![Entrance, Hombroich Museum, Langen Foundation.
In his design for the Langen Foundation’s Hombroich Museum (2004) we see a poetic composition of form and
material. The site, located in Neuss, near Dusseldorf, Germany, is a former NATO missile launch facility. The
use of concrete would, initially, seem to be in keeping with the site’s history. However, in Ando’s hands,
concrete, in conjunction with other materials, becomes a constituent of a new picturesque. The building initially
appears, through a concrete arch, as a composition of glass, concrete, and water. This plays transparency and
solidity against one another, as well as the real and the reflected.
The building is divided into two sections. Apair of concrete structures partially submerged into the site (thus the
26-foot [8-meter] height can only be experienced inside) house the modern collections. Their solidity is
contrasted by the Japanese gallery, which is a 140-foot-long (43-meter-long) concrete box enclosed in a steel-
and-glass box.
It is the Japanese gallery wing that gives the project its outstanding views and is representative of Ando’s use of
material and light. The enclosing of the concrete gallery space within glass and steel serves to reduce the
visual impact of the concrete within. Further, the play of shadows cast by the steel supports of the glass
enclosure animates the surface of the concrete with a rhythm of shadows.
While some architects become known for the daring of the forms they create, or the scale of projects they
complete, Ando has become a recognized master of material and light.](https://image.slidesharecdn.com/5688072-250528063313-e3eafc91/85/Architecture-An-Introduction-Portfolio-Geoffrey-Makstutis-30-320.jpg)
![animal" for " is human," even if I believe falsely that the Phoenix is
rational and immortal.
We give the name of "derived extensional function" to the function
constructed as above, namely, to the function: "There is a function having
the property and formally equivalent to ," where the original function
was "the function has the property ."
We may regard the derived extensional function as having for its
argument the class determined by the function , and as asserting of this
class. This may be taken as the definition of a proposition about a class. I.e.
we may define:
To assert that "the class determined by the function has the property
" is to assert that satisfies the extensional function derived from .
This gives a meaning to any statement about a class which can be made
significantly about a function; and it will be found that technically it yields
the results which are required in order to make a theory symbolically
satisfactory.[41]
[41]See Principia Mathematica, vol. I. pp. 75-84 and * 20.
What we have said just now as regards the definition of classes is
sufficient to satisfy our first four conditions. The way in which it secures
the third and fourth, namely, the possibility of classes of classes, and the
impossibility of a class being or not being a member of itself, is somewhat
technical; it is explained in Principia Mathematica, but may be taken for
granted here. It results that, but for our fifth condition, we might regard our
task as completed. But this condition—at once the most important and the
most difficult—is not fulfilled in virtue of anything we have said as yet. The
difficulty is connected with the theory of types, and must be briefly
discussed.[42]
[42]The reader who desires a fuller discussion should consult Principia
Mathematica, Introduction, chap. II.; also * 12.
We saw in Chapter XIII. that there is a hierarchy of logical types, and
that it is a fallacy to allow an object belonging to one of these to be
substituted for an object belonging to another. Now it is not difficult to
show that the various functions which can take a given object as argument
are not all of one type. Let us call them all -functions. We may take first](https://image.slidesharecdn.com/5688072-250528063313-e3eafc91/85/Architecture-An-Introduction-Portfolio-Geoffrey-Makstutis-65-320.jpg)
![world. Among "possible" worlds, in the Leibnizian sense, there will be
worlds having one, two, three, ... individuals. There does not even seem any
logical necessity why there should be even one individual[43]—why, in fact,
there should be any world at all. The ontological proof of the existence of
God, if it were valid, would establish the logical necessity of at least one
individual. But it is generally recognised as invalid, and in fact rests upon a
mistaken view of existence—i.e. it fails to realise that existence can only be
asserted of something described, not of something named, so that it is
meaningless to argue from "this is the so-and-so" and "the so-and-so exists"
to "this exists." If we reject the ontological argument, we seem driven to
conclude that the existence of a world is an accident—i.e. it is not logically
necessary. If that be so, no principle of logic can assert "existence" except
under a hypothesis, i.e. none can be of the form "the propositional function
so-and-so is sometimes true." Propositions of this form, when they occur in
logic, will have to occur as hypotheses or consequences of hypotheses, not
as complete asserted propositions. The complete asserted propositions of
logic will all be such as affirm that some propositional function is always
true. For example, it is always true that if implies and implies then
implies , or that, if all 's are 's and is an then is a . Such
propositions may occur in logic, and their truth is independent of the
existence of the universe. We may lay it down that, if there were no
universe, all general propositions would be true; for the contradictory of a
general proposition (as we saw in Chapter XV.) is a proposition asserting
existence, and would therefore always be false if no universe existed.
[43]The primitive propositions in Principia Mathematica are such as to
allow the inference that at least one individual exists. But I now view
this as a defect in logical purity.
Logical propositions are such as can be known a priori, without study of
the actual world. We only know from a study of empirical facts that
Socrates is a man, but we know the correctness of the syllogism in its
abstract form (i.e. when it is stated in terms of variables) without needing
any appeal to experience. This is a characteristic, not of logical propositions
in themselves, but of the way in which we know them. It has, however, a
bearing upon the question what their nature may be, since there are some
kinds of propositions which it would be very difficult to suppose we could
know without experience.](https://image.slidesharecdn.com/5688072-250528063313-e3eafc91/85/Architecture-An-Introduction-Portfolio-Geoffrey-Makstutis-79-320.jpg)
![It is clear that the definition of "logic" or "mathematics" must be sought
by trying to give a new definition of the old notion of "analytic"
propositions. Although we can no longer be satisfied to define logical
propositions as those that follow from the law of contradiction, we can and
must still admit that they are a wholly different class of propositions from
those that we come to know empirically. They all have the characteristic
which, a moment ago, we agreed to call "tautology." This, combined with
the fact that they can be expressed wholly in terms of variables and logical
constants (a logical constant being something which remains constant in a
proposition even when all its constituents are changed)—will give the
definition of logic or pure mathematics. For the moment, I do not know
how to define "tautology."[44] It would be easy to offer a definition which
might seem satisfactory for a while; but I know of none that I feel to be
satisfactory, in spite of feeling thoroughly familiar with the characteristic of
which a definition is wanted. At this point, therefore, for the moment, we
reach the frontier of knowledge on our backward journey into the logical
foundations of mathematics.
[44]The importance of "tautology" for a definition of mathematics was
pointed out to me by my former pupil Ludwig Wittgenstein, who was
working on the problem. I do not know whether he has solved it, or even
whether he is alive or dead.
We have now come to an end of our somewhat summary introduction to
mathematical philosophy. It is impossible to convey adequately the ideas
that are concerned in this subject so long as we abstain from the use of
logical symbols. Since ordinary language has no words that naturally
express exactly what we wish to express, it is necessary, so long as we
adhere to ordinary language, to strain words into unusual meanings; and the
reader is sure, after a time if not at first, to lapse into attaching the usual
meanings to words, thus arriving at wrong notions as to what is intended to
be said. Moreover, ordinary grammar and syntax is extraordinarily
misleading. This is the case, e.g., as regards numbers; "ten men" is
grammatically the same form as "white men," so that 10 might be thought
to be an adjective qualifying "men." It is the case, again, wherever
propositional functions are involved, and in particular as regards existence
and descriptions. Because language is misleading, as well as because it is
diffuse and inexact when applied to logic (for which it was never intended),
logical symbolism is absolutely necessary to any exact or thorough](https://image.slidesharecdn.com/5688072-250528063313-e3eafc91/85/Architecture-An-Introduction-Portfolio-Geoffrey-Makstutis-80-320.jpg)
Architecture An Introduction Portfolio Geoffrey Makstutis
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- 7. architecture
- 8. Introduction Architects and development / What is architecture? / What does an architect do? / Reasons to become an architect / Who is this book for? / How to use this book 1. The Setting History / Theory / The profession—history and theory 2. Education and Qualification How long will it take? / Getting there by other routes / What to expect from courses / Professional experience 3. The Client and The Brief The client / The brief 4. From Brief to Project Inception and feasibility / Scheme design/schematic design / From concept to proposal 5. The Project and The Process Outline proposals and scheme design / Design and production information / The tender/bidding process / Project planning and works on site / Completion / Post-construction and feedback 6. The Practice Types of practice / Roles within a practice / Production architects / The team / Workflow / Consultants / Interiors / Masterplanning/urban design / Health and safety 7. The Future Digital architectures / Professional development / Research / Embracing diversity / Changes in professional practice / Alternative careers / Thinking architecture / Sustainability and the future / Architecture for social change ... Glossary / Further reading / Online resources / Useful addresses / Index / Picture credits / Acknowledgements
- 9. For Dylan, Amelia, and, most of all, Sarah—without you there is nothing. Copyright © 2010 Central Saint Martins College of Art & Design, The University of the Arts London Published in 2010 by Laurence King Publishing in Association with Central Saint Martins College of Art & Design The content for this book has been produced by Central Saint Martins Book Creation, Southampton Row, London, WC1B 4AP, UK Laurence King Publishing Ltd 361–373 City Road London EC1V 1LR United Kingdom Tel: +44 20 7841 6900 Fax: +44 20 7841 6910 e-mail: enquiries@laurenceking.com www.laurenceking.com All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage or retrieval system, without permission in writing from the publisher. Acatalogue record for this book is available from the British Library. ISBN-13: 978-1-85669-623-4 Designed by Karen Wilks Printed in China Front cover: Winter view of Delta Shelter, Mazama, Washington, USA. Tom Kundig (Olson Sundberg Kundig Allen Architects) Back cover: Upper-level plan, Sagaponac House, Long Island, New York, USA. Reiser + Umemoto Frontispiece: View of roof terrace, COR, Miami, Florida, USA. Oppenheim Architecture + Design
- 10. Introduction
- 11. Architects and development The practice of architecture is not new. While the profession, as we recognize it today, has only been around since the seventeenth century, there is evidence of “designed” structures, and drawings of these, dating as far back as the sixth or seventh millennium BCE. As settlements became more complex it became necessary to plan developments rather than rely upon an ad hoc accretion of structures. This increasing complexity called for design— the ability to envision a future and mediate the building process to achieve that future. The architect was the individual who had the overview of the needs of development, the knowledge of building techniques, and the application of these to specific types of structures. This was the person who understood a set of graphic conventions, which could be used to explain to others the quantifiable elements necessary to make the “imagined” become the “built.”
- 12. Frank Gehry, Guggenheim Museum, Bilbao, Spain, 1997 Architecture makes real the aspirations and projected values of our culture. Whether a private home or a major arts institution, the forms and functions of our buildings define the way in which we communicate to a broader public. Çatalhöyük, Anatolia, Turkey, 7500 BCE Building can be the basic process of meeting immediate need, but architecture, and the actions of those who are called architects, transcend simple “building” in order to embody a set of ideas about our present and our future.
- 13. Marc-Antoine (Abbé) Laugier, The Primitive Hut, frontispiece from Essai sur l’architecture (“Essay on Architecture”), 1753 Architecture has long been seen as one of the highest expressions of humankind’s ability to structure the world around it; to move beyond merely the construction of shelter and provide order and meaning. What is architecture? Ask a handful of architects this question and the odds are we will get a handful of different answers. Why should this be, when the practice of designing the buildings and spaces that surround us has been carried out for millennia? As with the way in which we build—the techniques, materials, and processes—architecture is in continuous transition and transformation. In response to changes in the world around us, architecture and the role of the architect are constantly being redefined. Ernest Dimnet, a French-American author, is quoted as saying that “Architecture, of all the arts, is the one which acts the most slowly, but the most surely, on the soul.” Whether one likes or dislikes a particular building, public space, or room, the experience of interacting with it will forever be a part of oneself. Architecture can inspire or depress, uplift or oppress, and incite a myriad of other emotional responses. From the “primitive hut” of Abbé Laugier’s writings to the “intelligent buildings” of the twenty-first century, architecture plays a vital role in the way we express our
- 14. personal beliefs and values, and those of our public and private institutions. Cathedral or mosque, terraced house or mansion, corner shop or shopping mall, architecture is the reflection of our cultural identity made manifest in the built environment. Architecture is both a local and a global pursuit. Every culture has its specific “vernacular” expressions that will be apparent in the buildings that make up its urban and rural fabric. However, as we have become increasingly mobile and international in our outlook the local vernacular has become a part of a global aesthetic. Architects now practice across national boundaries and bring new influences to bear upon the work that they undertake. In this process architecture becomes an international language. European “International Style” became American “Modernism.” The simplicity of Japanese design became Western “minimalism.” We have translated and transplanted styles, techniques, and philosophies between countries and cultures. In this way each culture makes these fusions a part of the fabric of their built environment and the process goes on. Filippo Brunelleschi, the dome of Florence Cathedral, Florence, Italy, 1436 Architecture reveals and represents our cultural identity as well as our deepest beliefs and values.
