# AK2021 Foundational Issues in Mathematics 7.5 credits

#### Matematiska grundvalsfrågor

The subject matter of mathematics is -- or, at any rate, appears to be -- a universe of abstract objects such as numbers, functions, and sets. Such objects can hardly be located in space and time; nevertheless we can study them and get to know their properties. How is this possible, and how reliable is such knowledge? Do numbers even exist independently of us, or are they rather some kind of mental construction? Is mathematical truth the same thing as provability, or may there be mathematical facts beyond the scope of rational inquiry? And can we ever be confident that our mathematical theories are free of contradiction?

In the course, we will study how three schools of thought in the philosophy of mathematics -- logicism, intutionism, and finitism -- have approached these issues, from a conceptual/philosophical point of view as well as from a technical/mathematical one. We will also find reason to acquaint ourselves with the traditional set-theoretical construction of mathematical number systems.

### Offering and execution

Course offering missing for current semester as well as for previous and coming semesters

## Course information

### Content and learning outcomes

#### Course contents *

The subject matter of mathematics is â or, at any rate, appears to be â a universe of abstract objects such as numbers, functions, and sets. Such objects can hardly be located in space and time; nevertheless we can study them and get to know their properties. How is this possible, and how reliable is such knowledge? Do numbers even exist independently of us, or are they rather some kind of mental construction? Is mathematical truth the same thing as provability, or may there be mathematical facts beyond the scope of rational inquiry? And can we ever be confident that our mathematical theories are free of contradiction?

In the course, we will study how three schools of thought in the philosophy of mathematics â logicism, intutionism, and finitism â have approached these issues, from a conceptual/philosophical point of view as well as from a technical/mathematical one. We will also find reason to acquaint ourselves with the traditional set-theoretical construction of mathematical number systems.

#### Intended learning outcomes *

On completion of the course, the student should be able to

- account for, and with critical reflection discuss, concepts and problems from the philosophy of mathematics;

- recount and contrast the positions of central figures and schools of thought in the philosophy of mathematics; and

- handle technical concepts and methods relevant to foundational issues in mathematics.

#### Course Disposition

No information inserted

### Literature and preparations

#### Specific prerequisites *

Higher education equivalent to 120 credits (two whole years) or more.

#### Recommended prerequisites

No information inserted

#### Equipment

No information inserted

#### Literature

No information inserted

### Examination and completion

A, B, C, D, E, FX, F

#### Examination *

• TEN1 - Exam 1, 4.0 credits, Grading scale: A, B, C, D, E, FX, F
• TEN2 - Exam 2, 3.5 credits, Grading scale: A, B, C, D, E, FX, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

#### Other requirements for final grade *

- Examination 1 (TEN1; 4.0 credits), grading scale A-F

- Examination 2 (TEN2; 3.5 credits), grading scale A-F

#### Opportunity to complete the requirements via supplementary examination

No information inserted

#### Opportunity to raise an approved grade via renewed examination

No information inserted

### Further information

#### Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web AK2021

ABE/Philosophy

Mathematics

Second cycle