Michele Friend

I present a comparative and holistic method for qualitatively measuring sound ecological practice in chemistry. I consider chemicals developed and used by man from cradle to grave, that is, from the moment they are extracted from the earth, biomass, water or air, to their transportation, purification, mixing and elaboration in a factory, to their distribution by means of the market, to waste products both from the factory, packaging, transportations and by the consumer. I divide the locations of the ‘life’ of the chemical into four spatio-temporal areas accordingly. I then use the ‘instituional compass’ method to determine a qualitative reading of the ecological soundness of the practice, where practice means the research, the adoption by industry and the distribution at scale on the market. The qualitative reading is in the form of an arrow on a trisected circle. The arrow holistically represents a table of data. The data can be economic, social or environmental. The arrow has a measurement: a degree and a length. The degrees, represent qualities spaning through: harmony, discipline and excitement. The length represents the importance, momentum or amplitude with which the quality is present. We use the compass method to compare the same product over time, or inter-substitutable, chemicals developed in different places, using different equipment or processes. In the conclusion, I discuss objectivity and science as they apply to the compass.

Hilbert’s axiomatic approach to the sciences was characterized by a dynamic methodology tied to scientific and mathematical fields under investigation. In particular, it is an analytic art for choosing axioms but, at the same time, it has to include dynamically synthetic procedures and meta-theoretical reflections. Axioms have to be useful, or capture something, or help as part of explanations. The Andréka-Németi group use several formal axiomatic theories together to re-capture, predict, recover or explain the phenomena of special relativity, general relativity and classical kinematics. Their scientific methodology is close to Hilbert’s conceptions of how science should be done ideally. In this paper, we compare Formica’s reading of Hilbert’s axiomatic method to the method used by the Andréka-Németi group.

The phrase ‘mathematical foundation’ has shifted in meaning since the end of the nineteenth century. It used to mean a consistent general theory in mathematics, based on basic principles and ideas to which the rest of mathematics could be reduced. There was supposed to be only one foundational theory and it was to carry the philosophical weight of giving the ultimate ontology and truth of mathematics. Under this conception of ‘foundation’ pluralism in foundations of mathematics is a contradiction.More recently, the phrase has come to mean a perspective from which we can see, or in which we can interpret, much of mathematics; it has lost the realist-type metaphysical, essentialist importance. The latter has been replaced with an emphasis on epistemology. The more recent use of the phrase shows a lack of concern for absolute ontology, truth, uniqueness and sometimes even consistency. It is only under the more modern conception of ‘foundation’ that pluralism in mathematical foundations is conceptually possible.Several problems beset the pluralist in mathematical foundations. The problems include, at least: paradox, rampant relativism, loss of meaning, insurmountable complexity and a rising suspicion that we can say anything meaningful about truth and objectivity in mathematics. Many of these are related to each other, and many can be overcome, explained, accounted for and dissolved by concentrating on crosschecking, fixtures and rigour of proof. Moreover, apart from being a defensible position, there are a lot of advantages to pluralism in foundations. These include: a sensitivity to the practice of mathematics, a more faithful account of the objectivity of mathematics and a deeper understanding of mathematics.The claim I defend in the paper is that we stand to learn more, not less, by adopting a pluralist attitude. I defend the claim by looking at the examples of set theory and homotopy type theory, as alternative viewpoints from which we can learn about mathematics. As the claim is defended, it will become apparent that ‘pluralism in mathematical foundations’ is neither an oxymoron, nor a contradiction, at least not in any threatening sense. On the contrary, it is the tension between different foundations that spurs new developments in mathematics. The tension might be called ‘a fruitful meta-contradiction’.I take my prompts from Kauffman’s idea of eigenform, Hersh’s idea of thinking of mathematical theories as models and from my own philosophical position: pluralism in mathematics. I also take some hints from the literature on philosophy of chemistry, especially the pluralism of Chang and Schummer.