- 15. Ludwig Mies van der Rohe, Barcelona Pavilion, Barcelona, Spain, 1929 The simplicity of modern architecture was not simply a stripping away of ornament, but a drive to explore the very basis of architectural form. No longer an expression of structural prowess, space was released to be considered as an entity rather than simply the emptiness between walls.
- 16. Traditional Japanese House Architecture is both a local and a global pursuit. Every culture has its specific “vernacular” expressions that will be apparent in the buildings that make up its urban and rural fabric. Architecture continues to be one of the most popular and challenging disciplines. Each generation brings a new vision to the way that we can transform the world in which we live. What does an architect do? Once again, asking a group of architects what they do will most likely result in a range of answers. Some responses will be affected by the scale of projects that are undertaken. Architects involved in large projects will often work in more specialized areas, whereas those working on smaller projects may carry out a much broader range of activities. Differences will also arise due to variations in the types of project. An architect who works primarily on private residential projects will have a very different view of the profession to one who works on commercial projects.
- 17. Site Meeting While it may be easy to find a dictionary definition of what an architect is, defining what an architect does is often a more difficult task. As the nature of the profession of architecture expands and diversifies, the role of the architect becomes increasingly broad. This divergence in the characteristics of what an architect does should not be seen as a weakness of the profession or an inability to define the practice of architecture. Rather, it is one of the unique qualities of the architectural profession. Because architecture is a broad field there are many opportunities to find either a specialism that suits one’s particular talents and interests or to choose a more generalist approach. The world needs architects in all shapes and sizes, because the projects themselves come in all shapes and sizes. One dictionary definition of an architect is “a person who designs buildings and advises in their construction.” If we had to take a single definition this would probably suffice to cover what most people would agree the role of an architect to be. As we shall see later in this book, architecture is not simply about buildings—nor is it necessarily about the design and construction of them. But, for most people the role of an architect is connected closely with the process of design and construction. Reasons to become an architect Like any profession, there are many reasons why one might choose to become an
- 18. architect. Some enjoy the interrelation between design and engineering. Others may find it exciting to see something that they have designed become real during the process of building. Yet others may find a challenge in the more theoretical aspects of architecture and become involved in writing and research. The list is endless. Film Still, The Fountainhead, 1949 Stereotypes of architects abound, but the reality is far more complex. Architects come from all walks of life and all corners of the globe. For many people there is a stylized image of an architect. Whether it is the round glasses and bow tie of the 1970s or the black-clad, goatee-wearing young men of the 1990s, such stereotypes belie the fact that architects do not conform to a “type.” However, there remains among some people a view of an architect as (usually) a well-dressed man, in a hard hat, with a roll of drawings, standing on a building site. Similarly, the architects of film and television are often portrayed as characters conjuring up the larger-than-life, possibly arrogant figure reminiscent of Gary Cooper in The Fountainhead. But these are stereotypes, and seldom does an architect fit neatly into those molds. There are many myths about architecture as a profession and architects in practice—it would be useful to get some of these out of the way from the outset.
- 19. Skidmore, Owings & Merrill, Freedom Tower, New York, USA, 2006– There are numerous myths about architects. Whatever the perception of what an architect is or does, for most practitioners it is their work that speaks for them. Whether it is a small house or the tallest building in the world, architects undertake each project as a new challenge. Architects are rich Some architects are indeed rich, but most are not. Architecture is not a profession that naturally leads to wealth. According to the US Department of Labor, for the years 2006–7 the average salary for an architect was $46–79,000. Similarly, in 2007, according to the Royal Institute of British Architects (RIBA), the average annual salary for a qualified architect in the UK with 3–5 years experience (after qualification) was £34–42,000 ($50–60,000). Compare this to the average earnings of a family doctor, which in 2002 were calculated as $150,267 in the USA and £80–120,000 in
- 20. the UK. Architects are famous For every architect whom a member of the general public could name (probably relatively few), there are thousands of architects who will never be known beyond a very small number of people (family, friends, clients, colleagues). Architects are not like football players or rock stars. Even the most well-known contemporary architects walk through crowded streets without being mobbed by fans. Architects command universal respect Historically architects have not really been valued that highly by either patron or public. In fact, during the Renaissance architects were considered more as tradesmen than professionals. However, we should keep in mind that the greatest painters of the Italian Renaissance were generally considered to be tradesman as well. These days good architects are respected as professionals in their field, just as good doctors or lawyers are respected for their work. Architects just draw/design things As we have mentioned above, architecture is an extremely varied profession. Some architects will be engaged in highly specialized activities—they may be designers and not be tackling the more technical aspects of a given project, while others may be working as technicians or drafting staff and not really be doing much designing. However, it is seldom the case that an architect just draws or just designs. Even if one is engaged as designer, there will be a need to understand and integrate technical issues during the creative process. Equally, the person who is developing technical drawings for a project will often be designing as the drawings progress, in order to work out issues of detail and construction. If there is one thing that can be agreed upon, it is that architecture is never just anything. So, we’ve addressed some of the myths that surround architects and the practice of architecture, but what are the realities? Being an architect is hard work As we will see in subsequent chapters, becoming an architect requires a serious commitment. Equally, once an architect has completed their education and professional qualification there is still a considerable amount of time to be spent developing as a professional. Further, the professional development of an architect is ongoing—it does not stop once qualification is achieved. To be successful, and to maintain professional standing, it is necessary to engage in continuing professional development—to stay abreast of changes to legislation, materials, products, processes, etc. This is all in addition to the day-to-day activities of professional
- 21. practice. Bennetts Associates, Façade Testing Rig, London, UK Architects are increasingly called upon to work in ways that are unique to the profession. Beyond simply “drawing buildings,” they also play a role in developing new ideas and new ways of building. Architecture is an occupation Note that we have not written that “architecture is just a job.” An occupation is something that becomes “the business of one’s life”—that is to say, it becomes more than just a job. For those who succeed in architecture, the world is a different place. As they walk down the street they experience the world around them differently from others, because they see through the eyes of an architect. The same might be said for artists or engineers—they see the world as an artist or an engineer. However, we should also recognize that there are very job-like aspects to the practice of architecture. If one is to earn a living as an architect, there is a necessity to be down-to-earth and realistic about the way in which one approaches projects,
- 22. meets deadlines, and is aware of one’s clients’ needs as well as one’s own personal goals. This is often one of the greatest challenges for architects—to balance their occupation and their job. Christopher David George Nicholson, Design for a Studio for Augustus John, Fryern Court, Hampshire, UK, 1934 The “blueprint” is no longer the prevailing method of reproducing drawings. However, the heart of the activity of architecture has always constituted more than simply drawing. Architecture involves a lot more than drawing For many the image of an architect (besides perhaps the round glasses and the bow tie) is the person with the roll of blueprints under their arm, and this is closely allied to the notion that all architects “draw buildings.” However, although in many cases there is an element of truth to this, it belies the complexity of architectural practice. In a small practice the principal architect may be engaged in almost every aspect of a project, from initial sketches, through technical drawings, to site management. However, in larger practices, where specialization is more common, you will find some architects who do “draw” in the traditional sense. They may be designers, who spend most of their time sketching ideas, or they may be involved in specification writing; others may specialize in site administration and spend their time working closely with contractors and builders. Architecture is complex—it incorporates many processes and (in the case of large projects) can involve many people, each carrying out a specific part of the work.
- 23. Drawing is a part of this process—a very important part—but it is not the only thing that an architect does. Architecture is rewarding While it is the case that any person who enjoys their job will find it rewarding, architects are often passionate about what they do. For anyone who feels that having a well-designed environment is a vitally important factor in the way that individuals and groups of people feel, then architecture will certainly provide a sense of reward. For anyone who is excited by the prospect of combining creativity, philosophy, science, and engineering (and much more), then architecture can provide a career full of challenges. Architectural projects, as we have seen, can require many different skills and many types of specialist. This means that there are numerous opportunities for people to become involved in the profession in different ways. Whether you wish to be a designer, a model-maker, a technical drafting specialist, or a site administrator, you can be a part of a profession that is at once ancient and constantly renewing itself. Who is this book for? If you are moving toward a career in architecture there are many things you might wish to consider. Architecture is not a profession that you can easily “walk into” without some level of specialist education. Even to be employed in architecture as a technical draftsperson usually requires training in the specific use of CAD software and architectural/building training. In order to choose the right path toward the aspect of architecture that you wish to pursue, you should explore the subject widely. This book is intended to help you in that exploration. How to use this book The seven chapters of this book represent a journey from the first principles of architecture through to professional practice and possible future careers. Chapter 1 “The Setting” provides a grounding in the past and begins the discussion of how one becomes an architect. This includes a historical overview of the profession and a consideration of how theories have informed the practice of architecture. Chapter 2 “Education and Qualification” explores different forms of education and the issues of professional qualification. The chapter provides an overview of what is required to
- 24. become a qualified architect. Chapter 3 “The Client and The Brief” is an exploration of the relationship between architect and client, as well as the process of defining the parameters of an architectural project— from what a client wants to how much an architect might charge for services. Chapter 4 “From Brief to Project” looks at the way in which architects approach the design phases of a project once a brief has been defined. This includes a review of the steps that are taken in architectural projects and what factors influence the design and production processes, including the environment. Chapter 5 From initial design ideas through to construction and beyond, an architectural project is a complex process. “The Project and The Process” takes the reader through the stages of an architectural project following on from the design phase; these include drawing, specification, tender, and construction. Chapter 6 “The Practice” is a review of the different ways in which architects practice and the structure of the teams involved in projects and companies. Just as a project is a multifaceted process, so the relationships between members of the professional team are complex as well. This chapter explores the different types of practice as well as the range of professions involved in architecture. Chapter 7 “The Future” considers the way in which architects and architectural practice are changing to meet the needs of a changing environment. Whether considering the impact of architecture and building on the planet or the challenges facing the profession in meeting the changing needs of the population, architects must consider the way that the industry is addressing a dynamic world. Appendices “Glossary” is a collection of common terms used in architecture, design, and professional practice. “Further Reading” details books that provide further insight into the subjects tackled in each chapter, along with a list of magazines and periodicals that are often found in architects’ offices. “Online Resources” emphasizes websites of professional and personal interest to architects, including blogging and networking sites, while “Useful Contacts” provides a list of some of the most important professional bodies and architectural-education institutions. www.ebook3000.com
- 25. 1. The Setting
- 26. In this chapter we will take a look at different facets of architecture—exploring the broad sweep of the discipline, the architect through the ages, and the way in which contemporary practitioners look to the past as inspiration for the present. Architecture is a highly dynamic profession. Every architectural project responds, in some way, to cultural, historic, economic, and theoretical contexts. The balance between these will vary depending upon the specifics of the project, but architects need to consider these aspects carefully. When one looks at large, internationally recognized projects it is easy to see that these issues can become very important. Consider the design of an Olympic stadium: because of worldwide media exposure and the international importance placed upon the Olympics, the buildings created become national symbols for their host country. Add to this the fact that many cities use such major events as drivers for change in their social and urban fabric, and it can be seen that in such cases history, theory, economics, and culture come together in order to shape architectural expression. Ludwig Mies van der Rohe, S.R. Crown Hall, Illinois Institute of Technology, Chicago, USA, 1956 Architects look to history to inform the present and the future. However, the expression of historic precedent does not always lie in copying an image from the past. Architects may employ the underlying principles of order, proportion, and arrangement, which echo through history, in ways that might be felt rather than immediately seen. History Why we look to the past Architecture does not reside simply in the way in which we build or the manner in which buildings are designed. The buildings that we construct— and, thereby, the communities that we develop—are expressions of culture, with architecture as the medium through which those expressions are made manifest. As a system of symbols, and the rules that govern the manipulation of those symbols, architecture can be used to express a vast range of ideas.
