User:Phlsph7/Mathematics - Definitions
Proposed definitions
[edit]Mathematics, also known by its short forms math and maths,[1][2] is a discipline studying abstract patterns associated with numbers, shapes, and other mathematical objects.[3][4][5][6] It is connected to the empirical world since these patterns can be used to describe and solve real-world problems in fields such as the natural sciences, engineering, and economics.[3][7][8] At the same time, it is a highly abstract form of inquiry that relies on deductive reasoning and can proceed without considering concrete applications.[9][8][5] For example, it investigates general relations between numbers independent of specific physical quantities they can represent.[8] It is sometimes labeled "queen of the sciences" due to its scientific rigor and influence on other sciences.[5][10]
Despite these general characteristics, there are deep disagreements about the precise definition of mathematics.[11][3][12] It is often seen as a formal science but has also been characterized as a natural science, an art, and a discipline of the humanities.[13][3][14] Some definitions emphasize the topics investigated in mathematics, focusing on the properties of and relationships between abstract objects like numbers, sets, functions, and shapes. Others are more interested in the methods of inquiry, highlighting the high standards of precision and the application of general rules. Definitions emphasizing formal aspects focus on the symbolic and exact notation to express ideas and theories.[10][15][8][5][16]
Characterizations based on usefulness highlight its connection to concrete reality in the process of creating and analyzing abstract models as a core feature.[10][8] A related definition views mathematics as the expression of a mental activity that follows logic and intuition to solve problems.[17][18][19] Based on modern attempts to provide a general foundation of mathematics, some theorists argue that mathematics is the same as set theory, understood as an extension of modern logic.[20][21] Because of the disagreements between the diverse definitions, mathematics is sometimes seen as a blanket term covering various topics and theories without a unifying principle. Another indirect approach describes mathematics as "what mathematicians do" or "what is taught in mathematics classes".[22][23][24][6]
The word mathematics comes from the Ancient Greek word máthēma (μάθημα), meaning 'something learned, knowledge, mathematics', and the derived expression mathēmatikḗ tékhnē (μαθηματικὴ τέχνη), meaning 'mathematical science'. It entered the English language during the Late Middle English period through French and Latin.[25][26] Over time, the meaning and scope of mathematics have evolved. Initially, the term referred to any subject studied in formal education, but its meaning narrowed to the study of arithmetic and geometry with the Pythagoreans in the 6th and 5th centuries BCE.[26][27] Aristotle described mathematics as "the science of quantity", a definition that was widely accepted until the 18th century. As various mathematical subfields emerged in the following period, it became increasingly difficult to provide a uniform definition.[27][28][29][6]
References
[edit]Notes
[edit]Citations
[edit]- ^ HarperCollins 2022.
- ^ HarperCollins 2022a.
- ^ a b c d Tobies 2012, pp. 9–10.
- ^ Ziegler & Loos 2017, p. 74.
- ^ a b c d Colyvan 2012, p. 1.
- ^ a b c Devlin 2018, p. 3.
- ^ Ziegler & Loos 2017, pp. 66–67, 74.
- ^ a b c d e Aleksandrov, Kolmogorov & Lavrent’ev 1999, § 1.1 Abstractions, proofs, applications.
- ^ Ziegler & Loos 2017, pp. 66–67.
- ^ a b c Mura 1993, pp. 379–381.
- ^ Gowers, Barrow-Green & Leader 2010, p. 1.
- ^ Ziegler & Loos 2017, pp. 66–67, 75.
- ^ Fitzpatrick 2022, § Foundations of Modern-day Mathematics.
- ^ Gamwell 2016, p. 509.
- ^ Brown & Porter 1995, pp. 324–326.
- ^ MacLane 2012, pp. 409–410, 415–418.
- ^ Mura 1993, pp. 375–376, 380.
- ^ Courant, Robbins & Stewart 1996, § What is Mathematics?.
- ^ Ziegler & Loos 2017, p. 75.
- ^ Strauss 2011, pp. 3–4.
- ^ Buium 2014, pp. xi–xii, 67.
- ^ Tanton 2005, p. vi.
