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How To Study Mathematics, Lawrence N. Stout
How To Study Mathematics, Lawrence N. Stout
Lawrence N. Stout
In high school mathematics much of your time was spent learning algorithms and manipulative techniques which you were expected to be able to apply in certain well-defined situations. This limitation of material and expectations for your performance has probably led you to develop study habits which were appropriate for high school mathematics but may be insufficient for college mathematics. This can be a source of much frustration for you and for your instructors. My object in writing this essay is to help ease this frustration by describing some study strategies which may help you channel your abilities and energies in …
When Does A Category Built On A Lattice With A Monoidal Structure Have A Monoidal Structure?, Lawrence Stout
When Does A Category Built On A Lattice With A Monoidal Structure Have A Monoidal Structure?, Lawrence Stout
Lawrence N. Stout
In a word, sometimes. And it gets harder if the structure on L is not commutative. In this paper we consider the question of what properties are needed on the lattice L equipped with an operation * for several different kinds of categories built using Sets and L to have monoidal and monoidal closed structures. This works best for the Goguen category Set(L) in which membership, but not equality, is made fuzzy and maps respect membership. Commutativity becomes critical if we make the equality fuzzy as well. This can be done several ways, so a progression of categories is considered. …
Recursion, Infinity, And Modeling, Lawrence Stout, Hans-Jorg Tiede
Recursion, Infinity, And Modeling, Lawrence Stout, Hans-Jorg Tiede
Lawrence N. Stout
Hauser, Chomsky, and Fitch (2002) claim that a core property of the human language faculty is recursion and that this property "yields discrete infinity" (2002: 1571) of natural languages. On the other hand, recursion is often motivated by the observation that there are infinitely many sentences that should be generated by a finite number of rules. It should be obvious that one cannot pursue both arguments simultaneously, on pain of circularity. The main aim of this paper is to clarify both conceptually and methodologically the relationship between recursion and infinity in language. We want to argue that discrete infinity is …
Categorical Approaches To Non-Commutative Fuzzy Logic, Lawrence Stout
Categorical Approaches To Non-Commutative Fuzzy Logic, Lawrence Stout
Lawrence N. Stout
In this paper we consider what it means for a logic to be non-commutative, how to generate examples of structures with a non-commutative operation * which have enough nice properties to serve as the truth values for a logic. Inference in the propositional logic is gotten from the categorical properties (products, coproducts, monoidal and closed structures, adjoint functors) of the categories of truth values. We then show how to extend this view of propositional logic to a predicate logic using categories of propositions about a type A with functors giving change of type and adjoints giving quantifiers. In the case …
A Categorical Semantics For Fuzzy Predicate Logic, Lawrence N. Stout
A Categorical Semantics For Fuzzy Predicate Logic, Lawrence N. Stout
Lawrence N. Stout
The object of this study is to look at categorical approaches to many valued logic, both propositional and predicate, to see how different logical properties result from different parts of the situation. In particular, the relationship between the categorical fabric I introduced at Linz in 2004 and the Fuzzy Logics studied by Hajek (2003) [5], Esteva et al. (2003) [1], and Hajek (1998) [4], comes from restricting the kind of structures used for truth values. We see how the structure of the various kinds of algebras shows up in the categorical logic, giving a variant on natural deduction for these …
Sequences, Series, And Function Approximation, Lawrence Stout
Sequences, Series, And Function Approximation, Lawrence Stout
Lawrence N. Stout
Sequences are important in approximation: the usual representation of real numbers using decimals is in fact the process of giving a sequence of rational numbers approximation the real number in question successively better as more decimal places are given. These decimal approximation sequences are actually rather special: successive decimal approximations never get smaller (so the sequence is monotone nondecreasing) and two approximations which agree to the kth decimal place differ by at most 10-k (so the sequence is a Cauchy sequence: to make two values in the sequence close to each other all you need to do is take them …
Open Problems From The Linz2000 Closing Session, Lawrence N. Stout
Open Problems From The Linz2000 Closing Session, Lawrence N. Stout
Lawrence N. Stout
No abstract provided.
