In the paper we discuss the problem of limitations of freedom in mathematics and search for criteria which would differentiate the new concepts stemming from the historical ones from the new concepts that have opened unexpected ways of thinking and reasoning. We also investigate the emergence of category theory and its origins. In particular we explore the origins of the term functor and present the strong evidence that Eilenberg and Carnap could have learned the term from Kotarbiński and Tarski.

Saunders Mac Lane famously remarked that "Bourbaki just missed" formulating adjoints in a 1948 appendix (written no doubt by Pierre Samuel) to an early draft of Algebre--which then had to wait until Daniel Kan's 1958 paper on adjoint functors. But Mac Lane was using the orthodox treatment of adjoints that only contemplates the object-to-object morphisms within a category, i.e., homomorphisms. When Samuel's treatment is reconsidered in view of the treatment of adjoints using heteromorphisms or hets (object-to-object morphisms between (...) objects in different categories), then he, in effect, isolated the concept of a left representation solving a universal mapping problem. When dualized to obtain the concept of a right representation, the two halves only need to be united to obtain an adjunction. Thus Samuel was only a now-simple dualization away for formulating adjoints in 1948. Apparently, Bodo Pareigis' 1970 text was the first and perhaps only text to give the heterodox "new characterization" (i.e., heteromorphic treatment) of adjoints. Orthodox category theory uses various relatively artificial devices to avoid formally recognizing hets--even though hets are routinely used by the working mathematician. Finally we consider a "philosophical" question as to whether the most important concept in category theory is the notion of an adjunction or the notion of a representation giving a universal mapping property (where adjunctions arise as the special case of a bi-representation of dual universal mapping problems). (shrink)

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one (...) that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks. (shrink)

We are used to seeing foundations linked to the mainstream mathematics of the late nineteenth century: the arithmetization of analysis, non-Euclidean geometry, and the rise of abstract structures in algebra. And a growing number of case studies bring a more philosophy-of-science viewpoint to the latest mathematics, as in [Carter, 2005; Corfield, 2006; Krieger, 2003; Leng, 2002]. Mac Lane's autobiography is a valuable bridge between these, recounting his experience of how the mid- and late-twentieth-century mainstream grew especially through Hilbert's school.An (...) autobiography at age 94 obviously has a lot of ground to cover. Mac Lane entered Yale in 1926 to study chemistry as a good practical career field but by the end of his first year he had won a $50 prize in mathematics and learned that you could make a living with it—as an actuary. He decided to do that. The next year he was excited to learn from his philosophy professor F.S.C. Northrop that you can also pursue new mathematical discoveries! Northrop had studied with A.N. Whitehead and sold Mac Lane on the excitement of Principia Mathematica. In a pattern that foreshadowed his career Mac Lane bought and annotated a copy of the first volume of Principia but his planned tutorial study of the book turned into a study of [Hausdorff, 1914] applying set theory to topology and analysis. As a graduate student at Chicago he was powerfully influenced by E.H. Moore, who taught the new axiomatic methods as tools for unifying and advancing the most classical analysis . Then he went to Göttingen. He heard Hilbert lecture on philosophy. He studied mathematics and its philosophy with Weyl. He followed Noether's lectures on abstract algebra for number theory. He had gone there to study logic and he wrote a …. (shrink)

Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions . Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered . Two derivations of the same sequent are equivalent if and only if (...) all their interpretations in K are equal. In fact, the assignment of values to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the “abstract coherence theorem” found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture also proves the correctness of the modified description. (shrink)

