How Do We Know That 2 + 2 = 4?

This is a survey chapter about issues in the epistemology of elementary arithmetic. Given the title of this volume, it is worth noting right at the outset that the classification of arithmetic as science is itself philosophically debatable, and that this debate overlaps with debates about the epistemology of arithmetic. It is also important to note that a survey chapter should not be mistaken for a comprehensive, definitive, or unbiased introduction to all that is important about its topic. It is rather an exercise in curation: a selection of material is prepared for display, and the selection process is influenced not only by the author’s personal opinions as to what is interesting and/or worthy, but also by various contingencies of her training, and my survey reflects my training in Anglo-American analytic philosophy of mathematics. Although I’m surveying an area of epistemology, I will classify approaches by metaphysical outlook. The reason for this is that the epistemology and metaphysics of arithmetic are so intimately intertwined that I have generally found it difficult to understand the shape of the epistemological terrain except by reference to the corresponding metaphysical landmarks. For instance, it makes little sense to say that arithmetical knowledge is a kind of “maker’s knowledge” unless arithmetic is in some way mind-dependent, or to classify it as a subspecies of logical knowledge unless arithmetical truth is a species of logical truth. I will be discussing 2+2=4 as an easily-graspable example of an elementary arithmetical truth, our knowledge of which stands in need of philosophical explanation. While some of the surveyed approaches to this explanatory demand proceed by rejecting the presumed explanandum—i.e. by denying that 2+2=4 is known (or even true)—for clarity and ease of expression I will proceed as if 2+2=4 is a known truth except when discussing these approaches. The rest of this chapter proceeds as follows. In the next section, I identify two key challenges for an epistemology of simple arithmetic, and then adduce two constraints on what should count as a successful response. Next, I discuss ways of addressing these challenges, grouped according to their corresponding metaphysical outlook. The subsequent sections survey non-reductive Platonist approaches, look at reductions (often better labelled “identifications”), and consider an array of anti-realist strategies. I conclude with a brief summary, returning to the question of arithmetic’s status as science.