Piece Of Mathematics Research Articles

A noetic account of explanation in mathematics

Abstract We defend a noetic account of intramathematical explanation. On this view, a piece of mathematics is explanatory just in case it produces understanding of an appropriate type. We motivate the view by presenting some appealing features of noeticism. We then discuss and criticize the most prominent extant version of noeticism, due to Inglis and Mejía Ramos, which identifies explanatory understanding with the possession of well-organized cognitive schemas. Finally, we present a novel noetic account. On our view, explanatory understanding arises from meeting specific explanatory objectives. We defend a cluster-concept account of explanatory objectives and identify four important subfamilies within the relevant network of resemblance relations. The resulting view is objectivist (in the sense that it takes explanatory success to be a matter of observer-independent fact), broader in scope than why-question-based accounts, compatible with empirical findings on experts’ explanatory judgments, and capable of generalizing (with appropriate provisos) to scientific explanation as a whole. It thus fulfills Friedman’s half-century-old demand for a general and objectivist theory, which accounts for the link between explanation and understanding.

Structure and applied mathematics

‘Mapping accounts’ of applied mathematics hold that the application of mathematics in physical science is best understood in terms of ‘mappings’ between mathematical structures and physical structures. In this paper, I suggest that mapping accounts rely on the assumption that the mathematics relevant to any application of mathematics in empirical science can be captured in an appropriate mathematical structure. If we are interested in assessing the plausibility of mapping accounts, we must ask ourselves: how plausible is this assumption as a working hypothesis about applied mathematics? In order to do so, we examine the role played by mathematics in the multiscalar modelling of sea ice melting behaviour and examine whether we can indeed capture the mathematics involved in the kind of mathematical structure employed by the mapping account. Along the way, we note that the cases of applied mathematics that appear in discussions of mapping accounts almost exclusively involve the employment of a single clearly circumscribed mathematical field or domain (e.g. ‘the use of arithmetic in counting physical objects’). While the core assumption of mapping accounts may appear plausible in such situations, we ultimately suggest that the mapping account is not able to handle the important added complexities involved in our sea ice case study. In particular, the notion of mathematical structure around which such accounts are framed does not seem to be able to capture the way in which some applications of mathematics require that very different pieces of mathematics be related to one another on the basis of both mathematical and empirical information.

When science mirrors life: on the origins of the Price equation.

The Price equation was a piece of abstract mathematics. What kind of a connection could it possibly have had to George Price's personal life and biography? Here, I will argue that the initial impetus for Price's foray into mathematical population genetics stemmed from a preoccupation with the origins of family, one that was born following a divorce from his wife and the abandonment of their two young girls. What is special about the Price equation is the way in which it associates statistically between two groups, a 'mother' and 'daughter' population. The association need not mean genetic relatedness in the narrow sense of direct descent, and it allows us to see selection working at different levels simultaneously, a fact that was not lost on William Hamilton. Hamilton was one of the few friends who desperately tried to save Price from falling into the abyss of depression and homelessness in the period following the publication of 'Selection and covariance' (Price 1928 Nature227, 520-521 (doi:10.1038/227520a0)). Viewed in this light, the Price equation assumes new meaning. This article is part of the theme issue 'Fifty years of the Price equation'.

A comparative performance analysis of different machine learning techniques

Over the past many decades, Machine Learning (ML) has advanced from the undertaking of few PC fans misusing the likelihood of PCs figuring out how to play diversions, and a piece of Mathematics (Statistics) that only here and there measured computational methodologies, to an autonomous research obedience that has not just given the essential base to measurable computational standards of learning systems, yet in addition has created different calculations that are routinely utilized for content translation, design acknowledgment, and a numerous other business purposes and has prompted a different research enthusiasm for information mining to recognize shrouded regularities or abnormalities in social information that developing by second. This paper centers around clarifying the idea and development of Machine Learning, a portion of the well known Machine Learning calculations and endeavor to think about three most prevalent calculations dependent on some essential thoughts. Sentiment140dataset was utilized and execution of every calculation as far as preparing time, forecast time and precision of expectation have been reported and analyzed.

Proof of van der Waerden’s Theorem in Nine Figures

This note contains a proof of van der Waerden’s theorem, “one of the most elegant pieces of mathematics ever produced,” in nine figures. The proof follows van der Waerden’s original idea to establish the existence of what are now called van der Waerden numbers by using double induction. It also contains ideas and terminology introduced by I. Leader and T. Tao.

On Linear Ordinary Differential Equation With Orthogonal Polynomial Solutions

: The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. In this paper we give a survey of the orthogonal polynomial solutions of second-order and fourth-order linear ordinary differential equation, on the generated self-adjoint differential operators .

Climate Shock: The Economic Consequences of a Hotter Planet. By Gernot Wagner and Martin L. Weitzman. Princeton (New Jersey): Princeton University Press. $27.95. xiii + 250 p.; ill.; index. ISBN: 978-0-691-15947-8. 2015.

