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They will also enhance their knowledge and understanding of the fundamental theorem of calculus. Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column. The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic eigenvectors and eigenvalues.

The student will learn how to express a linear transformation of the real Euclidean space using a matrix, from which they will be able to determine whether it is singular or not and obtain its characteristic equation and eigenspaces. The student is introduced to logic and mathematical proofs, with emphasis placed more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.

The language and structure of mathematical proofs will be explained, highlighting how logic can be used to express mathematical arguments in a concise and rigorous manner. The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials. Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics.

This module will introduce students to some simple combinatorics, set theory and the axioms of probability. Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.

To enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems, the module begins with a brief overview of statistics in science and society. It then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.

Students will develop a strategic understanding of statistics and the use of associated software, which will underpin the skills needed for all subsequent statistical modules of the degree. This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions. Here, students will select a small number of properties which these and other examples have in common, and use them to define a group.

They will also consider the elementary properties of groups. By looking at maps between groups which 'preserve structure', a way of formalizing and extending the natural concept of what it means for two groups to be 'the same' will be discovered.

Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives students a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials; several less familiar examples will also be explored. Complex Analysis has its origins in differential calculus and the study of polynomial equations.

In this module, students will consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. They will use integral calculus of complex functions to find elegant and important results and will also use classical theorems to evaluate real integrals.

The first part of the module reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The module then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by way of a bisection argument. These analytic ideas are used to prove the fundamental theorem of algebra, that every non-constant complex polynomial has a root. Finally, the theory is employed to evaluate some definite integrals.

The module ends with basic discussion of harmonic functions, which play a significant role in physics. Students will gain a solid understanding of computation and computer programming within the context of maths and statistics. This module expands on five key areas:. Under these headings, students will study a range of complex mathematical concepts, such as: data structures, fixed-point iteration, higher dimensions, first and second derivatives, non-parametric bootstraps, and modified Euler methods.

Throughout the module, students will gain an understanding of general programming and algorithms. They will develop a good level of IT skills and familiarity with computer tools that support mathematical computation. Over the course of this module, students will have the opportunity to put their knowledge and skills into practice. Workshops, based in dedicated computing labs, allow them to gain relatable, practical experience of computational mathematics.

Students will be provided with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year Two, and the more specialised Year Three modules. This module will give students the opportunity to study vector spaces, together with their structure-preserving maps and their relationship to matrices.

They will consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of their study will also involve looking at the concepts of length and angle with regard to vector spaces. Basic concepts from the first year probability module will be revisited and extended to these to encompass continuous random variables, with students investigating several important continuous probability distributions.

Commonly used distributions are introduced and key properties proved, and examples from a variety of applications will be used to illustrate theoretical ideas. Students will then focus on transformations of random variables and groups of two or more random variables, leading to two theoretical results about the behaviour of averages of large numbers of random variables which have important practical consequences in statistics.

Project Skills is a module designed to support and develop a range of key technical and professional skills that will be valuable for all career paths. Covering five major components, this module will guide students through and explore:. Students will gain an excellent grasp of LaTeX, learning to prepare mathematical documents; display mathematical symbols and formulae; create environments; and present tables and figures. Scientific writing, communication and presentations skills will also be developed.

Students will work on short and group projects to investigate mathematical or statistical topics, and present these in written reports and verbal presentations. A thorough look will be taken at the limits of sequences and convergence of series during this module.

Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at A-level. Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions rather than just numbers , which again has many applications and is central to more advanced analysis. Next, the notion of integration will be put under the microscope.

Once it is properly defined via limits students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in A-level, such as estimations of discrete sums of series. Statistics is the science of understanding patterns of population behaviour from data.

In the module, this topic will be approached by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated. The focus will be on likelihood-based parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and linear regression techniques within the statistical modelling framework will also be considered.

Students will come to recognise the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions. Bayesian statistics provides a mechanism for making decisions in the presence of uncertainty. The purpose of the data collection exercise is expressed through a utility function, which is specific to the client or user.

It defines what is to be gained or lost through taking particular actions in the current environment. Actions are continually made or not made depending on the expectation of this utility function at any point in time.

Bayesians admit probability as the sole measure of uncertainty. Thus Bayesian reasoning is based on a firm axiomatic system. In addition, since most people have an intuitive notion about probability, Bayesian analysis is readily communicated. Combinatorics is the core subject of discrete mathematics which refers to the study of mathematical structures that are discrete in nature rather than continuous for example graphs, lattices, designs and codes.

While combinatorics is a huge subject - with many important connections to other areas of modern mathematics - it is a very accessible one. In this module, students will be introduced to the fundamental topics of combinatorial enumeration sophisticated counting methods , graph theory graphs, networks and algorithms and combinatorial design theory Latin squares and block designs.

They will also explore important practical applications of the results and methods. They will be introduced to two new classes of integral domains called Euclidean domains, where they have a counterpart of the division algorithm, and unique factorisation domains, in which an analogue of the Fundamental Theorem of Arithmetic holds. How well-known concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorisation of polynomials, carry over to this new setting, will also be explored.

Questions relating to linear ordinary differential equations will be considered during this module. Differential equations arise throughout the applications of mathematics, and consequently the study of them has always been recognised as a fundamental branch of the subject. The module aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations, and theoretical results about the nature of solutions.

While explicit solutions can only be found for special types of equations, some of the ideas of real analysis allow us to answer questions about the existence and uniqueness of solutions to more general equations, as well as allowing us to study certain properties of these solutions. The concept of generalised linear models GLMs , which have a range of applications in the biomedical, natural and social sciences, and can be used to relate a response variable to one or more explanatory variables, will be explored.

The response variable may be classified as quantitative continuous or discrete, i. Students will come to understand the effect of censoring in the statistical analyses and will use appropriate statistical techniques for lifetime data. They will also become familiar with the programme R, which they will have the opportunity to use in weekly workshops. The topic of smooth curves and surfaces in three-dimensional space is introduced.

The various geometrical properties of these objects, such as length, area, torsion and curvature, will be explored and students will have the opportunity to discover the meaning of these quantities. They will then use a variety of examples to calculate these values, and will use those values to apply techniques from calculus and linear algebra.

A number of well-known concepts will be encountered, such as length and area, and some new ideas will be introduced, including the curvature and torsion of a curve, and the first and second fundamental forms of a surface. Students will learn how to compute these quantities for a variety of examples, and in doing so will develop their geometric intuition and understanding.

The study of graphs - mathematical objects used to model pairwise relations between objects - is a cornerstone of discrete mathematics. As a result, students will develop an appreciation for a range of discrete mathematical techniques while undertaking this module. Throughout the module, students will also learn about structural notions, such as connectivity, and will explore trees, minor closed families of graphs, matrices related to graphs, the Tutte polynomial of small graphs, and planar graphs and analogues.

While studying these areas, students will gain experience of following and constructing mathematical proofs, and correctly and coherently using mathematical notation. Students will develop the knowledge of groups that they gained in second year during the Groups and Rings module. They will learn several quite natural ways to do this, both for vectors of any dimension and for functions.

Focus will then be on the more special notion of an inner product which generalises angles at the same time as distances. Once these concepts have been established, students will have the opportunity to study geometrical ideas like orthogonality, which can be seen to apply to much more general spaces than Euclidean spaces of three or even n dimensions, notably to infinite dimensional spaces of functions.

For example, Hilbert space theory shows how Fourier series are really another case of expressing an element in terms of a basis, and how people can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, students will look at some of the main results of linear algebra, which generalise very nicely to linear operators between Hilbert spaces. Introducing the Lebesgue integral for functions on the real line, this module features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral.

The bounded convergence theorem is used to prove the monotone and dominated convergence theorems, and the results are illustrated in classical convergence problems including Fourier integrals. Among the range of topics addressed on this module, students will become familiar with Lebesgue's definition of the integral, and the integral of a step function.

There will be an introduction to subsets of the real line, including open sets and countable sets. Students will measure of an open set, and will discover measurable sets and null sets. Additionally, the module will focus on integral functions, along with Lebesgue's integral of a bounded measurable function, his bounded convergence theorem and the integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem. Other topics on the module will include applications of the convergence theorems and Wallis's product for P.

Gaussian integral, along with some classical limit inversion results and the Fourier cosine integral. Students will develop an understanding of Dirichlet's comb function, Archimedes' axiom and Cantor's uncountability theorem, and will learn to prove the structure theorem for open sets. In addition, students will be able to prove covering lemmas for open sets, as well as understanding the statement of Heine—Bore theorem, as well as understanding the concept and proving basic properties of outer measure.

As well as understanding inner measure. Finally, students will be expected to prove Lebesgue's theorem on countable additivity of measure. Statistical inference is the theory of the extraction of information about the unknown parameters of an underlying probability distribution from observed data. Consequently, statistical inference underpins all practical statistical applications.

This module reinforces the likelihood approach taken in second year Statistics for single parameter statistical models, and extends this to problems where the probability for the data depends on more than one unknown parameter. Students will also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?

The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module. The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics modules and provide opportunities for further study.

The theory of linear systems is engineering mathematics. In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.

Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard A,B,C,D model. These include electrical appliances, heating systems and economic processes. The module shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables students to classify A,B,C,D models and describe their properties in terms of quantities which are relatively easy to compute.

The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilise a A,B,C,D system. Using the classical problem of data classification as a running example, this module covers mathematical representation and visualisation of multivariate data; dimensionality reduction; linear discriminant analysis; and Support Vector Machines. While studying these theoretical aspects, students will also gain experience of applying them using R.

An appreciation for multivariate statistical analysis will be developed during the module, as will an ability to represent and visualise high-dimensional data. Students will also gain the ability to evaluate larger statistical models, apply statistical computer packages to analyse large data sets, and extract and evaluate meaning from data.

Students will be given an opportunity to consider key issues in the teaching and learning of mathematics during this module. Whilst it is an academic study of mathematics education and not a training course for teachers, it does provide an excellent foundation for a PGCE especially in preparing students to write academically. Having studied mathematics for many years, students are well-placed to reflect upon that experience and attempt to make sense of it in the light of theoretical frameworks developed by researchers in the field.

This module will help them with this process so that as mathematics graduates they will be able to contribute knowledgeably to future debate about the ways in which this subject is treated within the education system. Students will have the opportunity to take part in classroom observation and assistance, the development of classroom resources, the provision of one-on-one or small group support and possibly even teaching sections of lessons to the class as a whole.

This module formally introduces students to the discipline of financial mathematics, providing them with an understanding of some of the maths that is used in the financial and business sectors. Students will begin to encounter financial terminology and will study both European and American option pricing.

The module will cover these in relation to discrete and continuous financial models, which include binomial, finite market and Black-Scholes models. Students will also explore mathematical topics, some of which may be familiar, specifically in relation to finance.

These include:. Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated risk-neutral probability; performing calculations with the Black-Scholes formula; and proving various steps in the derivation of the Black-Scholes formula. They will also be able to describe basic concepts of investment strategy analysis, and perform price calculations for stocks with and without dividend payments.

In addition, to these subject specific skills and knowledge, students will gain an appreciation for how mathematics can be used to model the real-world; improve their written and oral communication skills; and develop their critical thinking. The aim is to introduce students to the study designs and statistical methods commonly used in health investigations, such as measuring disease, causality and confounding.

Students will develop a firm understanding of the key analytical methods and procedures used in studies of disease aetiology, appreciate the effect of censoring in the statistical analyses, and use appropriate statistical techniques for time to event data. They will look at both observational and experimental designs and consider various health outcomes, studying a number of published articles to gain an understanding of the problems they are investigating as well as the mathematical and statistical concepts underpinning inference.

An introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications, is given during this module. Studying this module will give students a deeper understanding of continuity as well as a basic grounding in abstract topology. With this grounding, they will be able to solve problems involving topological ideas, such as continuity and compactness.

They will also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply their knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

Many numbers are special in some sense, eg. Which numbers can be expressed as the sum of two squares? What is special about the number ? Are there short cuts to factorizing large numbers or determining whether they are prime this is important in cryptography? Questions like these are easy to ask, and to describe to the non-specialist, but vary hugely in the amount of work needed to answer them.

To answer questions about the natural numbers, we sometimes use rational, real and complex numbers, as well as any ideas from algebra and analysis that help, including groups, integration, infinite series and even infinite products. This module introduces some of the central ideas and problems of the subject, and some of the methods used to solve them, while constantly illustrating the results with exercises and examples involving actual numbers.