While Saunders Mac Lane studied for his D.Phil in Göttingen, he heard David Hilbert's weekly lectures on philosophy, talked philosophy with Hermann Weyl, and studied it with Moritz Geiger. Their philosophies and Emmy Noether's algebra all influenced his conception of category theory, which has become the working structure theory of mathematics. His practice has constantly affirmed that a proper large-scale organization for mathematics is the most efficient path to valuable specific results—while he sees that the question of which results (...) are valuable has an ineliminable philosophic aspect. His philosophy relies on the ideas of truth and existence he studied in Göttingen. His career is a case study relating naturalism in philosophy of mathematics to philosophy as it naturally arises in mathematics. Introduction Structures and Morphisms Varieties of Structuralism Göttingen Logic: Mac Lane's Dissertation Emmy Noether Natural Transformations Grothendieck: Toposes and Universes Lawvere and Foundations Truth and Existence Naturalism Austere Forms of Beauty. (shrink)

This note describes Saunders Mac Lane as a philosopher, and indeed as a paragon naturalist philosopher. He approaches philosophy as a mathematician. But, more than that, he learned philosophy from David Hilbert’s lectures on it, and by discussing it with Hermann Weyl, as much as he did by studying it with the mathematically informed Göttingen Philosophy professor Moritz Geiger.

Byrne & Russon's account of program imitation in primates involves propositional attitudes (expectations and goals), which limits its falsifiability. Yet their account of priming shows exactly how imitation without attitudes would look. The challenge is to upgrade the notion of priming to give an account of low-level program imitation without invoking propositional attitudes.

This article addresses some useful coordinates for the analysis of the relationship between philosophy and taxation. My argument aims to show the relevance of a philosophy of taxation and thus draw attention to civic education required to shape a democratic public reason, truthful, critical and attentive. I emphasize the character of “donation” that is the rule of law within a democratic constitutional regime, which depends on tax payments for the purposes proposed, and particularly those of a social distributive justice. I (...) distinguish between charity and justice and political obligation to maintain that by implementing an effective "rule of law" citizens “donate” ourselves mechanisms and procedures for advancing a system of justice as fairness. This grant does not pertain to the field of natural duties, charity or sacrifice of supererogatory acts, but is expressed in institutional programs whose implementation and maintenance required from us pay taxes, but always based on personal commitment. Thus, close to John Rawls I emphasize the value of freedom. (shrink)

Rousseau argued forcefully for the superiority of a life lived in accordance with “the simplest impulses of nature,” but his complex understanding of the relationship between humans and “nature” is rarely cited as a source of inspiration by those seeking to reform the human relationship with the natural world. We argue that the complexities of Rousseau's political thought illuminate important connections between his works and the programs put forth by deep ecology. In Part One, we explore the theoretical connections between (...) Rousseau's account of the human fall from nature and major works of radical environmentalism. In Part Two, we offer suggestions for a reconsideration of Rousseau's work that may illuminate the paradoxical political requirements of deep ecology's recommendations for a more ecological human life. We hope to illustrate howa careful reading of Rousseau'swork may serve as the basis for fruitful questioning of environmentalist thought. (shrink)

Rousseau argued forcefully for the superiority of a life lived in accordance with “the simplest impulses of nature,” but his complex understanding of the relationship between humans and “nature” is rarely cited as a source of inspiration by those seeking to reform the human relationship with the natural world. We argue that the complexities of Rousseau's political thought illuminate important connections between his works and the programs put forth by deep ecology. In Part One, we explore the theoretical connections between (...) Rousseau's account of the human fall from nature and major works of radical environmentalism. In Part Two, we offer suggestions for a reconsideration of Rousseau's work that may illuminate the paradoxical political requirements of deep ecology's recommendations for a more ecological human life. We hope to illustrate howa careful reading of Rousseau'swork may serve as the basis for fruitful questioning of environmentalist thought. (shrink)

Rousseau argued forcefully for the superiority of a life lived in accordance with "the simplest impulses of nature," but his complex (some would say contradictory) understanding of the relationship between humans and "nature" is rarely cited as a source of inspiration by those seeking to reform the human relationship with the natural world. We argue that the complexities of Rousseau's political thought illuminate important connections between his works and the programs put forth by deep ecology. In Part One, we explore (...) the theoretical connections between Rousseau's account of the human fall from nature and major works of radical environmentalism. In Part Two, we offer suggestions for a reconsideration of Rousseau's work that may illuminate the paradoxical political requirements of deep ecology's recommendations for a more ecological human life. We hope to illustrate how a careful reading of Rousseau's work may serve as the basis for fruitful questioning of environmentalist thought. (shrink)

