Beauty, even in Maths, can exist in the eye of the beholder. That might sound a little surprising, when, after all, what could be more objective than mathematics when thinking about truth, and what, therefore, could be more natural than for beauty and goodness, the twin accomplices to truth, to be co-joined ?
In the 70 odd years since Samuel Eilenberg and Saunders Mac Lane published their now infamous paper “A General Theory of Natural Equivalences“, the pursuit of maths by professionals (I use here the reference point definition of Michael Harris – see his recent publication “Mathematics without Apologies“) has become ever more specialised. I, for one, don’t doubt cross disciplinary excellence is alive and sometimes robustly so, but the industrially specialised silos that now create, produce and then sustain academic tenure are formidable within the community of mathematicians.
Beauty, in the purest sense, does not need to be captured in a definition but recognised through intuition. Whether we take our inspiration from Hardy or Dirac, or whether we experience a gorgeous thrill when encountering an austere proof that may have been confronted thousands of times before, the confluence of simplicity and beauty in maths may well be one of the few remaining places where the commonality of the “eye” across a spectrum of different beholders remains at its strongest.
Neither Eilenberg nor Mac Lane could have thought that Category theory, which was their attempt to link topology and algebra, would become so pervasive or so foundational in its influence when they completed and submitted their paper in those dark days of WW 2. But then neither could Cantor, have dreamt about his work on Set theory being adopted as the central pillar of “modern” mathematics so soon after his death. Under attack from establishment figures such as Kronecker during his lifetime, Cantor would not have believed that set theory would become the central edifice around which so much would be constructed.
Of course that is exactly what has happened. Set theory and the ascending magnitude of infinities that were unleashed through the crack in the door that was represented by Cantor’s diagonal conquered all before them.
Until now, that is.
In an article in Science News, Julie Rehmeyer describes Category Theory as “perhaps the most abstract area of all mathematics” and “where math is the abstraction of the real world, category theory is an abstraction of mathematics”.
Slowly, without fanfare, and with an alliance built with the emergent post transistor age discipline of computer science, Category theory looks set to become the dominant foundational basis for all mathematics. It could, in fact, already have achieved that status through stealth. After all, if sets are merely an example of a category, they become suborned without question or query. One might even use the description ‘subsumed’.
There is, in parallel, a wide ranging discussion in mathematics about the so called Univalent Foundation that is most widely associated with Voevodsky which is not the same. The text book produced for the year long univalence programme iniated at the IAS that was completed in 2013 Homotopy type theory – Univalent Foundations Programme states:
“The univalence ax-iom implies, in particular, that isomorphic structures can be identified, a principle that mathematicians have been happily using on workdays, despite its incompatibility with the “official”doctrines of conventional foundations..”
before going on to present the revelatory exposition that Univalent Foundations are the real unifying binding agent around mathematics.
I prefer to think of Voevodsky’s agenda as being narrower in many crucial respects than Category Theory, although both owe a huge amount to the over-arching reach of computational advances made through the mechanical aid proffered through the development of computers, particularly if one shares Voevodsky’s view that proofs will eventually have to be subject to mechanical confirmation.
In contrast, the journey, post Russell, for type theory based clarificatory approaches to formal logic continues in various ways, but Category theory brings a unifying effort to the whole of mathematics that had to wait almost two decades after Eilenberg and Mac Lane’s paper when a then virtually unknown mathematician, William Lawvere published his now much vaunted “An Elementary Theory of the Category of Sets” in 1964. This paper, and the revolutionary work of Grothendieck (see below) brought about a depth and breadth of work which created the environment from which Category Theory emerged through the subsequent decades until the early 2000’s.
Lawvere’s work has, at times, been seen as an attempt to simply re-work set theory in Category theoretic terms. This limitation is no longer prevalent, indeed the most recent biographical reviews of Grothendieck, following his death, assume that the unificatory expedient that is the essential feature of Category theory (and I should say here not just ETCS) is taken for granted, axiomatic, even. Grothendieck eventually went much further than defining Category theory in set theoretic terms, with both Algebraic Topology and Mathematical Physics being fields that now could not be approached without a foundational setting that is Category theory. The early language and notation of Category Theory where categories ‘C’ are described essentially as sets whose members satisfy the conditions of composition, morphism and identity eventually gave way post Lawvere and then Lambek to a systematic adoption of the approach we now see where any and all deductive systems can be turned into categories. Most standard histories give due credit to Eilenberg and Mac Lane as well as Lawvere (and sometimes Cartan), but it is Grothendieck’s ‘Sur quelques points d’algebre homologique’ in 1957 that is now seen as the real ground breaker.
