamental concepts of naive thinking, of the sort that were common in "natural philosophy" before the dawn of physical science. The most extreme forms of infinity ever formulated are embodied in the branch of mathematics known as abstract set theory, which forms the accepted foundation for all of mathematics. Each of these theories embodies the most extreme forms of infinity ever formulated, in the following sense. ZFC, and even extensions of ZFC (...) with the so called large cardinal axioms, are mutually interpretable with these theories. This is an extended abstract. Proofs of the claims will appear elsewhere. (shrink)

We axiomatize EFA in strictly mathematical terms, involving only the ring operations, without extending the language by either exponentiation, finite sets of integers, or polynomials.

In mathematics, we back up our discoveries with rigorous deductive proofs. Mathematicians develop a keen instinctive sense of what makes a proof rigorous. In logic, we strive for a *theory* of rigorous proofs.

We now fix A ⊆ Q. We study a fundamental class of digraphs associated with A, which we call the A-digraphs. An A,kdigraph is a digraph (Ak,E), where E is an order invariant subset of A2k in the following sense. For all x,y ∈ A2k, if x,y have the same order type then x ∈ E ↔ y ∈ E.

We focus on two formal methods contexts which generate investigations into decision problems for finite strings.

Since then we have been engaged in the development of such results of greater relevance to mathematical practice. In January, 1997 we presented some new results of this kind involving what we call “jump free” classes of finite functions. This Jump Free Theorem is treated in section 2.

Context: Distributed language and interactivity are central members of a set of concepts that are rapidly developing into rigorous, exciting additions to 4E cognitive science. Because they share certain assumptions and methodological commitments with enactivism, the two have sometimes been confused; additionally, while enactivism is a well-developed paradigm, interactivity has relied more on methodological development and on a set of focal examples. Problem: The goal of this article is to clarify the core conceptual commitments of both interactivity-based and enactive approaches (...) to cognitive science by contrasting the two and highlighting their differences in assumptions, focus, and explanatory strategies. Method: We begin with the shared commitments of interactivity and enactivism - e.g., antirepresentationalism, naturalism, interdisciplinarity, the importance of biology, etc. We then give an overview of several important varieties of enactivism, including sensorimotor and anti-representationalist enactivism, and then walk through the history of the “core” varieties, taking care to contrast Maturana’s approach with that of Varela and the current researchers following in Varela’s footsteps. We then describe the differences between this latter group and interactivity-based approaches to cognitive science. Results: We argue that enactivism’s core concepts are explanatorily inadequate in two ways. First, they mis-portray the organization of many living systems, which are not operationally closed. Second, they fail to realize that most epistemic activity depends on engagement with non-local resources. Both problems can be dealt with by adopting an interactivity-based perspective, in which agency and cognition are fundamentally distributed and involve integration of non-local resources into the local coupling of organism and environment. Implications: The article’s primary goal is theoretical clarification and exposition; its primary implication is that enactive concepts need to be modified or extended in some way in order to explain fully many aspects of cognition and directed biological activity. Or, read another way, the article’s primary implication is that interactivity already provides a rich set of concepts for doing just that, which, while closely allied with enactivism in several ways, are not enactivist concepts. Constructivist content: The article consists entirely of a comparison between two constructivist fields of theory. Key Words: Interactivity, enactivism, distributed language, radical embodied cognitive science, ecological psychology, autonomy. (shrink)

It has been accepted since the early part of the Century that there is no problem formalizing mathematics in standard formal systems of axiomatic set theory. Most people feel that they know as much as they ever want to know about how one can reduce natural numbers, integers, rationals, reals, and complex numbers to sets, and prove all of their basic properties. Furthermore, that this can continue through more and more complicated material, and that there is never a real problem.

We show the algorithmic unsolvability of a number of decision procedures in ordinary two dimensional Euclidean geometry, involving lines and integer points. We also consider formulations involving integral domains of characteristic 0, and ordered rings. The main tool is the solution to Hilbert's Tenth Problem. The limited number of facts used from recursion theory are isolated at the beginning.

We say that R is strictly dominating if and only if for all x,yŒ[1,n], if R(x,y) then max(x) 3k ¥ [1,n], there exists A Õ [1,n] such that R = A. Furthermore, A Õ [1,n] is unique.

To prove this, we fix P(x) to be any polynomial of degree ≥ 1 with a positive and negative value. We define a critical interval to be any nonempty open interval on which P is strictly monotone and where P is not strictly monotone on any larger open interval. Here an open interval may not have endpoints in F, and may be infinite on the left or right or both sides. Obviously, the critical intervals are pairwise disjoint.

