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It is often desirable to solve mathematical problems as a limit of simplerÌýproblems. However, are such techniques always guaranteed to work? ForÌýinstance, the problem of finding roots of polynomials of degree higher thanÌýtwo was only solved in the 1980s (Newton's method isn't guaranteed toÌýconverge)! Doyle and McMullen showed that this is only possible if oneÌýallows for multiple independent limits to be taken, not just one. Theycalled such structures "Towers of Algorithms". In this talk I will applyÌýthis idea to other problems (such as computational quantum mechanics,Ìýinverse problems, spectral analysis), show that Towers of Algorithms are anecessary tool, and introduce the Solvability Complexity Index -- aÌýmeasurement of the complexity of a given problem. An important consequenceÌýis that solutions to some problems can never be obtained as a limit offinite dimensional approximations (and hence can never be solvedÌýnumerically). If time permits, I will mention connections with analogousÌýnotions in logic and theoretical computer science.ÌýThis is joint work with Anders Hansen (Cambridge), Olavi Nevalinna (Aalto)Ìýand Markus Seidel (Zwickau).