Postmodern mathematics
| Postmodernism |
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| Preceded by Modernism |
| Postmodernity |
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| Criticism of postmodernism |
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Postmodern mathematics is a philosophical theory developed as an adjunct to postmodernism. The theory asserts that there is no such thing as ‘absolutism’ or ultimate truth in mathematics.[1] It also declares that the term ‘mathematics’ can't be used to define a specific object.[1] This concept emerged from the postmodernistic idea of the relativity of truth.[1]. The postulate also maintains that the ideas in mathematics are subjective and that there is no ‘right’ answer to a mathematical question.[1]
Ideas
The conceptual framework affirmed by adherents in Postmodern Mathematics can be summed up as follows:
Plurality
The term 'Mathematics', in its singular form, cannot be used to describe any object.[2] However, mathematics, in its plural form, describes the multiplicity and flexibility of this term based upon context. All these various definitions are not necessarily true or false and can be right or wrong based on the context and the observer's viewpoint in which they are defined.[2] It is a medium of expression, with inherent relativity and ambiguity.
Rejection of 'universality' and 'absolutism'
Postmodernism rejects the idea of universality in mathematics.[2] It claims that there is no such thing as 'absolutism' and asserts the notion of multiplicity in mathematics.[1] This comes from the postmodernist idea that rejects notions of absolutism and the claim that the knowledge we attain as ‘true’ is only a representation constructed by society which is not truer than any other representations.[1] Postmodernists also reject the absolutist idea of mathematics being “human free” and objective and instead assert a “humanistic” view of mathematics which is filled with subjective and humane values.[3] This view sees mathematics as:
imbued with moral and social values which play a significant role in the development and applications of mathematics.[3]
Rejection of certainty
Certainty in mathematics is not attainable. The answers proposed are perceived as probabilities and not certainties.[1] Postmodernism in mathematics opposes Aristotelian and modernist concepts of an object being either true or false and instead asserts the concept of a “degree of truth”. Proclaiming therewith the notion of partial truths and uncertainty.[1] According to the Postmodernist thought, theories in mathematics accepted as 'true' and 'certain' are products of society as the criteria for 'certainty' and 'truth' are socially constructed norms. Therefore, postmodernism rejects the notion of certainty and absolute truth in mathematics, asserting instead that certainties in mathematics are not static and can change throughout time and societies.[4] This is significant as mathematics is usually perceived as the "most certain part of human knowledge." A fortiori, questioning the certainty of this part of knowledge rejects certainty from any form of human knowledge.[3]
Fallibility in mathematics
Mathematics is a corrigible discipline as it is subject to the notion of fallibility and is in a state of constant change.[5] This theory was first proposed by Lakatos in his book "Proofs and Refutation". It also constitutes the hypothetical-deductive system of mathematics which declares that the discipline of mathematics is fallible and corrigible, the development of theorems require the falsification of “falsifiers” and transmission to hypothetical knowledge. In other words, in order to develop theorems, one must first falsify the premises under which the theorem would be falsified.[3] This would further constitute that mathematics is connected as a part of the broad human knowledge, including the culture, language and perspectives.[3]
Social constructivism in mathematics
As theories and laws proclaimed in mathematics are representations made by society, mathematics is constructed by the needs of societies.[5] Hence, it holds no other value other than being representations of cultures and societies.[4]
History
Postmodernism was a movement that started against modernism,[1] and its alleged claims to value the existence of an objective and universal truth.[6] For modernists[who?], knowledge is the use of empirical methodologies in order to discover the ‘ultimate truth’. Postmodernism counters that knowledge isn't as empirical and logical as modernism persists, rather it is subjective and is open to change.[7] For postmodernists, knowledge is a construct of society and is subject to change over time and place.[5]
In mathematics, modernists[who?] assert notions of Platonism (the view that theories in mathematics are unchangeable and perceives mathematics as perfect and eternal),[4] Logicism (the view that perceives mathematics as a part of Logic)[4] and Formalism.[6] These concepts were accepted universally before the introduction of postmodernism. This change in thoughts brought about by the introduction of postmodernism caused a shift of mathematics research and practice from “logical theories” and challenged the notion of objectivity, universality and certainty in mathematics.[7]
