Philosophy of Mathematics: An Introduction

An Introduction to Philosophy of Mathematics
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Except, perhaps, one little thing: if mathematics amounts to deductive reasoning using the axioms and rules of set theory, then to ground the subject all we need to do is to figure out what sort of entities sets are, how we can know things about them, and why that particular kind of knowledge tells us anything useful about the world. Such questions about the nature of abstract objects have therefore been the central focus of the philosophy of mathematics from the middle of the 20th century to the present day.

In other branches of philosophy, where no neat story was available, philosophers had to deal with the inherently messy nature of language, science and thought. This required them to grapple with serious methodological issues.

What are Numbers? Philosophy of Mathematics

From the s on, philosophers of language engaged with linguists to make sense of the Chomskyean revolution in thinking about the structure of language and human capacities for understanding and generating speech. Philosophers of mind interacted with psychologists and computer scientists to forge cognitive science, the new science of the mind.

Philosophers of biology struggled with methodological issues related to evolution and the burgeoning field of genetics, and philosophers of physics worried about the coherence of the fundamental assumptions of quantum mechanics and general relativity. Meanwhile, philosophers of mathematics were chiefly concerned with the question as to whether numbers and other abstract objects really exist.

This fixation was not healthy.

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It has almost nothing to do with everyday mathematical practice, since mathematicians generally do not harbour doubts whether what they are doing is meaningful and useful — and, regardless, philosophy has had little reassurance to offer in that respect. Insofar as it is possible to provide compelling justification for doing mathematics the way we do, it will not come from making general pronouncements but, rather, undertaking a careful study of the goals and methods of the subject and exploring the extent to which the methods are suited to the goals. When we begin to ask why mathematics looks the way it does and how it provides us with such powerful means of solving problems and explaining scientific phenomena, we find that the story is rich and complex.

The problem is that set-theoretic idealisation idealises too much. Mathematical thought is messy.

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The collection by Field [] contains replies to some of these objections. The law of excluded middle and impredicative definitions are central items in the mathematician's toolbox—to the extent that many practitioners are not aware when these items have been invoked. So a promising strategy for linguists and philosophers of language was to start with the modelling of mathematical language, where the mechanics are more easily understood, and then adapt the models to accommodate a broader range of linguistic constructs. Some contemporary intuitionists, such as Michael Dummett [ , ] and Neil Tennant [ , ] , take a different route to roughly the same revisionist conclusion. Empiricism, Naturalism, and Indispensability 4. Share your experience with other students. The s also brought a clear mathematical analysis of the notion of computability.

When we dig beneath the neatly composed surface we find a great buzzing, blooming confusion of ideas, and we have a lot to learn about how mathematics channels these wellsprings of creativity into rigorous scientific discourse. But that requires doing hard work and getting our hands dirty. And so the call of the sirens is pleasant and enticing: mathematics is set theory! Just tell us a really good story about abstract objects, and the secrets of the Universe will be unlocked. This siren song has held the philosophy of mathematics in thrall, leaving it to drift into the rocky shores.

Given that the philosophy of mathematics has been closely aligned with logic for the past century or so, one would expect the fortunes of the two subjects to rise and fall in tandem. Over that period, logic has grown into a bona-fide branch of mathematics in its own right, and in Paul Cohen won a Fields Medal, the most prestigious prize in mathematics, for solving two longstanding open problems in set theory. Ideally, information should flow back and forth, with philosophical understanding informing implementation and practical results, and challenges informing philosophical study.

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So it makes sense to consider the role that logic has played in computer science as well. A rival approach, with its origins in the s, incorporates neural networks, a computational model whose state is encoded by the activation strength of very large numbers of simple processors connected together like neurons in the brain. The early decades of AI were dominated by the logic-based approach, but in the s researchers demonstrated that neural networks could be trained to recognise patterns and classify images without a manifest algorithm or encoding of features that would explain or justify the decision.

This gave rise to the field of machine learning. Improvements to the methods and increased computational power have yielded great success and explosive growth in the past few years. The approach, known as deep learning, is now all the rage. Logic has also lost ground in other branches of automated reasoning.

