Over the past few decades the notion of symmetry has played a major role in physics and in the philosophy of physics. Philosophers have used symmetry to discuss the ontology and seeming objectivity of the laws of physics. We introduce several notions of symmetry in mathematics and explain how they can also be used in resolving different problems in the philosophy of mathematics. We use symmetry to discuss the objectivity of mathematics, the role of mathematical objects, the unreasonable effectiveness of (...) mathematics and the relationship of mathematics to physics. (shrink)

Fefermans argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.

The concept of process is ubiquitous in science, engineering and everyday life. Category theory, and monoidal categories in particular, provide an abstract framework for modelling processes of many kinds. In this paper, we concentrate on sequential and parallel decomposability of processes in the framework of monoidal categories: We will give a precise definition, what it means for processes to be decomposable. Moreover, through examples, we argue that viewing parallel processes as coupled in this framework can be seen as a category (...) mistake or a misinterpretation. We highlight the suitability of category theory for a structuralistic interpretation of mathematical modelling and argue that for appliers of mathematics, such as engineers, there is a pragmatic advantage from this. (shrink)

In science as in mathematics, it is popular to know little and resent much about category theory. Less well known is how common it is to know little and like much about set theory. The set theory of almost all scientists, and even the average mathematician, is fundamentally different from the formal set theory that is contrasted against category theory. The latter two are often opposed by saying one emphasizes Substance, the other Form. However, in all known systems of mathematics (...) throughout history, mathematicians have moved fluidly between ideas conceived of as thing-like, property-like, and process-like. On the other hand one way to advance science is to better distinguish between thing, property, and process. All this constitutes a distracting background for those interested in, or distressed by, the possible application of category theory to science, and to mathematics as well. (shrink)

In science as in mathematics, it is popular to know little and resent much about category theory. Less well known is how common it is to know little and like much about set theory. The set theory of almost all scientists, and even the average mathematician, is fundamentally different from the formal set theory that is contrasted against category theory. The latter two are often opposed by saying one emphasizes Substance, the other Form. However, in all known systems of mathematics (...) throughout history, mathematicians have moved fluidly between ideas conceived of as thing-like, property-like, and process-like. On the other hand one way to advance science is to better distinguish between thing, property, and process. All this constitutes a distracting background for those interested in, or distressed by, the possible application of category theory to science, and to mathematics as well. (shrink)

(No abstract is available for this citation).

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most (...) one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

The reductionist/holist debate seems an impoverished one, with many participants appearing to adopt a position first and constructing rationalisations second. Here I propose an intermediate position of pragmatic holism, that irrespective of whether all natural systems are theoretically reducible, for many systems it is completely impractical to attempt such a reduction, also that regardless if whether irreducible `wholes' exist, it is vain to try and prove this in absolute terms. This position thus illuminates the debate along new pragmatic lines, and (...) refocusses attention on the underlying heuristics of learning about the natural world. (shrink)

A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.

In this paper we study the axiomatic system proposed by Bourbaki for the Theory of Sets in the Éléments de Mathématique. We begin by examining the role played by the sign \(\uptau \) in the framework of its formal logical theory and then we show that the system of axioms for set theory is equivalent to Zermelo–Fraenkel system with the axiom of choice but without the axiom of foundation. Moreover, we study Grothendieck’s proposal of adding to Bourbaki’s system the axiom (...) of universes for the purpose of considering the theory of categories. In this regard, we make some historical and epistemological remarks that could explain the conservative attitude of the Group. (shrink)

This dissertation makes two primary contributions. The first three chapters develop an interpretation of Carnap's Meta-Philosophical Program which places stress upon his methodological analysis of the sciences over and above the Principle of Tolerance. Most importantly, I suggest, is that Carnap sees philosophy as contiguous with science—as a part of the scientific enterprise—so utilizing the very same methods and subject to the same limitations. I argue that the methodological reforms he suggests for philosophy amount to philosophy as the explication of (...) the concepts of science through the construction and use of suitably robust meta-logical languages. My primary interpretive claim is that Carnap's understanding of logic and mathematics as a set of formal auxiliaries is premised upon this prior analysis of the character of logico-mathematical knowledge, his understanding of its role in the language of science, and the methods used by practicing mathematicians. Thus the Principle of Tolerance, and so Carnap's logical pluralism, is licensed and justified by these methodological insights. This interpretation of Carnap's program contrasts with the popular Deflationary reading as proposed in Goldfarb & Ricketts. The leading idea they attribute to Carnap is a Logocentrism: That philosophical assertions are always made relative to some particular language, and that our choice of syntactical rules for a language are constitutive of its inferential structure and methods of possible justification. Consequently Tolerance is considered the foundation of Carnap's entire program. My third chapter argues that this reading makes Carnap's program philosophically inert, and I present significant evidence that such a reading is misguided. The final chapter attempts to extend the methodological ideals of Carnap's program to the analysis of the ongoing debate between category- and set-theoretic foundations for mathematics. Recent criticism of category theory as a foundation charges that it is neither autonomous from set theory, nor offers a suitable ontological grounding for mathematics. I argue that an analysis of concepts can be foundationally informative without requiring the construction of those concepts from first principles, and that ontological worries can be seen as methodologically unfruitful. (shrink)

When modelling complex systems one can not include all the causal factors, but one has to settle for partial models. This is alright if the factors left out are either so constant that they can be ignored or one is able to recognise the circumstances when they will be such that the partial model applies. The transference of knowledge from the point of application to the point of learning utilises a combination of recognition and inference ­ a simple model of (...) the important features is learnt and later situations where inferences can be drawn from the model are recognised. Context is an abstraction of the collection of background features that are later recognised. Different heuristics for recognition and model formulation will be effective for different learning tasks. Each of these will lead to a different type of context. Given this, there are (at least) two ways of modelling context: one can either attempt to investigate the contexts that arise out of the heuristics that a particular agent actually applies (the `internal' approach); or (if this is feasible) one can attempt to model context using the external source of regularity that the heuristics exploit. There are also two basic methodologies for the investigation of context: a top-down (or `foundationalist') approach where one tries to lay down general, a priori principles and a bottom-up (or `scientific') approach where one can try and find what sorts of context arise by experiment and simulation. A simulation is exhibited which is designed to illustrate the practicality of the bottom-up approach in elucidating the sorts of internal context that arise in an artificial agent which is attempting to learn simple models of a complex environment. It ends with a plea for the cooperation of the AI and Machine Learning communities as both learning and inference is needed if context is to make complete sense. (shrink)

The aim of this paper is to re-emphasise that the purpose of formal systems is to provide something to map into and to stem the tide of unjustified formal systems. I start by arguing that expressiveness alone is not a sufficient justification for a new formal system but that it must be justified on pragmatic grounds. I then deal with a possible objection as might be raised by a pure mathematician and after that to the objection that theory can be (...) later used by more specific models. I go on to compare two different methods of developing new formal systems: by a priori principles and intuitions; and by post hoc generalisation from data and examples. I briefly describe the phenomena of “social embedding” and use it to explain the social processes that underpin “normal” and “revolutionary” science. This suggests social grounds for the popularity of normal science. I characterise the “foundational” and “empirical” approaches to the use of formal systems and situate these with respect to “normal” and “revolutionary” modes of science. I suggest that successful sciences (in the sense of developing relevant mappings to formal systems) bare either more tolerant of revolutionary ideas or this tolerance is part of the reason they are successful. I finish by enumerating a number of ‘tell-tale’ signs that a paper is presenting an unjustified formal system. (shrink)