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18
Hypercomputation: computing more than the Turing machine
, 2002
"... In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which al ..."
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Cited by 32 (5 self)
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In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which allows comparisons of the many approaches and results. To this I add several new results and draw out some interesting consequences of hypercomputation for several different disciplines. I begin with a succinct introduction to the classical theory of computation and its place amongst some of the negative results of the 20 th Century. I then explain how the ChurchTuring Thesis is commonly misunderstood and present new theses which better describe the possible limits on computability. Following this, I introduce ten different hypermachines (including three of my own) and discuss in some depth the manners in which they attain their power and the physical plausibility of each method. I then compare the powers of the different models using a device from recursion theory. Finally, I examine the implications of hypercomputation to mathematics, physics, computer science and philosophy. Perhaps the most important of these implications is that the negative mathematical results of Gödel, Turing and Chaitin are each dependent upon the nature of physics. This both weakens these results and provides strong links between mathematics and physics. I conclude that hypercomputation is of serious academic interest within many disciplines, opening new possibilities that were previously ignored because of long held misconceptions about the limits of computation.
Implementing WS1S via Finite Automata
"... It has long been known that WS1S is decidable through the use of finite automata. However, since the worst case running time has been proven to grow extremely quickly, few have explored the implementation of the algorithm. In this paper we describe some of the points of interest that have come up wh ..."
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Cited by 6 (0 self)
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It has long been known that WS1S is decidable through the use of finite automata. However, since the worst case running time has been proven to grow extremely quickly, few have explored the implementation of the algorithm. In this paper we describe some of the points of interest that have come up while coding and running the algorithm. These points include the data structures used as wekk as the special properties of the automata, which we can exploit to perform minimization very quickly in certain cases. We also present some data that enable us to gain insight into how the algorithm performs in the average case, both on random inputs ans on inputs that come from the use of Presburger Arithmetic (which can be converted to WS1S) in compiler optimization.
1 Pluralistic Modeling of Complex Systems
, 1007
"... The modeling of complex systems such as ecological or socioeconomic systems can be very challenging. Although various modeling approaches exist, they are generally not compatible and mutually consistent, and empirical data often do not allow one to decide what model is the right one, the best one, ..."
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Cited by 2 (1 self)
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The modeling of complex systems such as ecological or socioeconomic systems can be very challenging. Although various modeling approaches exist, they are generally not compatible and mutually consistent, and empirical data often do not allow one to decide what model is the right one, the best one, or most appropriate one. Moreover, as the recent financial and economic crisis shows, relying on a single, idealized model can be very costly. This contribution tries to shed new light on problems that arise when complex systems are modeled. While the arguments can be transferred to many different systems, the related scientific challenges are illustrated for social, economic, and traffic systems. The contribution discusses issues that are sometimes overlooked and tries to overcome some frequent misunderstandings and controversies of the past. At the same time, it is highlighted how some longstanding scientific puzzles may be solved by considering nonlinear models of heterogeneous agents with spatiotemporal interactions. As a result of the analysis, it is concluded that a paradigm shift towards a pluralistic or possibilistic modeling approach, which integrates multiple world views, is overdue. In this connection, it is argued that it can be useful to combine many different approaches to obtain a good picture of reality, even though they may be inconsistent. Finally, it is identified what would be profitable areas of collaboration between the socioeconomic, natural, and engineering sciences. 1
Our Prediction of the Near Future for the West.................... 8
, 2009
"... A very short pamphlet overviewing our principles and proposals which offer a highgrowth future for our civilisation in a time of ..."
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A very short pamphlet overviewing our principles and proposals which offer a highgrowth future for our civilisation in a time of
unknown title
"... Discuss the advantages and disadvantages of adopting a multiparadigm or multirationality approach to management and decision making This paper shall examine the consequences of a conscious adoption of a multiparadigm or multirationality approach. Firstly the limits of formal logic are examined, a ..."
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Discuss the advantages and disadvantages of adopting a multiparadigm or multirationality approach to management and decision making This paper shall examine the consequences of a conscious adoption of a multiparadigm or multirationality approach. Firstly the limits of formal logic are examined, and thus those of rationality, and what logic itself suggests should be done about it. Management is supposed to direct organisations through highlevel decision making, but this paper suggests its true purpose is actually to manage uncertainty 1 i.e.; organisations exist to manage uncertainty, usually through conversion into risk. The paper then concludes with case examples of how pathological the fear of uncertainty can become, but also how paradoxically the exact same mechanism can save us from the same morass. The Limits of Rationality More than anything else, the age of reason defines the modern age 2, and nothing is of more importance to rational thought than the systematic application of logic and mathematics. However, with the publication of Principia Mathematica in 19111913, the terrible reality that logic has limits reared its head 3. With Kurt Gödel’s 1931 paper on Principia Mathematica, the incompleteness of formal logical systems became incontrovertible truth. Put simply, Gödel’s Incompleteness Theorem means that for any consistent formal theory that proves basic arithmetical truths, one can construct an arithmetical statement that is true but not 1 Uncertainty throughout this paper is used as a technical term for the future.
UNIVERSITY OF PITTSBURGH FACULTY OF ARTS AND SCIENCES
, 2003
"... This dissertation was presented by ..."