- 27. Three Classical Orders (Doric, Ionic, Corinthian) The visual order of architecture sets the tone of the building. The Classical orders provided not just a coherent set of decorative motifs, but also defined a system of proportion and rules of usage for different building elements—essentially forming a visual “language.” Throughout history there have been architectural styles that were contemporary to their period, but there has also been a tendency to integrate past styles. This could stem from nostalgia or the architect’s desire to invoke specific aspects of the past. However, whether through overt reference to historical precedent (as with revival styles) or through more subtle allusions (for example, Classical proportioning or ordering), the past is constantly influencing, and is expressed in, the present. In the same way in which, when developing a new car, designers and engineers do not reinvent every aspect of the automobile, architects equally do not “start from scratch” with every project. There is always some reference to historical precedent— whether obvious or not. Theory While history plays a clear role in the way that we design and understand architecture, there is also a discourse of architecture. Architectural theory considers
- 28. the principles and concepts that form the basis on which architecture is thought about. At the most basic level, the consideration of order and proportion can be thought of as architectural theory. At a deeper level, theories relating to the way in which meaning is derived from architecture bring to bear a wide range of other theories and concepts, from philosophy to semiotics and linguistics. How does theory inform practice? One might think of architecture as a language (a system of signs, the use of which is governed by a set of rules—or grammar—which, in turn, allows us to read and understand), and building as being the writing of that language. The way in which we deploy that language will determine the meaning that is derived. It is often said that architectural theory is about “thinking a new architecture,” and that in thinking a new architecture we may move toward a position of being able to make a new architecture. So, practice can be influenced by the process of theorizing about architecture—we may come to new ways of expressing meaning through architecture by questioning the way in which we read the language of architecture. Material, Light, Poetry: Tadao Ando, Hombroich Museum, Langen Foundation, Neuss, Germany
- 29. Gallery block, Hombroich Museum, Langen Foundation. Materials play a crucial role in the way that we view and understand architecture. Throughout history the development of new materials and methods has allowed architects to achieve new designs that embody the spirit of their time. They also allow us to have different relationships to architecture. The use of brick, for example, often creates buildings that seem more human in scale, because of our ability to see the smaller module of the brick rather than a single continuous surface. The use of wooden siding on a rural retreat can create a sense of a building closely related to nature, particularly as the wood weathers to suggest the passage of time. The selection, specification, and use of materials in architecture is one of the most important aspects of a design. Concrete is an old material, the Romans being the best known of the ancients to use it, most notably in the dome of the Pantheon. Despite this, many people feel it affords little other than a cold, harsh, industrial feel. We often see concrete used primarily for its structural properties, in highway embankments, retaining walls, and foundations. But for some architects concrete becomes a material of expression and poetry. Tadao Ando has become known for his use of concrete in buildings ranging from churches to hotels. In most cases his use of the material is very simple: untreated, cast-in-place, and perfectly executed. But concrete is not the only material that Ando uses in his work. Often concrete combined with glass, timber, and (most importantly) light gives Ando’s work its beauty and poetry. In projects like the Church of the Light (1989) we see this combination of material and light in its most striking form. The church, located in Ibaraki (a suburb of Osaka, Japan), is modest in scale. From the exterior we see a plain concrete box; a steel cross gives the only indication of its religious nature. Inside, however, the relationship between concrete and light creates a space of quiet serenity. The precision with which the large cross, formed by a gap between four concrete wall panels, meets the joints between panels on the flank walls speaks volumes about the attention to detail Ando requires. We see how the material and immaterial come together to create something that transcends the physicality of the space. Church of the Light, Ibaraki, Osaka, Japan.
- 30. Entrance, Hombroich Museum, Langen Foundation. In his design for the Langen Foundation’s Hombroich Museum (2004) we see a poetic composition of form and material. The site, located in Neuss, near Dusseldorf, Germany, is a former NATO missile launch facility. The use of concrete would, initially, seem to be in keeping with the site’s history. However, in Ando’s hands, concrete, in conjunction with other materials, becomes a constituent of a new picturesque. The building initially appears, through a concrete arch, as a composition of glass, concrete, and water. This plays transparency and solidity against one another, as well as the real and the reflected. The building is divided into two sections. Apair of concrete structures partially submerged into the site (thus the 26-foot [8-meter] height can only be experienced inside) house the modern collections. Their solidity is contrasted by the Japanese gallery, which is a 140-foot-long (43-meter-long) concrete box enclosed in a steel- and-glass box. It is the Japanese gallery wing that gives the project its outstanding views and is representative of Ando’s use of material and light. The enclosing of the concrete gallery space within glass and steel serves to reduce the visual impact of the concrete within. Further, the play of shadows cast by the steel supports of the glass enclosure animates the surface of the concrete with a rhythm of shadows. While some architects become known for the daring of the forms they create, or the scale of projects they complete, Ando has become a recognized master of material and light.
- 31. View toward entrance, Hombroich Museum, Langen Foundation. Concrete and glass junction, Hombroich Museum, Langen Foundation. The profession—history and theory When considering the history of the profession of architecture we must recognize that the term “architect,” referring to someone who holds a specific position related to a specific set of activities in a professional capacity, has only been in use since the seventeenth century. Prior to this there was no profession of architecture, but simply
- 32. the making of architecture. In reality, the vast majority of building throughout history has not involved the architect as a professional—the individual we call “architect” was a craftsman or the leader of craftsmen. However, for clarity we will continue to refer to these individuals as architects—and we are thereby able to understand the relationship that the architects and their work had to broader society and culture in these periods. The Pyramid of Djoser (The Stepped Pyramid), Saqqara, Egypt, 2700–2600 BCE The role of architecture in ancient Egypt found its boldest expression in the funerary monuments of the ruling elite. In a culture steeped in religious ritual, the designer of the pyramids took on a role closely allied to that of a priest.
- 33. Bronze Statue of Imhotep, 332–330 BCE Often cited as the first architect, Imhotep was probably a functionary within the Egyptian governing body; his position as the designer of Djoser’s pyramid was probably only one of his roles. He was clearly a very important individual within this religious/political structure—so much so that he was later venerated as one of the many Egyptian gods. The priest and the pyramid The Egyptian pyramids are among the most well-known structures on the planet. Scientists have long pondered, studied, and proposed ways in which they might have been constructed. There is little actual evidence to suggest the building techniques employed in the erection of these monumental funerary buildings. While we do not know the dates of his birth or death, and much of the information available is indirect, it is held that Imhotep was the designer of the Pyramid of Djoser (also called the Stepped Pyramid) in Saqqara. Dating from the Third Dynasty (approx. 2600–2700 BCE), Imhotep is considered to be the first architect (as well as the first physician)— even if not in the sense that we know the profession today. We may assume that there was some status associated with his position. The pyramid was not simply a place to bury a pharaoh; it represented one of the most
- 34. important aspects of the life and afterlife of the most powerful individual in the kingdom. The burial of a pharaoh was a highly ritualized and profound process; those involved were of the highest standing, and the designer of the pyramid would have, of necessity, been held in high esteem by both the ruling and priestly castes. This may also be attested by the fact that Imhotep was given the honor of becoming a deity in his own right following his death. Sir Robert Smirke, The British Museum, London, UK, 1827 The influence of the Parthenon can be seen in many of the cultural, financial, and government buildings of the world’s major cities. There is little evidence that anyone beyond the Egyptian ruling elite employed professionals to design for them. This is not to say that all construction was undertaken in an ad hoc manner. It is likely that individuals were charged with the management of construction, and probably had some role in design, but this would most likely not have been a role that carried with it any particular social standing. Imhotep’s role would seem to have been relatively unusual for the period—the titles that he held (Chancellor of the King of Lower Egypt, High Priest of Heliopolis, Builder, Chief Carpenter, Chief Sculptor, and Maker of Vases in Chief) would suggest not only a professional, but a clearly noble status. The architect and the state During the period commonly referred to as Classical Antiquity, encompassing the height of the Greek (approximately 1000 BCE to 146 BCE) and Roman (approximately 753 BCE to 476 CE) empires, architects are still found in service to the ruling classes, and being commissioned to design and manage the erection of funerary structures as well as an expanding array of other building types. We have clear records regarding those credited with the design of some of the major buildings of ancient Greece. The Parthenon, the centerpiece of the Athenian Acropolis, is said to have been the product of a collaboration between Iktinos and Kallikrates. Initiated during a major building program undertaken by Pericles—the military leader, orator, and statesman of the Athenian city-state during the fifth
- 35. century BCE—the Acropolis became the cultural center of both the Athenian city-state and the broader Greek collection of city-states, the Delian League. It was to be the template for important public and private buildings for centuries to follow. The Parthenon, Athens, Greece, fifth century BCE Buildings such as the Parthenon have become templates; many important works have followed the form and language that was expressed in ancient Greece.
- 36. The Pantheon, Rome, Italy, 120–6 CE The architecture of ancient Rome retains iconic status not only because it remains standing but because it represents a bringing together of the language of architecture into a coherent work that has stood the test of time. Similarly we have some sense of those who were the figures of note in the development of Roman architecture. Much of what Roman architects achieved in
- 37. terms of design can be clearly traced to developments in Greece. There was an obvious sense in which the younger Roman Republic and Empire saw value in the appropriation and integration of long-established Greek styles. In part this was a political as well as a cultural endeavor. By taking up the styles and cultural trappings of Greece, Rome was suggesting that it too was a culture worthy of longevity and importance on the world stage. Leonardo da Vinci, The Vitruvian Man, c.1490 Vitruvius saw the proportions of architecture as being related to the proportions of the human body. His writings give us some of the first theories of architecture, and define the language of architecture. Marcus Pollio Vitruvius, best known simply as Vitruvius, is often cited as the first architectural theorist. In reality we can say only that he was the first Roman architect to leave extant writings on the discipline. His treatise De Architectura (“The Ten Books of Architecture”) is a wide-ranging collection of writings dating from around 27 BCE; some are specifically architectural in content; others cover subjects such as materials, plumbing, water, and more. While we know little of Vitruvius’ actual work as an architect, his writings set out clear principles for architecture. Set out in De Architectura, his notions of firmitas, utilitas, venustas (firmness, commodity, and beauty) were some of the earliest articulations of those things that must be present in order to constitute architecture (as opposed, say, to building). It was also Vitruvius’ position that architecture sought to imitate nature and strove for the expression of proportion and order. These, in turn, allowed for a reflection on the order and proportion of the human body. To explain this
- 38. relationship Vitruvius defined the order and proportion of the body through a series of relationships, with the body inscribed within a circle. The Pantheon in Rome is, again, one of the most iconic and recognizable buildings of the Classical period. It is a unique structure in its design, construction, and political context. The building is essentially a temple dedicated to the seven main gods of the Roman state religion. While it is by no means definite, the design of the Pantheon is credited to Apollodorus of Damascus. While individuals such as Apollodorus cannot be considered architects in the professional sense that we know, they were referred to as architects. This we know from the many funerary inscriptions that can be found throughout the former Roman Empire. That individuals were recognized at their death by reference to their occupation (“ARCHITECTVS”) suggests a change in the role that architecture and its practitioners played within society in general. Although no longer associated with the priestly class, the practice remained the preserve of the upper social orders. The master builder and the cathedral During the Middle Ages the power of the Roman Catholic Church and the rise in the monastic orders brought about a change to the role of architectural practitioners. There is little evidence of individuals called architects being involved in the design and construction of the great cathedrals of Europe, but there is a considerable amount of information that shows that the practice was active and well established. The growth of the Gothic style was as much a change in the conceptual relationship between mankind and God as it was between religion and building. It is difficult to identify clearly who conceived of this new form of architectural expression. It is generally held that Abbot Suger (a Cistercian monk) set out a new vision for the cathedral of St Denis in a treatise of the mid-twelfth century. In his Liber de rebus in administratione sua gestis (“Book of what was done under his administration”), he offered a theory that set out to define an architecture that was representative of a new notion of the relationship between humanity and God. While many of the features that were combined at St Denis—the pointed arch, ribbed vaulting, flying buttresses, and the ambulatory with radiating chapels—had been developed during the Romanesque period, by bringing them together with an overarching logic and relationship to a physical manifestation of the Heavenly Jerusalem, Suger defined a new theory of architecture.
