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Notes
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Though perhaps not the issue of beauty, which shares with understanding both an elusive quality and a psychological connotation.
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Though, as I have said, not a theory with necessary and sufficient conditions.
- 5.
I do not specifically address how structure relates to understanding the different aspects of mathematics because I think that understanding definitions vs. theorems vs. axiom systems, etc., can only be addressed in terms of abilities – though I do not argue for this here.
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That is, understanding of any sort is hard to explain; mathematical understanding presents special challenges given its technical nature and abstract subject matter.
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Frege 1884, introduction.
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This is not always an explicit assumption, but it leads to a dismissive attitude one finds in Frege and some of his defenders.
- 9.
Avigad 2008 focuses on abilities: those behind understanding mathematical proofs.
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This is a paper on the philosophy of mathematics education (rather than mathematics education); nevertheless, I should note that its intended audience still seems different from that of mainstream philosophy of mathematics.
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Later in the article he helps himself to a wider use of “knowledge”, as in “know-how” or “knowledge of”; however, this is a problem that I will not address.
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Assuming this is what Gettier examples show. Porteous agrees that these examples challenge the traditional account, but maintains that for mathematics justified true belief is close enough.
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For a detailed look at some specific relationships between explanation and understanding, see Tappenden 2005. This is an interesting, useful paper; but like Kitcher, Steiner and Friedman, Tappenden focuses on theories or frameworks that promote understanding. Though more sketchy, my project here has a broader (even broader!) aim.
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The volume Mancosu et al. (2005) shows that there is some progress and interest; nevertheless, most of its articles are programmatic in nature.
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For recent work on the concept of grounding in mathematics, start with Šebestik 2016.
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This is a limitation, not a criticism. Steiner is explicit about the narrowness of his project: “We have not analyzed mathematical explanation, but explanation by proof; there are other kinds of mathematical explanation.” (147, emphasis in original)
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For example, Hafner and Mancosu 2005 “maintain that mathematical explanations are heterogeneous” (222) – so no single theory – and that the topic is even “treacherous.” (241)
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Avigad 2010 also recommends a “bottom up” approach in order to – at least first – better articulate the data that a theory of understanding would explain. (See conclusion.) I am grateful to an anonymous referee for pointing me to Avigad’s work on this topic; although his work resonates a great deal with my views I unfortunately discovered it too late to follow his recommendation!
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I say “just” because one can adopt the structuralist perspective and still think mathematical objects exist. That is, one can retain the view that arithmetic is about the numbers 3, 4, 5, etc., while maintaining that important arithmetic results typically concern the natural number structure rather than individual numbers. The perspective is thus agnostic on philosophical issues about the priority or independence of structures from mathematical objects and systems.
- 23.
The idea that it is relations and relation-types that “matter” more to mathematics is a core view of mathematical structuralism. For example, according to Shapiro, “The theme of mathematical structuralism is that what matters to a mathematical theory is not the internal nature of its objects, such as its numbers, functions, sets, or points, but how those objects relate to each other.” (Shapiro undated, opening sentence.) Despite this core, the general idea of “structure” is quite hard to pin down; in addition it is ambiguous. With regard to physical objects, it can be used to refer to a particular, as in “The bank is in that big brick structure down the road,” though it can also mean a type, as in “The A-frame structure is visually striking”. When it comes to explaining what a mathematical structure is, the notion of abstraction is often used. For example, Shapiro explains that a mathematical system is a “collection of objects together with certain relations on those objects…. A structure is the abstract form of a system, which ignores or abstracts away from any features of the objects that do not bear on the relations. So, the natural number structure is the form common to all of the natural number systems.” (Shapiro undated, section 1.) The idea is that by bracketing, or removing from consideration, certain features of two mathematical systems their underlying common form, or structure, can be unveiled. This process of ignoring some features to get at a common, usually more abstract, underlying property or form is roughly what “abstraction” means. However, the notorious problems of what philosophers mean by an abstract structure (or any kind of universal), the activity of abstraction or, especially perhaps, the relation between the two are well beyond the purview of this paper.
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See for example Maclaurin 1742, Introduction.
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See again Maclaurin 1742, Book I, Chapter I, especially opening paragraph.
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I hope to provide more argument for this elsewhere. Here I am just addressing how or why images carry epistemic weight in mathematics, not arguing that they can do so.
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Tappenden (2005) also appeals to visualization in connection with mathematical understanding, but, as mentioned above, his focus is on whole theories. For him, a theory that lends itself to visualization tends to be more understandable and fruitful. (150) He also notes that visualizability counts in favor of the “naturalness” of a theory or framework. (180) Here I focus on a neighboring but different question of how and why visual images function as tools for understanding, which Tappenden brackets (155) to pursue his other goals.
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This is not to say I accept such diagrams as full proofs or as better than proofs. My point is merely that visual information can assist understanding; my question is why and how.
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There are of course some issues. One is to explain how a single diagram can support a general theorem. Another is to connect the geometric and the symbolic expressions of the theorem – to explain how the spatial object supports the numerical equation, x2 + y2 = z2. For this latter in particular I think the structuralist lens is important; I focus on another example to make my point more obvious.
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The “object” in question can be seen as the set of natural numbers and its “characterizing property” as induction. (There is of course literature debating whether inductive proofs can explain. For example, Lange 2009 and responses.)
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For example, if adding a new “column”, one must add it next to the previously “tallest” column and one must make it exactly one block higher. One cannot add that taller column to the middle or the other end of the staircase. If iterating via “layers” one must put a block on top of each existing step, and then add one to the “floor” next to the lowest end or new stack of two. These rules can easily be taught to a child.
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Again, such pictures are in my view neither genuine proofs nor better than traditional proofs, except insofar as they are more widely accessible. They do not supplant the equation or its traditional proof. But they are tools that promote understanding, and as such they function like partial justifications. Further, this function explains their normativity (why we are correct to believe the theorem on the basis of the picture), and their epistemic contributions (why they lead to understanding rather than mere belief).
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Contrary to the views of Nelsen, Brown and others.
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I believe other examples can be analyzed similarly to explain why such pictures carry justifying weight. For example, a similar analyses can be given regarding the standard spatial representation of 1 + 3 + 5 … + 2n-1 = n2, or that of the Zeno series (though these further examples are beyond the scope of this paper). As with the case here analyzed, I would argue that the normative status of the diagram depends on shared structural features: that key properties of the picture match key elements of the relevant equations. The shared structure is what explains why they increase our understanding as well as why we take them to reveal the facts (rather than taking them to be mere causes of beliefs that perhaps happen to be true).
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I am grateful to an anonymous referee for some constructive feedback on a written draft of this paper, though I fear I have sidestepped rather than answered the hard questions. I am also indebted to participants at two conferences, for the opportunity to try out some of these ideas: the Logic Colloquium 2015 (joint meeting of the Association for Symbolic Logic and the Conference of Logic, Methodology and Philosophy of Science), at the University of Helsinki, and FilMat 2016 (second international conference of the Italian network for the philosophy of mathematics), at the University of Chieti-Pescara. Finally, thanks go to the editors of this volume for all of their hard work.
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Folina, J. (2018). Towards a Better Understanding of Mathematical Understanding. In: Piazza, M., Pulcini, G. (eds) Truth, Existence and Explanation. Boston Studies in the Philosophy and History of Science, vol 334. Springer, Cham. https://doi.org/10.1007/978-3-319-93342-9_8
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