The chapters in part I are essentially the text of a brief talk presented at a workshop on “Limits to Scientific Knowability,” held at the Santa Fe Institute (SFI) in 1994, May 24 to 26. As described to me, the workshop was intended to explore the impacts (if any) of the famous Gödel Incompleteness results in mathematics upon the sciences. The workshop’s tone was to be informal and exploratory, aimed at determining whether a more extensive effort by the Institute along these dimensions was warranted.

Accordingly, the workshop consisted primarily of roundtable discussion, with no formal papers delivered. However, the...

Erwin Schrödinger’s essay What Is Life?, which first appeared in print in 1944, was based on a series of public lectures delivered the preceding year in Dublin. Much has happened, both in biology and in physics, during the half century since then. Hence, it might be appropriate to reappraise the status of Schrödinger’s question, from a contemporary perspective, at least as I see it today. This I shall attempt herein.

I wonder how many people actually read this essay nowadays. I know I have great difficulty in getting my students to read anything more than five years old, their approximate...

The following remarks are intended to address two problems: (a) the role of contemporary physics in dealing with the nature and properties of living systems, and (b) the role of mimetic approaches (usually prefaced by the adjective artificial) in dealing with these same matters. Both approaches are offered as (quite distinct) ways of making biology scientific, or objective, by in effect making it something other than biology. And they are both, in a historical sense, ancient strategies; in their separate ways, they appear to embody a mechanistic approach to biological phenomena, whose only alternative seems to be a discredited, mystical,...

In this chapter, I had intended to consider only the legitimacy of identifying two very differently defined things: a “Mendelian gene,” defined in indirect functional terms via its manifestations in larger systems (phenotypes), and an intrinsic structural feature (sequence or primary structure) of a polymeric molecule. As it turned out, this question could not be readily separated from its own, deeper context, one that goes to the very heart of reductionism itself. I have touched already on many of these issues in Life Itself (1991), in dealing with the theoretical basis of organism, when I argued that reductionism provides a...

The mind-brain problem is somewhat apart from my direct line of inquiry, but it is an important collateral illustration of the circle of ideas I have developed to deal with the life-organism problem. I do not deny the importance of the mind-brain problem; it was simply less interesting to me personally, if for no other reason than that one has to be alive before one is sentient. Life comes before mind, and anything I could say about mind and brain would be a corollary of what I had to say about life. That is indeed the way it has turned...

It is my contention that mathematics took a disastrous wrong turn some time in the sixth century B.C. This wrong turn can be expressed as an ongoing attempt, since then, to identify effectiveness with computability. That identification is nowadays associated with the name of Alonzo Church and is embodied in Church’s Thesis. But Church was only among the latest in a long line going back to the original culprit. And that was no less than Pythagoras himself.

From that original mistake, and the attempts to maintain it, have grown a succession of foundation crises in mathematics. The paradoxes of Zeno...

The discussion to follow will focus on the obvious irreconcilability of the two statements below. The first of them is taken from a rhapsodic paean to reductionism; it was penned over two decades ago, but could have been written yesterday:

Life can be understood in terms of the laws that govern and the phenomena that characterize the inanimate, physical universe, and indeed, at its essence, life can be understood only in the language of chemistry.… Only two truly major questions remain shrouded in a cloak of not quite fathomable mystery: (1) the origin of life …, and (2) the mind-body...

Phenotype is a uniquely biological concept. Etymologically, it derives from the Greek verb phainein, to show, the same root from whence derives the word phenomenon.

Phenotype is a rather recent word, but phenomenon is old. The dictionary tells us that phenomenon means “any fact, circumstance or experience that is apparent to the senses, and that can be scientifically described or appraised.” That is, phenomena devolve on cognition, and hence on direct (or sometimes indirect) sensory experience. From this follows another characterization of phenomenon, typically regarded as synonymous with the one I have given: the appearance or observed features of something...

In chapter 5, I discussed the mind-brain problem (and the closely related life-organism problem) as attempts to express subjectivities (mind and life) in presumably scientific, objective terms (brain and organism). I drew parallels with the notorious measurement problem in quantum mechanics on the one hand, and attempts to root the objectivity of mathematical truth entirely in syntax on the other. I roughly concluded this: Any attempt to reconcile the contradictions between subjective and objective on entirely objective grounds must fail as long as we persist in identifying objectivity with machines, and with syntax, alone. Indeed, not only can the mind-brain...