Unlike modernist ideas of singularity, postmodernism denotes the idea of plurality and polycentrism.[8] Postmodernism also promotes the idea of collaboration between the teachers and students of mathematics as it argues for the idea of everyone being an explorer of mathematics.[8] Modernism, on the other hand, proposes no question for the nature of mathematics or education as they contend that the two entities are not related to one another.[6] In this thought, students in search of education, mathematics in this context, are exploring an “independent world”. This independent world remains unhinged by the student's exploration and is not changed by the students’ encounter as they are two separate entities.[6] Postmodernism, on the other hand, encourages the transformation of knowledge by the student as they believe that the two only exist “relative to one another”.[6]
Notable figures
Ludwig Wittgenstein
Ludwig Wittgenstein (1889–1951) developed a notion against the modernist idea of rationality and claimed that it isn't as evident and clear as perceived by modernists.[5] He claimed that certainty in mathematics is “a collection of language games” and truth and false are based on their following of “the rules of linguistic games”.[3]
It is what human beings say that is true and false: and they agree in the language they use.[9]
Karl Popper
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Karl Popper (1902–1994) introduced the notion of falsification and asserted that to prove theories in sciences, researchers should strive to disprove them.[5] He also asserted that the laws in mathematics are subject to being proven false and hence he rejected the idea of ultimate truths.[need quotation to verify]
Imre Lakatos
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Imre Lakatos (1922–1974) was inspired by Karl Popper's idea of falsification in mathematics and wrote his thesis on the fallibility of mathematics and mathematical theorems. In his book, titled ‘Proofs and refutations’, Lakatos claimed that all the theorems in mathematics are fallible and the theories that are accepted as ‘true’ or ‘perfect’ are only accepted because there hasn't been any theorem that counters them. He claimed that theories in mathematics have no basis for certainty as they are products of assumptions made by humans and are hence refutable.[4]
Paul Ernest
Paul Ernest has extensive works on the philosophy of mathematics where he draws upon the works of other postmodern philosophers including Popper and Lakatos. His works mainly reflect his belief in the social constructivism of mathematics which illustrates that mathematical theories are a construct of society and are thus changeable. He also published the "Philosophy of Mathematics Education Journal" which contributes to discussions of the philosophy of mathematics and mathematics education and has extensive articles on postmodern mathematics.[10]
Wilkinson
He introduced the idea of 'fuzzy logic'. This idea rejects the Aristotelian Law of Truth and fallibility of an object (the notion that an object is either true or false). He instead proposed the existence of 'degrees of truth', the idea that an object is characterised into various degrees of truth.[5]
Thomas Kuhn
Thomas Kuhn (1922–1996) introduced the idea of paradigm shifts in sciences. he asserted that the paradigms and theories in sciences are not fixed but are instead on a continuum and hence are subject to transformations. He describes his theory of paradigms as:
.. each paradigm will be shown to satisfy more or less the criteria that it dictates for itself and to fall short of a few of those dictated by its opponent. .. no paradigm ever solves all the problems it defines .. [11]
According to Kuhn, education and knowledge strives to interpret and represent rather than provide an objective explanation.[5] These representations are examples of social constructivism and are products of societies throughout time.[5]
Postmodern mathematics in education
In the postmodernist view, mathematics is learnt not for knowledge but for utility.[1] According to this thought, teachers are not only an ‘authoritative figure’ but are also co-explorers and learners of mathematics[2] or as described by Ernest, “ringmasters of the mathematics circus”.[2] They also assert on the role of students in the teaching mathematics and their role in the shaping of curriculum.[3] Postmodern mathematics asserts the diminishing of the notion of objectivity and absolutism in mathematics thus asserting that a “notion of 2 + 2 = 1” can be true given certain subjective circumstances.[12] However, it doesn't mean that the notion can be used based on any ‘personal situation’ rather it is relative to the mathematical context and situation it is used in such that 2 +2 = 1 is true in mod (3) arithmetic but it would never be true in any other mathematical context.[5] Hence, mathematical theories are corrigible based on the mathematical contexts they are used in.
