Truth Through Proof: A Formalist Foundation for Mathematics


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The Foundations of Mathematics

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Refresh and try again. Open Preview See a Problem? Details if other :. Thanks for telling us about the problem. Return to Book Page. Truth Through Proof defends an anti-platonist philosophy of mathematics derived from game formalism. Classic formalists claimed implausibly that mathematical utterances are truth-valueless moves in a game. Alan Weir aims to develop a more satisfactory successor to game formalism utilising a widely accepted, broadly neo-Fregean framework, in which the proposition expressed Truth Through Proof defends an anti-platonist philosophy of mathematics derived from game formalism.

Alan Weir aims to develop a more satisfactory successor to game formalism utilising a widely accepted, broadly neo-Fregean framework, in which the proposition expressed by an utterance is a function of both sense and background circumstance. This framework allows for sentences whose truth-conditions are not representational, which are made true or false by conditions residing in the circumstances of utterances but not transparently in the sense. Applications to projectivism and fiction pave the way for the claim that mathematical utterances are made true or false by the existence of concrete proofs or refutations, though these truth-making conditions form no part of their sense or informational content.

The position is compared with rivals, an account of the applicability of mathematics developed, and a new account of the nature of idealisation proffered in which it is argued that the finitistic limitations Godel placed on proofs are without rational justification. Finally a non-classical logical system is provided in which excluded middle fails, yet enough logical power remains to recapture the results of standard mathematics.


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Get A Copy. Hardcover , pages. More Details Friend Reviews. To see what your friends thought of this book, please sign up. To ask other readers questions about Truth Through Proof , please sign up. Lists with This Book. Three schools, formalism , intuitionism , and logicism , emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Notions of axiom , proposition and proof , as well as the notion of a proposition being true of a mathematical object see Assignment mathematical logic , were formalized, allowing them to be treated mathematically. The Zermelo—Fraenkel axioms for set theory were formulated which provided a conceptual framework in which much mathematical discourse would be interpreted. In mathematics, as in physics, new and unexpected ideas had arisen and significant changes were coming. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led Hilbert to call such study metamathematics or proof theory.

At the middle of the century, a new mathematical theory was created by Samuel Eilenberg and Saunders Mac Lane , known as category theory , and it became a new contender for the natural language of mathematical thinking. Hilary Putnam summed up one common view of the situation in the last third of the century by saying:. When philosophy discovers something wrong with science, sometimes science has to be changed— Russell's paradox comes to mind, as does Berkeley 's attack on the actual infinitesimal —but more often it is philosophy that has to be changed.

I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need.

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.

Notre Dame Journal of Formal Logic

Mathematical realism , like realism in general, holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same.

Philosophy of Mathematics

In this point of view, there is really one sort of mathematics that can be discovered; triangles , for example, are real entities, not the creations of the human mind. Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects. Certain principles e. Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them.

Major forms of mathematical realism include Platonism. Mathematical anti-realism generally holds that mathematical statements have truth-values, but that they do not do so by corresponding to a special realm of immaterial or non-empirical entities. Major forms of mathematical anti-realism include formalism and fictionalism.

Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers. Both Plato's cave and Platonism have meaningful, not just superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular Pythagoreans of ancient Greece, who believed that the world was, quite literally, generated by numbers. A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them?

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Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is the Ultimate Ensemble , a theory that postulates that all structures that exist mathematically also exist physically in their own universe. This view bears resemblances to many things Husserl said about mathematics, and supports Kant 's idea that mathematics is synthetic a priori.

Philosophy of mathematics

Davis and Hersh have suggested in their book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism. Full-blooded Platonism is a modern variation of Platonism, which is in reaction to the fact that different sets of mathematical entities can be proven to exist depending on the axioms and inference rules employed for instance, the law of the excluded middle , and the axiom of choice.

It holds that all mathematical entities exist, however they may be provable, even if they cannot all be derived from a single consistent set of axioms. Set-theoretic realism also set-theoretic Platonism [8] a position defended by Penelope Maddy , is the view that set theory is about a single universe of sets. Max Tegmark 's mathematical universe hypothesis or mathematicism goes further than Platonism in asserting that not only do all mathematical objects exist, but nothing else does.

Tegmark's sole postulate is: All structures that exist mathematically also exist physically. That is, in the sense that "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world".

Logicism is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic. In this view, logic is the proper foundation of mathematics, and all mathematical statements are necessary logical truths.

Truth Through Proof: A Formalist Foundation for Mathematics - Oxford Scholarship

Rudolf Carnap presents the logicist thesis in two parts: [14]. Gottlob Frege was the founder of logicism. In his seminal Die Grundgesetze der Arithmetik Basic Laws of Arithmetic he built up arithmetic from a system of logic with a general principle of comprehension, which he called "Basic Law V" for concepts F and G , the extension of F equals the extension of G if and only if for all objects a , Fa equals Ga , a principle that he took to be acceptable as part of logic.

Frege's construction was flawed. Russell discovered that Basic Law V is inconsistent this is Russell's paradox.


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  • Frege abandoned his logicist program soon after this, but it was continued by Russell and Whitehead. They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it.

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    In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex form for example, there were different natural numbers in each type, and there were infinitely many types. They also had to make several compromises in order to develop so much of mathematics, such as an " axiom of reducibility ". Even Russell said that this axiom did not really belong to logic. Modern logicists like Bob Hale , Crispin Wright , and perhaps others have returned to a program closer to Frege's.

    Truth Through Proof: A Formalist Foundation for Mathematics Truth Through Proof: A Formalist Foundation for Mathematics
    Truth Through Proof: A Formalist Foundation for Mathematics Truth Through Proof: A Formalist Foundation for Mathematics
    Truth Through Proof: A Formalist Foundation for Mathematics Truth Through Proof: A Formalist Foundation for Mathematics
    Truth Through Proof: A Formalist Foundation for Mathematics Truth Through Proof: A Formalist Foundation for Mathematics
    Truth Through Proof: A Formalist Foundation for Mathematics Truth Through Proof: A Formalist Foundation for Mathematics
    Truth Through Proof: A Formalist Foundation for Mathematics Truth Through Proof: A Formalist Foundation for Mathematics
    Truth Through Proof: A Formalist Foundation for Mathematics Truth Through Proof: A Formalist Foundation for Mathematics
    Truth Through Proof: A Formalist Foundation for Mathematics Truth Through Proof: A Formalist Foundation for Mathematics
    Truth Through Proof: A Formalist Foundation for Mathematics

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