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What is the justification for topological arguments in philosophy? For instance, it is common to say that we are not the only conscious animals because there is a continuity between humans and animals by evolution. But how do we know what topological structure is appropriate in a certain philosophical situation? It seems wholly arbitrary to say that I am more like Sam than I am like an Amoeba (I guess that's getting more into geometry there though).

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I don't think topological arguments are appropriate in the context of evolution. Sure you can say a human is homeomorphic (continuously deformable) to an amoeba, but the same can be said about a piece of coal in relation to a human. I think you are using the word continuity in the wrong sense. Note that the same word can be used in different ways or contexts. In the context of evolution the word continuity can be used to indicate a gradual progression from amoeba to man like a ramp or incline (but we know that the path of evolution is more like a discrete or staircase progression due to mutations). In the context of math or topology, continuity has a specific definition.

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Topology as a purely mathematical notion came a long time after the idea of continuity was understood. Its not really appropriate in the sense that you are speaking of in evolution.

When one looks at mammals one notices a lot of similarities - they have two eyes and four limbs and one head - why not three heads and six limbs and twenty four eyes? On the face of it this regularity is inexplicable and improbable so demands an explanation.

This similarity makes one suppose that they are linked in some way - and this is at the root of the theory of evolution. That is most philosophers who have reflected on this have supposed there to be some continuity between the animal world and the human world. When one looks at ancient mythologies this notion becomes apparent - centaurs and fauns, falcon headed gods and native-american mythology.

The real advance in evolution theory is to propose a mechanism to explain specifically how evolution works - genes.

Now that we have that mechanism at hand, and databanks of actual gene sequences of animals which is specific, concrete and discrete information about a species one can use topological methods on this information. The problem will be to find a natural, or useful topology - and I expect this is an intensive and open research problem.

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