Abstract
In this short paper, we analyse whether assuming that mathematical objects are "thin" in Linnebo's sense simplifies the epistemology of mathematics. Towards this end, we introduce the notion of transparency and show that not all thin objects are transparent. We end by arguing that, far from being a weakness of thin objects, the lack of transparency of some thin objects is a fruitful characteristic mark of abstract mathematics.
Highlights
An object is thin when “little or nothing is required” (Linnebo, 2018, p. xi) for its existence
If the existence of certain objects does not make a substantial demand on the world, knowledge of such objects will be comparatively easy to attain. ... [It] might well be the only way to reconcile the need for an ontology of mathematical objects with the need for a plausible epistemology” (Linnebo, 2018, p. xi)
As a matter of fact, if mathematical objects are thin, any epistemology that we find reasonable and that helps us navigate the mathematical world would comply more with the very light requirements imposed on the existence of mathematical objects
Summary
An object is thin when “little or nothing is required” (Linnebo, 2018, p. xi) for its existence. An object is thin when “little or nothing is required” As Linnebo acknowledges, the idea that mathematical objects are thin “holds great philosophical promise. If the existence of certain objects does not make a substantial demand on the world, knowledge of such objects will be comparatively easy to attain. The doctrine of thin objects might make it easy to prove that mathematical objects a and b exist but hard to know whether they are the same object or not. We argue, this is not a problem but rather a result to welcome
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