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We investigate a new research area: we are interested in the ultimate thermodynamic cost of computing from x to y. Other than its fundamental importance, such research has implications for future miniaturiza-tion of VLSI chips reducing the energy dissipation below kT (thermal noise), and the similarity distance problem in pattern recognition. It turns out that the theory of thermodynamic cost of computation can be axiomatically developed. Our fundamental theorem connects physics to mathematics , providing the key that makes such a theory possible. It establishes optimal upper and lower bounds on the ultimate thermodynamic cost of computation. By computing longer and longer, the amount of dissipated energy gets closer to these limits. In fact, one can trade time for energy: there is a provable time-energy trade-oo hierarchy. The fundamental theorem also induces a thermodynamic distance metric. The topological properties of this metric show that neighborhoods are sparse, and get even sparser if they are centered on random elements. The proofs use Symmetry of Information in a basic way. These notions also nd an application in pattern recognition. People have been looking without success for an objective notion of cognitive distance to account for the intuitive notion of`similarity' of pictures. Thermodynamic considerations lead to a re-cursively invariant notion of cognitive distances. It turns out that the thermodynamic distance is a universal cognitive distance which discovers all eeective features used by any cognitive distance whatsoever.