My own pathway to Category theory has been via my interest in Lie Groups, and more broadly, in Quantum Computing, and it was only by accident (the best things really are those that come about by accident !) that I decided I had better learn the language of Category theory when I found Lawvere’s paper misleadingly familiar but annoyingly distant when, in common with most people, I assumed that my working knowledge of notation in logic and in set theory would map smoothly across to Category theory. That, of course, is not the case, and it was only after I gained some grounding in this new language that I realised just how and why Category theory has an impact far beyond computer science. It is this journey that also brings me face to face with a growing appreciation of the natural intersection between Category theory and a Wittgensteinian approach to the Philosophy of Mathematics. Wittgenstein’s disdain for Cantor is well documented (this short note is not an attempt to justify, using Category theory, a Wittgensteinian criticism of set theory). More specifically however, it was Abramsky and Coecke’s “Categorical Quantum Mechanics” that helped me to discern more carefully the links between Category Theory and Quantum Computing. They describe Category Theory as the ‘language of modern structural mathematics’ and use it as the tool for building a mathematical representation of quantum processes, and their paper is a thought provoking nudge in the ribs for anyone who is trying to make sense of the current noise that surrounds Quantum mechanics.
Awodey and Spivak are the two most impressive contemporary mathematicians currently working on Category Theory in my view, and whilst it is asking for trouble to choose one or two selected works as exemplars of their approach, I would have to say that Spivak’s book on Category Theory for the Sciences is the standout work of recent times (incidentally the section in this book on ‘aspects’ bears close scrutiny with Wittgenstein’s well known work on ‘family resemblances’).
Awodey’s 2003 paper is as good a recent balance between a mathematical and philosophical exposition of the importance of category theory as exists whilst his textbook is often referred to as the standard entry point for working mathematicians.
Going back to beauty, which is how I started this short note. Barry Mazur wrote an article in memory of Saunders Mac Lane titled ‘When is one thing equal to another‘ which is a gem of rare beauty, and the actual catalyst for this short note. If you read only one document in the links from this article, then I hope it is Mazur’s paper.
I’ll also give him a shout out for being a mathematician with a fledgling blog: Rick’s Ramblings.
The Postdoctoral Experience Revisited builds on the 2000 report Enhancing the Postdoctoral Experience for Scientists and Engineers. That ground-breaking report assessed the postdoctoral experience and provided principles, action points, and recommendations to enhance that experience. Since the publication of the 2000 report, the postdoctoral landscape has changed considerably. The percentage of PhDs who pursue postdoctoral training is growing steadily and spreading from the biomedical and physical sciences to engineering and the social sciences. The average length of time spent in postdoctoral positions seems to be increasing. The Postdoctoral Experience Revisited reexamines postdoctoral programs in the United States, focusing on how postdocs are being guided and managed, how institutional practices have changed, and what happens to postdocs after they complete their programs. This book explores important changes that have occurred in postdoctoral practices and the research ecosystem and assesses how well current practices meet the needs of these fledgling scientists and engineers and of the research enterprise. The Postdoctoral Experience Revisited takes a fresh look at current postdoctoral fellows - how many there are, where they are working, in what fields, and for how many years. This book makes recommendations to improve aspects of programs - postdoctoral period of service, title and role, career development, compensation and benefits, and mentoring. Current data on demographics, career aspirations, and career outcomes for postdocs are limited. This report makes the case for better data collection by research institution and data sharing. A larger goal of this study is not only to propose ways to make the postdoctoral system better for the postdoctoral researchers themselves but also to better understand the role that postdoctoral training plays in the research enterprise. It is also to ask whether there are alternative ways to satisfy some of the research and career development needs of postdoctoral researchers that are now being met with several years of advanced training. Postdoctoral researchers are the future of the research enterprise. The discussion and recommendations of The Postdoctoral Experience Revisited will stimulate action toward clarifying the role of postdoctoral researchers and improving their status and experience.
The National Academy of Sciences has published a (free) book: The Postdoctoral Experience (Revisited) discussing where we’re at and some ideas for a way forward.