• Wright Brothers made a two mile flight • Wright Brothers made a 42 mile flight • Want to ship goods • Want to move lots of passengers • Want reliability and safety • Want low cost • ... Modern aviation • Each major advance spawns reasonable demands for more and more • Excruciating difficulties overcome • Armies of people over decades or more • Same story for any practically any epoch breaking advance in anything..

The use of x[y,z,w] rather than the more usual y Πx has many advantages for this work. One of them is that we have found a convenient way to eliminate any need for axiom schemes. All axioms considered are single sentences with clear meaning. (In one case only, the axiom is a conjunction of a manageable finite number of sentences).

We present two forms of “sentential reflection”, which are shown to be mutually interpretable with Z2 and ZFC, respectively.

Neuroethological investigations of mammalian and avian auditory systems have documented species-specific specializations for processing complex acoustic signals that could, if viewed in abstract terms, have an intriguing and striking relevance for human speech sound categorization and representation. Each species forms biologically relevant categories based on combinatorial analysis of information-bearing parameters within the complex input signal. This target article uses known neural models from the mustached bat and barn owl to develop, by analogy, a conceptualization of human processing of consonant plus (...) vowel sequences that offers a partial solution to the noninvariance dilemma the locus equations orderly output constraint”) is hypothesized based on the notion of an evolutionarily conserved auditory-processing strategy. High correlation and linearity between critical parameters in the speech signal that help to cue place of articulation categories might have evolved to satisfy a preadaptation by mammalian auditory systems for representing tightly correlated, linearly related components of acoustic signals. (shrink)

This paper was referred to in the Introduction in our paper [Fr97a], “The Axiomatization of Set Theory by Separation, Reducibility, and Comprehension.” In [Fr97a], all systems considered used the axiom of Extensionality. This is appropriate in a set theoretic context.

Provide a formal system that is a conservative extension of PA for Π02 sentences, and even a conservative extension of HA, that supports the worry free smooth development of constructive analysis in the style of Errett Bishop.

I don’t remember if I got as high as 2-390, but I distinctly remember taking my first logic course - as a Freshman - with Hartley Rogers, in Fall 1964 - here in 2-190. Or was it in 2-290?

This topic has been discussed earlier on the FOM email list in various guises. The common theme is: big numbers and long sequences associated with mathematical objects. See..

Two countable well orderings are weakly comparable if there is an order preserving injection of one into the other. We say the well orderings are strongly comparable if the injection is an isomorphism between one ordering and an initial segment of the other. In [5], Friedman announced that the statement “any two countable well orderings are strongly comparable” is equivalent to ATR 0 . Simpson provides a detailed proof of this result in Chapter 5 of [13]. More recently, Friedman has (...) proved that the statement “any two countable well orderings are weakly comparable” is equivalent to ATR 0 . The main goal of this paper is to give a detailed exposition of this result. (shrink)

The kind of unknowability I will discuss concerns the count of certain natural finite sets of objects. Even the situation with regard to our present strong formal systems is rather unclear. One can just profitably focus on that, putting aside issues of general unknowability.

1. Transfer principles from N to On. A. Mahlo cardinals. B. Weakly compact cardinals. C. Ineffable cardinals. D. Ramsey cardinals. E. Ineffably Ramsey cardinals. F. Subtle cardinals. G. From N to (...) 4. Decidability of statements on N. 5. Decidability of statements on shrink)

It turns out, time and time again, in order to make serious progress in f.o.m., we need to take actual reasoning and actual development into account at precisely the proper level. If we take these into account too much, then we are faced with information that is just too difficult to create an exact science around - at least at a given state of development of f.o.m. And if we take these into account too little, our findings will not have (...) the relevance to mathematical practice that could be achieved. (shrink)

The most frequent criticism of the target article is the lack of clear separability of human speech data relative to neuroethological data. A rationalization for this difference was sought in the tinkered nature of such new adaptations as human speech. Basic theoretical premises were defended, and new data were presented to support a claim that speakers maintain a low-noise relationship between F2 transition onset and offset frequencies for stops in pre-vocalic positions through articulatory choices. It remains a viable and testable (...) hypothesis that the phenomenon described by the locus equation is a functional adaptation of production mechanisms to processing preferences of the auditory system. (shrink)

NOTE: This is an expanded version of my lecture at the special session on reverse mathematics, delivered at the Special Session on Reverse Mathematics held at the Atlanta AMS meeting, on January 6, 2005.

Normal mathematical culture is overwhelmingly concerned with finite structures, finitely generated structures, discrete structures (countably infinite), continuous and piecewise continuous functions between complete separable metric spaces, with lesser consideration of pointwise limits of sequences of such functions, and Borel measurable functions between complete separable metric spaces.

C. To what extent, and in what sense, is the natural hierarchy of logical strengths rep resented by familiar systems ranging from exponential function arithmetic to ZF + j:V Æ V robust?

