Campbell S. R. (2003) Reconnecting mind and world: Enacting a (new) way of life. In: Lamon S. J., Parker W. A. & Houston K. (eds.) Mathematical modelling\>Mathematical modelling. Woodhead Publishing, Sawston: 245–253.

A common assumption in teaching mathematical modelling and applications is that mind and world are ontologically distinct. This dualist view give rise to an explanatory gap as to how these two realms connect. An alternate view where mind and world are ontologically identical is explored here. This alternate view, grounded in a monist ontology of embodied cognition, undermines and attempts to fill this explanatory gap. Embodied cognition presents challenges of its own, but it also presents new pedagogical opportunities.

Gash H. (2013) Fixed or probable ideas. Foundations of Science 19(3): 283–284. https://cepa.info/894

This commentary concerns Nescolarde-Selva and Usó-Doménech’s paper on a semiotic model of ideologies. The commentary raises questions about the dynamic versus static nature of the model proposed, and in addition asks whether the model might be used to explain ethical flexibility and rigidity. The semiotic model is a mathematical one and a constructivist approach is evident. Relevance: This approach is clearly constructivist and concerns mathematical modelling of semiotics.

Problem: Evidence is quantified by statistical methods such as p-values and Bayesian posterior probabilities in a routine way despite the fact that there is no consensus about the meanings and implications of these approaches. A high level of confusion about these methods can be observed among students, researchers and even professional statisticians. How can a constructivist view of mathematical models and reality help to resolve the confusion? Method: Considerations about the foundations of statistics and probability are revisited with a constructivist attitude that explores which ways of thinking about the modelled phenomena are implied by different approaches to probability modelling. Results: The understanding of the implications of probability modelling for the quantification of evidence can be strongly improved by accepting that whether models are “true” or not cannot be checked from the data, and the use of the models should rather be justified and critically discussed in terms of their implications for the thinking and communication of researchers. Implications: Some useful questions that researchers can use as guidelines when deciding which approach and which model to choose are listed in the paper, along with some implications of using frequentist p-values or Bayesian posterior probability, which can help to address the questions. It is the – far too often ignored – responsibility of the researchers to decide which model is chosen and what the evidence suggests rather than letting the results decide themselves in an “objective way.”

To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality (to which there is no direct access), personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement about ‘truth’, but the assignment of mathematics to informal reality is not itself formally analysable, and it is dependent on social and personal construction processes. On these levels, absolute agreement cannot be expected. Starting from this point of view, repercussion of mathematical on social and personal reality, the historical development of mathematical modelling, and the role, use and interpretation of mathematical models in scientific practice are discussed.