- 39. Possible self-portrait of Villard de Honnecourt, c.1230 The Master Builders of the Gothic period were often itinerant workers, but their eclectic education and professional experience made them sought after by a range of potential patrons. Where we do have evidence of specific individuals associated with the design and building of the great cathedrals they are most often referred to as Master Builders. These were often people with a set of skills that allowed them to undertake the geometric, mathematical, and engineering tasks necessary to facilitate the cathedral’s construction. There is no evidence of any specific education, and it is likely that any such learning would have been via an apprenticeship—most often as a mason. Many were probably itinerant, traveling from town to town where cathedrals were being planned or were under construction, offering their services to the church or monastery concerned. It is difficult to assess the full role that such individuals would have played in the design and construction process—in part owing to the fact that there were no “sets of drawings” produced for the cathedrals. Much of their design and development seems to have happened on site. As this was a new form of building (both in concept and in construction), it would have been necessary to establish solutions to structural and construction issues as they arose. What makes the Gothic practitioner unique is the fact that theirs was a vocation that still grew from the process of making (stone-cutting or carving). There was still a very definite connection between the act of conceiving a design and that of making.
- 40. We might imagine that the Master Builder could, when walking through the building site, be equally as comfortable with hammer and chisel in hand as with pencil and rule. Patron and artist The “discovery” of a manuscript of Vitruvius’ treatise, De Architectura, in the library of the St Gall Monastery in 1415—part of the general reconsideration of “antiquity” known as the Renaissance—provided a clear indication of the qualities and image of Classical architecture, but also a definitive description of the role of an architect that bore little resemblance to that of the Master Builder. For Vitruvius, an architect must possess the ability to practice as well as theorize: Wherefore the mere practical architect is not able to assign sufficient reasons for the forms he adopts; and the theoretic architect also fails, grasping the shadow instead of the substance. He who is theoretic as well as practical, is therefore doubly armed; able not only to prove the propriety of his design, but equally so to carry it into execution. (Vitruvius, De Architectura, Book 2, Passage 2.)
- 41. Chartres Cathedral, France, 1194–1230 For many, the great cathedrals of Europe represent the pinnacle of architectural achievement. The designers of these fantastic structures—“Master Builders,” as they were known—would not be architects by our own modern definition. However, the complexity of the work that they developed and managed remains inspiring.
- 42. Filippo Brunelleschi, Foundling Hospital, Florence, Italy, 1419–45 There is no specific evidence of Brunelleschi’s transition from goldsmith to architect, but we do know that the commission for the Ospedale degli Innocenti (Foundling Hospital) in 1419 came to Brunelleschi from the same guild as that of his original apprenticeship. Based on Vitruvius’ definition it became possible, within the intellectual and social structures of the time, for the architect to be seen as on a par with other artists, and also as the possessor of a level of theoretical and Classical knowledge. The broader study of Classical Antiquity carried into architecture as well. Sketchbooks extant from a great many artists and architects show an appreciation of the visual detail and complexity of Classical form and ornament, but also a distinct study of the quantification of those forms. Many have detailed studies, with dimensions, of architectural forms and details. There is, then, a sense in which new “pattern books” were being developed, but from a theoretical as well as an aesthetic position. During the Renaissance, the architect was often also an artist. Apprentices would receive educations in the humanities as well as the arts, augmenting basic reading, writing, and mathematical studies. We should remember that in the fifteenth century most artists would have been seen as craftsmen—it was only the master artist who had social standing of note. Filippo Brunelleschi is a good example of the artist’s transition to architect. We have solid evidence of Brunelleschi’s work, both by his own hand and through various biographies. He trained as a goldsmith, becoming a master in the Silkmakers’ Guild (which included metalsmiths) in about 1398. In Brunelleschi we see an individual who was possessed of a certain amount of practical knowledge (although not directly linked to the building process) coupled with a level of theoretical knowledge. We may assume that, beyond a natural talent, his training in the arts and humanities gave Brunelleschi the ability to lend his hand to architecture. His study of Classical architecture gave him the basis for his later reinterpretation of Classical form in his own projects. Thus the architect is now
- 43. engaged in a practical pursuit, but from a position of scholarly study of geometry, mathematics, and aesthetics—theory and practice are beginning to come together. Another change was in the relationship between the architect and the commissioning body, whether this was an organization or individual. The Renaissance saw a rise in secular bodies (both in terms of social classes and commercial enterprises) who were in a financial position to employ artists, designers, and architects. The role of the architect, from being one largely allied to the ruling elite or religious orders, therefore expanded to include private clients. There was also a formalization of the role of the architect: by the middle of the sixteenth century—and particularly in northern Italy—treatises began to appear that articulated both the development of the architect as a profession and also a reasoning of the difference between the architect and the master mason or other skilled craftsman. The sixteenth-century French architect Philibert Delorme wrote that clients should hire an architect because other practitioners were schooled only in the art of manual labor, whereas the architect was possessed of both theory and practice. To rely upon other craftsmen to design a pleasing building would result only in “a shadow of a real building.” Delorme also conceived of a self-regulating profession with standards of education, responsibility, and practice—foreshadowing the professional arrangements found in architecture today. Perhaps the most marked change in the nature of architectural practice during this period was the shift toward design as a service in itself. Some architects became so busy with the process of designing for a large number of clients and projects that they were little, if at all, involved in the actual building process. This stands in marked contrast to the earlier periods, in which the practice was clearly weighted toward those who were closely engaged in the building process. The architect By the latter half of the sixteenth century we begin to see practitioners solely engaged in architecture as a design discipline. While it was still the case that many apprenticed within a traditional trade, the bulk of their career now lay in designing, drawing, and managing the process of building, rather than being directly engaged in construction.
- 44. Andrea Palladio, Villa Almerico-Capra (“Villa Rotunda”), Vicenza, Italy, 1591 Trained specifically as an architect, Palladio became influential in his own time as well as down the ages. His villas, built for the wealthy elite of Venice and intended as a reflection of the taste and intelligence of his clients, became a new model for architects’ relationship to their patron.
- 45. Andrea Palladio, I Quattro Libri dell’Architettura (“The Four Books of Architecture”), 1570 As well as being instrumental in changing the relationship between architect and client, Palladio continued to develop new architectural theories. His Four Books of Architecture offered a set of rules that defined the parameters for architecture. Andrea di Pietro della Gondola, or Palladio, was born in Padua in 1508, and later apprenticed as a stonemason until he ran away to continue his studies in Vicenza. It was not until his mid-thirties that he was “discovered” by Count Gian Giorgio Trissino, who sent his protégé to Rome to study geometry, proportion, Vitruvius, and the Roman monuments. In essence Palladio (despite a background in the building craft) was educated, specifically, to be an architect. Palladio went on to become one of the most influential architects. His work, during a career that lasted 40 years, came to represent a new direction in the fledgling profession. His success (although he was never a wealthy man) was due to the fact that his designs resonated with the social aspirations of his clients. His designs for rural villas in northern Italy combined a reinterpretation of Roman Classical tradition with new forms and layouts. By having a villa design that not only referenced the Classical but also used it as a new expressive language, Palladio was highlighting the intelligence and quality of his patrons. Palladio’s I Quattro Libri dell’Architettura (“The Four Books of Architecture”), of 1570, set out nine rule-sets that defined the principles and regulations upon which architecture should be based. In some cases these rule-sets are based on the construction process, while others are grounded in geometry, shape, and proportion. Through this combination of rules, building upon those set out by Vitruvius, Palladio set out to create a clear and coherent “grammar” for architecture. Inigo Jones, the first architect in England to have followed the late-Renaissance model of an education directed specifically toward architecture, became Surveyor of
- 46. the King’s Works in 1613. Through the Office of Works, architects such as Jones, Christopher Wren, Robert Adam, and many more were engaged in the design and construction of many of the most prominent buildings in England during the seventeenth and eighteenth centuries. John Smith after Sir Godfrey Kneller, Sir Christopher Wren, 1711–13 Although there were many who were being trained specifically as architects, the tradition continued of the “gentleman architect.” Christopher Wren, for instance, as well as being an architect, was a geometer, astronomer, and member of the Royal Society.
- 47. Sir Christopher Wren, St Paul’s Cathedral, London, UK, 1711 Although only one of a total of 53 churches designed by Wren, St Paul’s is considered to be the greatest of his achievements and still dominates London’s urban landscape.
- 48. Inigo Jones, Queen’s House, Greenwich, London, UK, 1616 While the relationship between architect and client continued to evolve through the seventeenth and eighteenth centuries, the expression of an architectural language continued to rely upon reference to a Classical tradition.
- 49. William Hogarth, Inigo Jones, 1757–8 The Office of Works, part of the Royal Household in England, became a hothouse for the development of architects. As one of the first places to develop a systematic program of education specifically for architects, it was to be the proving ground for Inigo Jones and for many others. From the eighteenth century onward there was a continued growth in the importance of the professional architect, and also a continued move away from the patronage model. With the increase in prosperity, based on the growth of industrial production, the middle class was becoming more affluent, and there was a greater demand for design and building services to support the expression of this in their business and domestic environments. Architects were now in greater competition with each other; therefore, the role of the architect developed to include a more business- oriented view of practice. Client and consultant Our modern conception of the professional architect, although a development from as early as the fifteenth century, is largely defined by changes during the nineteenth century. From the eighteenth century, as the profession became more distinct in its scope and practices, there had been attempts to establish organizations to protect the interests of practitioners, improve their standing, and develop a coherent and formal educational canon. In France we see the Académie de l’Architecture (formed in 1671 by Jean-Baptiste Colbert) and the later atelier system within the École des
- 50. Arts (1743), both of which served to formalize architects’ education. Similar developments were found in Rome. In England, however, it remained largely a process of apprenticeship within an established office. Some joined the Society of Artists, set up in 1761, and later the Royal Academy, but architects’ membership within these societies was relatively low and had more to do with social standing than a concerted effort to formalize the profession. The Institute of British Architects was formed in 1834, for “promoting and facilitating the acquirement of the knowledge of the various arts and sciences connected therewith; it being an art esteemed and encouraged in all enlightened nations, as tending greatly to promote the domestic convenience of citizens, and the public improvement and embellishment of towns and cities ....” In the United States the American Institute of Architects (AIA) was formed in 1857 along much the same lines. Giovanni Battista Piranesi, Plate IX of I Carceri (“Prisons”), c.1749–60 The formalizing of the profession of architecture created opportunities for architecture to be considered in a broader context. The Classical Revival styles also opened up possibilities for considering the nature of the language of architecture, and how it might allow alternative views of Classical tradition.
- 51. Étienne-Louis Boullée, Cénotaphe à Newton (“Monument to Newton”), 1784 Architecture has a long tradition of ambitious, unbuilt projects. Such theoretical works offer ways in which practitioners can think of and explore architectures that defy contemporary reality. We should not, however, assume that architecture was thus established as a coherent profession. In fact, the membership of these organizations represented only a small fraction of those practicing within the industry. The need for such organizations was seen as increasingly important, as the complexity of projects and the relationships between client and architect, and architect and builder, became more specialized. Under such circumstances the role of the architect carried increasing levels of responsibility. Sir John Soane, the first President of the Institute of British Architects, said in 1788: The business of the architect is to make the designs and estimates, to direct the works, and to measure and value the different parts; he is the intermediate agent between the employer, whose honor and interest he is to study, and the mechanic, whose rights he is to defend. His situation implies great trust; he is responsible for the mistakes, negligences, and ignorances of those he employs; and above all, he is to take care that the workmen’s bills do not exceed his own estimates. Where the membership of the nascent professional organizations had often been dependent upon the amount of time one had been in practice, or on the payment of a fee, it is only toward the close of the nineteenth century that we begin to see examination become a requirement. And it would be only in the twentieth century that legislation was enacted formally to define the architectural profession. In 1931 legislation was introduced in Britain resulting in a closed profession, with the title of
- 52. “Architect” conferred upon those who had achieved specific qualification through education and professional practice. Sir John Soane, Breakfast Room, Soane Museum, London, UK, 1824
- 53. As the first President of the Institute of British Architects Sir John Soane presided over the early stages of moves toward a defined profession of “architect,” as well as designing the inventive townhouse that was to become his own museum. In the United States the formalization of qualification and professional standing came somewhat earlier. Programs of education directed specifically toward architecture were more widespread than in Europe, and by the end of the nineteenth century there were a number of large institutions with established architecture programs. Based on the French École des Beaux-Arts tradition, these were to become the model for architectural education for much of the twentieth century. The first laws regulating the profession were established in Illinois in 1897. However, it was not until 1951 that all 50 states had achieved licensing laws for architects. In many countries there is now a rigorous process by which professional qualification is achieved, which can take a number of years of specialized study followed by a period of directed professional practice. Kelmscott Manor (home of William Morris), Lechdale, Gloucestershire, UK, c.1570s For Morris, and the Arts and Crafts Movement, the Classical Revivalist aesthetic and theories did not reflect the value of the handmade object. In architecture, arts, literature, and more, this move away from Classical ideals represented the first step toward Modernism.