This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities. First the notion of a probability space will be examined through simple examples featuring both discrete and continuous sample spaces. Random variables and the expectation will be used to develop a probability calculus, which can be applied to achieve laws of large numbers for sums of independent random variables. Students will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.

Important examples of stochastic processes, and how these processes can be analysed, will be the focus of this module. As an introduction to stochastic processes, students will look at the random walk process. Historically this is an important process, and was initially motivated as a model for how the wealth of a gambler varies over time initial analyses focused on whether there are betting strategies for a gambler that would ensure they won.

The focus will then be on the most important class of stochastic processes, Markov processes of which the random walk is a simple example. Students will discover how to analyse Markov processes, and how they are used to model queues and populations.

Modern statistics is characterised by computer-intensive methods for data analysis and development of new theory for their justification. In this module students will become familiar with topics from classical statistics as well as some from emerging areas. Time series data will be explored through a wide variety of sequences of observations arising in environmental, economic, engineering and scientific contexts.

Time series and volatility modelling will also be studied, and the techniques for the analysis of such data will be discussed, with emphasis on financial application. Another area the module will focus on is some of the techniques developed for the analysis of multivariates, such as principal components analysis and cluster analysis.

Lastly,students will spend time looking at Change-Point Methods, which include traditional as well as some recently developed techniques for the detection of change in trend and variance. Mathematics and Statistics graduates are very versatile, having in-depth specialist knowledge and a wealth of skills. Through this degree, you will graduate with a comprehensive skill set, including data analysis and manipulation, logical thinking, problem-solving and quantitative reasoning, as well as adept knowledge of the discipline.

As a result, mathematicians and statisticians are sought after in a range of industries, such as business and finance, defence, education, infrastructure and power, and IT and communications. The starting salary for many maths graduate roles is highly competitive, and career options include:. A degree in this discipline can also be useful for roles such as Finance Manager, Insurance Underwriter, Meteorologist, Quantity Surveyor, Software Developer, and many more.

Alternatively, you may wish to undertake postgraduate study at Lancaster and pursue a career in research and teaching. Our annual tuition fee is set for a month session, starting in the October of your year of study. The UK government has announced that students who will begin their course in will no longer be eligible to receive the same fee status and financial support entitlement as UK students.

This also applies to those who have deferred entry until Lancaster University has confirmed that students from EU Member States in and later, will now be charged the same tuition fees as other non-UK students. UK fees are set by the UK Government annually. For more information about tuition fees, including fees for Study Abroad and Work Placements, please visit our undergraduate tuition fees page. Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man.

You can find more information about this on our Island Fees page. For full details of the University's financial support packages including eligibility criteria, please visit our fees and funding page. Students also need to consider further costs which may include books, stationery, printing, photocopying, binding and general subsistence on trips and visits.

Following graduation it may be necessary to take out subscriptions to professional bodies and to buy business attire for job interviews. Graduate Statistician status is awarded by the Royal Statistical Society to successful graduates of accredited degrees. Exhibiting mass-produced objects as art objects, Duchamp exposes the conditions of possibility of art through the readymades.

Rather than postulating art as an expression of the object, of its formal and material qualities, Duchamp uncovers the fact that art inheres less in an object than in the institutional context that frames it and makes it legible. The ready-mades make visible the provisional and transitional status of art as they switch back and forth, undecidably, between art and nonart. By documenting this transition, Duchamp demystifies the art object at the same time that he reactivates the position of the spectator, as critical to both the reception and production of works of art.

Rather than being restricted to the ready-mades as objects or gestures, this study seeks to inquire into their nominal properties. I argue that the legibility of the ready-mades relies not merely on their visual appearance but on their nominal properties, since their titles pack in networks of puns and poetic associations.

As literal reproductions of objects, the ready-mades become legible as puns, as relays of signification, as switches that enable the spectator to discover mechanically the creative potential of language. Just as mechanical reproduction ensures the production of commercial prototypes, so do linguistic and social conventions ensure the production and circulation of puns.

Culturally generated and reproduced, puns func-. Duchamp's ready-mades make us stumble on the surprising discovery that linguistic puns are also ready-mades; that is, they are mechanisms whose venues for generating meaning are technically spelled out in the dictionary. The legibility of Duchamp's puns thus depends less on the spectator's imagination than on his or her ability to reactivate the puns by becoming aware and engaging with their potential meanings.

In this context the dictionary becomes a technical manual of sorts that makes visible the conceptual subtext that underlies the visual and nominal appearance of the ready-mades. Unfolding Duchamp's ready-mades as three-dimensional puns requires concerted attention to the interplay of language and image, as each system of reference intervenes to generate or undermine the production of meaning. Unpacking Duchamp's ready-mades, therefore, refers less to the handling of objects proper than to theunderstanding of the way they function as utterances in context.

As bearers of speech or cultural mouthpieces, the ready-mades capture the dilemma of an art that postpones its pictorial becoming and thus the finality of its attainment to become art. As the "plastic equivalent of a pun" to use Octavio Paz's terms , the ready-made stages the gratuitous conversion of an ordinary object into a work of art, while undermining through this very gesture the notion of an art object.

Chapter 4 is an examination of how the ready-made functions as a critique of classical notions of value. Instead of assuming the autonomy of art from the social and economic sphere, the focus is on how Duchamp rethinks the question of artistic value by redefining it as a function of its economic and social currency.

Instead of condemning Duchamp's forays into commercial ventures in art, I argue that Duchamp is redefining art according to a speculative model, whose conceptual implications liberally draw upon and expend classical economics. Ranging from checks and bonds to numismatic coins, Duchamp's artworks mimic economic currency and exchange only to undermine the notion of both artistic and monetary standards.

These works redefine artistic and economic forms of production byexploiting the speculative potential of reproduction. This study concludes with an examination of Duchamp's posthumously exhibited work Given: 1 the waterfall, 2 the illuminating gas, a work. Described as "startlingly gross and amateurish," Given "startles" by mirroring back the spectator's look.

In the context of the museum where everything is on display, however, the display of sexuality takes on an ironic tone. Having questioned the logic of the visible, does Duchamp's Given represent a continued challenge or a return to conventional modes of representation? I argue that despite this work's oven sexual display, or rather, because of its exaggeration as display, the equation of sexuality and vision is sundered. Duchamp undermines the logic of voyeurism by questioning the coincidence of sight with visual pleasure.

In doing so he moves away from equating sexuality with anatomical destiny, toward redefining it as a rhetorical operation. However, this effort to de-essentialize gender can be understood only in the framework of his attempts to experiment with genre. Just as Given fails to provide the spectator with a stable representation of sexuality, so does it also resist any generic classification.

Given is an installation, an assemblage of works that mimic artistic media such as painting, sculpture, and photography, without being reducible to a specific genre. The generic identity of this work, like the status of gender, remains transitive, resisting both fixity and closure. If Duchamp's works resist canonization, this is not simply because of their complexity or enigmatic character but rather because his works are by definition transitive.

They are like hinges, straddling the gap between vision and language, art and nonart, forms of artistic production and reproduction. Resembling Duchamp's elusive presence as an artist, his works are packages whose meaning continues to unfold in new and surprising ways. In the postscript is a brief assessment of Duchamp's impact on the history of Modernism. His redefinition of artistic modes of production through reproduction opens up the scope of Modernism to a notion of artistic production that is speculative, insofar as it reinvests rather than liquidates the legacy of tradition.

In doing so, Duchamp discovers within the experimental scope of Modernism a conceptual potential that becomes the terrain for the emergence of postmodernism. Having done away with. Just as Duchamp draws upon pictorial conventions to redefine the meaning of art, so does the legacy of his work open itself to appropriation by others. Is it then surprising to see artists such as J. Boggs's postmodern appropriation realizes a potential inscribed in Duchamp's postponed legacy of Modernism. If Duchamp's artistic life and his works are on credit, this credit can continue to be reinvested or spent.

The possibilities are unlimited, since, as Duchamp reminds us, "Posterity is a form of the spectator" DMD, Among Marcel Duchamp's gestures and artistic interventions, few have created as much controversy or been as puzzling as his putative abandonment of painting.

Although Duchamp starts to exhibit his work publicly in , the earliest works of his that are considered significant date to At the age of twenty-three, his enriched pictorial and compositional vocabulary is deployed in a figurative context, where nudes and group scenes dominate. By , Duchamp's use of abstraction demonstrates shared affinities with Cubism, insofar as it brings into question the figurative identity of the body through its spatial fragmentation and its serial deployment.

At this time he also begins to expand the meaning of the pictorial image by trying to find new ways of illuminating it, either through experiments with gaslight or by exploring how the title may have an impact on the nominal expectations of painting. At the end of and culminating in , Duchamp irrevocably establishes his authority as a painter through his signatory work, Nude. Descending a Staircase, No. It is during this same period that Duchamp begins to incorporate machine imagery and morphology in his paintings, leading to his mechanomorphic paintings.

This three-year trajectory that establishes Duchamp's creative identity and credibility as a painter renders his abandonment in of conventional painting and drawing all the more surprising, if not altogether shocking. At issue is neither Duchamp's failure nor, ironically, his success as a painter but rather his challenge of the limits of pictorial practice.

Thus within this three-year span —13 , Duchamp establishes himself as an internationally renowned painter, one who moves decisively from figuration to abstraction, only to begin to question painting altogether. In Duchamp practically gives up conventional forms of painting, but this does not mean that he stops working. Instead, he begins to experiment with chance as a way of getting away from the traditional methods of expression generally associated with art.

He lets pieces of string fall and records the shapes they generate; when his work on glass cracks he accepts the cracks as part of the work. During this period, Duchamp experiments with mechanical drawings, painted renderings, and notations that serve as studies for his seminal work, The Bride Stripped Bare by Her Bachelors, Even. By , the idea that Duchamp did not just give up painting but art altogether comes into currency. As Joseph Masheck explains: "Duchamp never discouraged it and seems to have enjoyed the mysterious notoriety.

Duchamp was said to have taken up a decided antiart position, abandoning art in favor of playing chess. How do we explain these radical transitions, from figuration to abstraction leading to the abandonment of painting, and ultimately art, given the speed at which these gestures succeed one another?

Can these transitions be illuminated by particular events in Duchamp's life, and more specifically, how are they manifest when considering works from this period? A small number of biographical details may prove to be significant to our discussion of Duchamp's pictorial origins. Born in a solid French bourgeois family on 28 July and following in the footsteps of his two brothers Jacques Villon and Raymond Duchamp-Villon and his sister Suzanne, Duchamp also became interested in art.

Far more significant in his professional formation was his apprenticeship as a printer in Rouen in , in lieu of doing military service. As an "art worker" he received exemption from military service after one year, having passed a juried exam based on the reprints of his grandfather's engravings.

Thus, in addition to his early exposure and family background in art, both engraving and cartooning. The influence of Duchamp's exposure to engraving and cartooning impacted on his efforts to discover alternative ways of conceiving painting and art. Unlike his siblings, Duchamp is not content to simply become a painter, for he will rapidly abandon painting in favor of activities that challenge the very meaning and definition of art.

When one considers attentively Duchamp's early works, one is invariably struck by his efforts to put into question the notion of pictorial image, by examining its relation to other frames of reference, the title, or the nominal expectations of the public. Moreover, his early explorations of serial works, or multiples, attest to his efforts to challenge the uniqueness and autonomy of the pictorial image.

Engraving and cartooning thus enabled Duchamp to conceive the plastic image in new terms, whose technical and intellectual content opened up the possibility of redefining the notion of artistic creativity as a form of production based on reproduction. Duchamp did not becomean engraver nor a cartoonist. He did, however, draw on the intellectual and speculative potential of these two media, in order to redefine not only painting as a medium but also art itself.

The fact that Duchamp began his artistic career as an "art worker" is significant, insofar as it enabled Duchamp to question the creative function of the artist and the meaning of art as a form of making:. I don't believe in the creative function of the artist. He's a man like any other. It's his job to do certain things, but the businessman does certain things also, you understand? On the other hand the word "art" interests me very much. If it comes from the Sanskrit, as I've heard, it signifies "making.

Formerly, they were called craftsmen, a term I prefer. We're all craftsmen, in civilian or military life. DMD, In the pages that follow, Duchamp's effort to question the meaning of art as pictorial practice, as an institution, and as a profession will be at issue. The notion of art as "making" enlarges the meaning of artistic activity to forms of production that include not only artisanal efforts but also conceptual insights.

