Socratic philosophy treats logical axioms as "self-evident truths" (i.e. I think, therefore I am).
I read the article. It's interesting (I liked the thing about pegs and strings), but I don't think the guy's (nor you) read a lot of actual Greek philosophy. I don't mean that as an attack (why would you want to, after all?), but it makes some of his, and your claims a little strange.
Socrates, in the Platonic dialogues, is unwilling to take the law of non-contradiction as an axiom. There just aren't any axioms in Socratic philosophy, just discussions. No proofs, just conversations. Plato (and certainly not Socrates) doesn't have doctrines, and Plato is totally and intentionally merciless with people who try to find Platonic doctrines.
Also, Plato and Socrates predate, for most purposes, logic.
Mathematics treats logical axioms as "propositions", and uses logic to see where those propositions lead
Right, Aristotle largely invented (or discovered) that trick. Aristotle's logic is consistant and strongly complete (i.e. it's not axiomatic, and relies on no external logical concepts). Euclid picked up on it, and produced a complete and consistant mathematics. So (some) Greek philosophy certainly shares this idea with modern mathematics.
Scientists treat logical axioms as "hypotheses", and logical "conclusions" as testable statements that can determine whether those axioms are true or not
I don't think scientists treat logical axioms as hypotheses. Logical axioms aren't empirical claims, and aren't really subject to testing. But Aristotle's work on biology, meteorology, etc. forwards plenty of empirical hypotheses, along with empirical evidence for them. Textual evidence suggests Aristotle performed lots of experiments, mostly in the form of vivisection of animals. He was wrong about pretty much everything, but his method was empirical.
This is to say nothing of contemporary philosophy, which certainly doesn't take very much as 'self-evident truth'. I can assure you, no one gets anywhere with that phrase anymore, in any study.
I believe Wei_Lai would say that the first approach, treating ethical axioms as "self-evident truths" is problematic due to the fact that a lot of hypothetical situations (like my example before) can create a lot of contradictions between various ethical axioms (i.e. choosing between telling a lie and letting terrorists blow up the planet).
Not if those ethical axioms actually are self-evident truths. Then hypothetical situations (no matter how uncomfortable they make us) can't disrupt them. But we might, on the basis of these situations, conclude that we don't have any self-evident moral axioms. But, as you neatly argue, we don't have any self-evident mathematical axioms either.
Thanks for taking the time to read and respond to the article, and for the critique; you are correct in that I am not well-versed in Greek philosophy. With that being said, allow me to try to expand my framework to explain what I'm trying to get at:
Example: "I think that heat is transferred between two objects via some sort of matter that I...
What do I mean by "morality isn't logical"? I mean in the same sense that mathematics is logical but literary criticism isn't: the "reasoning" we use to think about morality doesn't resemble logical reasoning. All systems of logic, that I'm aware of, have a concept of proof and a method of verifying with high degree of certainty whether an argument constitutes a proof. As long as the logic is consistent (and we have good reason to think that many of them are), once we verify a proof we can accept its conclusion without worrying that there may be another proof that makes the opposite conclusion. With morality though, we have no such method, and people all the time make moral arguments that can be reversed or called into question by other moral arguments. (Edit: For an example of this, see these posts.)
Without being a system of logic, moral philosophical reasoning likely (or at least plausibly) doesn't have any of the nice properties that a well-constructed system of logic would have, for example, consistency, validity, soundness, or even the more basic property that considering arguments in a different order, or in a different mood, won't cause a person to accept an entirely different set of conclusions. For all we know, somebody trying to reason about a moral concept like "fairness" may just be taking a random walk as they move from one conclusion to another based on moral arguments they encounter or think up.
In a recent post, Eliezer said "morality is logic", by which he seems to mean... well, I'm still not exactly sure what, but one interpretation is that a person's cognition about morality can be described as an algorithm, and that algorithm can be studied using logical reasoning. (Which of course is true, but in that sense both math and literary criticism as well as every other subject of human study would be logic.) In any case, I don't think Eliezer is explicitly claiming that an algorithm-for-thinking-about-morality constitutes an algorithm-for-doing-logic, but I worry that the characterization of "morality is logic" may cause some connotations of "logic" to be inappropriately sneaked into "morality". For example Eliezer seems to (at least at one point) assume that considering moral arguments in a different order won't cause a human to accept an entirely different set of conclusions, and maybe this is why. To fight this potential sneaking of connotations, I suggest that when you see the phrase "morality is logic", remind yourself that morality isn't logical.