Previous articleNext article FreeConservation BiologyClimate Shock: The Economic Consequences of a Hotter Planet. By Gernot Wagner and Martin L. Weitzman. Princeton (New Jersey): Princeton University Press. $27.95. xiii + 250 p.; ill.; index. ISBN: 978-0-691-15947-8. 2015.Richard S. J. TolRichard S. J. TolEconomics, University of Sussex, Brighton, United Kingdom and Institute for Environmental Studies, Vrije Universiteit, Amsterdam, The Netherlands Search for more articles by this author Economics, University of Sussex, Brighton, United Kingdom and Institute for Environmental Studies, Vrije Universiteit, Amsterdam, The NetherlandsPDFPDF PLUSFull Text Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinked InRedditEmailQR Code SectionsMoreMartin Weitzman’s papers are eloquent, precise, and insightful. He excels at taking pieces of mathematics and reinterpreting them to shine a new light on old questions. In Climate Shock, the eminent Harvard scholar teams up with Gernot Wagner, an environmental activist. Weitzman’s later papers are turned into a long essay, replacing the succinct and precise mathematics of the originals with woolly words. The style is eclectic. Parts of the book follow the standard in popular science writing, but Chapter 2 does not work without the hyperlinks and Chapter 7 starts with a poorly written script for a Bond movie. Fact checking was not a priority. The chronology of the discovery of the greenhouse effect is mixed up. Nixon’s amendments to the Clean Air Act are two years off. The first carbon tax was in Norway not Sweden. The volume argues for cap-and-trade and against a carbon tax, as if Weitzman’s seminal 1974 paper on that question was never written.The book barely scratches the surface on the subject of its subtitle, the economic consequences of climate change. Uncertainty dominates the discussion in the first half of the volume. We should proceed with caution if the effects of our decisions are uncertain. Climate Shock emphasizes the uncertainties about climate change and its impacts, but hardly discusses the uncertainties about the impacts of climate policy. It ignores that negative climate surprises are more likely than positive ones, and that climate change is irreversible. Both are good reasons to be extra cautious and reduce greenhouse gas emissions faster. Instead, the focus is on Weitzman’s Dismal Theorem, which states that the uncertainties about climate change are so large that it is not possible to do a cost-benefit analysis of greenhouse gas emission reduction policy.The Dismal Theorem was first published in 2009. We have since learned that this striking result comes apart if Weitzman’s analysis is generalized to include mitigation policy or the feedbacks of climate change on economic growth, or if his constant relative risk aversion is replaced by more realistic hyperbolic absolute risk aversion. Project evaluation methods that are robust to large uncertainties do not recommend policies that are very different from cost-benefit analysis. Unfortunately, Climate Shock omits these critiques.The second half of the book is about geoengineering. It is short on the technical aspects, instead focusing on one key characteristic: global climate can be geoengineered by a single country or even a mid-sized organization. The international policy response should therefore focus on stopping countries that deploy geoengineering against the wishes of the rest of world. Climate Shock correctly diagnoses the issue, but fails to suggest a solution.Altogether, the volume disappoints. If you want to understand Weitzman’s work, his papers are superior. If you want a popular discussion of the economics of climate change, William Nordhaus’ books are unsurpassed. Previous articleNext article DetailsFiguresReferencesCited by The Quarterly Review of Biology Volume 92, Number 1March 2017 Published in association with Stony Brook University Article DOIhttps://doi.org/10.1086/690865 For permission to reuse, please contact [email protected]PDF download Crossref reports no articles citing this article.

Contemporary PDEs between theory and applications

This special issue of Discrete and Continuous Dynamical Systems is devoted to some recent developments in some important fields of partial differential equations. The aim is to bring together several contributions in different fields that range from classical to modern topics with the intent to present new research perspectives, innovative methods and challenging applications. Though it was of course impossible to take into account all the possible lines of research in PDEs, we tried to present a wide spectrum, hoping to capture the interest of both the general mathematical audience and the specialized mathematicians that work in differential equations and related fields. We think that the Authors put a great effort to write their contributions in the clearest possible language. We are indeed grateful to all the Authors that contributed to this special issue, donating beautiful pieces of mathematics to the community and promoting further developments in the field. We also thank the Managing Editor for his kind invitation to act as an editor of this special issue. Also, we express our gratitude to all the Referees who kindly agreed to devote their time and efforts to read and check all the papers carefully, providing useful comments and recommendations. Indeed, each paper was submitted to the meticulous inspection of two independent and anonymous Experts, whose observations were fundamental to the final outcome of this special issue. Finally, we would like to wish a `Happy reading!' to the Reader. This volume is for Her (or Him), after all.

Engineers Do It, Scientists Do It, Mathematicians?

There is a magnificent mathematical gem missing from most numerical analysis curricula. A literature search of numerical analysis and mathematical modeling texts indicates its absence. What is missing from the numerical analysis toolbox and why isn't it there? The missing tool is the Kalman filter. The Kalman filter requires a modest knowledge of statistics. Is this why it is missing from the toolbox? Read and reach your own conclusion whether this piece of mathematics should be part of a numerical analysis or mathematical modeling curriculum.