In his interview with Marcel Duchamp, Pierre Cabanne asks him to explain the key event of his life: his abandonment of painting. Duchamp's response identifies Nude Descending a Staircase, No. While serving to establish his reputation, the initial rejection of the work alerts him to the norms and strictures that define not just conventional art but also contemporary art movements, such as Cubism:.

In the most advanced group of the period, certain people had extraordinary qualms, a sort of fear! People like Gleizes, who were, nevertheless, extremely intelligent, found this "Nude" wasn't in the line that they had predicted. Cubism had lasted two or three years, and they already had an absolutely clear, dogmatic line on it, foreseeing everything that might happen.

DMD , Duchamp is less concerned with the rejection of the painting than the fact it embodies a doctrinal gesture—one where a work of art is defined by living up to its nominal expectations. By failing to fall into line, that is to conform to a set of pregiven rules, Nude Descending a Staircase, No.

For Duchamp, the turning point that the Nude represents is not merely its challenge to the public but also to his peers, whose artistic and intellectual expectations define the work's conditions of possibility. Was Duchamp's dramatic gesture an expression of his "distrust of systematization," of his inability to contain himself to "accept established formulas" DMD , 26? Duchamp rejects Cubism not just as an artistic movement but as a discipline with a set aesthetic program: "Now, we have a lot of little Cubists, monkeys following the motion of a leader.

Their favorite word is discipline. It means everything to them and nothing. Coming in the wake of a series of representational nudes in , Nude Descending a Staircase, No. As this study will demonstrate, however, Duchamp's passage through abstraction involves the speculative goal of getting away from "the physical aspect of painting" by putting "painting once again to the service of the mind.

Duchamp's adoption of the nude as pictorial genre did not have entirely auspicious beginnings. It is also interesting to recall that he failed the Ecole des Beaux-Arts competition over a test that involved doing a nude in charcoal DMD , In , when Duchamp turns to the genre of the nude. Schwarz notes that in Nude with Black Stockings Nu aux bas noirs; , the "use of heavy black lines—characteristic of the Fauves' reaction to the Impressionists' careful avoidance of black—is freely adopted.

It reflects an understanding of the extent to which an artistic movement may be defined by its strategic response to the aesthetic tenets of a previous, or even contemporary, movement. Duchamp's use of heavy black lines to outline the body, as in Nude Seated in a Bathtub Nu assis dans une bagnoire; fig. The black lines emphatically reframe the successive color shadings, thus. In the Red Nude, color as one of the constitutive elements of painting is deployed in a manner that reveals its affinity to engraving.

The red shadings and black lines compete as color templates that redefine the pictorial appearance of the nude as a successive set of impressions or imprints. If these nudes are graphic, it is in their treatment of painting and not in their ostensible subject matter. When comparing Duchamp's Nude with Black Stockings with Gustave Courbet's Woman with White Stockings Femme aux bas blancs; , one is struck by its unerotic demeanor that resists voyeuristic appropriation as an image.

Rather than emphasizing and framing genitality, as the white stockings do in Courbet's painting, the black stockings dismember the body by erasing it from the. Duchamp's cropping of the nude body displaces the viewer's attention from the frontality of sex to the pictorial frame that cuts the body off—a feature shared by other works, such as Two Nudes Deux Nus; , and Red Nude.

The effort to draw the spectator's attention to framing devices is deliberately underlined in Red Nude , where the profile of the crouching red nude breaking out of the frame of the painting also cuts into the frame of another painting. Located in the upper lefthand corner of the image, this painting is further disfigured by the painter's signature cutting across the head of a female figure.

The authorial signature is displaced into a position where its nominal content interferes with the visual content and consumption of the image. Rather than merely stripping the nude, Duchamp begins to strip away the visual conventions that define the nude as a pictorial genre.

By , Duchamp's exploration of the nude enters a new phase, one where issues of pictorial abstraction are reframed by their interplay with nominal expectations triggered by the title. Loosely identified as his "Symbolist" phase because of its visual affinities to the works of Paul Gauguin and Pierre Girieud, Duchamp's works betray the Symbolist conceit of combining word and image.

The doubling of female nudes in The Bush Le Buisson ; fig. For Lawrence Steefel, The Bush "seems to point towards the ultimate goal of turning the world inside out. Duchamp trivializes the visual referent by his puns on the title "bush," thereby defying the nominal expectations of the spectator as voyeur.

In Paradise Le Paradis; fig. There is no illumination nor spiritual "Ascension" here. The title Paradise contradicts the viewer's expectations, unless it is interpreted literally, as a pun on the French word paradis, which means no radiance, to be struck out, canceled, or just broken. The lack of radiance in Paradise. While Duchamp admits in his interview with Cabanne: "I don't know where I had been to pick up on this hieratic business" DMD , 23 , this statement should not discourage us from considering this question.

This halo effect or aura can be found in another work of this period entitled Portrait of Dr. Dumouchel Portrait du Dr. Dumouchel; fig. Referring to this painting, Duchamp wrote in a letter to his patrons Louise and Walter Arensberg: "The portrait is very colorful red and green and has a note of humor which indicated my future direction to abandon mere retinal painting. The word figure means figure, shape, or form, but its use by Duchamp suggests that it refers to Dumouchel's appearance: it is a reflection on the way he looks, his "air," or "aura.

This pun on color blindness in the context of painting foreshadows, as it were, Duchamp's denunciation and subsequent aban-. For Duchamp, the hieratic aura associated with Symbolist painting becomes the locus of investigation of the interplay of word and image, not under the guise of symbols but as puns. This "halo" effect or aura continues to reappear throughout Duchamp's works, either as an analogy to smell in such works as Fountain [] and Beautiful Breath, Veil Water [Belle Haleine, Eau de Voilette; ] , or as an analogy for electricity in Bec Auer [a gas lamp circa ]; The Large Glass [—23]; Given: 1 the waterfall, 2 the illuminating gas —66 ; and in a set of prints entitled The Bec Auer [].

Considered from this perspective, Duchamp's early experiments with the hieratic can be understood as an allusion to the history of painting. This was at a time when the appearance of the nude, like painting itself, attained value by virtue of its religious, philosophical, and moral function, and was thus in excess of visual semblance. If painting exuded an "aura," this is because its significance was originally defined by its social rather than cultural function.

The loss of painting's "aura" in the age of mechanical reproduction heralds the end of painting as a purely manual and visual event and its conceptual rebirth as a practice stripped of the hallowed echoes of visual semblance. As Duchamp explains to Cabanne:. Since Courbet, it's been believed that painting is addressed to the retina. That was everyone's error.

The retinal shudder! Before, painting had other functions: it could be religious, philosophical, moral. If I had the chance to take an antiretinal attitude, it unfortunately hasn't changed much; our whole century is completely retinal, except for the Surrealists, who tried to go outside it somewhat. And still, they didn't go so far!

If Nude Descending a Staircase, No. Described as an "explosion in a shingle factory," Nude Descending a Staircase, No. The word rude appropriately captures the impact of the Nude, its deliberate disregard for the artistic conventions of the genre. This work scandalized not only the general public but also the avant-garde circles of the time. As Duchamp explains, the title plays a significant role in explaining the particular interest and impact of this work:.

What contributed to the interest provoked by the canvas was its title. One just doesn't do a nude woman coming down the stairs, that's ridiculous. It doesn't seem ridiculous now, because it's been talked about so much, but when it was new, it seemed scandalous. A nude should be respected.

DMD, 44; emphasis added. Duchamp's comments indicate that the reception of this painting was being filtered through a set of expectations, whose nominal character was staged by the title. The abstract nature of this work and thus its failure to provide a visual referent for the title only increased the public's disappointment.

Instead of reclining passively, Duchamp's fractured nude is actively descending a staircase. The scandal surrounding the exhibition of Nude. In his book The Nude, Kenneth Clark maintains that the nude is not the starting point of a painting but a way of seeing that the painting achieves. Constructed as the subject of desire from the Renaissance to the late nineteenth century, the nude as a pictorial genre involves a structure of spectatorship that relies upon the objectification of the female body.

This interplay of visual and nominal expectations staged by the nude as a pictorial genre was put into question by painters such as Edouard Manet, who in Olympia and Le Dejeuner sur l'herbe —63 challenged the inscription of the desiring look of the spectator.

The Nude. The splintering of vision into a series of frames that fragment and abstract both the identity of the nude and the process of movement inscribe into the painting an interval, a temporal dimension. Functioning neither descriptively nor prescriptively, the title Nude Descending a Staircase inscribes a temporal delay that interferes with the visual consumption of the image.

This strategy of delay also redefines and defers notions of visual reference that are traditionally associated with photography. While appealing to techniques of mechanical reproduction, such as photography, to redefine the pictorial medium and its subject matter, Duchamp succeeds in redefining painting itself as a process whose plasticity includes temporal considerations.

In the nude itself. To do a nude different from the classic reclining or standing nude, and to put it into motion. There was something funny there, but it wasn't funny when I did it. Movement appeared like an argument that made me decide to do it.

In the "Nude Descending a Staircase," I wanted to create a static image of movement: movement is an abstraction, a deduction articulated within the painting, without our knowing if a real person is or isn't descending an equally real staircase. Fundamentally, movement is in the eye of the spectator, who incorporates it into the painting.

The picture presents the viewer with a "vertigo of delay," to use Paz's term, rather than one of acceleration. The staggered motion of the "nude" demonstrates an analysis of movement rather than the Futurist seduction with the dynamics of movement. But why is the nude descending?

A network of visual puns connects Nude Descending a Staircase, No. Just as Laforgue's poem denounces the idealist aspirations of Symbolist poetry by pointing out that the stellar image of the sun is undermined by its ordinary and pockmarked appearance, so does Duchamp transform the idealism that underlies pictorial praxis into a mere stair, a pun on the notion of descent understood both literally and figuratively. The ambiguous title of the Nude nu, in French gives no particular indication as to the referent's gender, although critics have identified it generically as female, de rigueur.

Duchamp's Nude Descending a Staircase, No. Unable to incarnate the nominal expectations of the spectator, the nude visually fractures the spectator's gaze by setting it into a spiraling motion. In doing so Duchamp points both to the title and to the spectator's gaze as the sites on which hinges the facticity of gender. This resistance to the equation of spectatorship with visual consumption and delectation is explicitly thematized in Duchamp's later works.

The Nude 's descent thus functions as an index of Duchamp's strategic displacement and rethematization of the nude as a pictorial genre and its declension from the spectator's nominal expectations. The descent of the Nude is not merely the mark of a genealogical decline but also the legal index of the passage of an estate through inheritance.

Is it surprising then that Nude Descending a Staircase, No. As Joseph Masheck notes: "Typical of Duchamp is this work's self-illustrative and self-reproductive function, as well as the fact that as an actual photograph it returns to one of the technical sources of the 'original' painting. However, this reproductive industry did not stop short with the fullsized versions of the Nude.

This work was further reproduced as a miniature pencil-and-ink drawing Nude. This doll-sized version of the Nude was followed by further miniature reproductions in The Box in a Valise — From a single work that is by definition a multiple, insofar as it is part of a series, Duchamp generates an entire corpus.

By discovering the self-productive and self-reproductive potential of the Nude, Duchamp redefines the nude as a medium of and for reproduction. Eroticism in this context no longer refers to the visual appearance of the nude but instead functions as an index of its proliferation as modes of appearance.

Duchamp challenges the eroticism traditionally associated with spectatorship and voyeurism by proposing an alternative eroticism whose speculative, technical, and humorous character restages through reproduction the notion of artistic creativity and production.

Given Duchamp's explicit rejection of the equation of vision and eroticism, how are we to explain his interest in the nude as pictorial genre? It seems that the entire trajectory of his life's work is defined by the arching movement from Nude Descending a Staircase, No. While Duchamp maintains that eroticism is the only -ism he believes in, it is. Eroticism in the figurative arts is most commonly represented as the relationship between clothing and nudity, and thus, as Mario Perniola suggests, it is conditional on the possibility of movement or transition from one state to another.

Now we begin to understand the conceptual import of both engraving and cartooning in Duchamp's work. Engraving is one of the earliest forms of mechanical reproduction that involves a different way of conceiving. Not only is the appearance of the engraved image the result of multiple reproductions but its very identity is defined as a technical process, involving multiple impressions or imprints.

An engraving is a template, a sculptural mold that functions like a photographic negative. Engraving as a medium challenges the autonomy of the pictorial image, insofar as the image acts as a temporal record of multiple impressions.