Connected Mathematics  Building . . . with Mathematical Knowledge
, 1993
"... The context for this thesis is the conflict between two prevalent ways of viewing mathematics. The first way is to see mathematics as primarily a formal enterprise, concerned with working out the syntactic/formal consequences of its definitions. In the second view, mathematics is a creative enterpri ..."
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The context for this thesis is the conflict between two prevalent ways of viewing mathematics. The first way is to see mathematics as primarily a formal enterprise, concerned with working out the syntactic/formal consequences of its definitions. In the second view, mathematics is a creative enterprise concerned primarily with the construction of new entities and the negotiation of their meaning and value. Among teachers of mathematics the formal view dominates. The consequence for learners is a shallow brittle understanding of the mathematics they learn. Even for mathematics that they can do, in the sense of calculating an answer, they often can't explain why they're doing what they're doing, relate it to other mathematical ideas or operations, or connect the mathematics to any idea or problem they may encounter in their lives. The aim of this thesis is to develop alternative ways of teaching mathematics which strengthen the informal, intuitive and creative in mathematics. This research develops an approach to learning mathematics called "connected mathematics" which emphasizes learners’ negotiation of mathematical meaning. I have
The Failure of Leibniz’s Infinite Analysis view of Contingency
"... Abstract: In this paper, it is argued that Leibniz’s view that necessity is grounded in the availability of a demonstration is incorrect and furthermore, can be shown to be so by using Leibniz’s own examples of infinite analyses. First, I show that modern mathematical logic makes clear that Leibniz’ ..."
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Abstract: In this paper, it is argued that Leibniz’s view that necessity is grounded in the availability of a demonstration is incorrect and furthermore, can be shown to be so by using Leibniz’s own examples of infinite analyses. First, I show that modern mathematical logic makes clear that Leibniz’s "infinite analysis " view of contingency is incorrect. It is then argued that Leibniz's own examples of incommensurable lines and convergent series undermine, rather than bolster his view by providing examples of necessary mathematical truths that are not demonstrable. Finally, it is argued that a more modern view on convergent series would, in certain respects, help support some claims he makes about the necessity of mathematical truths, but would still not yield a viable theory of necessity due to remaining problems with other logical, mathematical, and modal claims. From his early metaphysical writings, such as “On Freedom and Possibility ” to his later writings such as “The Monadology”, Leibniz distinguished between those propositions which are necessary and those that are contingent. A central problem for Leibniz is to explain how there could be such a distinction. Since all truths necessarily follow from God’s choice to actualize this world, (truths that Leibniz calls hypothetically or morally necessary), it seems that since God
“Common Grace Social Capital ” Investments for Sustaining Ethical Conduct in New and Emerging Economies” replaces original title below: Synergistic Development of Sustainable Ethical Processes when Pursuing Opportunities with Unpredictable Emerging Econom
"... Ten “common grace ” social capital investments are developed as normative propositions for sustaining ethical conduct with a variety of new economy and emerging economy challenges. Drawing primarily upon Clay Christensen and Geoff Moore’s findings regarding adoption risks with “disruptive technologi ..."
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Ten “common grace ” social capital investments are developed as normative propositions for sustaining ethical conduct with a variety of new economy and emerging economy challenges. Drawing primarily upon Clay Christensen and Geoff Moore’s findings regarding adoption risks with “disruptive technologies ” authenticity/integrity and competence issues are explored within the larger perspectives of true excellence and goodness derived from Morris ’ “If Aristotle Ran G.M.”. The cognitive psychological limits to moral development such as the illusion of ethical superiority and inoculation effects attending legalistic judgmentalism constitute some of the more serious impediments that suggest the need for a combination of trust and gratitude resolutions. When practical applications of trust and gratitudebased mentoring are synthesized with the “social capital ” guidelines from Prusak & Cohen we begin to see the desirability of a “common grace ” social capital construct workedout in terms of five trust and five gratitude guidelines for sustainable ethical conduct robust enough to handle extraordinary pressures from “new economy pathologies ” and emerging economies.
A WEAK QUANTUMLIKE INDETERMINACY PRINCIPLE IN LOGIC
"... We resort to a logical phenomenon related to paradoxes and possibly to other logical facts, like limitation theorems and transfinite set theory, to shed light upon the meaning of Heisenberg’s Indeterminacy Principle. Our aim is to show how critical realism as opposed to naïve realism might be consis ..."
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We resort to a logical phenomenon related to paradoxes and possibly to other logical facts, like limitation theorems and transfinite set theory, to shed light upon the meaning of Heisenberg’s Indeterminacy Principle. Our aim is to show how critical realism as opposed to naïve realism might be consistent with this principle and its empirical and theoretical consequences. Call (1) the sentencetoken “(1) expresses no true proposition”. (1) has no truth value, so it describes no stateofaffairs. We partly state this by means of the sentencetoken (2) which says “(1) expresses no true proposition”. (2) describes an actual stateofaffairs and is true. This is possible only because (1) and (2) are uttered in different logical contexts. Since (2) is based upon a previous assessment of (1), we say that (2) stands in a logically posterior instant. There is a stateofaffairs that is an available stateofaffairs in the logical instant to which (2) belongs but is not yet such in the logical instant in which (1) stands. Only the introduction of this kind of logical temporality makes the set of available statesofaffairs relative to the logical context, thereby rendering possible the solution to the paradox. This reveals that some logical objects are distributed along some logical temporality. The reason of this relativity resides in a fact that can be couched in the terms of Husserl’s