Discussions of the mind-brain problem are inseparable from the capabilities and limitations of reductionism. All problems of this type possess the same basic form: Given a material system x, such as an organism or a brain, we want to answer a question of the form, Is x alive? or Is x sentient? or, in general, Does x manifest or realize or instantiate a property P? That is, we want to treat P as a predicate, or adjective, of a given referent x, just as we do, for example, specific gravity, or reactivity, or the shape of an orbit. Stated otherwise,...

The chapters in this part are loosely organized around the concept of genericity. I devoted chapter 2 of Life Itself to this concept, to what is special and what is general, in connection with the widely held belief that biology is only a special case of the general laws of contemporary physics (i.e., that biology reduces to physics, as presently understood).

It was a main thrust of Life Itself that what is more generic does not merely reduce to what is less so—it is more the other way around. In particular, I argued that mechanistic systems are nongeneric in...

The appearance of René Thom’s Stabilité Structurelle et Morphogénèse in 1972 (translated into English in 1976) was a watershed event in mathematics, in theoretical biology, and for the philosophy of science generally. The questions he raised in that book, both directly and by implication, were deeply disquieting to most practicing, empirical scientists, making their dogmatic slumbers untenable. Their predictable responses took several forms: (1) outrage and indignation; (2) violent but irrelevant counterattacks; (3) pretense that Thom did not exist, and hence that his ideas did not need to be addressed at all; and (4) a distortion of his views into...

The crucial feature of natural languages is simply that they are about something outside the languages themselves. It is their essence to express things about external referents. These referents convey meanings on expressions in a language, and, most important, it is entirely by virtue of these meanings that the expressions become true or false.

The province of meanings is the realm of semantics. What is left of a language when the referents are stripped away constitutes syntax. Syntactic aspects of a language seem to take the form of a limited number of rules for manipulating symbols and expressions, and they...

The primary distinction between “pure” and “applied” mathematics is that the latter is overtly directed toward extramathematical referents. That is, in employing mathematics as a tool to study other things, it is not enough to prove theorems; it is at least equally important that the conditions imposed by these extramathematical referents be respected. Such conditions are, by their very nature, informal; that is why mathematical modeling is an art and is in many ways harder than mathematics itself.

For example, ecological population dynamics becomes mathematical ecology when it is expressed as some kind of mass-action law imposed on a state...

A property of a mathematical object is often called generic if a sufficiently small but otherwise arbitrary perturbation of the object produces another object with the same property. In other words, if something is generic with respect to a given property, it is to that extent indistinguishable from any of its immediate neighbors.

For instance, let us consider the set of n × n real matrices. The property of invertibility (i.e., nonvanishing of the determinant, and hence of any of the eigenvalues) is generic; singularity is not. Possession of real or complex eigenvalues is generic; possession of pure imaginary eigenvalues...

The class of behaviors collectively called chaos represents a milestone for mathematics in general and for the theory of dynamical systems in particular. However, the significance of chaos, as a tool for the study of material phenomena, is by no means as clear-cut. Particularly is this true in biology, where it is increasingly argued (e.g., see West 1990) that “healthy” biological behavior is not a manifestation of homeostasis, but rather of chaos—indeed, that homeostasis is pathological. Similar arguments rage in the physical sciences, as for instance in the role of chaos as a paradigm for turbulence, and even for...

The chapters in this part cover a variety of more special topics and applications, especially to biological form and to morphogenesis. Chapter 14 was prepared originally as a tribute to Richard Bellman, who was perhaps best known as the creator of a powerful optimization technique that he called dynamic programming. This technique is closely related to the Hamilton-Jacobi equation, which in turn arises out of the mechano-optical analogy of William Rowan Hamilton, one of the most profound innovations of nineteenth-century physics. It was based on the observation that both the paths of mechanical particles in configuration space, and the paths...

Optimality is the study of superlatives. Nowadays, optimality plays two distinct but interrelated roles in science: (1) in pure science, its impact is primarily analytic or explanatory; we use it to characterize the way in which a natural process does occur, out of all the ways it could occur; and (2) in applied science (technology or engineering in the broadest sense), we use it to decide how we should do something, out of all the possible ways in which we could do it (here the emphasis is primarily on design or synthesis).