Unlike the modernist idea of teachers being the authoritative figure and “echo-phrasing” the textbooks, postmodernism encourages collaborative work between the students and the teachers which again relates to the notion of ‘truth’ and ‘knowledge’ being subjective.[8] This would also emphasise the notion that students should be given the opportunity to exercise alternate methods and solutions to the ones determined by the teachers.[5]
If mathematics education is reshaped according to the postmodernist thought, it opens up windows for the implementation of mathematics by the students to their everyday life, culture and language. It enables them to reshape their knowledge of mathematics and concepts of mathematics.[3]
Mathematics becomes responsible for its uses and consequences, in education and society. Those of us in education have a special reason for wanting this more human view of mathematics. Anything else alienates and dis-empowers learners.[3]
Postmodern education also encourages the use of technology and computers in the discovery of new ideas in mathematics.[5] Although these ideas are subject to uncertainty as denoted by postmodernism, they can still have the same or higher ‘degree of truth’ as those proven and discovered by classical methods.[5]
The rejection of uncertainty also asserts researchers of mathematics and other knowledge to strive to achieve a higher degree of truth for human knowledge and hence encourages further research and study. This in turn expands the human knowledge vastly.[3]
Postmodern education encourages teachers, students and researchers of mathematics to focus on criticism of the known mathematical facts rather than evaluation.[4] It asserts that mathematicians should focus not on what ‘mathematics’ is but rather what it could be and what it might be. It encourages constant criticism of the mathematical theories and trying to disprove the theories in order to get a higher degree of truth in mathematics education.[4] Jean-François Lyotard describes this as
Reality (certainty in this context) is not expressed by a phrase like X is such but by one like X is such and not such.[13]
He also claims that to prove the ‘reality’ or the ‘truth’ of a description of an entity, which in this context would be mathematics, the negation of another description is needed.[5]
See also
References
- ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 Moslehian, M. S. (November 2005). "Postmodern view of humanistic mathematics". Resonance. 10 (11): 98–105. doi:10.1007/bf02837651. ISSN 0971-8044.
- ↑ 2.0 2.1 2.2 2.3 2.4 Ernest, Paul (2015). "Postmodern Mathematics". The Proceedings of the 12th International Congress on Mathematical Education. Springer International Publishing. pp. 605–608. doi:10.1007/978-3-319-12688-3_68. ISBN 9783319106854.
- ↑ 3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 Ernest, Paul (1991). The philosophy of mathematics education. London: Falmer Press. ISBN 978-0203224236. OCLC 52100009. Search this book on
- ↑ 4.0 4.1 4.2 4.3 4.4 4.5 4.6 Moslehian, Mohammad (2003). "A glance at Postmodern pedagogy of mathematics". Philosophy of Mathematics Education Journal. 17.
- ↑ 5.00 5.01 5.02 5.03 5.04 5.05 5.06 5.07 5.08 5.09 5.10 5.11 5.12 Ernest, Paul; Izmirli, Ilhan (2011). "Does a Postmodernist Philosophy of Mathematics Make Sense? Is "2 + 2 = 5" Correct as Long as One's Personal Situation or Perspective Requires It?". Philosophy of Mathematics Education Journal.
- ↑ 6.0 6.1 6.2 6.3 6.4 Peck, Frederick; Sriraman, Bharath (2017-06-15). "Breaking the Constraints of Modernist Psychologizing: Mathematics Education Flirts with the Postmodern". Interchange. 48 (4): 351–362. doi:10.1007/s10780-017-9306-1. ISSN 0826-4805.
- ↑ 7.0 7.1 Ernest, Paul (1997). "Chapter 3: The Epistemological Basis of Qualitative Research in Mathematics Education: A Postmodern Perspective". Journal for Research in Mathematics Education. Monograph. 9: 22–177. doi:10.2307/749945. JSTOR 749945.
- ↑ 8.0 8.1 8.2 Tarp, Allan (2018). "Modern and Postmodern Critical Research". socialsciences.exeter.ac.uk. Retrieved 2018-10-16.
- ↑ Wittgenstein, Ludwig (2001). Philosophical investigations : the German text, with a revised English translation. Anscombe, G. E. M. (Gertrude Elizabeth Margaret) (3rd ed.). Oxford: Blackwell. p. 88. ISBN 978-0631231271. OCLC 49055907. Search this book on
- ↑ Ernest, Paul. "Philosophy of Mathematics Education Journal".
- ↑ Kuhn, Thomas (1970). The structure of scientific revolutions (Second, enlarged ed.). Chicago, Illinois: University of Chicago Press. ISBN 978-0226458038. OCLC 93075. Search this book on
- ↑ Bertonneau, Thomas (1997). Declining Standards at Michigan Public Universities. How Changes in Curricula and Teaching Affect Academic Achievement, Career Preparation, and K-12 Education (Second ed.). Mackinac Center for Public Policy. pp. 56–59. ISBN 978-0-9647703-9-3. Search this book on
- ↑ Lyotard, Jean-François (1988). The differend : phrases in dispute. Minneapolis: University of Minnesota Press. p. 48. ISBN 978-0816616114. OCLC 17621488. Search this book on
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