The Philosophy of Mathematics

Introduction

Over the years mathematics has been part of people’s life. Different scholars have in turn participated trying to understand how mathematics varies and impacts human life through its different fields. From the philosophy of mathematics, scholars have studied assumptions foundations as well as implications of mathematics and in turn attempting to provide a detailed viewpoint with reference to mathematics methodology not to mention the efforts to understand and categorize place of mathematics in human life. From the contemporary schools of thoughts with reference to mathematics, there comes schools such as Platonism, empiricism, logicism and structuralism among others. According to (Berry 2018, p.206), mathematical structuralism assumes that mathematic is fundamentally related to structures or with the related objects in mathematics in an effort to bear to each other in the essence of belonging to some structure. This thought is affiliated with the ontological claim; as mathematical objectives are referred as mere ‘positions in structures’. Structuralism is viewed presently as one of the real and more promising mathematics philosophies as the claim of mathematics as the study of structures looks to be supported by the mathematics practice. This essay addresses structuralism as a mathematic philosophy focusing on its development, application and implication to people’s relation with mathematics.

Menzel’s Article on Structuralism in Mathematics

Menzel (2018, p.92) in his article identified that, structuralism in mathematics is a theory with reference to mathematical philosophy which holds on to the idea that, mathematical theories describe structures of mathematical objects. More so, structuralism maintains that, there are no intrinsic properties with reference to mathematical objects, but rather their external relation to the external environment relations within a system. According to MacBride (2008, p.160), structuralism in mathematics is regarded as an epistemological realistic view, in the aspect that mathematical statements carry with them an objective truth value. Drawn from the non-traditional version of the Platonism and developed by Resnik and Shapiro, structuralism refers to the ideas that, the real objects of study in the field of mathematics are patterns or structures in nature. For example things such as infinite series, geometric spaces and also the idea that, individual objects in mathematics such numbers are not objects but rather positioned within structures or patterns (Shapiro 2004, p.22). The idea of structuralism can first be deduced by one thinking of non-mathematical patterns.

History of Structuralism in Mathematics

Structuralists are against the idea that, there exists a reality that simply describes as well as defines in an immediate and transparent fashion. In simplified form, structuralism is defined as a field in mathematics that is concerned with deep structures which in turn shapes how people understand and classify the globe. The field studies structures rather than things in themselves. Fredric Jameson describes structuralism as the search ‘for the permanent structures of the mind itself, the organizational categories and forms through which the mind is able to experience the world, or to organize a meaning in what is essentially in itself meaningless (Fredric 1972, p.109).’ Structuralists refer to the structure they mean the overall system from which particular utterances are created

History of Structuralism in Mathematics

Structuralism in Mathematics goes back to Bourbaki when it was realized that the important thing about mathematical objects is not only its structure but also structure preserving maps. Structuralism in mathematics draws its roots from the fundamental field of ontology. It has been a debate among philosophers since medieval times as whether ontology of mathematics entails abstract objects. With reference to the mathematics, philosophy, an abstract object is defined by; existing independently of the mind, exists independent of the empirical world and finally it possesses eternal and unchangeable properties. The traditional Platonism in mathematics holds that, some set of mathematical elements such as real numbers, natural numbers, functions, and systems are abstract objects. However, this is an aspect that is denied by mathematics nominalism. In the turn of 19th century, anti-Platonist garnered popularity among the scholars. These include schools such as intuitionism, formalism, and predicativism. However, by the turn of mid-20th century, the anti-Platonist views developed their own related issues a development that led to resurgence of interest in Platonism. With this turn of events, there was the in turn development of structuralism development. In 1965, Paul Benacerraf contributed in the field by arguing that, Platonism could not be adopted as a philosophical theory in the field of mathematics. He argued that, the Platonic approaches does not satisfy the ontological test.