- 54. Frank Lloyd Wright, Kaufman Residence (Falling Water), Bear Run, Pennsylvania, USA, 1935 Prior to the establishment of legislation related to architecture, anyone could call themselves an architect. For some famous architects, such as Frank Lloyd Wright, time spent as an apprentice in the offices of established architects offered both training and the skill to act as an architect. As legislation came into force it was no longer possible to simply “act like an architect.” In the United States, by the middle of the twentieth century, many states had enshrined the role of the architect in law. This protected the term “architect” and clearly identified those who practiced as specialists in their field. Contemporary practice—contemporary theory The role of the architect in the professional sphere continues to grow and diversify. In many cases an architect may be part of a large multidisciplinary team. Different modes of practice parallel a diversification of the theories that affect architectural design. New ways of working and new ways of thinking about architecture often go hand in hand. The upheaval following the First World War led many architects to see a new role for architecture and the expression of society and culture through their work—a new social order was now possible, and the idea was that architecture should express a new relationship among people that was no longer about aristocracy and class, but about equality and democracy. For others, new materials and technologies, mass production, and economic realities meant that there was a new range of possibilities for designers to exploit. It is likely that it was a combination of these and other factors that led to the rise of Modernism.
- 55. Other documents randomly have different content
- 56. two conditions together are defined as giving the meaning of "the author of Waverly exists." We may now define "the term satisfying the function exists." This is the general form of which the above is a particular case. "The author of Waverly" is "the term satisfying the function ' wrote Waverly.'" And "the so-and-so" will always involve reference to some propositional function, namely, that which defines the property that makes a thing a so-and-so. Our definition is as follows:— "The term satisfying the function exists" means: "There is a term such that is always equivalent to ' is .'" In order to define "the author of Waverly was Scotch," we have still to take account of the third of our three propositions, namely, "Whoever wrote Waverly was Scotch." This will be satisfied by merely adding that the in question is to be Scotch. Thus "the author of Waverly was Scotch" is: "There is a term such that (1) ' wrote Waverly' is always equivalent to ' is ,' (2) is Scotch." And generally: "the term satisfying satisfies " is defined as meaning: "There is a term such that (1) is always equivalent to ' is ,' (2) is true." This is the definition of propositions in which descriptions occur. It is possible to have much knowledge concerning a term described, i.e. to know many propositions concerning "the so-and-so," without actually knowing what the so-and-so is, i.e. without knowing any proposition of the form " is the so-and-so," where " " is a name. In a detective story propositions about "the man who did the deed" are accumulated, in the hope that ultimately they will suffice to demonstrate that it was who did the deed. We may even go so far as to say that, in all such knowledge as can be expressed in words—with the exception of "this" and "that" and a few other words of which the meaning varies on different occasions—no names, in the strict sense, occur, but what seem like names are really descriptions. We may inquire significantly whether Homer existed, which we could not do if "Homer" were a name. The proposition "the so-and-so exists" is significant, whether true or false; but if is the so-and-so (where " " is a name), the
- 57. words " exists" are meaningless. It is only of descriptions—definite or indefinite—that existence can be significantly asserted; for, if " " is a name, it must name something: what does not name anything is not a name, and therefore, if intended to be a name, is a symbol devoid of meaning, whereas a description, like "the present King of France," does not become incapable of occurring significantly merely on the ground that it describes nothing, the reason being that it is a complex symbol, of which the meaning is derived from that of its constituent symbols. And so, when we ask whether Homer existed, we are using the word "Homer" as an abbreviated description: we may replace it by (say) "the author of the Iliad and the Odyssey." The same considerations apply to almost all uses of what look like proper names. When descriptions occur in propositions, it is necessary to distinguish what may be called "primary" and "secondary" occurrences. The abstract distinction is as follows. A description has a "primary" occurrence when the proposition in which it occurs results from substituting the description for " " in some propositional function ; a description has a "secondary" occurrence when the result of substituting the description for in gives only part of the proposition concerned. An instance will make this clearer. Consider "the present King of France is bald." Here "the present King of France" has a primary occurrence, and the proposition is false. Every proposition in which a description which describes nothing has a primary occurrence is false. But now consider "the present King of France is not bald." This is ambiguous. If we are first to take " is bald," then substitute "the present King of France" for " " and then deny the result, the occurrence of "the present King of France" is secondary and our proposition is true; but if we are to take " is not bald" and substitute "the present King of France" for " " then "the present King of France" has a primary occurrence and the proposition is false. Confusion of primary and secondary occurrences is a ready source of fallacies where descriptions are concerned. Descriptions occur in mathematics chiefly in the form of descriptive functions, i.e. "the term having the relation to ," or "the of " as we may say, on the analogy of "the father of " and similar phrases. To say "the father of is rich," for example, is to say that the following propositional function of : " is rich, and ' begat ' is always equivalent to ' is ,'" is
- 58. "sometimes true," i.e. is true for at least one value of . It obviously cannot be true for more than one value. The theory of descriptions, briefly outlined in the present chapter, is of the utmost importance both in logic and in theory of knowledge. But for purposes of mathematics, the more philosophical parts of the theory are not essential, and have therefore been omitted in the above account, which has confined itself to the barest mathematical requisites.
- 59. CHAPTER XVII CLASSES IN the present chapter we shall be concerned with the in the plural: the inhabitants of London, the sons of rich men, and so on. In other words, we shall be concerned with classes. We saw in Chapter II. that a cardinal number is to be defined as a class of classes, and in Chapter III. that the number 1 is to be defined as the class of all unit classes, i.e. of all that have just one member, as we should say but for the vicious circle. Of course, when the number 1 is defined as the class of all unit classes, "unit classes" must be defined so as not to assume that we know what is meant by "one"; in fact, they are defined in a way closely analogous to that used for descriptions, namely: A class is said to be a "unit" class if the propositional function "' is an ' is always equivalent to ' is '" (regarded as a function of ) is not always false, i.e., in more ordinary language, if there is a term such that will be a member of when is but not otherwise. This gives us a definition of a unit class if we already know what a class is in general. Hitherto we have, in dealing with arithmetic, treated "class" as a primitive idea. But, for the reasons set forth in Chapter XIII., if for no others, we cannot accept "class" as a primitive idea. We must seek a definition on the same lines as the definition of descriptions, i.e. a definition which will assign a meaning to propositions in whose verbal or symbolic expression words or symbols apparently representing classes occur, but which will assign a meaning that altogether eliminates all mention of classes from a right analysis of such propositions. We shall then be able to say that the symbols for classes are mere conveniences, not representing objects called "classes," and that classes are in fact, like descriptions, logical fictions, or (as we say) "incomplete symbols."
- 60. The theory of classes is less complete than the theory of descriptions, and there are reasons (which we shall give in outline) for regarding the definition of classes that will be suggested as not finally satisfactory. Some further subtlety appears to be required; but the reasons for regarding the definition which will be offered as being approximately correct and on the right lines are overwhelming. The first thing is to realise why classes cannot be regarded as part of the ultimate furniture of the world. It is difficult to explain precisely what one means by this statement, but one consequence which it implies may be used to elucidate its meaning. If we had a complete symbolic language, with a definition for everything definable, and an undefined symbol for everything indefinable, the undefined symbols in this language would represent symbolically what I mean by "the ultimate furniture of the world." I am maintaining that no symbols either for "class" in general or for particular classes would be included in this apparatus of undefined symbols. On the other hand, all the particular things there are in the world would have to have names which would be included among undefined symbols. We might try to avoid this conclusion by the use of descriptions. Take (say) "the last thing Cæsar saw before he died." This is a description of some particular; we might use it as (in one perfectly legitimate sense) a definition of that particular. But if " " is a name for the same particular, a proposition in which " " occurs is not (as we saw in the preceding chapter) identical with what this proposition becomes when for " " we substitute "the last thing Cæsar saw before he died." If our language does not contain the name " " or some other name for the same particular, we shall have no means of expressing the proposition which we expressed by means of " " as opposed to the one that we expressed by means of the description. Thus descriptions would not enable a perfect language to dispense with names for all particulars. In this respect, we are maintaining, classes differ from particulars, and need not be represented by undefined symbols. Our first business is to give the reasons for this opinion. We have already seen that classes cannot be regarded as a species of individuals, on account of the contradiction about classes which are not members of themselves (explained in Chapter XIII.), and because we can prove that the number of classes is greater than the number of individuals.
- 61. We cannot take classes in the pure extensional way as simply heaps or conglomerations. If we were to attempt to do that, we should find it impossible to understand how there can be such a class as the null-class, which has no members at all and cannot be regarded as a "heap"; we should also find it very hard to understand how it comes about that a class which has only one member is not identical with that one member. I do not mean to assert, or to deny, that there are such entities as "heaps." As a mathematical logician, I am not called upon to have an opinion on this point. All that I am maintaining is that, if there are such things as heaps, we cannot identify them with the classes composed of their constituents. We shall come much nearer to a satisfactory theory if we try to identify classes with propositional functions. Every class, as we explained in Chapter II., is defined by some propositional function which is true of the members of the class and false of other things. But if a class can be defined by one propositional function, it can equally well be defined by any other which is true whenever the first is true and false whenever the first is false. For this reason the class cannot be identified with any one such propositional function rather than with any other—and given a propositional function, there are always many others which are true when it is true and false when it is false. We say that two propositional functions are "formally equivalent" when this happens. Two propositions are "equivalent" when both are true or both false; two propositional functions , are "formally equivalent" when is always equivalent to . It is the fact that there are other functions formally equivalent to a given function that makes it impossible to identify a class with a function; for we wish classes to be such that no two distinct classes have exactly the same members, and therefore two formally equivalent functions will have to determine the same class. When we have decided that classes cannot be things of the same sort as their members, that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than symbolic fictions. And if we can find any way of dealing with them as symbolic fictions, we increase the logical security of our position, since we avoid the need of assuming that there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from
- 62. both assumptions. This is an example of Occam's razor, namely, "entities are not to be multiplied without necessity." But when we refuse to assert that there are classes, we must not be supposed to be asserting dogmatically that there are none. We are merely agnostic as regards them: like Laplace, we can say, "je n'ai pas besoin de cette hypothèse." Let us set forth the conditions that a symbol must fulfil if it is to serve as a class. I think the following conditions will be found necessary and sufficient:— (1) Every propositional function must determine a class, consisting of those arguments for which the function is true. Given any proposition (true or false), say about Socrates, we can imagine Socrates replaced by Plato or Aristotle or a gorilla or the man in the moon or any other individual in the world. In general, some of these substitutions will give a true proposition and some a false one. The class determined will consist of all those substitutions that give a true one. Of course, we have still to decide what we mean by "all those which, etc." All that we are observing at present is that a class is rendered determinate by a propositional function, and that every propositional function determines an appropriate class. (2) Two formally equivalent propositional functions must determine the same class, and two which are not formally equivalent must determine different classes. That is, a class is determined by its membership, and no two different classes can have the same membership. (If a class is determined by a function , we say that is a "member" of the class if is true.) (3) We must find some way of defining not only classes, but classes of classes. We saw in Chapter II. that cardinal numbers are to be defined as classes of classes. The ordinary phrase of elementary mathematics, "The combinations of things at a time" represents a class of classes, namely, the class of all classes of terms that can be selected out of a given class of terms. Without some symbolic method of dealing with classes of classes, mathematical logic would break down. (4) It must under all circumstances be meaningless (not false) to suppose a class a member of itself or not a member of itself. This results from the contradiction which we discussed in Chapter XIII.
