H2020Индивидуална стипендия2016–2018

SLMK · The Scope and Limits of Mathematical Knowledge

„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“

Период: 2016-10-01 → 2018-09-30
Финансиране от ЕС: 159 461 €
Участници: 1
Схема: MSCA-IF-EF-ST

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Накратко на български

Разликата между строго формалните доказателства и начина, по който математиците мислят в практиката, се анализира чрез теорията на изчислимостта. Това помага да се разбере дали човешкият ум притежава възможности, които надхвърлят тези на компютрите.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

The Scope and Limits of Mathematical Knowledge

Digital computers are based on a mathematical idea (Turing machines) originally developed to resolve a question in the foundations of mathematics: whether all mathematical problems could be solved by repeated application of a set of rules. The negative answer to this question led to a debate about whether the human mind was more powerful than a computer, and what the implications were for the scope of mathematical knowledge. In order to achieve progress in this debate, we must develop a more precise account of what an idealised mathematician can know in principle by means of rigorous mathematical proof. In the SLMK project, the idealisations embedded in the notions of formal proof (proof within an axiomatic system with a fixed language and fixed rules of inference, from a fixed set of basic principles) and informal proof (proof as carried out in mathematical practice, using any linguistic and inferential means the mathematician deems necessary) were investigated using an interdisciplinary methodology, applying formal tools from computability theory, proof theory, and model theory, and analysing case studies from contemporary and historical mathematics. The project objectives were as follows: (i) to investigate the relationship between formal and informal arithmetical reasoning in order to identify a formal system that captures informal reasoning formally, allowing the investigation of the scope of mathematical knowledge; (ii) to achieve substantial progress on the debate on the mechanisation of the mind; and (iii) to use formal tools to obtain a naturalistic account of informal mathematical reasoning and the mechanisms of justification in mathematical practice. The result of the project is a novel and comprehensive view of mathematical methodology and the mechanisms of justification in mathematical practice that is historically and mathematically informed, and compatible with a broadly empiricist view of science and scientific justification.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

A fundamental philosophical question is whether the mind can be mechanised. Attempts to answer it so far have been inconclusive; I argue that with the tools of mathematical logic this question can be sharpened and addressed in a framework where genuine progress can be achieved.I will consider a disjunctive thesis proposed by Gödel (known as Gödel's Disjunction) as a precise version of this question. Once sharpened, the question becomes whether a Turing machine (an idealised computer) can output exactly the statements that are 'absolutely provable'—i.e. the mathematical statements that can be proved in principle by an idealised mathematician not bound by limitations of time and cognitive resources. Gödel's Disjunction states that either the powers of the human mind exceed those of a Turing machine, or there are true but unprovable mathematical statements—i.e. mathematical statements that are beyond the reach of human reason. My proposed research will provide a novel account of 'absolute provability' or 'provability in principle' by developing a formal framework that overcomes the philosophical and technical shortcomings of the previous approaches. Having formulated the correct framework for absolute provability and uncovered its underlying mechanisms, I will be able to determine the status of Gödel’s disjunction. This will shed considerable light on the question of whether mind can be mechanised, a question central to philosophy of mind and artificial intelligence, and on the scope and limits of mathematical knowledge.

Оригинален текст от CORDIS (на английски).

Участници

LUDWIG-MAXIMILIANS-UNIVERSITAET MUENCHEN · PlaneggКоординаторГермания