# Philosophy of Mathematics

## Øystein Linnebo

- 216 pagine
- English
- ePUB (disponibile sull'app)
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# Philosophy of Mathematics

## Øystein Linnebo

## Informazioni sul libro

A sophisticated, original introduction to the philosophy of mathematics from one of its leading contemporary scholars Mathematics is one of humanity's most successful yet puzzling endeavors. It is a model of precision and objectivity, but appears distinct from the empirical sciences because it seems to deliver nonexperiential knowledge of a nonphysical reality of numbers, sets, and functions. How can these two aspects of mathematics be reconciled? This concise book provides a systematic yet accessible introduction to the field that is trying to answer that question: the philosophy of mathematics.Written by Øystein Linnebo, one of the world's leading scholars on the subject, the book introduces all of the classical approaches to the field, including logicism, formalism, intuitionism, empiricism, and structuralism. It also contains accessible introductions to some more specialized issues, such as mathematical intuition, potential infinity, the iterative conception of sets, and the search for new mathematical axioms. The groundbreaking work of German mathematician and philosopher Gottlob Frege, one of the founders of analytic philosophy, figures prominently throughout the book. Other important thinkers whose work is introduced and discussed include Immanuel Kant, John Stuart Mill, David Hilbert, Kurt Gödel, W. V. Quine, Paul Benacerraf, and Hartry H. Field.Sophisticated but clear and approachable, this is an essential introduction for all students and teachers of philosophy, as well as mathematicians and others who want to understand the foundations of mathematics.

## Domande frequenti

## Informazioni

*a priori*, in the sense that it doesn’t rely on sense experience or on experimentation. Truths are arrived at by reflection alone, without any sensory observation. Second, the knowledge is concerned with truths that are

*necessary*, in the sense that things could not have been otherwise. It is therefore safe to appeal to these truths when reasoning not only about how the world

*actually is*but also when reasoning about how it

*would have been*had things been otherwise. Third, the knowledge is concerned with objects that are not located in space or time, and that don’t participate in causal relationships. Such objects are said to be

*abstract*.

*Meno*, where Plato describes a slave boy who has been taught no mathematics but is nevertheless able to discover “out of his own head” an interesting geometrical truth about squares, namely that the square of the diagonal is two times the square of each side. In the dialogue, Socrates asks the slave boy some carefully chosen questions, which prompt the boy to reflect on geometry and discover some simple geometrical truths and eventually reason his way to the mentioned fact about squares.

*a priori*and can be known without relying on experience for one’s justification. It may be objected that the slave boy relies on experience in order to understand Socrates’ questions. Of course he does! But this experience serves only to

*trigger*the process that results in geometrical knowledge and doesn’t itself constitute

*evidence*for this knowledge.

^{1}

*a priori*mathematical knowledge. How can this be? The usual answer from rationalistically inclined philosophers has been that our “faculty of reason” is the source of such concepts and knowledge. Until more has been said about this faculty and its workings, however, this answer is little more than a pompous redescription of what we set out to explain. Plato, to his credit, recognizes the need to say more. In the

*Meno*, he therefore proposes—or at least entertains—an explanation.

*Meno*, 81cd)

*necessarily*true, that is, true not only as things actually are, but true no matter how things might have been.

^{2}Presumably, the domain of “everything thinkable” includes everything that is possible in the sense explained above.

^{3}Had intelligent life never existed, there would have been no laws, contracts, or marriages—yet the mathematical truths would have remained the same. These truths can thus be assumed by us actually existing intelligent agents when we reason about this sad intelligence-free scenario.

*abstract*, we recall, if it lacks spatiotemporal location and is causally inefficacious; otherwise it is said to be

*concrete*. While this distinction may not be entirely sharp, it suffices for our present purposes.

^{4}

*seems*that mathematics is concerned with abstract objects. Mathematical texts brim with talk about numbers, sets, functions, and more exotic objects yet, and these objects seem nowhere to be found in space and time.

^{5}It is useful to “factor” the third feature of mathematics into two distinct claims.

**Object realism**. There are mathematical objects.

**Abstractness**. Mathematical objects are abstract.

^{6}

*is*true, as anyone who possesses even basic arithmetical competence will confirm. It follows that ‘Eleven’ must succeed in referring to an object, and hence there are mathematical objects.