It has been convincingly argued that computer simulation modeling differs from traditional science. If we understand simulation modeling as a new way of doing science, the manner in which scientists learn about the world through models must also be considered differently. This article examines how researchers learn about environmental processes through computer simulation modeling. Suggesting a conceptual framework anchored in a performative philosophical approach, we examine two modeling projects undertaken by research teams in England, both aiming to inform flood risk (...) management. One of the modeling teams operated in the research wing of a consultancy firm, the other were university scientists taking part in an interdisciplinary project experimenting with public engagement. We found that in the first context the use of standardized software was critical to the process of improvisation, the obstacles emerging in the process concerned data and were resolved through exploiting affordances for generating, organizing, and combining scientific information in new ways. In the second context, an environmental competency group, obstacles were related to the computer program and affordances emerged in the combination of experience-based knowledge with the scientists’ skill enabling a reconfiguration of the mathematical structure of the model, allowing the group to learn about local flooding. (shrink)

This study is concerned with processes for discovering new theories in science. It considers a computational approach to scientific discovery, as applied to the discovery of theories in cognitive science. The approach combines two ideas. First, a process-based scientific theory can be represented as a computer program. Second, an evolutionary computational method, genetic programming, allows computer programs to be improved through a process of computational trialand-error. Putting these two ideas together leads to a system that can automatically generate and (...) improve scientific theories. The application of this method to the discovery of theories in cognitive science is examined. Theories are built up from primitive operators. These are contained in a theory language that defines the space of possible theories. An example of a theory generated by this method is described. These results support the idea that scientific discovery can be achieved through a heuristic search process, even for theories involving a sequence of steps. However, this computational approach to scientific discovery does not eliminate the need for human input. Human judgment is needed to make reasonable prior assumptions about the characteristics of operators used in the theory generation process, and to interpret and provide context for the computationally generated theories. (shrink)

James Franklin aspires to a realist view of mathematical objects as concrete, rather than abstract, objects. It is shown that he fails to carry out his program but is forced to revert to Platonism.