Most might agree that our educational system is far less than ideal, but few pay attention to significant problems at the highest levels of academia which are holding back a great deal of our national “innovation machinery”. The National Academy of Sciences has published a (free) book: The Postdoctoral Experience (Revisited) discussing where we’re at and some ideas for a way forward. There are some interesting ideas here, but we’ve still got a long way to go.
Information theory provides a constructive criterion for setting up probability distributions on the basis of partial knowledge, and leads to a type of statistical inference which is called the maximum-entropy estimate. It is the least biased estimate possible on the given information; i.e., it is maximally noncommittal with regard to missing information. If one considers statistical mechanics as a form of statistical inference rather than as a physical theory, it is found that the usual computational rules, starting with the determination of the partition function, are an immediate consequence of the maximum-entropy principle. In the resulting "subjective statistical mechanics," the usual rules are thus justified independently of any physical argument, and in particular independently of experimental verification; whether or not the results agree with experiment, they still represent the best estimates that could have been made on the basis of the information available.
It is concluded that statistical mechanics need not be regarded as a physical theory dependent for its validity on the truth of additional assumptions not contained in the laws of mechanics (such as ergodicity, metric transitivity, equal a priori probabilities, etc.). Furthermore, it is possible to maintain a sharp distinction between its physical and statistical aspects. The former consists only of the correct enumeration of the states of a system and their properties; the latter is a straightforward example of statistical inference.
We describe the evolution of macromolecules as an information transmission process and apply tools from Shannon information theory to it. This allows us to isolate three independent, competing selective pressures that we term compression, transmission, and neutrality selection. The first two affect genome length: the pressure to conserve resources by compressing the code, and the pressure to acquire additional information that improves the channel, increasing the rate of information transmission into each offspring. Noisy transmission channels (replication with mutations) gives rise to a third pressure that acts on the actual encoding of information; it maximizes the fraction of mutations that are neutral with respect to the phenotype. This neutrality selection has important implications for the evolution of evolvability. We demonstrate each selective pressure in experiments with digital organisms.
To be published in J. theor. Biology 222 (2003) 477-483
This is the third in a series of three papers devoted to energy flow and entropy changes in chemical and biological processes, and their relations to the thermodynamics of computation. The previous two papers have developed reversible chemical transformations as idealizations for studying physiology and natural selection, and derived bounds from the second law of thermodynamics, between information gain in an ensemble and the chemical work required to produce it. This paper concerns the explicit mapping of chemistry to computation, and particularly the Landauer decomposition of irreversible computations, in which reversible logical operations generating no heat are separated from heat-generating erasure steps which are logically irreversible but thermodynamically reversible. The Landauer arrangement of computation is shown to produce the same entropy-flow diagram as that of the chemical Carnot cycles used in the second paper of the series to idealize physiological cycles. The specific application of computation to data compression and error-correcting encoding also makes possible a Landauer analysis of the somewhat different problem of optimal molecular recognition, which has been considered as an information theory problem. It is shown here that bounds on maximum sequence discrimination from the enthalpy of complex formation, although derived from the same logical model as the Shannon theorem for channel capacity, arise from exactly the opposite model for erasure.
This is the second in a series of three papers devoted to energy flow and entropy changes in chemical and biological processes, and to their relations to the thermodynamics of computation. In the first paper of the series, it was shown that a general-form dimensional argument from the second law of thermodynamics captures a number of scaling relations governing growth and development across many domains of life. It was also argued that models of physiology based on reversible transformations provide sensible approximations within which the second-law scaling is realized. This paper provides a formal basis for decomposing general cyclic, fixed-temperature chemical reactions, in terms of the chemical equivalent of Carnot's cycle for heat engines. It is shown that the second law relates the minimal chemical work required to perform a cycle to the Kullback–Leibler divergence produced in its chemical output ensemble from that of a Gibbs equilibrium. Reversible models of physiology are used to create reversible models of natural selection, which relate metabolic energy requirements to information gain under optimal conditions. When dissipation is added to models of selection, the second-law constraint is generalized to a relation between metabolic work and the combined energies of growth and maintenance.
