How does all this apply to generalizing from visual thinking about an arbitrary instance? Take the example of the visual route to the formula for triangular numbers using the diagram of Figure 2. The diagram reveals that the formula holds for the 6th triangular number. The generalisation to all triangular numbers is justified only if the visuo-spatial method used is applicable to the \(n\)th triangular number for all positive integers \(n\), that is, provided that the method used does not depend on a property not shared by all positive integers. A conscious, explicit check that this proviso is met requires making explicit the method exemplified for 6 and proving that the method is applicable for all positive integers in place of 6. (For a similar idea in the context of automating visual arguments, see Jamnik 2001). This is not done in practice when thinking visually, and so if we accept the strict option for thinking through a proof involving generalisation, we would have to accept that the visual route to the formula for triangular numbers does not amount to thinking through a proof of it; and the same would apply to the familiar visual routes to other general positive integer formulas, such as that \(n^2 =\) the sum of the first \(n\) odd numbers.