Mathematical Beauty, Understanding, and Discovery

In a very influential paper Rota stresses the relevance of mathematical beauty to mathematical research, and claims that a piece of mathematics is beautiful when it is enlightening. He stops short, however, of explaining what he means by ‘enlightening’. This paper proposes an alternative approach, according to which a mathematical demonstration or theorem is beautiful when it provides understanding. Mathematical beauty thus considered can have a role in mathematical discovery because it can guide the mathematician in selecting which hypothesis to consider and which to disregard. Thus aesthetic factors can have an epistemic role qua aesthetic factors in mathematical research.

An example from the world of tsetse flies

In biomathematics, communication between mathematicians and biologists is crucial. This matter is illustrated using studies aimed at estimating mortality rates of tsetse flies (Glossina spp.). Examples are provided of apparently sound pieces of mathematics which, when applied to real data, provide obviously erroneous results. More serious objections arise when mathematical models make no attempt to address the real world in such a way that they can be tested. Unless models account for the known biology of the problem under investigation, and are challenged with data, the existence and nature of imperfections in the models will likely not be detected.

What is the most symmetric quadrilateral?

What is the most symmetric quadrilateral you can think of? Most people would say ‘a square’, but having given this (or any other) answer, can you create your own definition of a ‘quadrilateral’, and of ‘most symmetric’, so that you can justify your answer as a rigorous piece of mathematics? This is a challenging investigation because the answer may not be the obvious one.Often the main difficulty in solving a problem lies in finding appropriate definitions; here we must first consider the questions ‘what is a quadrilateral?’, and ‘what is a symmetry of a quadrilateral?’.

Some odd permutations

I recently stumbled on an identity (Theorem 2) that related two types of permutations. It looked like one of those serendipitous pieces of mathematics that led to a simple enumerative result; mostly when this happens it leads joyously to an enumerative proof that was ‘blindingly’ obvious. Except in this case, it wasn’t, I don’t think.

Collective Mathematical Understanding as Improvisation

This article explores the phenomenon of mathematical understanding, and offers a response to the question raised by Martin (2001) at the Annual Meeting of the Psychology of Mathematics Education Group (North American Chapter) about the possibility for and nature of collective mathematical understanding. In referring to collective mathematical understanding, we point to the kinds of learning and understanding we may see occurring when a group of learners work together on a piece of mathematics. We characterize the growth of collective mathematical understanding as a creative and emergent improvisational process and illustrate how this can be observed in action. In doing this, we demonstrate how a collective perspective on mathematical understanding can more fully explain its growth. We also discuss how considering the growth of mathematical understanding as a collective process has implications for classroom practice and in particular for the setting of mathematical tasks.

Answer to Question #31. Does any piece of mathematics exist for which there is no application whatsoever in physics?

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What goes round comes round

At school I used to love the geometry of the circle, but in my own career I had little opportunity to teach it. Now in semi-retirement my work with Teasers in the Sunday Times brings me into contact with many entertaining aspects of mathematics, not least being the occasional piece of geometry. In this short article I will quote four potted versions of recent Teasers, each of which relies on some interesting piece of mathematics, and I shall sketch in each case how that property can be deduced neatly using circles.

Inspiring exercises for undergraduates

There are two ways to learn mathematics: to discover (research) and to rediscover. Inspiring exercises are used to guide students at all levels to rediscover the essential meaning of various individual pieces of mathematics. Their role has become all the more important at atime when Bourbakization is the dominating fashion. While mathematics is decorated with a thick layer of cosmetics, such as utmost generalities, excessive formalism and rigour, subtle proofs, etc., its natural beauty (intuitive ways of thinking, simplicity of ideas, etc.) is obscured. From apedagogical point of view, to say the least, it would be most desirable if a theorem, or an idea, could be fully explained by one or two simple examples. These kinds of illustrative examples actually exist everywhere, and at all levels. We can use them as inspiring exercises. In this paper, five sets of examples are given, beginning with asimple one on Abel's identity, followed by examples on Hensel's lemma, finitely generated Abelian groups, Baire's category theorem and the Weierstrass preparation theorem. Solutions, hints and discussions are provided in the final section.

Answer to Question #31. Does any piece of mathematics exist for which there is no application whatsoever in physics?

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Question ♯31. Does any piece of mathematics exist for which there is no application whatsoever in physics?

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Knot physics

It is a curious fact that the structures invented by physicists to provide a framework to describe the basic laws of the Universe are often of interest to mathematicians. This is strange because the aims of the two disciplines are very different. In physics, there is a wish to describe, at least in principle, all phenomena as economically viable, at least in simple testable cases. If alternative schemes exist, then elegance may play a role in deciding between them, but generally, experiment provides strong constraints. Thus, a theory which is both elegant and fits the facts is rare. When such a theory is discovered it is rich in predictive power and can often be generalised beyond the hopes of its discoverers. It will last for a long time. Examples of these are Newtonian mechanics, general relativity, quantum mechanics and quantum field theory, the latter providing the modern framework for describing all the fundamental interactions between elementary particles. Mathematics, on the other hand, is more abstract and free from the need to explain or find relationship between real-world phenomena. Nevertheless, in mathematics, fruitful concepts will apply to more than one branch of the subject and can provide unifying principles, bringing together pieces of mathematics which appeared to be unrelated.