Duchamp's pictorial series of Nude Descending a Staircase, as a multiple that undergoes extensive reproduction, illustrates the logic of engraving operative in his works. This is not to say that these works are engravings, since they are clearly paintings; rather, the conditions of production and reproduction evidenced in these series suggest conceptual processes akin to those involved in the technical reproduction of engravings.

You may ask how cartoons inform Duchamp's oeuvre? The answer by now is clear. Regarded as a form of popular art associated with the print medium, cartoons are images that are constructed like rebuses, as composites of language and image. Their humor is not just visual but intellectual.

They are often visual analogues of linguistic puns. This is not to suggest, however, that Nude Descending a Staircase, No. Rather, Duchamp's use of the title as nominal intervention in order to restage the expectations of the spectator reframes the reception of this work as an intellectual, instead of a purely visual, experience.

Consequently, despite its mechanomorphic character, Duchamp's Nude can also be seen an an "anti-machine. Their relation to utility is the same as that of delay to movement;they are without sense and meaning. They are machines that distill criticism of themselves. As this study has suggested, Duchamp's humor lies in redefining the visual image as a serial imprint, as a construct where appearance does not refer to an external reality but to a mode of production whose logic is reproductive.

Duchamp doubly displaces painting: first, by redefining it through the logic of engraving, as a print medium, and second, by draw-. Instead, it becomes the rhetorical interplay between language and vision, which constructs the facticity of gender as a pun. While all artists are not chess players, all chess players are artists. When asked by Katherine Kuh, one of his interviewers, which of his works he considers to be the most important, Marcel Duchamp replied:.

That was really when I tapped the mainspring of my future. In itself it was not an important work of art, but for me it opened the way—the way to escape from those traditional methods of expression long associated with art. I didn't realize at that time exactly what I had stumbled on.

When you tap something you don't always recognize the sound. That's apt to come later. For me the Three Stoppages was a first gesture liberating me from the past. As Duchamp subsequently explains, the idea of letting a piece of thread fall on a canvas was accidental, but "from this accident came a carefully planned work. Was it the idea of chance, or its plastic deployment and embodiment as an event or work?

Duchamp's interest in chance as a way of redefining conventional forms of artistic expression appears early on in his paintings and is tied to his interest in chess. For Duchamp, chess is not merely a pastime or an ordinary game because its intellectual character represents for him a plastic. As a strategic game that requires the interplay of two opponents, chess provides Duchamp with a new way of envisioning art in its dialogue with the tradition.

The analogy of art and chess enables Duchamp to appropriate chance and redefine its plastic impact in a field of already given determinations. Starting with The Chess Game Le jeu d' hecs; fig. Duchamp's lifetime interest and preoccupation with chess is well known, but its significance and precise impact on his art is less recognized. II , it is clear that he is already playing another game DMD, The checkered pattern of the board, however, is also an allusion to another set of rules, those of Albertian perspective that have guided the development of painting.

If we pursue Duchamp's analogies in The Chess Game, art no less than chess emerges as a strategic, rather than purely plastic, domain. Both chess and perspective are systems whose normative standards prescribe and determine the nature of representation. What had been originally conceived as an arbitrary relation between painting and the world is now revealed to be a strategic, albeit conventional game, a chess game.

The answer lies in his understanding of chess as a plastic, rather than a purely intellectual, game. As Duchamp's comments to James Johnson Sweeney indicate, playing. This plasticity, however, is not in the realm of the visible but in the abstraction of the movement of pieces on the board. In his interview with Francis Roberts, Duchamp explains how the strategic and positional nature of chess generates plastic effects:.

In my life chess and art stand at opposite poles, but do not be deceived. Chess is not merely a mechanical function. It is plastic, so to speak. Each time I make a movement of the pawns on the board, I create a new form, a new pattern, and in this way I am satisfied by the always changing contour. Not to say that there is no logic in chess. Chess forces you to be logical. The logic is there, but you just don't see it.

The plasticity that Duchamp ascribes to chess is not aesthetic in the visual sense but rather intellectual. The movement of the pieces on the board creates patterns and forms whose contours are constantly shifting. This moving geometry is described by Duchamp as "a drawing" or as a "mechanical reality" DMD, As Duchamp elaborates: "In chess there are some extremely beautiful things in the domain of movement, but not in the visual domain.

It's the imagining of the movement or the gesture that makes the beauty, in this case. It's completely in one's gray matter" DMD, The beauty that Duchamp appeals to is not one based on aesthetic categories, on visual appearance and artistic self-expression. Rather, the beauty in question is defined by the plasticity of the imagination, by the poetry of its ever changing contours.

The analogy of chess and art is one that is mediated by an allusion to the abstract nature of both music and poetry. As Duchamp explains:. Objectively, a game of chess looks very much like a pen-and-ink drawing, with the difference, however, that the chess player paints with black-and-white forms already prepared instead of having to invent forms as does the artist.

The design thus formed on the chessboard has apparently no visual aesthetic value, and it is more like a score for music, which can be played again and again. Beauty in chess is closer to beauty in poetry; the chess pieces are the block.

Actually, I believe that every chess player experiences a mixture of two aesthetic pleasures, first the abstract image akin to the poetic idea of writing, second the sensuous pleasure of the ideographic execution of that image of the chessboards.

Relying on analogies to the media of music and poetry, Duchamp uses chess as a way of expanding the meaning of art. No longer bound to the creation or invention of visual forms, the chess player "paints" with already given black-and-white forms. The interest of the exercise lies in the composition of the design, a visual score that is open to multiple performances, for the nature and value of chess exists only as a performance, a duet where two interpreters put their heads together, so to speak.

The chess pieces in this game function as linguistic elements already given by convention, but ready to be redeployed poetically in new ways. While subject to particular rules governing the possibility of movement, the mechanisms generated, as the "ideographic execution of that image," are always open to further reinterpretation.

Thus Duchamp uncovers within chess a paradigm for the reinterpretation of aesthetic pleasure as a pleasure derived neither from invention nor the sensuality of the pieces themselves, but from their recomposition and poetic deployment as a game. Rather than interpreting this tearing as Cubist dislocation, Duchampis reinterpreting Cubism itself conceptually from the perspective of chess, as a game whose. More specifically, the serial fragmentation and multiplication of the protagonists into shards, while depersonalizing them into mechanical patterns, illuminates painting in a new light.

Duchamp visibly draws on Cubism, only to redefine its logic as a representation: the dislocations visible in the image are but the diagrams of movements ideographically transposed from chess. While such an interpretation seems to force Duchamp's hand, as it were, it is important to recall Duchamp's comment to Cabanne regarding the fact that Portrait of Chess Players was painted not in ordinary light, but by gaslight:.

This "Chess Players," or rather "Portrait of Chess Players," is more finished, and was painted by gaslight. It was a tempting experiment. You know, that gaslight from the old Aver jet is green; I wanted to see what the changing of colors would do. When you paint in green light and then, the next day, you look at it in day light, it's a lot more mauve, gray, more like what the Cubists were painting at the time.

It was an easy way of getting a lowering of tones, a grisaille. Duchamp's innovative use of a green gaslight as a way of lowering the color tones of this painting grisaille, in French and in English also func-. Rather than merely seeking to reproduce Cubist colors, Duchamp uses the "gas" light as a pun that reframes the notion of pictorial "gaze.

Duchamp's portrait pourtraire, in French, from Latin, pro [forth] and trahere [to draw] draws upon and restages earlier versions of chess games and chess players by redefining the meaning of pictorial representation as a practice that is not merely visual but also mental. If the sequence of paintings and sketches from The Chess Game to Portrait of Chess Players represents Duchamp's artist brothers playing chess, in later paintings the chess pieces themselves become the subject matter of art.

They take over the board, as it were. Having uncovered the affiliation between art and chess by revealing their shared strategic and positional nature, Duchamp now proceeds to examine the plastic dimension of the mechanics of strategy. As Duchamp observes: "In chess, as in art, we find a form of mechanics, since chess could be described as the movement of pieces eating one another.

Whether his adversaries may be his own brothers, who were also artists, or artistic movements such as Cubism, Duchamp's understanding of art as a strategic game enables him to redefine the notion of artistic creativity as a form of production based on reproduction. Duchamp's redefinition of art in terms of chess involves, on the one hand, accepting one's affiliation to traditions, that is, the readymade character of pictorial convention, and on the other hand, the effort to redraw the board in order to radically rethink the terms of the game.

Consequently, artistic innovation is freed from the "anxiety of influence," since tradition itself provides conventional or ready-made elements that Duchamp can redeploy in a strategic fashion. Among the most significant are The King and Queen. According to Duchamp, the title "'King and Queen' was once again taken from chess, but the players of my two brothers have been eliminated and replaced by the chess figures of the King and Queen. In some card games, pinochle for instance, the term marriage refers to the combination of the king and queen of the same suit.

In these paintings we are witnessing the passage from chess analogies into works whose diagrammatic character begins to challenge the very limits of art. The King and Queen Surrounded by Swift Nudes was painted on the back of an earlier painting, Paradise, which was turned upside down. Commenting on this work to Katherine Kuh, Duchamp notes:. You know this was a chess king and queen—and the picture became a combination of many ironic implications connected with the words "king and queen.

The use of nudes completely removed any chance of suggesting an actual scene or an actual king and queen. The irony in question here involves the substitution of chess pieces for the traditional subject matter of painting, which as reproductions put into question the referential status of painting.

His choice of renaming the nudes in terms of their quality or agility of movement was "literary play," a way of transposing sport terminology into painting. The introduction of speed into these paintings is not merely a concession to Futurism but rather the affirmation of the affinity between art and chess. If playing chess is like designing or constructing a mechanism, then art becomes a way of setting this mechanism into motion strategically.

Instead of visually representing motion, Duchamp, following Marey's chronophotographic techniques, maps it as a "system of dots delineating different movements" DMD , He thus reduces the intervention of vision by demonstrating that all the visible traces are but forms of ideographic mapping. By interpreting painting as a conceptual intervention that converts visible objects into cartographic networks, Duchamp redefines painting as a philosophical enterprise.

As Octavio Paz observes:. In these canvases the human form has disappeared completely. Its place is not taken by abstract forms but by transmutations of the human being into delirious pieces of mechanism. The object is reduced to its most simple elements: volume becomes line; the line a series of dots.

Painting is converted into a symbolic cartography; the object into idea. It is not the philosophy of painting but painting as philosophy. The painting The passage from the Virgin to the Bride fig. The resonant tension that this work sustains between abstraction and figuration is amplified by the punning echoes of the title.

Itself a transitional work, between two preliminary sketches Virgin No. The "passage" in question here refers both to the intermediary status of this work and, more importantly, to the redefinition of painting itself as a transitional activity, a "rite of passage" of sorts. If engraving has enabled Duchamp to conceive painting itself as a transitional activity, as a set of impressions or imprints, chess enables him to redefine it strategically.

From the chess Queen of King and Queen Surrounded by Swift Nudes, Duchamp now stages the process of becoming painting, the passage from visual "Virgin" to conceptual "Bride" painting. As Thierry de Duve observes: "the painting does what it says, and says what it does," but in so doing it opens up the space for a new kind of passage, away from painting altogether.

His job is not that of an ordinary groom—merely to equip, prepare, or dress up painting—but rather, equipped with the insights that painting provides, he draws on painting in order to redefine the meaning of art. Given Duchamp's exploration of the limits of painting as a kind of end game to use an expression that is common in chess , it is not surprising that by he begins to envisage leaving it behind altogether.

He starts experimenting with chance operations, which, while drawing on pictorial traditions, also announce his future discovery of the ready-mades. As Duchamp himself noted, Three Standard Stoppages fig. This work is an assemblage a semi-ready-made consisting of a croquet box that contains three separate measuring devices that were individually formed by chance operations. Three different threads, one meter in length, were dropped or allowed to fall freely, depending on one's perspective onto a canvas painted Prussian blue, and glued into place.

The resulting impressions, capturing the curved outline of their chance configurations, were then permanently affixed to glass plate strips. These plates served as imprints for the preparation of three wood templates. Designated as an instance of "canned chance," this work is also categorized by Duchamp as "a joke about the meter. Naumann concludes that "the central aim of this work was to throw into question the accepted authority of the meter," as a standard unit of measurement.

Not only does Three Standard Stoppages distort the length of the meter through curvature but in doing so, it demonstrates the recognition that the meter itself as a unit of length is generated through approximation: the straightening out, as it were, of a curved meridian. While this work may parody the authority of the meter, and thus by extension challenge notions of authority in general, it is less clear whether it reflects Duchamp's effort to champion individual rights, as Naumann claims. Duchamp's invocation of chance is not a strategy to personalize the laws of physics but merely to ironically "strain" its laws in order to reveal their instability.