This is the first of three papers analyzing the representation of information in the biosphere, and the energetic constraints limiting the imposition or maintenance of that information. Biological information is inherently a chemical property, but is equally an aspect of control flow and a result of processes equivalent to computation. The current paper develops the constraints on a theory of biological information capable of incorporating these three characterizations and their quantitative consequences. The paper illustrates the need for a theory linking energy and information by considering the problem of existence and reslience of the biosphere, and presents empirical evidence from growth and development at the organismal level suggesting that the theory developed will capture relevant constraints on real systems. The main result of the paper is that the limits on the minimal energetic cost of information flow will be tractable and universal whereas the assembly of more literal process models into a system-level description often is not. The second paper in the series then goes on to construct reversible models of energy and information flow in chemistry which achieve the idealized limits, and the third paper relates these to fundamental operations of computation.
The most significant legacy of philosophical skepticism is the realization that our concepts, beliefs and theories are social constructs. This belief has led to epistemological relativism, or the thesis that since there is no ultimate truth about the world, theory preferences are only a matter of opinion. In this book, William Harms seeks to develop the conceptual foundations and tools for a science of knowledge through the application of evolutionary theory, thus allowing us to acknowledge the legacy of skepticism while denying its relativistic offspring.
No one can escape a sense of wonder when looking at an organism from within. From the humblest amoeba to man, from the smallest cell organelle to the amazing human brain, life presents us with example after example of highly ordered cellular matter, precisely organized and shaped to perform coordinated functions. But where does this order spring from? How does a living organism manage to do what nonliving things cannot do--bring forth and maintain all that order against the unrelenting, disordering pressures of the universe? In The Touchstone of Life, world-renowned biophysicist Werner Loewenstein seeks answers to these ancient riddles by applying information theory to recent discoveries in molecular biology. Taking us into a fascinating microscopic world, he lays bare an all-pervading communication network inside and between our cells--a web of extraordinary beauty, where molecular information flows in gracefully interlaced circles. Loewenstein then takes us on an exhilarating journey along that web and we meet its leading actors, the macromolecules, and see how they extract order out of the erratic quantum world; and through the powerful lens of information theory, we are let in on their trick, the most dazzling of magician's acts, whereby they steal form out of formlessness. The Touchstone of Life flashes with fresh insights into the mystery of life. Boldly straddling the line between biology and physics, the book offers a breathtaking view of that hidden world where molecular information turns the wheels of life. Loewenstein makes these complex scientific subjects lucid and fascinating, as he sheds light on the most fundamental aspects of our existence.
This highly interdisciplinary book discusses the phenomenon of life, including its origin and evolution (and also human cultural evolution), against the background of thermodynamics, statistical mechanics, and information theory. Among the central themes is the seeming contradiction between the second law of thermodynamics and the high degree of order and complexity produced by living systems. This paradox has its resolution in the information content of the Gibbs free energy that enters the biosphere from outside sources, as the author shows. The role of information in human cultural evolution is another focus of the book. One of the final chapters discusses the merging of information technology and biotechnology into a new discipline — bio-information technology.
Information Theory, Evolution and the Origin of Life presents a timely introduction to the use of information theory and coding theory in molecular biology. The genetical information system, because it is linear and digital, resembles the algorithmic language of computers. George Gamow pointed out that the application of Shannon's information theory breaks genetics and molecular biology out of the descriptive mode into the quantitative mode and Dr Yockey develops this theme, discussing how information theory and coding theory can be applied to molecular biology. He discusses how these tools for measuring the information in the sequences of the genome and the proteome are essential for our complete understanding of the nature and origin of life. The author writes for the computer competent reader who is interested in evolution and the origins of life.
The transmission of genomic information from coding sequence to protein structure during protein synthesis is subject to stochastic errors. To analyze transmission limits in the presence of spurious errors, Shannon's noisy channel theorem is applied to a communication channel between amino acid sequences and their structures established from a large-scale statistical analysis of protein atomic coordinates. While Shannon's theorem confirms that in close to native conformations information is transmitted with limited error probability, additional random errors in sequence (amino acid substitutions) and in structure (structural defects) trigger a decrease in communication capacity toward a Shannon limit at 0.010 bits per amino acid symbol at which communication breaks down. In several controls, simulated error rates above a critical threshold and models of unfolded structures always produce capacities below this limiting value. Thus an essential biological system can be realistically modeled as a digital communication channel that is (a) sensitive to random errors and (b) restricted by a Shannon error limit. This forms a novel basis for predictions consistent with observed rates of defective ribosomal products during protein synthesis, and with the estimated excess of mutual information in protein contact potentials.