We present several selection theorems for Borel relations, involving only Borel sets and functions, all of which can be obtained as consequences of closely related theorems proved in [DSR 96,99,01,01X] involving coanalytic sets. The relevant proofs given there use substantial set theoretic methods, which were also shown to be necessary. We show that none of our Borel consequences can be proved without substantial set theoretic methods. The results are established for Baire space. We give equivalents of some of the main (...) results for the reals. (shrink)

We have been recently engaged in this search, and have announced a long series of successively simpler and more convincing examples. See [Fr09-10].

Reflection, in the sense of [Fr03a] and [Fr03b], is based on the idea that a category of classes has a subclass that is “similar” to the category. Here we present axiomatizations based on the idea that a category of classes that does not form a class has extensionally different subclasses that are “similar”. We present two such similarity principles, which are shown to interpret and be interpretable in certain set theories with large cardinal axioms.

We present some new set and class theoretic independence results from ZFC and NBGC that are particularly simple and close to the primitives of membership and equality. They are shown to be equivalent to familiar small large cardinal hypotheses. We modify these independendent statements in order to give an example of a sentence in set theory with 5 quantifiers which is independent of ZFC. It is known that all 3 quantifier sentences are decided in a weak fragment of ZF without (...) power set. (shrink)

The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen. The concepts were extended to that of k-subtle, k-almost ineffable, and k-ineffable cardinals in 1975 by J. Baumgartner. In this paper we give a self contained treatment of the basic facts about this level of the large cardinal hierarchy, which were established by J. Baumgartner. In particular, we give a proof that the k-subtle, k-almost ineffable, and k-ineffable cardinals define three (...) properly intertwined hierarchies with the same limit, lying strictly above “total indescribability” and strictly below “arrowing ω”. The innovation here is presented in Section 2, where we take a distinctly minimalist approach. Here the subtle cardinal hierarchy is characterized by very elementary properties that do not mention closed unbounded or stationary sets. This development culminates in a characterization of the hierarchy by means of a striking universal second-order property of linear orderings. (shrink)

Extrapolating from the work of Mahlo , one can prove that given any pair of countable closed totally bounded subsets of complete separable metric spaces, one subset can be homeomorphically embedded in the other. This sort of topological comparability is reminiscent of the statements concerning comparability of well orderings which Friedman has shown to be equivalent to ATR0 over the weak base system RCA0. The main result of this paper states that topological comparability is also equivalent to ATR0. In Section (...) 1, the pertinent subsystems of second-order arithmetic and results on well orderings are reviewed. Sections 2 and 3 overview the encoding of metric spaces and homeomorphisms in second-order arithmetic. Section 4 contains a proof of the topological comparability result in ATR0. Section 5 contains the reversal, a derivation of ATR0 from the topological comparability result. In Section 6, additional information about the structure of the embeddings is obtained, culminating in an application to closed subsets of the real numbers. (shrink)

In the Foundational Life, philosophy is commonly used as a method for choosing and analyzing fundamental concepts, and mathematics is commonly used for rigorous development. The mathematics informs the philosophy and the philosophy informs the mathematics.

This distinction between logic and mathematics is subject to various criticisms and can be given various defenses. Nevertheless, the division seems natural enough and is commonly adopted in presentations of the standard foundations for mathematics.

We consider intuitionistic number theory with recursive infinitary rules . Any primitive recursive binary relation for which transfinite induction schema is provable is in fact well founded. Its ordinal is less than ε 0 if the transfinite induction schema is intuitionistically provable in elementary number theory. These results are provable intuitionistically. In fact, it suffices to consider transfinite induction with respect to one particular number-theoretic property.

i. Proofless text is based on a variant of ZFC with free logic. Here variables always denote, but not all terms denote. If a term denotes, then all subterms must denote. The sets are all in the usual extensional cumulative hierarchy of sets. There are no urelements.

Upshot: We agree with commenters that enactivism incorporates a broad variety of methodologies, metaphysical stances, concepts, and investigative approaches, and that this is a good thing. However, we remain concerned that autonomy and sense-making are problematic concepts for post-Varelian enactivism, and that they form the foundations of a conceptual framework that may hamper the development of effective explanations for cognitive activity, as well as the paradigmatic aspirations of this particular enactivist approach.

We begin by presenting the language L(N,℘N,℘℘N). This is the standard language for presenting third order sentences, using its intended interpretation.

Russell’s way out of his paradox via the impredicative theory of types has roughly the same logical power as Zermelo set theory - which supplanted it as a far more flexible and workable axiomatic foundation for mathematics. We discuss some new formalisms that are conceptually close to Russell, yet simpler, and have the same logical power as higher set theory - as represented by the far more powerful Zermelo-Frankel set theory and beyond. END.