Lecture

Philosophy of Mathematics: Ontology and Structures

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In course

HUM-417: Philosophical perspectives on the exact sciences I

The course considers central themes in the philosophy of science. Starting from the debate between Leibniz and Newton about space and time, we move on to the transition from classical to quantum physics, the explanatory role of mathematics and philosophica ...

Description

This lecture delves into the philosophical perspectives on the exact sciences, exploring the existence of mathematical objects, the truth of mathematical propositions, and our knowledge about them. It discusses various theories such as Platonism, Intuitionism, Structuralism, Nominalism, Logicism, and Formalism. The lecture also covers significant concepts like the two constructions of natural numbers, Russell's Paradox, Gödel's Theorem, and the Axiom of Choice, shedding light on the nature of mathematics and its relationship with logic and reality.

Official source

Ontological neighbourhood

Logic

Classical logic: First-order logic

Related concepts (32)

Philosophy of mathematics

The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics makes this branch of philosophy broad and unique. The philosophy of mathematics has two major themes: mathematical realism and mathematical anti-realism. The origin of mathematics is of arguments and disagreements.

Foundations of mathematics

Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (set, function, geometrical figure, number, etc.

Logicism

In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.

Mathematical beauty

Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as beautiful or describe mathematics as an art form, (a position taken by G. H. Hardy) or, at a minimum, as a creative activity. Comparisons are made with music and poetry. Mathematicians describe an especially pleasing method of proof as elegant.

Logic

Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language.

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