- ^ Ziegler & Loos 2017, p. 65.
- ^ Mura 1993, pp. 381, 384.
- ^ Cresswell 2021, § Mathematics.
- ^ a b Perisho 1965, p. 64.
- ^ a b Franklin 2009, p. 104.
- ^ Franklin 2014, pp. 31–32.
- ^ Gandon 2012, pp. 135–136.
Sources
[edit]- Perisho, Margaret W. (1965). "The Etymology of Mathematical Terms". Pi Mu Epsilon Journal. 4 (2): 62–66. ISSN 0031-952X. JSTOR 24338341. LCCN 58015848. OCLC 1762376.
- Franklin, James (2009). "Aristotelian Realism". In Irvine, Andrew D. (ed.). Philosophy of Mathematics. Elsevier. ISBN 978-0-08-093058-9.
- Cresswell, Julia (2021). Oxford Dictionary of Word Origins (3 ed.). Oxford University Press. ISBN 978-0-19-886875-0.
- Brown, Ronald; Porter, Timothy (1995). "The Methodology of Mathematics". The Mathematical Gazette. 79 (485). doi:10.2307/3618304.
- Buium, Alexandru (2014). Mathematics: A Minimal Introduction. CRC Press. ISBN 978-1-4822-1600-4.
- Strauss, Danie (2011). "Defining mathematics". Acta academica. 43 (4). doi:10.38140/aa.v43i4.1328. ISSN 0587-2405.
- Mura, Roberta (1993). "Images of mathematics held by university teachers of mathematical sciences". Educational Studies in Mathematics. 25 (4). doi:10.1007/bf01273907.
- Tobies, Renate (2012). Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer Science & Business Media. ISBN 978-3-0348-0251-2.
- Courant, Richard; Robbins, Herbert; Stewart, Ian (1996). What is Mathematics? An Elementary Approach to Ideas and Methods (2 ed.). Oxford University Press. ISBN 0-19-510519-2.
- Ziegler, Günter M.; Loos, Andreas (2017). “What is Mathematics?” and why we should ask, where one should experience and learn that, and how to teach it. Springer International Publishing. ISBN 978-3-319-62597-3.
- Tanton, James Stuart (2005). Encyclopedia of Mathematics. Facts On File. ISBN 0-8160-5124-0.
- Colyvan, Mark (2012). An Introduction to the Philosophy of Mathematics. Cambridge University Press. ISBN 978-0-521-82602-0.
- Aleksandrov, A. D.; Kolmogorov, A. N.; Lavrent’ev, M. A. (1999). Mathematics: Its Content, Methods and Meaning. Dover Publications. ISBN 978-0-486-15787-0.
- Gowers, Timothy; Barrow-Green, June; Leader, Imre, eds. (2010). The Princeton Companion to Mathematics. Princeton University Press. ISBN 978-1-4008-3039-8.
- HarperCollins (2022). "Math". The American Heritage Dictionary entry. HarperCollins.
- HarperCollins (2022a). "Maths". The American Heritage Dictionary entry. HarperCollins.
- MacLane, Saunders (2012). Mathematics Form and Function. Springer Science & Business Media. ISBN 978-1-4612-4872-9.
- Franklin, James (2014). "Elementary Mathematics: The Science of Quantity". An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure. Palgrave Macmillan. ISBN 978-1-137-40073-4.
- Gandon, S. (2012). Russell's Unknown Logicism: A Study in the History and Philosophy of Mathematics. Springer. ISBN 978-1-137-02465-7.
- Devlin, Keith (2018). Sets, Functions, and Logic: An Introduction to Abstract Mathematics, Third Edition (3 ed.). CRC Press. ISBN 978-1-4822-8602-1.
- Fitzpatrick, Thora L. (2022). "Formal Sciences". In Mattingly, James (ed.). The SAGE Encyclopedia of Theory in Science, Technology, Engineering, and Mathematics. SAGE Publications. ISBN 978-1-5063-5328-9.
- Gamwell, Lynn (2016). Mathematics and Art: A Cultural History. Princeton University Press. ISBN 978-0-691-16528-8.