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This is a discussion of some themes in Max Tegmark’s recent book, Our Mathematical Universe. It was written as a review for Plus Magazine, the online magazine of the UK’s national mathematics education and outreach project, the Mathematics Millennium Project (http://plus.maths.org/content/). Since some of the discussion---about symmetry breaking, and Pythagoreanism in the philosophy of mathematics---went beyond reviewing Tegmark’s book, the material was divided into three online articles. This version combines those three articles, and adds some other material, in particular a brief defence of quidditism about properties. It also adds some references, to other Plus articles as well as academic articles. But it retains the informal style of Plus.