As he explains:"I was satisfied with the idea of not being responsible for the form taken by chance. Duchamp's experiment with "canned chance" not only questions scientific authority but also notions of artistic authority, which by their embedded and ingrained character reduce art to a mechanical practice. Roberts's question whether the dependence on chance betrays a disdain for the mechanics of art, leads Duchamp to respond:. I don't think the public is prepared to accept it. This depending on coincidence is too difficult for them.

They think everything has to be done on purpose by complete deliberation and so forth. In time they will come to accept chance as a possibility to produce things. In fact, the whole world is based on. The notion of coincidence challenges through its arbitrary logic both personal and institutional forms of determinism. More important still, it puts into question the voluntaristic and intentional logic that defines the creative act and the identity of the artist.

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The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves. This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.

Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.

An introduction to the basic ideas and notations involved in describing sets and their functions will be given. This module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, it is said that one is bigger than another if it contains more elements.

What about infinite sets? Are some infinite sets bigger than others? Students will develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e. The module will also consider the connections between objects, leading to the study of graphs and networks — collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.

This course extends ideas of MATH from functions of a single real variable to functions of two real variables. The notions of differentiation and integration are extended from functions defined on a line to functions defined on the plane. Partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes. In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure and temperature, and the direction of the wind.

To study functions of several variables, we introduce rates of change with respect to several quantities. We learn how to find maxima and minima. Applications include the method of least squares. Finally, we investigate various methods for solving differential equations of one variable.

The main focus of this module is vectors in two and three-dimensional space. Starting with the definition of vectors, students will discover some applications to finding equations of lines and planes, then they will consider some different ways of describing curves and surfaces via equations or parameters. Partial differentiation will be used to determine tangent lines and planes, and integration will be used to calculate the length of a curve. In the second half of the course, the functions of several variables will be studied.

When attempting to calculate an integral over one variable, one variable is often substituted for another more convenient one; here students will see the equivalent technique for a double integral, where they will have to substitute two variables simultaneously. They will also investigate some methods for finding maxima and minima of a function subject to certain conditions. Finally, the module will explain how to calculate the areas of various surfaces and the volumes of various solids.

Building on the Convergence and Continuity module, students will explore the familiar topics of integration, and series and differentiation, and develop them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.

Applying the skills and knowledge gained from this module, students will tackle questions such as: can you sum up infinitely many numbers and get a finite number? They will also enhance their knowledge and understanding of the fundamental theorem of calculus. Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.

The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic eigenvectors and eigenvalues. The student will learn how to express a linear transformation of the real Euclidean space using a matrix, from which they will be able to determine whether it is singular or not and obtain its characteristic equation and eigenspaces.

The student is introduced to logic and mathematical proofs, with emphasis placed more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.

The language and structure of mathematical proofs will be explained, highlighting how logic can be used to express mathematical arguments in a concise and rigorous manner. The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.

Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability. Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.

To enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems, the module begins with a brief overview of statistics in science and society. It then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.

Students will develop a strategic understanding of statistics and the use of associated software, which will underpin the skills needed for all subsequent statistical modules of the degree. This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions.

Here, students will select a small number of properties which these and other examples have in common, and use them to define a group. They will also consider the elementary properties of groups. By looking at maps between groups which 'preserve structure', a way of formalizing and extending the natural concept of what it means for two groups to be 'the same' will be discovered.

Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives students a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials; several less familiar examples will also be explored.

Complex Analysis has its origins in differential calculus and the study of polynomial equations. In this module, students will consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. They will use integral calculus of complex functions to find elegant and important results and will also use classical theorems to evaluate real integrals.

The first part of the module reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The module then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by way of a bisection argument. These analytic ideas are used to prove the fundamental theorem of algebra, that every non-constant complex polynomial has a root.

Finally, the theory is employed to evaluate some definite integrals. The module ends with basic discussion of harmonic functions, which play a significant role in physics. Students will gain a solid understanding of computation and computer programming within the context of maths and statistics.

This module expands on five key areas:. Under these headings, students will study a range of complex mathematical concepts, such as: data structures, fixed-point iteration, higher dimensions, first and second derivatives, non-parametric bootstraps, and modified Euler methods. Throughout the module, students will gain an understanding of general programming and algorithms. They will develop a good level of IT skills and familiarity with computer tools that support mathematical computation.

Over the course of this module, students will have the opportunity to put their knowledge and skills into practice. Workshops, based in dedicated computing labs, allow them to gain relatable, practical experience of computational mathematics. Students will be provided with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year Two, and the more specialised Year Three modules.

This module will give students the opportunity to study vector spaces, together with their structure-preserving maps and their relationship to matrices. They will consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible.

Part of their study will also involve looking at the concepts of length and angle with regard to vector spaces. Basic concepts from the first year probability module will be revisited and extended to these to encompass continuous random variables, with students investigating several important continuous probability distributions.

Commonly used distributions are introduced and key properties proved, and examples from a variety of applications will be used to illustrate theoretical ideas. Students will then focus on transformations of random variables and groups of two or more random variables, leading to two theoretical results about the behaviour of averages of large numbers of random variables which have important practical consequences in statistics.

Project Skills is a module designed to support and develop a range of key technical and professional skills that will be valuable for all career paths. Covering five major components, this module will guide students through and explore:. Students will gain an excellent grasp of LaTeX, learning to prepare mathematical documents; display mathematical symbols and formulae; create environments; and present tables and figures.

Scientific writing, communication and presentations skills will also be developed. Students will work on short and group projects to investigate mathematical or statistical topics, and present these in written reports and verbal presentations. A thorough look will be taken at the limits of sequences and convergence of series during this module.

Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at A-level. Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions rather than just numbers , which again has many applications and is central to more advanced analysis.

Next, the notion of integration will be put under the microscope. Once it is properly defined via limits students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in A-level, such as estimations of discrete sums of series. Statistics is the science of understanding patterns of population behaviour from data. In the module, this topic will be approached by specifying a statistical model for the data.

Statistical models usually include a number of unknown parameters, which need to be estimated. The focus will be on likelihood-based parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and linear regression techniques within the statistical modelling framework will also be considered.

Students will come to recognise the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions. Bayesian statistics provides a mechanism for making decisions in the presence of uncertainty.

The purpose of the data collection exercise is expressed through a utility function, which is specific to the client or user. It defines what is to be gained or lost through taking particular actions in the current environment. Actions are continually made or not made depending on the expectation of this utility function at any point in time.

Bayesians admit probability as the sole measure of uncertainty. Thus Bayesian reasoning is based on a firm axiomatic system. In addition, since most people have an intuitive notion about probability, Bayesian analysis is readily communicated. Combinatorics is the core subject of discrete mathematics which refers to the study of mathematical structures that are discrete in nature rather than continuous for example graphs, lattices, designs and codes.

While combinatorics is a huge subject - with many important connections to other areas of modern mathematics - it is a very accessible one. In this module, students will be introduced to the fundamental topics of combinatorial enumeration sophisticated counting methods , graph theory graphs, networks and algorithms and combinatorial design theory Latin squares and block designs. They will also explore important practical applications of the results and methods.

They will be introduced to two new classes of integral domains called Euclidean domains, where they have a counterpart of the division algorithm, and unique factorisation domains, in which an analogue of the Fundamental Theorem of Arithmetic holds. How well-known concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorisation of polynomials, carry over to this new setting, will also be explored.

Questions relating to linear ordinary differential equations will be considered during this module. Differential equations arise throughout the applications of mathematics, and consequently the study of them has always been recognised as a fundamental branch of the subject. The module aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations, and theoretical results about the nature of solutions.

While explicit solutions can only be found for special types of equations, some of the ideas of real analysis allow us to answer questions about the existence and uniqueness of solutions to more general equations, as well as allowing us to study certain properties of these solutions. The concept of generalised linear models GLMs , which have a range of applications in the biomedical, natural and social sciences, and can be used to relate a response variable to one or more explanatory variables, will be explored.

The response variable may be classified as quantitative continuous or discrete, i. Students will come to understand the effect of censoring in the statistical analyses and will use appropriate statistical techniques for lifetime data. They will also become familiar with the programme R, which they will have the opportunity to use in weekly workshops.

The topic of smooth curves and surfaces in three-dimensional space is introduced. The various geometrical properties of these objects, such as length, area, torsion and curvature, will be explored and students will have the opportunity to discover the meaning of these quantities. They will then use a variety of examples to calculate these values, and will use those values to apply techniques from calculus and linear algebra. A number of well-known concepts will be encountered, such as length and area, and some new ideas will be introduced, including the curvature and torsion of a curve, and the first and second fundamental forms of a surface.

Students will learn how to compute these quantities for a variety of examples, and in doing so will develop their geometric intuition and understanding. The study of graphs - mathematical objects used to model pairwise relations between objects - is a cornerstone of discrete mathematics. As a result, students will develop an appreciation for a range of discrete mathematical techniques while undertaking this module.

Throughout the module, students will also learn about structural notions, such as connectivity, and will explore trees, minor closed families of graphs, matrices related to graphs, the Tutte polynomial of small graphs, and planar graphs and analogues.

While studying these areas, students will gain experience of following and constructing mathematical proofs, and correctly and coherently using mathematical notation. Students will develop the knowledge of groups that they gained in second year during the Groups and Rings module. They will learn several quite natural ways to do this, both for vectors of any dimension and for functions. Focus will then be on the more special notion of an inner product which generalises angles at the same time as distances.

Once these concepts have been established, students will have the opportunity to study geometrical ideas like orthogonality, which can be seen to apply to much more general spaces than Euclidean spaces of three or even n dimensions, notably to infinite dimensional spaces of functions.

For example, Hilbert space theory shows how Fourier series are really another case of expressing an element in terms of a basis, and how people can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, students will look at some of the main results of linear algebra, which generalise very nicely to linear operators between Hilbert spaces. Introducing the Lebesgue integral for functions on the real line, this module features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral.

The bounded convergence theorem is used to prove the monotone and dominated convergence theorems, and the results are illustrated in classical convergence problems including Fourier integrals. Among the range of topics addressed on this module, students will become familiar with Lebesgue's definition of the integral, and the integral of a step function. There will be an introduction to subsets of the real line, including open sets and countable sets.

Students will measure of an open set, and will discover measurable sets and null sets. Additionally, the module will focus on integral functions, along with Lebesgue's integral of a bounded measurable function, his bounded convergence theorem and the integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem.

Other topics on the module will include applications of the convergence theorems and Wallis's product for P. Gaussian integral, along with some classical limit inversion results and the Fourier cosine integral. Students will develop an understanding of Dirichlet's comb function, Archimedes' axiom and Cantor's uncountability theorem, and will learn to prove the structure theorem for open sets. In addition, students will be able to prove covering lemmas for open sets, as well as understanding the statement of Heine—Bore theorem, as well as understanding the concept and proving basic properties of outer measure.

As well as understanding inner measure. Finally, students will be expected to prove Lebesgue's theorem on countable additivity of measure. Statistical inference is the theory of the extraction of information about the unknown parameters of an underlying probability distribution from observed data. Consequently, statistical inference underpins all practical statistical applications. This module reinforces the likelihood approach taken in second year Statistics for single parameter statistical models, and extends this to problems where the probability for the data depends on more than one unknown parameter.

Students will also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'? The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.

The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics modules and provide opportunities for further study. The theory of linear systems is engineering mathematics.

In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.

Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard A,B,C,D model. These include electrical appliances, heating systems and economic processes. The module shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables students to classify A,B,C,D models and describe their properties in terms of quantities which are relatively easy to compute.

The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilise a A,B,C,D system. Using the classical problem of data classification as a running example, this module covers mathematical representation and visualisation of multivariate data; dimensionality reduction; linear discriminant analysis; and Support Vector Machines. While studying these theoretical aspects, students will also gain experience of applying them using R.

An appreciation for multivariate statistical analysis will be developed during the module, as will an ability to represent and visualise high-dimensional data. Students will also gain the ability to evaluate larger statistical models, apply statistical computer packages to analyse large data sets, and extract and evaluate meaning from data. Students will be given an opportunity to consider key issues in the teaching and learning of mathematics during this module. Whilst it is an academic study of mathematics education and not a training course for teachers, it does provide an excellent foundation for a PGCE especially in preparing students to write academically.

