The strawberry has those tiny miniscule seeds attached to the exocarp’s surface. They are referred to as achene and are produced by many species such as strawberries. Achenes are from one carpel and as they are indehiscent they do not open at maturity – so are they nuts? No they are not because they do not contain a composite of a fruit and seed.

In many species, what we think of as the “seed” is actually an achene, a fruit containing the seed. The seed-like appearance arises from the fact that the wall of the seed-vessel hardens and encloses the solitary seed.

http://en.wikipedia.org/wiki/Achene