- 63. (5) Lastly—and this is the condition which is most difficult of fulfilment, —it must be possible to make propositions about all the classes that are composed of individuals, or about all the classes that are composed of objects of any one logical "type." If this were not the case, many uses of classes would go astray—for example, mathematical induction. In defining the posterity of a given term, we need to be able to say that a member of the posterity belongs to all hereditary classes to which the given term belongs, and this requires the sort of totality that is in question. The reason there is a difficulty about this condition is that it can be proved to be impossible to speak of all the propositional functions that can have arguments of a given type. We will, to begin with, ignore this last condition and the problems which it raises. The first two conditions may be taken together. They state that there is to be one class, no more and no less, for each group of formally equivalent propositional functions; e.g. the class of men is to be the same as that of featherless bipeds or rational animals or Yahoos or whatever other characteristic may be preferred for defining a human being. Now, when we say that two formally equivalent propositional functions may be not identical, although they define the same class, we may prove the truth of the assertion by pointing out that a statement may be true of the one function and false of the other; e.g. "I believe that all men are mortal" may be true, while "I believe that all rational animals are mortal" may be false, since I may believe falsely that the Phoenix is an immortal rational animal. Thus we are led to consider statements about functions, or (more correctly) functions of functions. Some of the things that may be said about a function may be regarded as said about the class defined by the function, whereas others cannot. The statement "all men are mortal" involves the functions " is human" and " is mortal"; or, if we choose, we can say that it involves the classes men and mortals. We can interpret the statement in either way, because its truth- value is unchanged if we substitute for " is human" or for " is mortal" any formally equivalent function. But, as we have just seen, the statement "I believe that all men are mortal" cannot be regarded as being about the class determined by either function, because its truth-value may be changed by the substitution of a formally equivalent function (which leaves the class unchanged). We will call a statement involving a function an
- 64. "extensional" function of the function , if it is like "all men are mortal," i.e. if its truth-value is unchanged by the substitution of any formally equivalent function; and when a function of a function is not extensional, we will call it "intensional," so that "I believe that all men are mortal" is an intensional function of " is human" or " is mortal." Thus extensional functions of a function may, for practical purposes, be regarded as functions of the class determined by , while intensional functions cannot be so regarded. It is to be observed that all the specific functions of functions that we have occasion to introduce in mathematical logic are extensional. Thus, for example, the two fundamental functions of functions are: " is always true" and " is sometimes true." Each of these has its truth-value unchanged if any formally equivalent function is substituted for . In the language of classes, if is the class determined by , " is always true" is equivalent to "everything is a member of ," and " is sometimes true" is equivalent to " has members" or (better) " has at least one member." Take, again, the condition, dealt with in the preceding chapter, for the existence of "the term satisfying ." The condition is that there is a term such that is always equivalent to " is ." This is obviously extensional. It is equivalent to the assertion that the class defined by the function is a unit class, i.e. a class having one member; in other words, a class which is a member of 1. Given a function of a function which may or may not be extensional, we can always derive from it a connected and certainly extensional function of the same function, by the following plan: Let our original function of a function be one which attributes to the property ; then consider the assertion "there is a function having the property and formally equivalent to ." This is an extensional function of ; it is true when our original statement is true, and it is formally equivalent to the original function of if this original function is extensional; but when the original function is intensional, the new one is more often true than the old one. For example, consider again "I believe that all men are mortal," regarded as a function of " is human." The derived extensional function is: "There is a function formally equivalent to ' is human' and such that I believe that whatever satisfies it is mortal." This remains true when we substitute " is a rational
- 65. animal" for " is human," even if I believe falsely that the Phoenix is rational and immortal. We give the name of "derived extensional function" to the function constructed as above, namely, to the function: "There is a function having the property and formally equivalent to ," where the original function was "the function has the property ." We may regard the derived extensional function as having for its argument the class determined by the function , and as asserting of this class. This may be taken as the definition of a proposition about a class. I.e. we may define: To assert that "the class determined by the function has the property " is to assert that satisfies the extensional function derived from . This gives a meaning to any statement about a class which can be made significantly about a function; and it will be found that technically it yields the results which are required in order to make a theory symbolically satisfactory.[41] [41]See Principia Mathematica, vol. I. pp. 75-84 and * 20. What we have said just now as regards the definition of classes is sufficient to satisfy our first four conditions. The way in which it secures the third and fourth, namely, the possibility of classes of classes, and the impossibility of a class being or not being a member of itself, is somewhat technical; it is explained in Principia Mathematica, but may be taken for granted here. It results that, but for our fifth condition, we might regard our task as completed. But this condition—at once the most important and the most difficult—is not fulfilled in virtue of anything we have said as yet. The difficulty is connected with the theory of types, and must be briefly discussed.[42] [42]The reader who desires a fuller discussion should consult Principia Mathematica, Introduction, chap. II.; also * 12. We saw in Chapter XIII. that there is a hierarchy of logical types, and that it is a fallacy to allow an object belonging to one of these to be substituted for an object belonging to another. Now it is not difficult to show that the various functions which can take a given object as argument are not all of one type. Let us call them all -functions. We may take first
- 66. those among them which do not involve reference to any collection of functions; these we will call "predicative -functions." If we now proceed to functions involving reference to the totality of predicative -functions, we shall incur a fallacy if we regard these as of the same type as the predicative -functions. Take such an everyday statement as " is a typical Frenchman." How shall we define a "typical" Frenchman? We may define him as one "possessing all qualities that are possessed by most French men." But unless we confine "all qualities" to such as do not involve a reference to any totality of qualities, we shall have to observe that most Frenchmen are not typical in the above sense, and therefore the definition shows that to be not typical is essential to a typical Frenchman. This is not a logical contradiction, since there is no reason why there should be any typical Frenchmen; but it illustrates the need for separating off qualities that involve reference to a totality of qualities from those that do not. Whenever, by statements about "all" or "some" of the values that a variable can significantly take, we generate a new object, this new object must not be among the values which our previous variable could take, since, if it were, the totality of values over which the variable could range would only be definable in terms of itself, and we should be involved in a vicious circle. For example, if I say "Napoleon had all the qualities that make a great general," I must define "qualities" in such a way that it will not include what I am now saying, i.e. "having all the qualities that make a great general" must not be itself a quality in the sense supposed. This is fairly obvious, and is the principle which leads to the theory of types by which vicious-circle paradoxes are avoided. As applied to -functions, we may suppose that "qualities" is to mean "predicative functions." Then when I say "Napoleon had all the qualities, etc.," I mean "Napoleon satisfied all the predicative functions, etc." This statement attributes a property to Napoleon, but not a predicative property; thus we escape the vicious circle. But wherever "all functions which" occurs, the functions in question must be limited to one type if a vicious circle is to be avoided; and, as Napoleon and the typical Frenchman have shown, the type is not rendered determinate by that of the argument. It would require a much fuller discussion to set forth this point fully, but what has been said may suffice to make it clear that the functions which can take a given argument are of an infinite series of types. We could, by various technical devices, construct a variable which would run through the first of these types, where is finite, but we
- 67. cannot construct a variable which will run through them all, and, if we could, that mere fact would at once generate a new type of function with the same arguments, and would set the whole process going again. We call predicative -functions the first type of -functions; -functions involving reference to the totality of the first type we call the second type; and so on. No variable -function can run through all these different types: it must stop short at some definite one. These considerations are relevant to our definition of the derived extensional function. We there spoke of "a function formally equivalent to ." It is necessary to decide upon the type of our function. Any decision will do, but some decision is unavoidable. Let us call the supposed formally equivalent function . Then appears as a variable, and must be of some determinate type. All that we know necessarily about the type of is that it takes arguments of a given type—that it is (say) an -function. But this, as we have just seen, does not determine its type. If we are to be able (as our fifth requisite demands) to deal with all classes whose members are of the same type as , we must be able to define all such classes by means of functions of some one type; that is to say, there must be some type of - function, say the , such that any -function is formally equivalent to some -function of the type. If this is the case, then any extensional function which holds of all -functions of the type will hold of any - function whatever. It is chiefly as a technical means of embodying an assumption leading to this result that classes are useful. The assumption is called the "axiom of reducibility," and may be stated as follows:— "There is a type ( say) of -functions such that, given any -function, it is formally equivalent to some function of the type in question." If this axiom is assumed, we use functions of this type in defining our associated extensional function. Statements about all -classes (i.e. all classes defined by -functions) can be reduced to statements about all - functions of the type . So long as only extensional functions of functions are involved, this gives us in practice results which would otherwise have required the impossible notion of "all -functions." One particular region where this is vital is mathematical induction. The axiom of reducibility involves all that is really essential in the theory of classes. It is therefore worth while to ask whether there is any
- 68. reason to suppose it true. This axiom, like the multiplicative axiom and the axiom of infinity, is necessary for certain results, but not for the bare existence of deductive reasoning. The theory of deduction, as explained in Chapter XIV., and the laws for propositions involving "all" and "some," are of the very texture of mathematical reasoning: without them, or something like them, we should not merely not obtain the same results, but we should not obtain any results at all. We cannot use them as hypotheses, and deduce hypothetical consequences, for they are rules of deduction as well as premisses. They must be absolutely true, or else what we deduce according to them does not even follow from the premisses. On the other hand, the axiom of reducibility, like our two previous mathematical axioms, could perfectly well be stated as an hypothesis whenever it is used, instead of being assumed to be actually true. We can deduce its consequences hypothetically; we can also deduce the consequences of supposing it false. It is therefore only convenient, not necessary. And in view of the complication of the theory of types, and of the uncertainty of all except its most general principles, it is impossible as yet to say whether there may not be some way of dispensing with the axiom of reducibility altogether. However, assuming the correctness of the theory outlined above, what can we say as to the truth or falsehood of the axiom? The axiom, we may observe, is a generalised form of Leibniz's identity of indiscernibles. Leibniz assumed, as a logical principle, that two different subjects must differ as to predicates. Now predicates are only some among what we called "predicative functions," which will include also relations to given terms, and various properties not to be reckoned as predicates. Thus Leibniz's assumption is a much stricter and narrower one than ours. (Not, of course, according to his logic, which regarded all propositions as reducible to the subject-predicate form.) But there is no good reason for believing his form, so far as I can see. There might quite well, as a matter of abstract logical possibility, be two things which had exactly the same predicates, in the narrow sense in which we have been using the word "predicate." How does our axiom look when we pass beyond predicates in this narrow sense? In the actual world there seems no way of doubting its empirical truth as regards particulars, owing to spatio-temporal differentiation: no two particulars have exactly the same spatial and temporal relations to all other
- 69. particulars. But this is, as it were, an accident, a fact about the world in which we happen to find ourselves. Pure logic, and pure mathematics (which is the same thing), aims at being true, in Leibnizian phraseology, in all possible worlds, not only in this higgledy-piggledy job-lot of a world in which chance has imprisoned us. There is a certain lordliness which the logician should preserve: he must not condescend to derive arguments from the things he sees about him. Viewed from this strictly logical point of view, I do not see any reason to believe that the axiom of reducibility is logically necessary, which is what would be meant by saying that it is true in all possible worlds. The admission of this axiom into a system of logic is therefore a defect, even if the axiom is empirically true. It is for this reason that the theory of classes cannot be regarded as being as complete as the theory of descriptions. There is need of further work on the theory of types, in the hope of arriving at a doctrine of classes which does not require such a dubious assumption. But it is reasonable to regard the theory outlined in the present chapter as right in its main lines, i.e. in its reduction of propositions nominally about classes to propositions about their defining functions. The avoidance of classes as entities by this method must, it would seem, be sound in principle, however the detail may still require adjustment. It is because this seems indubitable that we have included the theory of classes, in spite of our desire to exclude, as far as possible, whatever seemed open to serious doubt. The theory of classes, as above outlined, reduces itself to one axiom and one definition. For the sake of definiteness, we will here repeat them. The axiom is: There is a type such that if is a function which can take a given object as argument, then there is a function of the type which is formally equivalent to . The definition is: If is a function which can take a given object as argument, and the type mentioned in the above axiom, then to say that the class determined by has the property is to say that there is a function of type , formally equivalent to , and having the property .