Like most good scientific ideas, optimality originated with...

One of the most ancient, and at the same time most current, fields of theoretical biology is that concerned with morphogenesis–the generation of pattern and form in biological systems. This chapter is devoted to the development of an integrated framework for treating morphogenetic problems, not only because they are of the greatest interest and importance in their own right but also because they tell us some important things about theoretical biology in general, and they help us articulate the position of biology vis-á-vis other scientific disciplines. I will first discuss theoretical biology in general, and then morphogenesis in particular....

Two basic characteristics of biological systems are the complex developmental processes which give rise to them, and the equally complex physiological processes which maintain them. These have been the subject of intensive study over the years because of their dramatic nature and their extreme practical importance. Not nearly so much attention, however, has been given to processes like senescence and mortality, which are in their way equally dramatic, equally important, and in a certain sense even more puzzling. Why should a physiological system, which is a collection of interacting homeostats designed precisely to buffer the organism against externally imposed stresses,...

My late colleague and friend James F. Danielli liked to say that the progress of human culture depends on the capacity to move from an “age of analysis” to an “age of synthesis” (Danielli 1974:1). He viewed such a progression, in our own day, as an immediate practical urgency. Even thirty years ago, he was arguing that the capacity to synthesize life provided perhaps the only road to our continued survival as a species; that our deep problems of pollution, resource exhaustion, and ultimate extinction require giving up the machine-based technologies that create these problems and going to biological modes...

The chapters in this part are of a different character from those preceding. They bear not so much on what we can learn about biology from other disciplines as on what we can learn about other disciplines from an understanding of biological modes of organization. Most particularly, they bear on technologies—how to solve problems. Here I shall use technology in the broadest sense, to include problems of an environmental and social nature, not just the fabrication of better mechanical devices, and to connote the execution of functions.

I have long believed, and argued, that biology provides us with a...

Perhaps the first lesson to be learned from biology is that there are lessons to be learned from biology. Nowadays, biologists would be among the first to deny this elementary proposition as nothing more than a vulgar anthropomorphism. To say there are lessons to be learned is, in their eyes, to impute a design to be discerned. And according to Jacques Monod (1971), one of the major spokesmen of contemporary biology, the absolute denial of design (he calls it the “Postulate of Objectivity”) is the cornerstone of science itself. Science lets us therefore learn about things, but not from them....

Many relations exist between biology and “the machine,” or, more generally, between biology and technology. Certainly, the machine and machine technology impact on the concept of organism (Rosen 1987a, 1988a, 1989), but the interaction goes both ways: the organism is increasingly a source, or a resource, for technology. The interplay between the two was regarded as the domain of an ill-defined area once called bionics, not much of which is left now, and even that is vastly different from its initial conception.

St. Augustine said that God placed animals on the earth for four reasons: to feed man, to instruct...

Medicine has been called an art and it has been called a science. Indeed, it must possess elements of both. But primarily, it is a craft in its practice, and a technology in its aspirations. It is applied science–primarily applied biology. In some of its aspects, it is even rather more applied science.

Science has always had philosophy associated with it; indeed, for a long time, science was called natural philosophy. The ancient Greeks were keenly interested in the way the world was put together and how it worked, and they had laid out the major alternatives in this...

In mythology, Chimera was a monster built up from parts of lions, goats, and serpents. According to Homer, it breathed fire. It may have represented a volcano in Lydia, where serpents dwelt at the base, goats along its slopes, and lions at the crest. Accordingly, chimera has come to mean any impossible, absurd, or fanciful thing–an illusion.

In biology, aside from being the name of a strange-looking fish, chimera refers to any individual composed of diverse genetic parts–any organism containing cell populations arising from different zygotes (McLaren 1976). Indeed, the fusion of two gametes to form a zygote...

I have been, and remain, entirely dedicated to the idea that modeling is the essence of science and the habitat of all epistemology. Although I have concentrated my efforts on biology and the nature of organism, I have also asserted for a long time that human systems and organisms are very much alike—i.e., they share, or realize, many common models. I was a pioneer in the wide deployment of mathematical ideas for purposes of modeling, particularly in the area of stability theory; indeed, I wrote perhaps the first modern text devoted to this purpose (1970).

What follows is a...