This work underlines the significance of reproduction in Duchamp's works, not merely as means of perpetuating his works through facsimile but as a means of redefining artistic production by reassembling its constitutive elements. Resuming the survey of Duchamp's artistic career, despite his suggestion that he had given up art in favor of chess, Duchamp did not stop making new works.

Starting in the late s he begins to produce artworks in which exaggerated realism and deliberate artificiality stand in contrast with the literal simplicity and technological character of the ready-mades. While appearing to return to figurative conventions, Duchamp redefines the notion of artistic production by using artistic conventions, rather than objects, as ready-mades.

These works are assemblages that parody the conventions of pictorial naturalism in order to demonstrate visibly their demise as literal death. These three-dimensional works, such as Female Fig Leaf Feuille de vigne femelle; , a plaster cast of female genitalia , and Dart-Object Objet-Dard; , a riblike phallus in galvanized plaster are puns on the use of artistic conventions to represent gender.

Following Duchamp's earlier exploration of the nude as a pictorial genre, these "realistic" plaster casts playfully reveal the artifices employed in both painting and sculpture to represent issues of sexual difference. During this period, unknown to his public and critics alike, Duchamp was also working on. Devoid of modernist abstract tendencies, this work stunned the critics by its contrived naturalism, a tableau-vivant of a mock nude that displays herself in a dioramalike landscape.

Painstakingly assembled during a period of twenty years, this work shocked the public because Duchamp seemed to be returning to pictorial conventions and a concept of art that he had supposedly abandoned long before. Duchamp appeared to be coming back full circle to the nude, no longer as an abstract representation but as a grossly literal one. Duchamp's rapid abandonment not just of painting but of conventional art, and his subsequent return to works that mimic art but are not readily classifiable as such, raise significant questions regarding his paradoxical renunciation of and consistent dependence on pictorial and artistic conventions.

How is it possible to abandon both painting and traditional art, while continuing to evoke and strategically draw upon them? Duchamp's deliberate focus on reproduction, on the literal transposition or translation of a previously defined corpus, represents a literal pun on the task of painting to reproduce nature. To the extent that Duchamp's work relies on conventional painting, it displaces its priority by undermining it through reproduction.

As Duchamp suggests, reproduction dispenses with the originality of painting, substituting for it the playful verisimilitude of the facsimile: "Instead of painting something it was—use a reproduction of those paintings that I loved so much, into a small reducedform, in a small shape, and—how to do it—I thought of a book which I didn't like so— I thought of the idea of a box.

This literal play on painting generates a new kind of artwork, whose meaning as a facsimile undermines the logic of originality. Duchamp challenges the dependence of copies on originals by demonstrating that originals are multiples of sorts, to the extent that they embody an assemblage of already determined gestures and conventions. Displacing the priority of both the artwork and the intervention of the hand with the facsimile, whose reproduction is associated with laborious, time-consuming techniques, Duchamp redefines art by questioning its conditions of production.

Even when it seems that Duchamp is returning to a more conventional understanding of reproduction, works such as Given, by their hyperrealistic and contrived character, parody the notion of artistic reference. Thus, while appearing to return to legitimate works of art, Duchamp succeeds in questioning the legitimacy of art as a mimetic medium. Rather than accepting the traditional labels of art and the artist, Duchamp proceeds to systematically challenge these definitions.

Reacting against Romantic ideology that isolates the artist from the social and economic sphere and singles out art as a unique form of expression, Duchamp redefines the artist as a maker, rather than a creator. This is not because Duchamp denies the powers of either inspiration or creativity but because he recognizes that any creative act is embedded in a set of conventions and expectations that predetermine its outcome.

If Duchamp appropriates the notion of mechanical reproduction in order to redefine artistic creativity, this is neither for lack of inspiration nor for having run out of ideas for making new works. Rather, as I suggested earlier, mechanical reproduction becomes the paradigm for a new way of thinking about artistic production, one that recognizes that creativity operates in a field of givens, of ready-made rules.

By understanding the creative act in context, Duchamp redefines its meaning as a strategic intervention that derives its significance from its plasticity, its ability to generate new meanings by drawing upon already given terms. For the spectator completes the picture, as it were, interpretatively unpacking the work through the interplay of visual and verbal puns. From this perspective, originality emerges as a multiple gesture, one that generates, in turn, the illusion of multiple authorship.

Is it then surprising to discover Duchamp's playful use of multiple signatures and personas, as well as his reliance on the spectator as authorizing agency of his work? Duchamp, however, is not content to revolutionize the notion of artistic identity or the identity of the artwork.

His efforts to question the. The attempt to explore issues of genre coincides with the effort to rethink the notion of gender by destabilizing it referentially and de-essentializing it. Throughout his career, starting with Nude Descending a Staircase, No.

Rather than perpetuating traditional representations of nudity by automatically associating it with femininity and the removal of garments, he begins to treat the nude as a symptom of the problems embodied in pictorial representation in general. At issue is the reliance of pictorial representation on its visual, rather than intellectual, impact; hence the emphasis on spectatorship as voyeurism, on visual fascination and seduction.

Duchamp, however, is concerned with exploring the conceptual aspects of pictorial representation, with its conditions of possibility not merely as a visual medium but as a philosophical and institutional construct. Given the erosion of the traditional mimetic role of painting by mechanical reproduction and the emergence of new media, such as photography and cinema, which incorporate technological developments, it is not surprising to note Duchamp's desire to rethink the function and the representational modes that define painting and sculpture, and by extension, art in general.

While other modernist movements such as Cubism and Futurism turn to abstraction as a way of responding to social and technological changes, Duchamp turns to a conceptual investigation of the meaning and function of art. The technical precision and methodical nature of his interventions stand in contrast to contemporary Dadaist and Surrealist efforts to radicalize art through chance operations.

Chance in Duchamp's work is grounded in a field of preestablished determinations, so that its plasticity emerges from its strategic deployment and recontextualization. Following the trajectory of the lines of inquiry outlined above, on the one hand, this book unpacks Duchamp's works by focusing on the persistence of the nude as a pictorial genre as it passes from figuration to abstraction, through its generic decomposition and transposition in The Large Glass and its belated figurative reassemblage in Given.

On the other hand, it underlines the fact that Duchamp's representations of gen-. It is important to note that from its inception in Nude Descending a Staircase, No. Before stumbling on the discovery of the ready-mades, Duchamp elaborates the nude as a transitive genre whose logic is in the order of reproducibility.

Having begun to strip not the nude but the pictorial conventions that define it, in The Large Glass Duchamp goes on to strip art bare. He does not do so by leaving art behind but rather he draws upon it, restaging it in a manner that postpones its pictorial becoming.

Playing on mimesis, a definition of art as a copy of nature, Duchamp generates copies of art that through reproduction undermine notions of artistic intent. Despite the seemingly disparate nature of such works as The Large Glass, the ready-mades, The Box in a Valise, and Given, they all represent Duchamp's strategic interpretation of art as an assemblage whose productive logic is reproductive. These works are staged compendia of generic conventions that enable Duchamp to test the boundaries of art by exceeding and, therefore, postponing its artistic intent.

Thus, the structure of this book unfolds like a box, around the above-mentioned works as its organizing hinges. Whereas The Large Glass and the ready-mades hold up a mirror to painting and sculpture by de-realizing their artistic import, The Box in a Valise and Given by their mirrorical return to figurality unhinge art by reproducing and objectifying its generic conventions. In chapter 1, the focus is on Duchamp's ostensible abandonment of painting and his challenge of its generic and conceptual limits.

I argue that instead of relying on the notion of pictorial image and the conventions of painting, Duchamp redefines them both by rethinking them through other media, such as engraving and cartooning. As a mode of mechanical reproduction, engraving enables Duchamp to conceive the visual image in new terms—not as a unique entity but as a series of imprints whose temporal structure acts to delay the retinal impact of the image.

The technology of engraving and printing as media of mechanical reproduction becomes the source for intellectual insights that Duchamp deploys to redefine the identity and immediacy of the visual image. As a graphic and linguistic medium, cartooning enables Duchamp to redefine. If titles are significant in Duchamp's works, this is because they no longer function as mere captions or labels but instead as devices that reframe the retinal impact of images in terms of poetic or punning associations.

The visual opacity of the Large Glass attests to Duchamp's successful displacement of meaning away from the retinal and toward its active interplay with linguistic and poetic frames of reference. A reassemblage that reproduces his previous pictorial works on glass, the Large Glass makes manifest the recognition of the ready-made character of pictorial representation.

In chapters 2 and 3, Duchamp's work speaks eloquently and decisively about art as an institution, as a system for packaging and framing various objects and gestures. Exhibiting mass-produced objects as art objects, Duchamp exposes the conditions of possibility of art through the readymades. Rather than postulating art as an expression of the object, of its formal and material qualities, Duchamp uncovers the fact that art inheres less in an object than in the institutional context that frames it and makes it legible.

The ready-mades make visible the provisional and transitional status of art as they switch back and forth, undecidably, between art and nonart. By documenting this transition, Duchamp demystifies the art object at the same time that he reactivates the position of the spectator, as critical to both the reception and production of works of art. Rather than being restricted to the ready-mades as objects or gestures, this study seeks to inquire into their nominal properties.

I argue that the legibility of the ready-mades relies not merely on their visual appearance but on their nominal properties, since their titles pack in networks of puns and poetic associations. As literal reproductions of objects, the ready-mades become legible as puns, as relays of signification, as switches that enable the spectator to discover mechanically the creative potential of language.

Just as mechanical reproduction ensures the production of commercial prototypes, so do linguistic and social conventions ensure the production and circulation of puns. Culturally generated and reproduced, puns func-. Duchamp's ready-mades make us stumble on the surprising discovery that linguistic puns are also ready-mades; that is, they are mechanisms whose venues for generating meaning are technically spelled out in the dictionary.

The legibility of Duchamp's puns thus depends less on the spectator's imagination than on his or her ability to reactivate the puns by becoming aware and engaging with their potential meanings. In this context the dictionary becomes a technical manual of sorts that makes visible the conceptual subtext that underlies the visual and nominal appearance of the ready-mades. Unfolding Duchamp's ready-mades as three-dimensional puns requires concerted attention to the interplay of language and image, as each system of reference intervenes to generate or undermine the production of meaning.

Unpacking Duchamp's ready-mades, therefore, refers less to the handling of objects proper than to theunderstanding of the way they function as utterances in context. As bearers of speech or cultural mouthpieces, the ready-mades capture the dilemma of an art that postpones its pictorial becoming and thus the finality of its attainment to become art.

As the "plastic equivalent of a pun" to use Octavio Paz's terms , the ready-made stages the gratuitous conversion of an ordinary object into a work of art, while undermining through this very gesture the notion of an art object. Chapter 4 is an examination of how the ready-made functions as a critique of classical notions of value. Instead of assuming the autonomy of art from the social and economic sphere, the focus is on how Duchamp rethinks the question of artistic value by redefining it as a function of its economic and social currency.

Instead of condemning Duchamp's forays into commercial ventures in art, I argue that Duchamp is redefining art according to a speculative model, whose conceptual implications liberally draw upon and expend classical economics. Ranging from checks and bonds to numismatic coins, Duchamp's artworks mimic economic currency and exchange only to undermine the notion of both artistic and monetary standards.

These works redefine artistic and economic forms of production byexploiting the speculative potential of reproduction. This study concludes with an examination of Duchamp's posthumously exhibited work Given: 1 the waterfall, 2 the illuminating gas, a work. Described as "startlingly gross and amateurish," Given "startles" by mirroring back the spectator's look.

In the context of the museum where everything is on display, however, the display of sexuality takes on an ironic tone. Having questioned the logic of the visible, does Duchamp's Given represent a continued challenge or a return to conventional modes of representation? I argue that despite this work's oven sexual display, or rather, because of its exaggeration as display, the equation of sexuality and vision is sundered.

Duchamp undermines the logic of voyeurism by questioning the coincidence of sight with visual pleasure. In doing so he moves away from equating sexuality with anatomical destiny, toward redefining it as a rhetorical operation. However, this effort to de-essentialize gender can be understood only in the framework of his attempts to experiment with genre. Just as Given fails to provide the spectator with a stable representation of sexuality, so does it also resist any generic classification. Given is an installation, an assemblage of works that mimic artistic media such as painting, sculpture, and photography, without being reducible to a specific genre.

The generic identity of this work, like the status of gender, remains transitive, resisting both fixity and closure. If Duchamp's works resist canonization, this is not simply because of their complexity or enigmatic character but rather because his works are by definition transitive. They are like hinges, straddling the gap between vision and language, art and nonart, forms of artistic production and reproduction.

