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Can space be cognized by only verbal means or does it require experience to be understood? Let me show you what I am getting at. You could never imagine what the color red is from a description of it and I think most people see that as an intrinsic limitation on language. No matter how sophisticated the person listening/describing or how sophisticated the language used you would never know what red is without an experience of it. Is space equally ineffable when it comes to descriptions of it?

Imagine there's a pure, disembodied intellect, and you somehow have the ability to communicate with it. It's a very clever intellect, so it's perfectly receptive to abstract, a priori mathematics: but it has never had any experience of spatial things, and it wants you to explain space to it. How might you go about this? Well, first you might explain the number line. You invite it to consider an infinite set of objects (we'll call them 'real numbers'), all different from one another, but continuously ordered in two directions from a particular element that we'll call 'zero' (or 'the origin'): ever greater to a positive infinity, and ever less to a negative infinity. And now, with the number line in place, you invite the intellect to take three such lines. That is to say, you invite it to consider an infinity of ordered triples of the form <x, y, z>, where x, y and z are all real numbers from this same set, but are capable of varying independently of one another. Let's call each of these triples a ...

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