- 71. CHAPTER XVIII MATHEMATICS AND LOGIC MATHEMATICS and logic, historically speaking, have been entirely distinct studies. Mathematics has been connected with science, logic with Greek. But both have developed in modern times: logic has become more mathematical and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one. They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic. This view is resented by logicians who, having spent their time in the study of classical texts, are incapable of following a piece of symbolic reasoning, and by mathematicians who have learnt a technique without troubling to inquire into its meaning or justification. Both types are now fortunately growing rarer. So much of modern mathematical work is obviously on the border- line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premisses which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary.
- 72. In the earlier chapters of this book, starting from the natural numbers, we have first defined "cardinal number" and shown how to generalise the conception of number, and have then analysed the conceptions involved in the definition, until we found ourselves dealing with the fundamentals of logic. In a synthetic, deductive treatment these fundamentals come first, and the natural numbers are only reached after a long journey. Such treatment, though formally more correct than that which we have adopted, is more difficult for the reader, because the ultimate logical concepts and propositions with which it starts are remote and unfamiliar as compared with the natural numbers. Also they represent the present frontier of knowledge, beyond which is the still unknown; and the dominion of knowledge over them is not as yet very secure. It used to be said that mathematics is the science of "quantity." "Quantity" is a vague word, but for the sake of argument we may replace it by the word "number." The statement that mathematics is the science of number would be untrue in two different ways. On the one hand, there are recognised branches of mathematics which have nothing to do with number —all geometry that does not use co-ordinates or measurement, for example: projective and descriptive geometry, down to the point at which co- ordinates are introduced, does not have to do with number, or even with quantity in the sense of greater and less. On the other hand, through the definition of cardinals, through the theory of induction and ancestral relations, through the general theory of series, and through the definitions of the arithmetical operations, it has become possible to generalise much that used to be proved only in connection with numbers. The result is that what was formerly the single study of Arithmetic has now become divided into numbers of separate studies, no one of which is specially concerned with numbers. The most elementary properties of numbers are concerned with one-one relations, and similarity between classes. Addition is concerned with the construction of mutually exclusive classes respectively similar to a set of classes which are not known to be mutually exclusive. Multiplication is merged in the theory of "selections," i.e. of a certain kind of one-many relations. Finitude is merged in the general study of ancestral relations, which yields the whole theory of mathematical induction. The ordinal properties of the various kinds of number-series, and the elements of the theory of continuity of functions and the limits of functions, can be generalised so as no longer to involve any essential reference to numbers. It
- 73. is a principle, in all formal reasoning, to generalise to the utmost, since we thereby secure that a given process of deduction shall have more widely applicable results; we are, therefore, in thus generalising the reasoning of arithmetic, merely following a precept which is universally admitted in mathematics. And in thus generalising we have, in effect, created a set of new deductive systems, in which traditional arithmetic is at once dissolved and enlarged; but whether any one of these new deductive systems—for example, the theory of selections—is to be said to belong to logic or to arithmetic is entirely arbitrary, and incapable of being decided rationally. We are thus brought face to face with the question: What is this subject, which may be called indifferently either mathematics or logic? Is there any way in which we can define it? Certain characteristics of the subject are clear. To begin with, we do not, in this subject, deal with particular things or particular properties: we deal formally with what can be said about any thing or any property. We are prepared to say that one and one are two, but not that Socrates and Plato are two, because, in our capacity of logicians or pure mathematicians, we have never heard of Socrates and Plato. A world in which there were no such individuals would still be a world in which one and one are two. It is not open to us, as pure mathematicians or logicians, to mention anything at all, because, if we do so, we introduce something irrelevant and not formal. We may make this clear by applying it to the case of the syllogism. Traditional logic says: "All men are mortal, Socrates is a man, therefore Socrates is mortal." Now it is clear that what we mean to assert, to begin with, is only that the premisses imply the conclusion, not that premisses and conclusion are actually true; even the most traditional logic points out that the actual truth of the premisses is irrelevant to logic. Thus the first change to be made in the above traditional syllogism is to state it in the form: "If all men are mortal and Socrates is a man, then Socrates is mortal." We may now observe that it is intended to convey that this argument is valid in virtue of its form, not in virtue of the particular terms occurring in it. If we had omitted "Socrates is a man" from our premisses, we should have had a non- formal argument, only admissible because Socrates is in fact a man; in that case we could not have generalised the argument. But when, as above, the argument is formal, nothing depends upon the terms that occur in it. Thus we may substitute for men, for mortals, and for Socrates, where
- 74. and are any classes whatever, and is any individual. We then arrive at the statement: "No matter what possible values and and may have, if all 's are 's and is an , then is a "; in other words, "the propositional function 'if all 's are and is an , then is a ' is always true." Here at last we have a proposition of logic—the one which is only suggested by the traditional statement about Socrates and men and mortals. It is clear that, if formal reasoning is what we are aiming at, we shall always arrive ultimately at statements like the above, in which no actual things or properties are mentioned; this will happen through the mere desire not to waste our time proving in a particular case what can be proved generally. It would be ridiculous to go through a long argument about Socrates, and then go through precisely the same argument again about Plato. If our argument is one (say) which holds of all men, we shall prove it concerning " ," with the hypothesis "if is a man." With this hypothesis, the argument will retain its hypothetical validity even when is not a man. But now we shall find that our argument would still be valid if, instead of supposing to be a man, we were to suppose him to be a monkey or a goose or a Prime Minister. We shall therefore not waste our time taking as our premiss " is a man" but shall take " is an ," where is any class of individuals, or " " where is any propositional function of some assigned type. Thus the absence of all mention of particular things or properties in logic or pure mathematics is a necessary result of the fact that this study is, as we say, "purely formal." At this point we find ourselves faced with a problem which is easier to state than to solve. The problem is: "What are the constituents of a logical proposition?" I do not know the answer, but I propose to explain how the problem arises. Take (say) the proposition "Socrates was before Aristotle." Here it seems obvious that we have a relation between two terms, and that the constituents of the proposition (as well as of the corresponding fact) are simply the two terms and the relation, i.e. Socrates, Aristotle, and before. (I ignore the fact that Socrates and Aristotle are not simple; also the fact that what appear to be their names are really truncated descriptions. Neither of these facts is relevant to the present issue.) We may represent the general form of such propositions by " ," which may be read " has the relation to ." This general form may occur in logical propositions, but no particular instance of
- 75. it can occur. Are we to infer that the general form itself is a constituent of such logical propositions? Given a proposition, such as "Socrates is before Aristotle," we have certain constituents and also a certain form. But the form is not itself a new constituent; if it were, we should need a new form to embrace both it and the other constituents. We can, in fact, turn all the constituents of a proposition into variables, while keeping the form unchanged. This is what we do when we use such a schema as " ," which stands for any one of a certain class of propositions, namely, those asserting relations between two terms. We can proceed to general assertions, such as " is sometimes true"—i.e. there are cases where dual relations hold. This assertion will belong to logic (or mathematics) in the sense in which we are using the word. But in this assertion we do not mention any particular things or particular relations; no particular things or relations can ever enter into a proposition of pure logic. We are left with pure forms as the only possible constituents of logical propositions. I do not wish to assert positively that pure forms—e.g. the form " "—do actually enter into propositions of the kind we are considering. The question of the analysis of such propositions is a difficult one, with conflicting considerations on the one side and on the other. We cannot embark upon this question now, but we may accept, as a first approximation, the view that forms are what enter into logical propositions as their constituents. And we may explain (though not formally define) what we mean by the "form" of a proposition as follows:— The "form" of a proposition is that, in it, that remains unchanged when every constituent of the proposition is replaced by another. Thus "Socrates is earlier than Aristotle" has the same form as "Napoleon is greater than Wellington," though every constituent of the two propositions is different. We may thus lay down, as a necessary (though not sufficient) characteristic of logical or mathematical propositions, that they are to be such as can be obtained from a proposition containing no variables (i.e. no such words as all, some, a, the, etc.) by turning every constituent into a variable and asserting that the result is always true or sometimes true, or that it is always true in respect of some of the variables that the result is
- 76. sometimes true in respect of the others, or any variant of these forms. And another way of stating the same thing is to say that logic (or mathematics) is concerned only with forms, and is concerned with them only in the way of stating that they are always or sometimes true—with all the permutations of "always" and "sometimes" that may occur. There are in every language some words whose sole function is to indicate form. These words, broadly speaking, are commonest in languages having fewest inflections. Take "Socrates is human." Here "is" is not a constituent of the proposition, but merely indicates the subject-predicate form. Similarly in "Socrates is earlier than Aristotle," "is" and "than" merely indicate form; the proposition is the same as "Socrates precedes Aristotle," in which these words have disappeared and the form is otherwise indicated. Form, as a rule, can be indicated otherwise than by specific words: the order of the words can do most of what is wanted. But this principle must not be pressed. For example, it is difficult to see how we could conveniently express molecular forms of propositions (i.e. what we call "truth- functions") without any word at all. We saw in Chapter XIV. that one word or symbol is enough for this purpose, namely, a word or symbol expressing incompatibility. But without even one we should find ourselves in difficulties. This, however, is not the point that is important for our present purpose. What is important for us is to observe that form may be the one concern of a general proposition, even when no word or symbol in that proposition designates the form. If we wish to speak about the form itself, we must have a word for it; but if, as in mathematics, we wish to speak about all propositions that have the form, a word for the form will usually be found not indispensable; probably in theory it is never indispensable. Assuming—as I think we may—that the forms of propositions can be represented by the forms of the propositions in which they are expressed without any special word for forms, we should arrive at a language in which everything formal belonged to syntax and not to vocabulary. In such a language we could express all the propositions of mathematics even if we did not know one single word of the language. The language of mathematical logic, if it were perfected, would be such a language. We should have symbols for variables, such as " " and " " and " ," arranged in various ways; and the way of arrangement would indicate that something was being said to be true of all values or some values of the variables. We
- 77. should not need to know any words, because they would only be needed for giving values to the variables, which is the business of the applied mathematician, not of the pure mathematician or logician. It is one of the marks of a proposition of logic that, given a suitable language, such a proposition can be asserted in such a language by a person who knows the syntax without knowing a single word of the vocabulary. But, after all, there are words that express form, such as "is" and "than." And in every symbolism hitherto invented for mathematical logic there are symbols having constant formal meanings. We may take as an example the symbol for incompatibility which is employed in building up truth- functions. Such words or symbols may occur in logic. The question is: How are we to define them? Such words or symbols express what are called "logical constants." Logical constants may be defined exactly as we defined forms; in fact, they are in essence the same thing. A fundamental logical constant will be that which is in common among a number of propositions, any one of which can result from any other by substitution of terms one for another. For example, "Napoleon is greater than Wellington" results from "Socrates is earlier than Aristotle" by the substitution of "Napoleon" for "Socrates," "Wellington" for "Aristotle," and "greater" for "earlier." Some propositions can be obtained in this way from the prototype "Socrates is earlier than Aristotle" and some cannot; those that can are those that are of the form " ," i.e. express dual relations. We cannot obtain from the above prototype by term- for-term substitution such propositions as "Socrates is human" or "the Athenians gave the hemlock to Socrates," because the first is of the subject- predicate form and the second expresses a three-term relation. If we are to have any words in our pure logical language, they must be such as express "logical constants," and "logical constants" will always either be, or be derived from, what is in common among a group of propositions derivable from each other, in the above manner, by term-for-term substitution. And this which is in common is what we call "form." In this sense all the "constants" that occur in pure mathematics are logical constants. The number 1, for example, is derivative from propositions of the form: "There is a term such that is true when, and only when, is ." This is a function of , and various different propositions result from giving different values to . We may (with a little
- 78. omission of intermediate steps not relevant to our present purpose) take the above function of as what is meant by "the class determined by is a unit class" or "the class determined by is a member of 1" (1 being a class of classes). In this way, propositions in which 1 occurs acquire a meaning which is derived from a certain constant logical form. And the same will be found to be the case with all mathematical constants: all are logical constants, or symbolic abbreviations whose full use in a proper context is defined by means of logical constants. But although all logical (or mathematical) propositions can be expressed wholly in terms of logical constants together with variables, it is not the case that, conversely, all propositions that can be expressed in this way are logical. We have found so far a necessary but not a sufficient criterion of mathematical propositions. We have sufficiently defined the character of the primitive ideas in terms of which all the ideas of mathematics can be defined, but not of the primitive propositions from which all the propositions of mathematics can be deduced. This is a more difficult matter, as to which it is not yet known what the full answer is. We may take the axiom of infinity as an example of a proposition which, though it can be enunciated in logical terms, cannot be asserted by logic to be true. All the propositions of logic have a characteristic which used to be expressed by saying that they were analytic, or that their contradictories were self-contradictory. This mode of statement, however, is not satisfactory. The law of contradiction is merely one among logical propositions; it has no special pre-eminence; and the proof that the contradictory of some proposition is self-contradictory is likely to require other principles of deduction besides the law of contradiction. Nevertheless, the characteristic of logical propositions that we are in search of is the one which was felt, and intended to be defined, by those who said that it consisted in deducibility from the law of contradiction. This characteristic, which, for the moment, we may call tautology, obviously does not belong to the assertion that the number of individuals in the universe is , whatever number may be. But for the diversity of types, it would be possible to prove logically that there are classes of terms, where is any finite integer; or even that there are classes of terms. But, owing to types, such proofs, as we saw in Chapter XIII., are fallacious. We are left to empirical observation to determine whether there are as many as individuals in the
- 79. world. Among "possible" worlds, in the Leibnizian sense, there will be worlds having one, two, three, ... individuals. There does not even seem any logical necessity why there should be even one individual[43]—why, in fact, there should be any world at all. The ontological proof of the existence of God, if it were valid, would establish the logical necessity of at least one individual. But it is generally recognised as invalid, and in fact rests upon a mistaken view of existence—i.e. it fails to realise that existence can only be asserted of something described, not of something named, so that it is meaningless to argue from "this is the so-and-so" and "the so-and-so exists" to "this exists." If we reject the ontological argument, we seem driven to conclude that the existence of a world is an accident—i.e. it is not logically necessary. If that be so, no principle of logic can assert "existence" except under a hypothesis, i.e. none can be of the form "the propositional function so-and-so is sometimes true." Propositions of this form, when they occur in logic, will have to occur as hypotheses or consequences of hypotheses, not as complete asserted propositions. The complete asserted propositions of logic will all be such as affirm that some propositional function is always true. For example, it is always true that if implies and implies then implies , or that, if all 's are 's and is an then is a . Such propositions may occur in logic, and their truth is independent of the existence of the universe. We may lay it down that, if there were no universe, all general propositions would be true; for the contradictory of a general proposition (as we saw in Chapter XV.) is a proposition asserting existence, and would therefore always be false if no universe existed. [43]The primitive propositions in Principia Mathematica are such as to allow the inference that at least one individual exists. But I now view this as a defect in logical purity. Logical propositions are such as can be known a priori, without study of the actual world. We only know from a study of empirical facts that Socrates is a man, but we know the correctness of the syllogism in its abstract form (i.e. when it is stated in terms of variables) without needing any appeal to experience. This is a characteristic, not of logical propositions in themselves, but of the way in which we know them. It has, however, a bearing upon the question what their nature may be, since there are some kinds of propositions which it would be very difficult to suppose we could know without experience.
