What is mathematics?

To paraphrase the famous 16th/17th century scientist and mathematician, Galileo Galilei, mathematics is the language of the universe.

Whether we consciously realise it, mathematics is embedded in our everyday lives, especially with the rise of technology. Multiple disciplines, including economics, computer sciences, physics, chemistry, biology, astronomy, medicine rely on mathematics, and all the tools and technology we use in our everyday lives are underpinned by mathematics in some way.

Although we may all have a concept of mathematics in our heads, there is actually no agreed definition. A dictionary definition is the “science that deals with the measurement, properties, and relations of quantities, including arithmetic, geometry, algebra, etc.” (Macquarie dictionary publishers, 2020). This definition is derived from the thinking of the ancient Greek philosopher Aristotle; however, post-enlightenment views of mathematics expanded to include multiple ways to conceptualise what mathematics is.

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Snapper (1979) lists three approaches to defining mathematics:

• Logicism tends to think of mathematics in terms of the science of logic. Alfred North Whitehead and Bertrand Russell’s three volume Principia Mathmatica was able to reduce mathematics into a system of logical principles. Indeed, Russell summarised that “all mathematics is symbolic logic” (1903, p.5). In other words, mathematics is a set of logical principles that a person follows to get to the correct answer.
• Intuitionalism considers mathematics in terms of mental processes and constructs. Snapper (1979), in making sense of the intuitionist approach, defines mathematics as “the mental activity which consists in carrying out constructs one after the other” (p. 210). In effect, mathematics has to be constructed to exist and cannot exist independently of the mind that constructs it. A simple way of thinking about mathematics is that it exists because the individual is undertaking the calculations and constructing a mathematical solution.
• Formalism, in very simple terms, suggests that maths is a game with specific sets of rules. David Hilbert followed this approach in the 1920s when he tried to define the rules through formal language and axioms.

Alternatively, Ernest (2018) lists three hierarchical philosophies that underpin mathematics:

• Instrumentalist: mathematics is an accumulation of facts, rules and skills. It is a set of unrelated but utilitarian rules and facts.
• Platonist: mathematics is a static but unified body of certain knowledge. It is discovered, not created.
• Problem solving: mathematics is a cultural product and a dynamic, continually expanding field of human creation and invention. It is a process of enquiry, not a finished product, as its results remain open to revision.

Numeracy

It is a common belief that mathematics enables the numeracy capabilities necessary to navigate different life domains, but what exactly is numeracy?

The Australian Curriculum Authority writes that “Numeracy encompasses the knowledge, skills, behaviours and dispositions that students need to use mathematics in a wide range of situations. It involves students recognising and understanding the role of mathematics in the world and having the dispositions and capacities to use mathematical knowledge and skills purposefully” (ACARA 2017).

Additionally, Cockroft articulates numeracy as the “at homeness” with with numbers required in everyday life, an understanding of the information “presented in mathematical terms”, and an “appreciation and understanding of information which is presented in mathematical terms, for instance in graphs, charts or tables” (cited in Agnello & Agnello, 2019, p. 2).

Therefore, numeracy is the skill required to complete mathematical tasks. This is why it is important to not only understand and communicate mathematical concepts to students, but to teach the numeracy skills associated with it.

Agnello, Ellen C., and Kevin M. Agnello. “Crossing the Final Frontier: Exploring the Numeracy Demands of Texts Read in English Language Arts.” Numeracy 12, Iss. 2 (2019): Article 7. DOI: https://doi.org/10.5038/1936-4660.12.2.7

Australian Curriculum, Assessment and Reporting Authority [Internet]. Australian Curriculum, Assessment and Reporting Authority; 2017. What is numeracy?; [cited 2022 August 8]. Available from: https://www.australiancurriculum.edu.au/resources/national-literacy-and-numeracy-learning-progressions/national-numeracy-learning-progression/what-is-numeracy/

Ernest, P. (2018). The Ethics of Mathematics: Is Mathematics Harmful?. In: Ernest, P. (eds) The Philosophy of Mathematics Education Today. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-77760-3_12

Macquarie Dictionary [Internet]. Macquarie Dictionary Publishers; 2022. Mathematics; [cited 2022 July 6]. Available from: https://www.macquariedictionary.com.au/features/word/search/?search_word_type=Dictionary&word=mathematics

Snapper E. The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism. Mathematics Magazine. 1979 Sep; 52(4):207-216.