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Logic-based methods have yet to yield substantial success in automating mathematical practice, whereas statistical methods of drawing conclusions, especially those adapted to the analysis of extremely large data sets, are highly prized in industry and finance. Computational approaches to linguistics once involved mapping out the grammatical structure of language and then designing algorithms to parse down utterances to their logical form.

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These days, however, language processing is generally a matter of statistical methods and machine learning, which underwrite our daily interactions with Siri and Alexa. The structure of language is inherently amorphous. Concepts have fuzzy boundaries. Whereas mathematics seeks precise and certain answers, obtaining them in real life is often intractable or outright impossible.

In such circumstances, what we really want are algorithms that return reasonable approximations to the right answers in an efficient and reliable manner. Real-world models also tend to rely on assumptions that are inherently uncertain and imprecise, and our software needs to handle such uncertainty and imprecision in robust ways. Evidence for a scientific theory is rarely definitive but, rather, supports the hypotheses to varying degrees.

If the appropriate scientific models in these domains require soft approaches rather than crisp mathematical descriptions, philosophy should take heed. We need to consider the possibility that, in the new millennium, the mathematical method is no longer fundamental to philosophy.

Philosophy of Mathematics

But the rise of soft methods does not mean the end of logic. Our conversations with Siri and Alexa, for instance, are never very deep, and it is reasonable to think that more substantial interactions will require more precise representations under the hood.

In an article in The New Yorker in , the cognitive scientist Gary Marcus provided the following assessment:.

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For some purposes, soft methods are blatantly inappropriate. If you go online to change an airline reservation, the system needs to follow the relevant policies and charge your credit card accordingly, and any imprecision is unwarranted. Computer programs themselves are precise artifacts, and the question as to whether a program meets a design specification is fairly crisp.

Getting the answer right is especially important when that software is used to control an airplane, a nuclear reactor or a missile launch site. Even soft methods sometimes call for an element of hardness. The question, then, is not whether the acquisition of knowledge is inherently hard or soft but, rather, where each sort of knowledge is appropriate, and how the two approaches can be combined.

Leslie Valiant, a winner of the celebrated Turing Award in computer science, has observed:. Give us serenity to accept the things we cannot understand, courage to analyse the things we can, and wisdom to know the difference.

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W hat about the role of mathematical thought, beyond logic, in our philosophical understanding? The influence of mathematics on science, which has only increased over time, is telling. Even soft approaches to acquiring knowledge are grounded in mathematics. Statistics is built on a foundation of mathematical probability, and neural networks are mathematical models whose properties are analysed and described in mathematical terms.

To be sure, the methods make use of representations that are different from conventional representations of mathematical knowledge. But we use mathematics to make sense of the methods and understand what they do.

Mathematics has been remarkably resilient when it comes to adapting to the needs of the sciences and meeting the conceptual challenges that they generate. The world is uncertain, but mathematics gives us the theory of probability and statistics to cope. Newton solved the problem of calculating the motion of two orbiting bodies, but soon realised that the problem of predicting the motion of three orbiting bodies is computationally intractable.

In response, the modern theory of dynamical systems provides a language and framework for establishing qualitative properties of such systems even in the face of computational intractability. At the extreme, such systems can exhibit chaotic behaviour, but once again mathematics helps us to understand how and when that happens.

Natural and designed artifacts can involve complex networks of interactions, but combinatorial methods in mathematics provide means of analysing and understanding their behaviour. Mathematics has therefore soldiered on for centuries in the face of intractability, uncertainty, unpredictability and complexity, crafting concepts and methods that extend the boundaries of what we can know with rigour and precision. In the s, the American theologian Reinhold Niebuhr asked God to grant us the serenity to accept the things we cannot change, the courage to change the things we can, and the wisdom to know the difference.

But to make sense of the world, what we really need is the serenity to accept the things we cannot understand , courage to analyse the things we can, and wisdom to know the difference.

When it comes to assessing our means of acquiring knowledge and straining against the boundaries of intelligibility, we must look to philosophy for guidance. Great conceptual advances in mathematics are often attributed to fits of brilliance and inspiration, about which there is not much we can say. But some of the credit goes to mathematics itself, for providing modes of thought, cognitive scaffolding and reasoning processes that make the fits of brilliance possible.

This is the very method that was held in such high esteem by Descartes and Leibniz, and studying it should be a source of endless fascination.