- 80. It is clear that the definition of "logic" or "mathematics" must be sought by trying to give a new definition of the old notion of "analytic" propositions. Although we can no longer be satisfied to define logical propositions as those that follow from the law of contradiction, we can and must still admit that they are a wholly different class of propositions from those that we come to know empirically. They all have the characteristic which, a moment ago, we agreed to call "tautology." This, combined with the fact that they can be expressed wholly in terms of variables and logical constants (a logical constant being something which remains constant in a proposition even when all its constituents are changed)—will give the definition of logic or pure mathematics. For the moment, I do not know how to define "tautology."[44] It would be easy to offer a definition which might seem satisfactory for a while; but I know of none that I feel to be satisfactory, in spite of feeling thoroughly familiar with the characteristic of which a definition is wanted. At this point, therefore, for the moment, we reach the frontier of knowledge on our backward journey into the logical foundations of mathematics. [44]The importance of "tautology" for a definition of mathematics was pointed out to me by my former pupil Ludwig Wittgenstein, who was working on the problem. I do not know whether he has solved it, or even whether he is alive or dead. We have now come to an end of our somewhat summary introduction to mathematical philosophy. It is impossible to convey adequately the ideas that are concerned in this subject so long as we abstain from the use of logical symbols. Since ordinary language has no words that naturally express exactly what we wish to express, it is necessary, so long as we adhere to ordinary language, to strain words into unusual meanings; and the reader is sure, after a time if not at first, to lapse into attaching the usual meanings to words, thus arriving at wrong notions as to what is intended to be said. Moreover, ordinary grammar and syntax is extraordinarily misleading. This is the case, e.g., as regards numbers; "ten men" is grammatically the same form as "white men," so that 10 might be thought to be an adjective qualifying "men." It is the case, again, wherever propositional functions are involved, and in particular as regards existence and descriptions. Because language is misleading, as well as because it is diffuse and inexact when applied to logic (for which it was never intended), logical symbolism is absolutely necessary to any exact or thorough
- 81. treatment of our subject. Those readers, therefore, who wish to acquire a mastery of the principles of mathematics, will, it is to be hoped, not shrink from the labour of mastering the symbols—a labour which is, in fact, much less than might be thought. As the above hasty survey must have made evident, there are innumerable unsolved problems in the subject, and much work needs to be done. If any student is led into a serious study of mathematical logic by this little book, it will have served the chief purpose for which it has been written.
- 82. INDEX Aggregates, 12 Alephs, 83, 92, 97, 125 Aliorelatives, 32 All, 158 ff. Analysis, 4 Ancestors, 25, 33 Argument of a function, 47, 108 Arithmetising of mathematics, 4 Associative law, 58, 94 Axioms, 1 Between, 38 ff., 58 Bolzano, 138 n. Boots and socks, 126 Boundary, 70, 98, 99 Cantor, Georg, 77, 79, 85 n., 86, 89, 95, 102, 136 Classes, 12, 137, 181 ff.; reflexive, 80, 127, 138; similar, 15, 16 Clifford, W. K., 76 Collections, infinite, 13 Commutative law, 58, 94 Conjunction, 147 Consecutiveness, 37, 38, 81 Constants, 202 Construction, method of, 73 Continuity, 86, 97 ff.; Cantorian, 102 ff.; Dedekindian, 101 ff.; in philosophy, 105; of functions, 106 ff. Contradictions, 135 ff.
- 83. Convergence, 115 Converse, 16, 32, 49 Correlators, 54 Counterparts, objective, 61 Counting, 14, 16 Dedekind, 69, 99, 138 n. Deduction, 144 ff. Definition, 3; extensional and intensional, 12 Derivatives, 100 Descriptions, 139, 144 Descriptions, 167 Dimensions, 29 Disjunction, 147 Distributive law, 58, 94 Diversity, 87 Domain, 16, 32, 49 Equivalence, 183 Euclid, 67 Existence, 164, 171, 177 Exponentiation, 94, 120 Extension of a relation, 60 Fictions, logical, 14 n., 45, 137 Field of a relation, 32, 53 Finite, 27 Flux, 105 Form, 198 Fractions, 37, 64 Frege, 7, 10, 25 n., 77, 95, 146 n. Functions, 46; descriptive, 46, 180; intensional and extensional, 186; predicative, 189; propositional, 46, 144; propositional, 155; Gap, Dedekindian, 70 ff., 99 Generalisation, 156 Geometry, 29, 59, 67, 74, 100, 145; analytical, 4, 86
- 84. Greater and less, 65, 90 Hegel, 107 Hereditary properties, 21 Implication, 146, 153; formal, 163 Incommensurables, 4, 66 Incompatibility, 147 ff., 200 Incomplete symbols, 182 Indiscernibles, 192 Individuals, 132, 141, 173 Induction, mathematical, 20 ff., 87, 93, 185 Inductive properties, 21 Inference, 148 Infinite, 28; of rationals, 65; Cantorian, 65; of cardinals, 77 ff.; and series and ordinals, 89 ff. Infinity, axiom of, 66 n., 77, 131 ff., 202 Instances, 156 Integers, positive and negative, 64 Intervals, 115 Intuition, 145 Irrationals, 66, 72 Kant, 145 Leibniz, 80, 107, 192 Lewis, C. I., 153, 154 Likeness, 52 Limit, 29, 69 ff., 97 ff.; of functions, 106 ff. Limiting points, 99 Logic, 159, 65, 194 ff.; mathematical, v, 201, 206 Logicising of mathematics, 7 Maps, 52, 60 ff., 80 Mathematics, 194 ff.
- 85. Maximum, 70, 98 Median class, 104 Meinong, 169 Method, vi Minimum, 70, 98 Modality, 165 Multiplication, 118 ff. Multiplicative axiom, 92, 117 ff. Names, 173, 182 Necessity, 165 Neighbourhood, 109 Nicod, 148, 149, 151 Null-class, 23, 132 Number, cardinal, 10 ff., 56, 77 ff., 95; complex, 74 ff.; finite, 20 ff.; inductive, 27, 78, 131; infinite, 77 ff.; irrational, 66, 72; maximum? 135; multipliable, 130; natural, 2 ff., 22; non-inductive, 88, 127; real, 66, 72, 84; reflexive, 80, 127; relation, 56, 94; serial, 57 Occam, 184 Occurrences, primary and secondary, 179 Ontological proof, 203 Order 29ff.; cyclic, 40 Oscillation, ultimate, 111 Parmenides, 138 Particulars, 140 ff., 173 Peano, 5 ff., 23, 24, 78, 81, 131, 163 Peirce, 32 n. Permutations, 50 Philosophy, mathematical, v, 1 Plato, 138
- 86. Plurality, 10 Poincaré, 27 Points, 59 Posterity, 22 ff., 32; proper, 36 Postulates, 71, 73 Precedent, 98 Premisses of arithmetic, 5 Primitive ideas and propositions, 5, 202 Progressions, 8, 81 ff. Propositions, 155; analytic, 204; elementary, 161 Pythagoras, 4, 67 Quantity, 97, 195 Ratios, 64, 71, 84, 133 Reducibility, axiom of, 191 Referent, 48 Relation numbers, 56 ff. Relations, asymmetrical 31, 42; connected, 32; many-one, 15; one-many, 15, 45; one-one, 15, 47, 79; reflexive, 16; serial, 34; similar, 52; squares of, 32; symmetrical, 16, 44; transitive, 16, 32 Relatum, 48 Representatives, 120 Rigour, 144 Royce, 80 Section, Dedekindian, 69 ff.; ultimate, 111 Segments, 72, 98 Selections, 117 Sequent, 98 Series, 29 ff.; closed, 103; compact, 66, 93, 100; condensed in itself, 102;
- 87. Dedekindian, 71, 73, 101; generation of, 41; infinite, 89; perfect, 102, 103; well-ordered, 92, 123 Sheffer, 148 Similarity, of classes, 15 ff.; of relations, 83; of relations, 52 Some, 158 ff. Space, 61, 86, 140 Structure, 60 ff. Sub-classes, 84 ff. Subjects, 142 Subtraction, 87 Successor of a number, 23, 35 Syllogism, 197 Tautology, 203, 205 The, 167, 172 ff. Time, 61, 86, 140 Truth-function, 147 Truth-value, 146 Types, logical, 53, 135 ff., 185, 188 Unreality, 168 Value of a function, 47, 108 Variables, 10, 161, 199 Veblen, 58 Verbs, 141 Weierstrass, 97, 107 Wells, H. G., 114 Whitehead, 64, 76, 107, 119 Wittgenstein, 205 n. Zermelo, 123, 129 Zero, 65
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