Resembling Duchamp's elusive presence as an artist, his works are packages whose meaning continues to unfold in new and surprising ways. In the postscript is a brief assessment of Duchamp's impact on the history of Modernism. His redefinition of artistic modes of production through reproduction opens up the scope of Modernism to a notion of artistic production that is speculative, insofar as it reinvests rather than liquidates the legacy of tradition.

In doing so, Duchamp discovers within the experimental scope of Modernism a conceptual potential that becomes the terrain for the emergence of postmodernism. Having done away with. Just as Duchamp draws upon pictorial conventions to redefine the meaning of art, so does the legacy of his work open itself to appropriation by others. Is it then surprising to see artists such as J. Boggs's postmodern appropriation realizes a potential inscribed in Duchamp's postponed legacy of Modernism.

If Duchamp's artistic life and his works are on credit, this credit can continue to be reinvested or spent. The possibilities are unlimited, since, as Duchamp reminds us, "Posterity is a form of the spectator" DMD, Among Marcel Duchamp's gestures and artistic interventions, few have created as much controversy or been as puzzling as his putative abandonment of painting.

Although Duchamp starts to exhibit his work publicly in , the earliest works of his that are considered significant date to At the age of twenty-three, his enriched pictorial and compositional vocabulary is deployed in a figurative context, where nudes and group scenes dominate.

By , Duchamp's use of abstraction demonstrates shared affinities with Cubism, insofar as it brings into question the figurative identity of the body through its spatial fragmentation and its serial deployment. At this time he also begins to expand the meaning of the pictorial image by trying to find new ways of illuminating it, either through experiments with gaslight or by exploring how the title may have an impact on the nominal expectations of painting.

At the end of and culminating in , Duchamp irrevocably establishes his authority as a painter through his signatory work, Nude. Descending a Staircase, No. It is during this same period that Duchamp begins to incorporate machine imagery and morphology in his paintings, leading to his mechanomorphic paintings. This three-year trajectory that establishes Duchamp's creative identity and credibility as a painter renders his abandonment in of conventional painting and drawing all the more surprising, if not altogether shocking.

At issue is neither Duchamp's failure nor, ironically, his success as a painter but rather his challenge of the limits of pictorial practice. Thus within this three-year span —13 , Duchamp establishes himself as an internationally renowned painter, one who moves decisively from figuration to abstraction, only to begin to question painting altogether. In Duchamp practically gives up conventional forms of painting, but this does not mean that he stops working.

Instead, he begins to experiment with chance as a way of getting away from the traditional methods of expression generally associated with art. He lets pieces of string fall and records the shapes they generate; when his work on glass cracks he accepts the cracks as part of the work. During this period, Duchamp experiments with mechanical drawings, painted renderings, and notations that serve as studies for his seminal work, The Bride Stripped Bare by Her Bachelors, Even.

By , the idea that Duchamp did not just give up painting but art altogether comes into currency. As Joseph Masheck explains: "Duchamp never discouraged it and seems to have enjoyed the mysterious notoriety. Duchamp was said to have taken up a decided antiart position, abandoning art in favor of playing chess. How do we explain these radical transitions, from figuration to abstraction leading to the abandonment of painting, and ultimately art, given the speed at which these gestures succeed one another?

Can these transitions be illuminated by particular events in Duchamp's life, and more specifically, how are they manifest when considering works from this period? A small number of biographical details may prove to be significant to our discussion of Duchamp's pictorial origins.

Born in a solid French bourgeois family on 28 July and following in the footsteps of his two brothers Jacques Villon and Raymond Duchamp-Villon and his sister Suzanne, Duchamp also became interested in art. Far more significant in his professional formation was his apprenticeship as a printer in Rouen in , in lieu of doing military service. As an "art worker" he received exemption from military service after one year, having passed a juried exam based on the reprints of his grandfather's engravings.

Thus, in addition to his early exposure and family background in art, both engraving and cartooning. The influence of Duchamp's exposure to engraving and cartooning impacted on his efforts to discover alternative ways of conceiving painting and art. Unlike his siblings, Duchamp is not content to simply become a painter, for he will rapidly abandon painting in favor of activities that challenge the very meaning and definition of art.

When one considers attentively Duchamp's early works, one is invariably struck by his efforts to put into question the notion of pictorial image, by examining its relation to other frames of reference, the title, or the nominal expectations of the public. Moreover, his early explorations of serial works, or multiples, attest to his efforts to challenge the uniqueness and autonomy of the pictorial image.

Engraving and cartooning thus enabled Duchamp to conceive the plastic image in new terms, whose technical and intellectual content opened up the possibility of redefining the notion of artistic creativity as a form of production based on reproduction. Duchamp did not becomean engraver nor a cartoonist. He did, however, draw on the intellectual and speculative potential of these two media, in order to redefine not only painting as a medium but also art itself.

The fact that Duchamp began his artistic career as an "art worker" is significant, insofar as it enabled Duchamp to question the creative function of the artist and the meaning of art as a form of making:. I don't believe in the creative function of the artist. He's a man like any other. It's his job to do certain things, but the businessman does certain things also, you understand?

On the other hand the word "art" interests me very much. If it comes from the Sanskrit, as I've heard, it signifies "making. Formerly, they were called craftsmen, a term I prefer. We're all craftsmen, in civilian or military life. DMD, In the pages that follow, Duchamp's effort to question the meaning of art as pictorial practice, as an institution, and as a profession will be at issue.

The notion of art as "making" enlarges the meaning of artistic activity to forms of production that include not only artisanal efforts but also conceptual insights. In his interview with Marcel Duchamp, Pierre Cabanne asks him to explain the key event of his life: his abandonment of painting. Duchamp's response identifies Nude Descending a Staircase, No.

While serving to establish his reputation, the initial rejection of the work alerts him to the norms and strictures that define not just conventional art but also contemporary art movements, such as Cubism:. In the most advanced group of the period, certain people had extraordinary qualms, a sort of fear! People like Gleizes, who were, nevertheless, extremely intelligent, found this "Nude" wasn't in the line that they had predicted. Cubism had lasted two or three years, and they already had an absolutely clear, dogmatic line on it, foreseeing everything that might happen.

DMD , Duchamp is less concerned with the rejection of the painting than the fact it embodies a doctrinal gesture—one where a work of art is defined by living up to its nominal expectations. By failing to fall into line, that is to conform to a set of pregiven rules, Nude Descending a Staircase, No. For Duchamp, the turning point that the Nude represents is not merely its challenge to the public but also to his peers, whose artistic and intellectual expectations define the work's conditions of possibility.

Was Duchamp's dramatic gesture an expression of his "distrust of systematization," of his inability to contain himself to "accept established formulas" DMD , 26? Duchamp rejects Cubism not just as an artistic movement but as a discipline with a set aesthetic program: "Now, we have a lot of little Cubists, monkeys following the motion of a leader. Their favorite word is discipline. It means everything to them and nothing. Coming in the wake of a series of representational nudes in , Nude Descending a Staircase, No.

As this study will demonstrate, however, Duchamp's passage through abstraction involves the speculative goal of getting away from "the physical aspect of painting" by putting "painting once again to the service of the mind. Duchamp's adoption of the nude as pictorial genre did not have entirely auspicious beginnings. It is also interesting to recall that he failed the Ecole des Beaux-Arts competition over a test that involved doing a nude in charcoal DMD , In , when Duchamp turns to the genre of the nude.

Schwarz notes that in Nude with Black Stockings Nu aux bas noirs; , the "use of heavy black lines—characteristic of the Fauves' reaction to the Impressionists' careful avoidance of black—is freely adopted.

It reflects an understanding of the extent to which an artistic movement may be defined by its strategic response to the aesthetic tenets of a previous, or even contemporary, movement. Duchamp's use of heavy black lines to outline the body, as in Nude Seated in a Bathtub Nu assis dans une bagnoire; fig. The black lines emphatically reframe the successive color shadings, thus. In the Red Nude, color as one of the constitutive elements of painting is deployed in a manner that reveals its affinity to engraving.

The red shadings and black lines compete as color templates that redefine the pictorial appearance of the nude as a successive set of impressions or imprints. If these nudes are graphic, it is in their treatment of painting and not in their ostensible subject matter. When comparing Duchamp's Nude with Black Stockings with Gustave Courbet's Woman with White Stockings Femme aux bas blancs; , one is struck by its unerotic demeanor that resists voyeuristic appropriation as an image.

Rather than emphasizing and framing genitality, as the white stockings do in Courbet's painting, the black stockings dismember the body by erasing it from the. Duchamp's cropping of the nude body displaces the viewer's attention from the frontality of sex to the pictorial frame that cuts the body off—a feature shared by other works, such as Two Nudes Deux Nus; , and Red Nude. The effort to draw the spectator's attention to framing devices is deliberately underlined in Red Nude , where the profile of the crouching red nude breaking out of the frame of the painting also cuts into the frame of another painting.

Located in the upper lefthand corner of the image, this painting is further disfigured by the painter's signature cutting across the head of a female figure. The authorial signature is displaced into a position where its nominal content interferes with the visual content and consumption of the image.

Rather than merely stripping the nude, Duchamp begins to strip away the visual conventions that define the nude as a pictorial genre. By , Duchamp's exploration of the nude enters a new phase, one where issues of pictorial abstraction are reframed by their interplay with nominal expectations triggered by the title. Loosely identified as his "Symbolist" phase because of its visual affinities to the works of Paul Gauguin and Pierre Girieud, Duchamp's works betray the Symbolist conceit of combining word and image.

The doubling of female nudes in The Bush Le Buisson ; fig. For Lawrence Steefel, The Bush "seems to point towards the ultimate goal of turning the world inside out. Duchamp trivializes the visual referent by his puns on the title "bush," thereby defying the nominal expectations of the spectator as voyeur.

In Paradise Le Paradis; fig. There is no illumination nor spiritual "Ascension" here. The title Paradise contradicts the viewer's expectations, unless it is interpreted literally, as a pun on the French word paradis, which means no radiance, to be struck out, canceled, or just broken. The lack of radiance in Paradise. While Duchamp admits in his interview with Cabanne: "I don't know where I had been to pick up on this hieratic business" DMD , 23 , this statement should not discourage us from considering this question.

This halo effect or aura can be found in another work of this period entitled Portrait of Dr. Dumouchel Portrait du Dr. Dumouchel; fig. Referring to this painting, Duchamp wrote in a letter to his patrons Louise and Walter Arensberg: "The portrait is very colorful red and green and has a note of humor which indicated my future direction to abandon mere retinal painting.

The word figure means figure, shape, or form, but its use by Duchamp suggests that it refers to Dumouchel's appearance: it is a reflection on the way he looks, his "air," or "aura. This pun on color blindness in the context of painting foreshadows, as it were, Duchamp's denunciation and subsequent aban-. For Duchamp, the hieratic aura associated with Symbolist painting becomes the locus of investigation of the interplay of word and image, not under the guise of symbols but as puns.

This "halo" effect or aura continues to reappear throughout Duchamp's works, either as an analogy to smell in such works as Fountain [] and Beautiful Breath, Veil Water [Belle Haleine, Eau de Voilette; ] , or as an analogy for electricity in Bec Auer [a gas lamp circa ]; The Large Glass [—23]; Given: 1 the waterfall, 2 the illuminating gas —66 ; and in a set of prints entitled The Bec Auer []. Considered from this perspective, Duchamp's early experiments with the hieratic can be understood as an allusion to the history of painting.

This was at a time when the appearance of the nude, like painting itself, attained value by virtue of its religious, philosophical, and moral function, and was thus in excess of visual semblance. If painting exuded an "aura," this is because its significance was originally defined by its social rather than cultural function.

The loss of painting's "aura" in the age of mechanical reproduction heralds the end of painting as a purely manual and visual event and its conceptual rebirth as a practice stripped of the hallowed echoes of visual semblance. As Duchamp explains to Cabanne:. Since Courbet, it's been believed that painting is addressed to the retina. That was everyone's error. The retinal shudder! Before, painting had other functions: it could be religious, philosophical, moral. If I had the chance to take an antiretinal attitude, it unfortunately hasn't changed much; our whole century is completely retinal, except for the Surrealists, who tried to go outside it somewhat.

And still, they didn't go so far! If Nude Descending a Staircase, No. Described as an "explosion in a shingle factory," Nude Descending a Staircase, No. The word rude appropriately captures the impact of the Nude, its deliberate disregard for the artistic conventions of the genre. This work scandalized not only the general public but also the avant-garde circles of the time.

