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One of the criticisms of Aristotle's final cause category is that if a thing has a purpose, there must exist some entity that has intention to set up that cause. Generally speaking, skeptical thinkers reject any sort of positive, universal teleology on the grounds of Occam's Razor: until we can show that someone has designed the universe, we can't say that anything in it has a proper end or purpose. In a sort of reverse Teleological Argument, the suggestion is that without a purposer, there can be no purpose.

However, Aristotle certainly believed that teleological causes existed and sometimes were required to answer "why" questions. There's also no evidence that he believed in a creator or designer. The Stanford Encyclopedia of Philosophy suggests this passage as part of Aristotle's defense of final causes:

For these [viz. teeth and all other parts of natural beings] and all other natural things come about as they do either always or for the most part, whereas nothing which comes about due to chance or spontaneity comes about always or for the most part. … If, then, these are either the result of coincidence or for the sake of something, and they cannot be the result of coincidence or spontaneity, it follows that they must be for the sake of something. Moreover, even those making these sorts of claims [viz. that everything comes to be by necessity] will agree that such things are natural. Therefore, that for the sake of which is present among things which come to be and exist by nature. (Phys. 198b32–199a8)

(I encourage you to read the Encyclopedia's article on Aristotelian Causation to be able to follow the full argument.)

The upshot of the argument is that there is regularity in natural things (especially animals) because the regularity allows for them to survive or thrive. Having teeth of certain types in certain places allows animals to efficiently digest their food. Critically, Aristotle does not appeal to a creator or designer, but observes that final causes are embedded in nature itself.

Leaving aside the speculation of whether Aristotle would have rejected teleological explanations if he could have known of the Darwinian model, was his position valid? Can there exist positive, teleological explanations without the existence of a designer?

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That's a great question.

It seems to me that a telos is an essence that would need to precede existence: to have a telos is to exist for some purpose, and that purpose is necessary for, and necessarily prior to, the existence of the entity.

It follows from that, it seems to me, that this particular essence must be external to the entity; if we were to say that a tooth is sharp so that an animal can efficiently digest its food, this purpose must have come from some entity outside of the tooth itself, as the purpose would have to precede the initial instance of the entity.

Now, we can say that this prior entity is impersonal, and not a god-- but it still must be (it seems to me) capable of intending; if we want to say that "nature" designed the tooth, then nature must be (in this case) motivated.

Put another way: if we wish to use a telos to answer a "why?" question, we are already assuming that there is an answer to the "why?", a cause, and not just an effect of an unmotivated process or regularity. I think the authors of the SEP article linked put their finger on it when they write "Final causality is here introduced as the best explanation for an aspect of nature which otherwise would remain unexplained." Rejecting the final cause is to abandon the possibility of a certain type of explanation, and to resign oneself to regularities of the type "things are the way they are because that's the way they are", or, in more Darwinian terms, the survival of the survivors.

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