Approaches to Structuralism

Structuralism came in three major schools, according to Shapiro which entails; ante REM, in re and post res. The Ante Rem is used to refer to before the thing or still, full realism variation of structuralism and possess a similar ontology to Platonism. With this school, structures are held like they do possess real but abstract/immaterial existence, and with such understanding there exists the standard epistemological issues as identified by Benacerraf whereby he explained the relation between abstract structures. On the other hand, there is the moderate realism, which by another name is known as In Re (in the thing). It is equivalent of Aristotelian realism. According to this view, structures are held to exist in as much as some tangible systems exemplifies them (Nodelman & Zalta 2014, p.45). Finally, there is the Post Res regarded as after things is the eliminating variant of structuralism is anti-realist regarding structures in a way that parallels nominalism. Post res approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure. The ante rem realist, for example, has a straightforward account of reference and semantics: the variables of a branch of mathematics range over the places in an ante rem structure; each singular term denotes one such place, etc. But the ante rem realist must account for how one obtains knowledge of structures, so construed, and for how statements about ante rem structures play a role in scientific theories of the physical world.

Reactions to Structuralism

Today structuralism has been superseded by approaches such as post-structuralism and deconstruction. There are many possible reasons for this. Structuralism has often been knocked for being ahistorical and for favoring deterministic structural forces over the ability of individual people to do. Every bit the political turbulence of the 1960s and 1970s (and especially the student uprisings of May 1968) began affecting academia, issues of power and political struggle moved to the heart of people’s attention. In the 1980s, deconstruction and its stress on the central ambiguity of language - rather than its crystalline logical structure - became popular (Nodelman & Zalta 2014, p.50). By the close of the century Structuralism was seen as a historically important school of thought, but it was the movements it spawned, rather than structuralism itself, which commanded attention.

The Modern Aspect of Mathematical Structuralism

With reference to the modern aspect of mathematical structuralism, they are placed in the latter period of the 19th century, when the axiomatic approach was adopting its current recognized shape. Some of the figures who were credited with the introduction of structuralism include; Richard Dedekind and David Hilbert. The work by Benacerraf’s acted as a motivator for other scholars to develop both nationalist and structuralism theories within the philosophy of mathematics.

Conclusion

In conclusion, it is evident that, there exists a general acceptance in the philosophy of mathematics that, mathematical thinking is structural. Opposition to this is almost on-existence with the few concerned views only adding to the idea that mathematics is people who are able to apply structural thinking in varying contexts. Structuralism in mathematics can in turn be summarized into five principal approaches which entail; formalist, implicit, and model, universals and modal structuralists. Category structuralism, while interpreting a limited font of all the previous, provides structuralists with a valuable tool of strictly structural external descriptions. The theme of mathematical structuralism is that what matters to a mathematical theory is not the internal nature of its objects, such as its numbers, functions, sets, or points, but how those objects relate to each other. Mathematical structuralism is a central position in contemporary philosophy of mathematics. It is the view that mathematical theories describe only abstract structures or structural attributes of their subject areas.

References

Berry, S 2018, ’Modal structuralism simplified’, Canadian Journal Of Philosophy, 48, 2, pp. 200-222, Academic Search Premier, EBSCOhost, viewed 11 July 2018.

Fredric Jameson ‘The Prison House of Language: a Critical Account of Structuralism and Formalism (Princeton: PUP, 1972), p.109.

MacBride, F. (2008). Can Ante Rem Structuralism Solve The Access Problem?. Philosophical Quarterly, 58(230), 155-164.

Menzel, C 2018, ’Haecceities and Mathematical Structuralism’, Philosophia Mathematica, 26, 1, pp. 84-111, Academic Search Premier, EBSCOhost, viewed 11 July 2018.

Nodelman, U, & Zalta, E 2014, ’Foundations for Mathematical Structuralism’, Mind, 123, 489, pp. 39-78, Academic Search Premier, EBSCOhost, viewed 11 July 2018.

Shapiro, S. (2004). Foundations of Mathematics: Metaphysics, Epistemology, Structure. Philosophical Quarterly, 54(214), 16-37.