As Duchamp explains, the title plays a significant role in explaining the particular interest and impact of this work:. What contributed to the interest provoked by the canvas was its title. One just doesn't do a nude woman coming down the stairs, that's ridiculous. It doesn't seem ridiculous now, because it's been talked about so much, but when it was new, it seemed scandalous.

A nude should be respected. DMD, 44; emphasis added. Duchamp's comments indicate that the reception of this painting was being filtered through a set of expectations, whose nominal character was staged by the title. The abstract nature of this work and thus its failure to provide a visual referent for the title only increased the public's disappointment. Instead of reclining passively, Duchamp's fractured nude is actively descending a staircase. The scandal surrounding the exhibition of Nude.

In his book The Nude, Kenneth Clark maintains that the nude is not the starting point of a painting but a way of seeing that the painting achieves. Constructed as the subject of desire from the Renaissance to the late nineteenth century, the nude as a pictorial genre involves a structure of spectatorship that relies upon the objectification of the female body.

This interplay of visual and nominal expectations staged by the nude as a pictorial genre was put into question by painters such as Edouard Manet, who in Olympia and Le Dejeuner sur l'herbe —63 challenged the inscription of the desiring look of the spectator. The Nude. The splintering of vision into a series of frames that fragment and abstract both the identity of the nude and the process of movement inscribe into the painting an interval, a temporal dimension.

Functioning neither descriptively nor prescriptively, the title Nude Descending a Staircase inscribes a temporal delay that interferes with the visual consumption of the image. This strategy of delay also redefines and defers notions of visual reference that are traditionally associated with photography.

While appealing to techniques of mechanical reproduction, such as photography, to redefine the pictorial medium and its subject matter, Duchamp succeeds in redefining painting itself as a process whose plasticity includes temporal considerations.

In the nude itself. To do a nude different from the classic reclining or standing nude, and to put it into motion. There was something funny there, but it wasn't funny when I did it. Movement appeared like an argument that made me decide to do it. In the "Nude Descending a Staircase," I wanted to create a static image of movement: movement is an abstraction, a deduction articulated within the painting, without our knowing if a real person is or isn't descending an equally real staircase.

Fundamentally, movement is in the eye of the spectator, who incorporates it into the painting. The picture presents the viewer with a "vertigo of delay," to use Paz's term, rather than one of acceleration. The staggered motion of the "nude" demonstrates an analysis of movement rather than the Futurist seduction with the dynamics of movement. But why is the nude descending? A network of visual puns connects Nude Descending a Staircase, No. Just as Laforgue's poem denounces the idealist aspirations of Symbolist poetry by pointing out that the stellar image of the sun is undermined by its ordinary and pockmarked appearance, so does Duchamp transform the idealism that underlies pictorial praxis into a mere stair, a pun on the notion of descent understood both literally and figuratively.

The ambiguous title of the Nude nu, in French gives no particular indication as to the referent's gender, although critics have identified it generically as female, de rigueur. Duchamp's Nude Descending a Staircase, No. Unable to incarnate the nominal expectations of the spectator, the nude visually fractures the spectator's gaze by setting it into a spiraling motion. In doing so Duchamp points both to the title and to the spectator's gaze as the sites on which hinges the facticity of gender.

This resistance to the equation of spectatorship with visual consumption and delectation is explicitly thematized in Duchamp's later works. The Nude 's descent thus functions as an index of Duchamp's strategic displacement and rethematization of the nude as a pictorial genre and its declension from the spectator's nominal expectations.

The descent of the Nude is not merely the mark of a genealogical decline but also the legal index of the passage of an estate through inheritance. Is it surprising then that Nude Descending a Staircase, No. As Joseph Masheck notes: "Typical of Duchamp is this work's self-illustrative and self-reproductive function, as well as the fact that as an actual photograph it returns to one of the technical sources of the 'original' painting. However, this reproductive industry did not stop short with the fullsized versions of the Nude.

This work was further reproduced as a miniature pencil-and-ink drawing Nude. This doll-sized version of the Nude was followed by further miniature reproductions in The Box in a Valise — From a single work that is by definition a multiple, insofar as it is part of a series, Duchamp generates an entire corpus. By discovering the self-productive and self-reproductive potential of the Nude, Duchamp redefines the nude as a medium of and for reproduction. Eroticism in this context no longer refers to the visual appearance of the nude but instead functions as an index of its proliferation as modes of appearance.

Duchamp challenges the eroticism traditionally associated with spectatorship and voyeurism by proposing an alternative eroticism whose speculative, technical, and humorous character restages through reproduction the notion of artistic creativity and production. Given Duchamp's explicit rejection of the equation of vision and eroticism, how are we to explain his interest in the nude as pictorial genre?

It seems that the entire trajectory of his life's work is defined by the arching movement from Nude Descending a Staircase, No. While Duchamp maintains that eroticism is the only -ism he believes in, it is. Eroticism in the figurative arts is most commonly represented as the relationship between clothing and nudity, and thus, as Mario Perniola suggests, it is conditional on the possibility of movement or transition from one state to another. Now we begin to understand the conceptual import of both engraving and cartooning in Duchamp's work.

Engraving is one of the earliest forms of mechanical reproduction that involves a different way of conceiving. Not only is the appearance of the engraved image the result of multiple reproductions but its very identity is defined as a technical process, involving multiple impressions or imprints.

An engraving is a template, a sculptural mold that functions like a photographic negative. Engraving as a medium challenges the autonomy of the pictorial image, insofar as the image acts as a temporal record of multiple impressions.

Duchamp's pictorial series of Nude Descending a Staircase, as a multiple that undergoes extensive reproduction, illustrates the logic of engraving operative in his works. This is not to say that these works are engravings, since they are clearly paintings; rather, the conditions of production and reproduction evidenced in these series suggest conceptual processes akin to those involved in the technical reproduction of engravings.

You may ask how cartoons inform Duchamp's oeuvre? The answer by now is clear. Regarded as a form of popular art associated with the print medium, cartoons are images that are constructed like rebuses, as composites of language and image. Their humor is not just visual but intellectual. They are often visual analogues of linguistic puns. This is not to suggest, however, that Nude Descending a Staircase, No. Rather, Duchamp's use of the title as nominal intervention in order to restage the expectations of the spectator reframes the reception of this work as an intellectual, instead of a purely visual, experience.

Consequently, despite its mechanomorphic character, Duchamp's Nude can also be seen an an "anti-machine. Their relation to utility is the same as that of delay to movement;they are without sense and meaning. They are machines that distill criticism of themselves.

As this study has suggested, Duchamp's humor lies in redefining the visual image as a serial imprint, as a construct where appearance does not refer to an external reality but to a mode of production whose logic is reproductive. Duchamp doubly displaces painting: first, by redefining it through the logic of engraving, as a print medium, and second, by draw-. Instead, it becomes the rhetorical interplay between language and vision, which constructs the facticity of gender as a pun.

While all artists are not chess players, all chess players are artists. When asked by Katherine Kuh, one of his interviewers, which of his works he considers to be the most important, Marcel Duchamp replied:. That was really when I tapped the mainspring of my future.

In itself it was not an important work of art, but for me it opened the way—the way to escape from those traditional methods of expression long associated with art. I didn't realize at that time exactly what I had stumbled on.

When you tap something you don't always recognize the sound. That's apt to come later. For me the Three Stoppages was a first gesture liberating me from the past. As Duchamp subsequently explains, the idea of letting a piece of thread fall on a canvas was accidental, but "from this accident came a carefully planned work. Was it the idea of chance, or its plastic deployment and embodiment as an event or work?

Duchamp's interest in chance as a way of redefining conventional forms of artistic expression appears early on in his paintings and is tied to his interest in chess. For Duchamp, chess is not merely a pastime or an ordinary game because its intellectual character represents for him a plastic.

As a strategic game that requires the interplay of two opponents, chess provides Duchamp with a new way of envisioning art in its dialogue with the tradition. The analogy of art and chess enables Duchamp to appropriate chance and redefine its plastic impact in a field of already given determinations.

Starting with The Chess Game Le jeu d' hecs; fig. Duchamp's lifetime interest and preoccupation with chess is well known, but its significance and precise impact on his art is less recognized. II , it is clear that he is already playing another game DMD, The checkered pattern of the board, however, is also an allusion to another set of rules, those of Albertian perspective that have guided the development of painting. If we pursue Duchamp's analogies in The Chess Game, art no less than chess emerges as a strategic, rather than purely plastic, domain.

Both chess and perspective are systems whose normative standards prescribe and determine the nature of representation. What had been originally conceived as an arbitrary relation between painting and the world is now revealed to be a strategic, albeit conventional game, a chess game. The answer lies in his understanding of chess as a plastic, rather than a purely intellectual, game.

As Duchamp's comments to James Johnson Sweeney indicate, playing. This plasticity, however, is not in the realm of the visible but in the abstraction of the movement of pieces on the board. In his interview with Francis Roberts, Duchamp explains how the strategic and positional nature of chess generates plastic effects:.

In my life chess and art stand at opposite poles, but do not be deceived. Chess is not merely a mechanical function. It is plastic, so to speak. Each time I make a movement of the pawns on the board, I create a new form, a new pattern, and in this way I am satisfied by the always changing contour. Not to say that there is no logic in chess.

Chess forces you to be logical. The logic is there, but you just don't see it. The plasticity that Duchamp ascribes to chess is not aesthetic in the visual sense but rather intellectual. The movement of the pieces on the board creates patterns and forms whose contours are constantly shifting.

This moving geometry is described by Duchamp as "a drawing" or as a "mechanical reality" DMD, As Duchamp elaborates: "In chess there are some extremely beautiful things in the domain of movement, but not in the visual domain. It's the imagining of the movement or the gesture that makes the beauty, in this case. It's completely in one's gray matter" DMD, The beauty that Duchamp appeals to is not one based on aesthetic categories, on visual appearance and artistic self-expression.

Rather, the beauty in question is defined by the plasticity of the imagination, by the poetry of its ever changing contours. The analogy of chess and art is one that is mediated by an allusion to the abstract nature of both music and poetry. As Duchamp explains:. Objectively, a game of chess looks very much like a pen-and-ink drawing, with the difference, however, that the chess player paints with black-and-white forms already prepared instead of having to invent forms as does the artist.

The design thus formed on the chessboard has apparently no visual aesthetic value, and it is more like a score for music, which can be played again and again. Beauty in chess is closer to beauty in poetry; the chess pieces are the block. Actually, I believe that every chess player experiences a mixture of two aesthetic pleasures, first the abstract image akin to the poetic idea of writing, second the sensuous pleasure of the ideographic execution of that image of the chessboards.

Relying on analogies to the media of music and poetry, Duchamp uses chess as a way of expanding the meaning of art. No longer bound to the creation or invention of visual forms, the chess player "paints" with already given black-and-white forms. The interest of the exercise lies in the composition of the design, a visual score that is open to multiple performances, for the nature and value of chess exists only as a performance, a duet where two interpreters put their heads together, so to speak.

The chess pieces in this game function as linguistic elements already given by convention, but ready to be redeployed poetically in new ways. While subject to particular rules governing the possibility of movement, the mechanisms generated, as the "ideographic execution of that image," are always open to further reinterpretation.

Thus Duchamp uncovers within chess a paradigm for the reinterpretation of aesthetic pleasure as a pleasure derived neither from invention nor the sensuality of the pieces themselves, but from their recomposition and poetic deployment as a game. Rather than interpreting this tearing as Cubist dislocation, Duchampis reinterpreting Cubism itself conceptually from the perspective of chess, as a game whose.

More specifically, the serial fragmentation and multiplication of the protagonists into shards, while depersonalizing them into mechanical patterns, illuminates painting in a new light. Duchamp visibly draws on Cubism, only to redefine its logic as a representation: the dislocations visible in the image are but the diagrams of movements ideographically transposed from chess.

While such an interpretation seems to force Duchamp's hand, as it were, it is important to recall Duchamp's comment to Cabanne regarding the fact that Portrait of Chess Players was painted not in ordinary light, but by gaslight:. This "Chess Players," or rather "Portrait of Chess Players," is more finished, and was painted by gaslight.

It was a tempting experiment. You know, that gaslight from the old Aver jet is green; I wanted to see what the changing of colors would do. When you paint in green light and then, the next day, you look at it in day light, it's a lot more mauve, gray, more like what the Cubists were painting at the time.

It was an easy way of getting a lowering of tones, a grisaille.

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