Most scientific theories are globally inconsistent. Chunk and Permeate is a method of rational reconstruction that can be used to separate, and identify, locally consistent chunks of reasoning or explanation. This then allows us to justify reasoning in a globally inconsistent theory. We extend chunk and permeate by adding a visually transparent way of guiding the individuation of chunks and deciding on what information permeates from one chunk to the next. The visual representation is in the form of bundle diagrams. We then extend the bundle diagrams to include not only reasoning in the presence of inconsistent information or reasoning in the logical sense of deriving a conclusion from premises, but more generally reasoning in the sense of trying to understand a phenomenon in science. This extends the use of the bundle diagrams in terms of the base space and the fibres. We then apply this to a case in physics, that of understanding binding energies in the nucleus of an atom using together inconsistent models: the liquid drop model and the shell model. We draw some philosophical conclusions concerning scientific reasoning, paraconsistent reasoning, the role of logic in science and the unity of science.

In the literature, there have been several methods and definitions for working out whether two theories are “equivalent” or not. In this article, we do something subtler. We provide a means to measure distances between formal theories. We introduce two natural notions for such distances. The first one is that of axiomatic distance, but we argue that it might be of limited interest. The more interesting and widely applicable notion is that of conceptual distance which measures the minimum number of concepts that distinguish two theories. For instance, we use conceptual distance to show that relativistic and classical kinematics are distinguished by one concept only.

"Second-order logic" is the name given to a formal system. Some claim that the formal system is a logical system. Others claim that it is a mathematical system. In the thesis, I examine these claims in the light of some philosophical criteria which first motivated Frege in his logicist project. The criteria are that a logic should be universal, it should reflect our intuitive notion of logical validity, and it should be analytic. The analysis is interesting in two respects. One is conceptual: it gives us a purchase on where and how to draw a distinction between logic and other sciences. The other interest is historical: showing that second-order logic is a logical system according to the philosophical criteria mentioned above goes some way towards vindicating Frege's logicist project in a contemporary context.

Realist philosophers of mathematics have accounted for the objectivity and robustness of mathematics by recourse to a foundational theory of mathematics that ultimately determines the ontology and truth of mathematics. The methodology for establishing these truths and discovering the ontology was set by the foundational theory. Other traditional philosophers of mathematics, but this time those who are not realists, account for the objectivity of mathematics by fastening on to: an objective account of: epistemology, ontology, truth, epistemology or methodology. One of these has to stay stable. Otherwise, it is traditionally thought, we have a rampant relativism where ‘anything goes’. Pluralism is a relatively new family of positions. The pluralist in mathematics who is pluralist in: epistemology, foundations, methodology, ontology and truth cannot account for the objectivity of mathematics in either the realist or in the other traditional ways. But such a pluralist is not a rampant relativist. In the paper, I look at what it is to be a pluralist in: epistemology, foundations, methodology, ontology and truth. I then give an account of the objectivity and robustness of mathematics in terms of rigour, borrowings, crosschecking and fixtures—all technical terms defined in the paper. This account is an alternative to the realist and traditional accounts of objectivity in mathematics.

In this paper, I compare how it is that inconsistencies are handled in mathematics to how they are handled in chemistry. In mathematics, they are very precisely formulated and identified, unlike in chemistry. So the chemists can learn from the precision and the very well-worked out strategies developed by logicians and deployed by mathematicians to cope with inconsistency. Some lessons can also be learned by the mathematicians from the chemists. Mathematicians tend to be intolerant towards inconsistencies. There are some philosophers of chemistry who attribute to chemists, collectively, a more tolerant attitude towards inconsistency, and so, mathematicians can learn from the chemists’ attitude to inform their choice of strategies.

What is mathematics about? Does the subject-matter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics but with some experience of formal logic it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics - Platonism, realism, logicism, formalism, constructivism and structuralism - as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy present-day philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upper-level undergraduate courses and the non-specialist

We answer three questions: 1. Can we give a wholly mathematical explanation of a physical phenomenon? 2. Can we give a wholly mathematical explanation for a whole physical theory? 3. What is gained or lost in giving a wholly, or partially, mathematical explanation of a phenomenon or a scientific theory? To answer these questions we look at a project developed by Hajnal Andréka, Judit Madarász, István Németi and Gergely Székely. They, together with collaborators, present special relativity theory in a three-sorted first-order formal language.

The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart from considering rigour of proof as a fixture, I discuss fixed models, invariant notions and fixed information about objects across theories. There are other fixtures, but it is enough to start with these