1. Rand's argument
Objection to i
The fact that Ayn Rand does not offer an argument for 1 does not make it false. This piece does not offer a counter argument. That besides, Ayn Rand bases Objectivism on axioms. Per definition, axioms are not proven but assumed to be true.
Objection to ii
The alternatives need to be there for either schools of value to be satisfied. Absolute good is in contrast with less good options, relative good is in contrast with other worse options. A million US dollars, or South African Rand, has to be compared with something of lesser value (a million Zimbabwean Dollars, for instance) to be considered valuable.
Objection to iii
Non-living things cannot make value judgements as far as we know. This is the point of the fact that they cannot face alternatives: they cannot choose to be living things. We can.
Objection to iv
These two interpretations, oddly enough, seem to reinforce the idea that value has to be of value to someone. Value as a verb implies an agent that values. Value as an absolute implies comparison with less valuable or more valuable things, which begs the question valuable to whom? This again implies an agent that compares.
Objection to v
The implicit premise 7 is derived incorrectly.
Rand's premise: 6. Every living thing acts to maintain its life, for its own sake.
The author's implied premise: 7. There is no other thing that they act to gain or keep for its own sake.
In other words, agent does x because of y. This does not mean that an agent can't do z because of y (or an agent can't do other things merely for their own sake). Rather, Rand's implied premise seems to be that an agent cannot do x because of z (namely an agent cannot act to maintain itself for the sake of someone else), not that there are no other things that agents do for their own sake.
The author focuses on this incorrect premise 7, thus v is a classic strawman argument. It is important to read the entire 6 and to understand that agents act to maintain themselves for their own sake. v claims that this implies innate values, which it does not. As per 2 and 3, Rand claims that agents choose among alternatives according to what they value. Thus, values cannot be innate as instinct. They are a result of volition. Rocks cannot act to satisfy value systems, regardless of the value their kinetic energy might have to us.
Objection to vi
This breaks up 6 and gets lost in semantics. Again, 6 claims agents maintain themselves for themselves. Nothing more.
Objection to vii
This again focuses on the incorrectly derived premise 10. Rand stated that an agent's life is its ultimate value, at the heart of its values. This is why Rand stated that life is valuable for its own sake. It is thus rational to act in order to promote life. The derived premise states that Rand claimed it is rational to act in order to promote one's values. This does not follow from Rand's statements (unless one's values are necessarily in tune with promoting one's life), as Rand claimed that values are volition.
You may choose to have any values whatsoever, and acting to satisfy them is not necessarily tantamount to promoting your life. Rand claimed that it is rational to act to promote your life, therefore you should choose values accordingly. NOT that you should act to promote your values. If this were true and your values were not chosen to promote your life, your values - and not your life - would have the ultimate value.
vii begs the question why is it rational to act to satisfy values, if they do not promote your own life? This, I belief, is the point Rand was trying to make. We should, therefore, act to promote our own lives, and allow others to act to promote their own lives. This is the real meaning of altruism: to allow agents to live for their own sake and not to require other agents to live for your own sake, or to live for the sake of others.
Objection to viii
Another strawman argument. Rand did not claim that living without 100% reason results in instant death. She merely claimed that we as agents should be allowed to use our ultimate survival tool, namely reason. Reason here refers to acting in your own best interest in order to maintain your life, and allowing others the same course of action. This because the assumption is that life has value for agents, for its own sake.
3. General arguments against ethical egoism
This argument does not follow from Objectivist ethics. This argument begs the question who is forcing you to torture 500 people for the benefit of a dime? This scenario implies the absence of volition, which means Objectivist values do not apply.
4. Attacking straw men
Rand never claimed that moral values are arbitrary. Rand claimed that values should be based on that which keeps you alive. This is hardly arbitrary.
By the metaphysical nature of man and of existence, man has to maintain his life by his own effort; the values he needs—such as wealth or knowledge—are not given to him automatically, as a gift of nature, but have to be discovered and achieved by his own thinking and work. - Ayn Rand, Wikiquote.
This contradicts what the author claims about Rand's values. Another strawman argument, based on the incorrectly derived premise 7, which as discussed was not Rand's implied premise but of the author's own invention.
5. Man qua man and fudge words
Rand never claimed that life is a state of not being dead. On the contrary, to claim this is to miss the entire point of egoism. With Man Qua Man, Rand claimed this:
Man—every man—is an end in himself, not a means to the ends of others; he must live for his own sake, neither sacrificing himself to others nor sacrificing others to himself; he must work for his rational self-interest, with the achievement of his own happiness as the highest moral purpose of his life. - Ayn Rand, Wikiquote.
The author again conducts a strawman argument by claiming Rand claims Life Qua Existence (which is the kind of Latin that Monty Python would probably not approve), and neglecting the basic purpose of Objectivist ethics, namely happiness.
Simply being alive is thus not the same as Rand's notions of acting in a rational manner to promote living in happiness.
6. Rand's intuitions
That is why there are no divergent views about mathematical theorems--when Euclid wrote his Elements, no one disagreed with it or presented arguments against it. That's because Euclid had actually proved his theorems.
Mathematics isn' t my field, but as far as I understand there is actually quite a bit of controversy concerning the axioms of mathematics. Particularly Euclid's axioms, which falls apart when gravity becomes too strong, becoming a mere approximation of physical space.
Besides, Euclid' s theorems are proved in exactly the same manner that Ayn Rand proved her Objectivist ethics, namely by defining axioms (self-evident truths) and arguing from them. Euclid never actually proved his axioms either, which is why they are called axioms.
If Euclidean geometry were so uncontroversial, why do we need Non-Euclidean geometry?
When we confront this sort of thing, it is imperative that we remember that Rand gave no argument for ethical egoism. She assumed egoism, discussed other propositions at some length, and then said that she proved it.
Again, not true. Rand proposed certain axioms and argued for egoism from these axioms. In this manner, Rand's theorems are proved just as much as Euclid' s theorems are proved, with the validity depending on the validity of the underlying axioms.
Now, I am not saying this means the concepts are illegitimate, nor does this, by itself, show that her argument is wrong (though the objections I raised in section 2 do).
The author does not claim that Rand was wrong, just that Rand was not right. What exactly does the author claim then? How do the objections raised in section 2 swing this either way?
The author in question focuses on absolute self-interest (more akin to selfishness), and neglects to consider the term 'rational self-interest' as a whole. The author does not actually critique Objectivist Ethics (which argues for rational self-interest) at all, but argues against irrational self-interest at the expense of another agent's rational self-interest. The irony is that altruism presumes it is noteworthy to act in the best interest of others, except if others want to act in their own best interest, as this would be selfish. Thus, under altruism, it is impossible to act in anybody's best interest, as nobody else can know your best interest, and acting in your own best interest is not selfless and thus not altruistic.