Lateral gene transfer (LGT) is often seen as a form of noise, obscuring the phylogenetic signal with which we might hope to reconstruct the evolution of a group of organisms, or indeed the history of all life (the Tree of Life). Such reconstruction might still be possible if the subset of genes conserved among all genomes in a group (or common to all genomes) comprise a core that is relatively refractory to LGT. Several papers designed to test this notion have recently appeared, and here we re-analyze one, which claims that the core of single-copy orthologs shared by all sequenced genomes of the gammaproteobacteria is essentially free of LGT. This conclusion is unfortunately premature, and it is very hard to determine what fraction of this core has been affected by LGT. We discuss other difficulties with the core concept and suggest that, although the core idea must remain part of our understanding of phylogenetic relationships, it should not be the sole basis for defining such relationships, because these are not exclusively tree-like. We suggest instead a more complex but more natural framework for classification, which we call the Synthesis of Life.