The question that it is the point of departure of this paper concerns the possibility of an “alternative” mathematics: is such a (form of) mathematics possible or conceivable? Or should one conclude that, in some sense or other, there can only be one kind of mathematics? On the one hand, one is tempted to answer that no such alternatives are possible or conceivable, since, e.g. 2 + 2 = 4, no matter what, where, when, how, who,… Once humankind “discovered” this elementary mathematical truth, it has remained unchanged as befits an eternal truth. Of course, one has to ignore artificial tricks and games: yes, one could claim that 2 + 2 = 5, provided one counts in this specific order: 1, 2, 3, 5, 4, 6, 7, …, but that is rather silly. In fact, they show that trying to find a counterexample leads almost necessarily to highly artificial proposals, hence the argument becomes self-refuting. On the other hand, one might equally be tempted to answer that there are alternatives all around us, so evidently they are possible or conceivable. Actually, they are quite easy to find as will be shown in this paper.

Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism (which is now lacking) and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, an evaluation of arguments and counterarguments is presented. Results: The main result is that strict finitism is indeed a viable option, next to other constructive approaches, in (the foundations of) mathematics. Implications: Arguing for strict finitism is more complex than is usually thought. For future research, strict finitist mathematics itself needs to be written out in more detail to increase its credibility. In as far as strict finitism is a viable option, it will change our views on such “classics” as the platonist-constructivist discussion, the discovery-construction debate and the mysterious applicability problem (why is mathematics so successful in its applications?). Constructivist content: Strict finitism starts from the idea that counting is an act of labeling, hence the mathematician is an active subject right from the start. It differs from other constructivist views in that the finite limitations of the human subject are taken into account.

Key words: mathematics, finite, largest number, infinite, limits, budget constraints

Open peer commentary on the article “Second-Order Science: Logic, Strategies, Methods” by Stuart A. Umpleby. Upshot: The author makes a strong plea for second-order science but somehow mathematics remains out of focus. The major claim of this commentary is that second-order science requires second-order mathematics.

Open peer commentary on the article “Religion: A Radical-Constructivist Perspective” by Andreas Quale. Upshot: An essential feature of Quale’s point of view is the strict distinction between the cognitive and the non-cognitive. I argue that this position is untenable and hence that a radical constructivist can discuss religious matters.

Open peer commentary on the article “Mathematical Work of Francisco Varela” by Louis H. Kauffman. Upshot: The aim of this commentary is to show that a new development in formal logic, namely paraconsistent logic, should be connected with the laws of form. This note also includes some personal history to serve as background.

Open peer commentary on the article “Eigenform and Reflexivity” by Louis H. Kauffman. Upshot: I show that the author underestimates the tricky matter of how to make a transition from the discrete, countable to the continuous, uncountable case.

Except in very poor mathematical contexts, mathematical arguments do not stand in isolation of other mathematical arguments. Rather, they form trains of formal and informal arguments, adding up to interconnected theorems, theories and eventually entire fields. This paper critically comments on some common views on the relation between formal and informal mathematical arguments, most particularly applications of Toulmin’s argumentation model, and launches a number of alternative ideas of presentation inviting the contextualization of pieces of mathematical reasoning within encompassing bodies of explicit and implicit, formal and informal background knowledge.

Key words: Contextual knowledge, Mathematical arguments, Mathematical practice, Toulmin, Rhetoric

The authors argue in favor of the view that in mathematics, which increasingly relies on computers to warrant mathematical results, the hunt for absolute certainty will become more and more expensive.

Context: As one of the major approaches within the philosophy of mathematics, constructivism is to be contrasted with realist approaches such as Platonism in that it takes human mental activity as the basis of mathematical content. Problem: Mathematical constructivism is mostly identified as one of the so-called foundationalist accounts internal to mathematics. Other perspectives are possible, however. Results: The notion of “meaning finitism” is exploited to tie together internal and external directions within mathematical constructivism. The various contributions to this issue support our case in different ways. Constructivist content: Further contributions from a multitude of constructivist directions are needed for the puzzle of an integrative, overarching theory of mathematical practice to be solved.

Key words: mathematical constructivism, conceptualism, internalism vs externalism, meaning finitism, mathematical practice