Categories Of Fuzzy Sets With Values In A Quantale Or Projectale, Lawrence Stout
Categories Of Fuzzy Sets With Values In A Quantale Or Projectale, Lawrence Stout
Lawrence N. Stout
Properties of the lattice L are reflected in the properties of the categories Set(L), Set(L)/(A,α), and the lattice U(A, α). The lattices U(A,α) best reflect the structures on the lattice if the structure is inherited by closed down segments and direct products. Operations at the level of the slice categories require distributivity too. The first object of this paper is to see how the additional structures given by an associative operation & which distributes over sups and hence has a right adjoint (a quantale in the sense of Rosenthal [7]) shows up at the three different levels for fuzzy sets. …
The Logic Of Unbalanced Subobjects In A Category With Two Closed Structures, Lawrence Stout
The Logic Of Unbalanced Subobjects In A Category With Two Closed Structures, Lawrence Stout
Lawrence N. Stout
No abstract provided.
Foundations Of Fuzzy Sets, Lawrence Stout, Ulrich Höhle
Foundations Of Fuzzy Sets, Lawrence Stout, Ulrich Höhle
Lawrence N. Stout
This paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice. It then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy AND connective. Within these categories weak forms of subobject representations provide an internal second order logic. An approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal; second order theory.
A Survey Of Fuzzy Set And Topos Theory, Lawrence Stout
A Survey Of Fuzzy Set And Topos Theory, Lawrence Stout
Lawrence N. Stout
This paper is a comparison and contrast of approaches to many-valued mathematics offered by Fuzzy Set theory and topos theory. It gives a survey of the categorical foundations of Fuzzy Set theory and related topoi. Topoi are not a basis for Fuzzy Set theory but they do suggest appropriate directions to go and questions to ask for a synthesis which does provide a foundation. One possible structure which has a topos-like internal logic and the, rich variety of logical connectives used in fuzzy sets is included.
Two Discrete Forms Of The Jordan Curve Theorem, Lawrence N. Stout
Two Discrete Forms Of The Jordan Curve Theorem, Lawrence N. Stout
Lawrence N. Stout
The Jordan curve theorem is one of those frustrating results in topology: it is intuitively clear but quite hard to prove. In this note we will look at two discrete analogs of the Jordan curve theorem that are easy to prove by an induction argument coupled with some geometric intuition. One of the surprises is that when we discretize the plane we get two Jordan curve theorems rather than one, a consequence of the interplay between two natural products in the category of graphs. Topology in this context has been studied by Farmer in [2]. To state the discrete versions, …
Problems, Lawrence Stout
Topoi And Categories Of Fuzzy Sets, Lawrence Stout
Topoi And Categories Of Fuzzy Sets, Lawrence Stout
Lawrence N. Stout
Let H be a completely distributive lattice and hence a Heyting algebra. Goguen's category of fuzzy sets Set(H) Eytan's logos Fuz(H) and the topos of sheaves on H, Sh(H), are interconnected by pairs of adjoint functors between them. Each of the categories has a predicate calculus. These predicate calculi are related through the functors between the categories. Change of base lattice gives rise to several functors which preserve or reflect specific kinds of statements in the predicate calculi. This paper gives details of the categories, the predicate calculi attached to them, the functors between the categories, and the preservation properties …
Laminations, Or How To Build A Quantum-Logic-Valued Model Of Set Theory, Lawrence Stout
Laminations, Or How To Build A Quantum-Logic-Valued Model Of Set Theory, Lawrence Stout
Lawrence N. Stout
An explicit construction of the colimit of a filtered diagram in the category of topoi and logical morphisms is given and then used to construct a family of topoi with a fixed Boolean algebra of truth values but with varying amounts of cocompleteness. This same construction, when applied to the diagram of complete Boolean algebras in a quantum logic Q gives a partial topos, a noncategory which is a close to being a model of set theory with algebra of truth values Q as a noncategory can be.
Independence Theories And Generalized Zero-One Laws, Lawrence Stout
Independence Theories And Generalized Zero-One Laws, Lawrence Stout
Lawrence N. Stout
In this paper an abstract characterization of the properties of independent events is given with examples from topology, probability, and Baire structures. Using this notion of independence, proofs of the Hewitt-Savage and Kolmogorov zero-one laws are given which include the probabilistic case and the topological cases considered by Oxtoby, Christensen, and K. P. S. and M. Bhaskara Rao.
A Topological Structure On The Structure Sheaf,, Lawrence Stout
A Topological Structure On The Structure Sheaf,, Lawrence Stout
Lawrence N. Stout
No abstract provided.