Against Quantitative Primitivism

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Submission Summary
In this paper, I introduce a novel approach to a problem that is, in the dominant literature, often thought to admit of only a partial solution. The problem of quantity is the problem of explaining why it is that certain properties and relations that we encounter in science and in everyday life, can be best represented using mathematical entities like numbers, functions, and vectors. We use a real number and a unit to refer to determinate magnitudes of mass or length (like 2kg, 7.5m etc.), and then appeal to the arithmetical relations between those numbers to explain certain physical facts. I cannot reach the coffee on the table because the shortest path between it and me is 3ft long, while my arm is only 2.2ft long, and 2.2< 3. The pan balance scale does not tilt because one pan holds a 90g tomato while the other holds two strawberries, of 38g and 52g respectively, and 38+52=90. While they provide a convenient way to express these explanations, the arithmetical less than relation, or the `+' and `x' operations on the real numbers are not really part of the physical explanations of these events. They just represent explanatorily relevant features inherent in the physical systems described. To solve the problem of quantity is to provide an account of this ``quantitative structure'', those physical properties and relations really doing the explaining. The vast majority of approaches in the literature have limited themselves to a much less ambitious project: Rather than explain quantitativeness in its entirety, they strive to leave ``only'' a small amount of quantitativeness unexplained. Primitivism about quantitativeness, or quantitative primitivism, is the position that some quantitative structure cannot be explained. I will argue that the problem of quantity, by its very nature, does not admit of any partial solutions. A reductive-explanatory account of quantitativeness is specifically one that provides an adequate explanation of quantitative struture without leaving any quantitative structure as an unexplained, primitive posit. This is done by reducing the quantitative structure to a more fundamental, non-quantitative base. Non-primitivist accounts also allow for a novel dissolution of a problem which has dominated contemporary debates about the metaphysics of quantity, the debate between ``absolutists'' who think that the fundamental quantitative notions are properties (like ``weighs 5g'' or ``is 2m long''), and and ``comparativists'' according to whom the fundamental notions are comparative relations (like ``is twice as massive as'' or ``is 2m shorter than''). This dispute, I argue, only makes sense from a primitivist perspective. For the non-primitivist, there is no debate to be had. There is no fundamental quantitative structure, and so there is no room for a dispute about what kind of fundamental quantitative structure we accept. The underlying intuitions which guide much of these debates (for example about whether things would be different if everything’s mass was doubled) can still be understood by the non-primitivist. Indeed, non-primitivist accounts can give a clearer and more explanatory judgement on these cases than any primitivist theory could.
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