In lieu of an abstract, here is a brief excerpt of the content:Black Horseman Lane: A ReflectionJanet PniewskiI felt a sinking feeling in the pit of my stomach upon getting the news this particular patient, let’s call him Stan, had burned through yet another nurse case manager and it would now be my responsibility to take charge of his care. As the medical director read aloud his patient profile, “Sixty–eight year old frail appearing Caucasian male with a terminal diagnosis (...) of...” I was already formulating excuses in my head as to why I would not be able to accept this charge. However, when she finished, I dutifully replied, “Ok, I’ll give it my best shot.” At the time, I was well aware that this assignment would test my resolve and commitment to providing the same high quality compassionate [End Page 117] care to every patient, but what I did not realize was how much I would grow personally and professionally in the process.Stan was well known in the organization for being “non–compliant” and “difficult.” Through his past actions he earned the label of racist and misogynist, having zero tolerance for people whom he perceived as different from him in any way. He was profoundly disrespectful toward women and was known to abuse his animals. Having been raised in a liberal household in New Jersey, surrounded by civil rights activists, animal activists and feminists, I was genetically programmed to abhor social injustice and raised to serve as an advocate for the disenfranchised and vulnerable. Since childhood, I’ve experienced visceral reactions to acts of injustice or cruelty of any kind. The very idea of someone like Stan was appalling.I found myself assigned to Stan’s case by default. He fired his first nurse because she was African American and painted her as dishonest and incompetent. He fired his second nurse because he was male and therefore must be gay. He fired his social worker because she was overweight, which certainly meant she was lazy. He fired the chaplain because he was not Catholic and thus unworthy of having any meaningful spiritual discourse. He would not accept the recommendations of our medical director because she was a woman. His family had given up on him a long time ago. His wife left him shortly after his youngest child had gone off to college, and all four of his children were estranged. Stan’s disease was progressing, despite therapy, and he was becoming too weak to care for himself at home without hospice support. His oncologist pleaded with us to give it one more try, even though we were running out of staff. These people that Stan had dismissed so unpleasantly were my colleagues and friends. The first nurse Stan dismissed served as my ever patient and knowledgeable preceptor when I was new to hospice and fumbling my way through unfamiliar protocols. The second nurse Stan rejected was a dedicated colleague and devoted friend, supporting me through several life challenges. I had witnessed our chaplain, judged as unworthy by Stan, tirelessly provide comfort to patients, family members and staff members in the darkest of hours. I knew them all as well–meaning, competent and compassionate team members and Stan’s disrespect for them offended me.Apparently, Stan was offended as well. His oncologist revealed to us that Stan was angered by his referral to hospice. He felt abandoned. I was Stan’s last hope; however, he didn’t see it that way. Neither Stan nor I was enthusiastic about our impending partnership.Stan was an engineer by trade, and apparently very successful in business. He lived on a sprawling plantation located at the end of his private road, which he named Black Horseman Lane. As I turned down the drive, aware of his reputation as a bigot, I contemplated how Stan might have arrived at this name. With the assistance of my smartphone, I was able to ascertain that there was, indeed, a Black Horseman of the Apocalypse, which was associated with famine. Delving further, into Wikipedia, my interpretation was that the Black Horseman’s famine decimated the grain crop, sparing the olive groves and grapevines, driving up the price of grain but leaving oil and wine... (shrink)

Large sets of elements interacting locally and producing specific architectures reliably form a category that transcends the usual dividing line between biological and engineered systems. We propose to call them morphogenetically architected complex systems. While taking the emergence of properties seriously, the notion of MACS enables at the same time the design of operational means that allow controlling and even, paradoxically, programming this emergence. To demonstrate our claim, we first show that among all the self-organized systems studied in the field (...) of Artificial Life, the specificity of MACS essentially lies in the close relation between their emergent properties and functional properties. Second, we argue that to be a MACS a system does not need to display more than weak emergent properties. Third, since the notion of weak emergence is based on the possibility of simulation, whether computational or mechanistic via machines, we see MACS as good candidates to help design artificial self-architected systems but also harness and redesign living ones. (shrink)

Large sets of elements interacting locally and producing specific architectures reliably form a category that transcends the usual dividing line between biological and engineered systems. We propose to call them morphogenetically architected complex systems (MACS). While taking the emergence of properties seriously, the notion of MACS enables at the same time the design (or “meta-design”) of operational means that allow controlling and even, paradoxically, programming this emergence. To demonstrate our claim, we first show that among all the self-organized systems studied (...) in the field of Artificial Life, the specificity of MACS essentially lies in the close relation between their emergent properties and functional properties. Second, we argue that to be a MACS a system does not need to display more than weak emergent properties. Third, since the notion of weak emergence is based on the possibility of simulation, whether computational or mechanistic via machines, we see MACS as good candidates to help design artificial self-architected systems (such as robotic swarms) but also harness and redesign living ones (such as synthetic bacterial films). (shrink)

This paper considers various objections to Carnap’s logical syntax definition of ’logical expression’, including those by Saunders Mac Lane and W. V. O. Quine. While the specific objections of these two authors can be answered, if necessary by a slight modification of Carnap’s definition, there are other objections that I do not see how to meet. I also consider the proposal by Denis Bonnay for avoiding the objections to Carnap’s definition. In light of the unresolved problems with Carnap’s definition, (...) I go on to consider what Carnap’s assumptions must have been in framing his definition and to assess how much damage is caused by this failure of Carnap’s definition. This damage is not as much as might be assumed. (shrink)