If we define purpose as the reason for which something exists.
Can there be an objective purpose for something?
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This answer has an objective purpose.
It exists to answer the question you asked.
The same can be said of a large number of artifacts (here meaning things that people make). Some objects will have compound objective purposes.
E.g., the chair exist so that someone can sit in it. (that's a type of reason). But it also exists because the chair maker made it to make money (a different type of reason).
Arguably this answer fulfills multiple and complex purposes as well. I answer it because I enjoy answering questions. I answer them on SE because SE has a good system where the answers are free to see. I enjoy answering questions about philosophy, because I have 4 degrees in it. I enjoy answering this question because I think a good deal about related issues in Hegel and Aristotle.
A harder variant on your question would be, can something exist which has only one purpose for its existence? Here, many artifacts will have trouble qualifying since they fulfill multiple purposes. And to get a definitive answer to this question is going to be difficult and somewhat opinion-based, because (a) atheists are going to reject that anything exists for a reason God gave it, (b) we can debate whether someone gave something a singular purpose or always has multiple purposes (see for instance the debate about whether one can have a pure maxim of the will -- Marcia Baron, Kantian Ethics Almost without Apology and Hegel's various critiques of Kant), (c) whether we accept teleology or natural kinds, and (d) whether we grant evolutionary purposes as objective purposes (why does the flying squirrel have membranes? because they enable it to glide from tree to tree and thus survive while getting food).
can humans act for a **single** objectively *verifiable* objective?
is no, but this is not the question actually asked above and largely seems to be a question we could not resolve on the SE.
What does "objective" mean?
But, before we deal with that, it is necessary to remark that this sentence,
If we define purpose as the reason for which something exists.
is ambiguous, and one of its meanings is false.
It is raining. Why? because water has condensated in the cooler upper atmosphere in the form of clouds. That's the reason why it is raining. It is not, however, the purpose of rain.
So let's assume that you mean "purpose as the reason for which something exists" as the intent why something exists. It is raining. Why? Because God has planned it so that the plants won't die. That would be the "purpose" of rain.
Now, back to "objective".
What is an "objective purpose"?
Again, this is ambiguous.
John intends to study Astronomy. This is a purpose. It is "objective" in the sense that it exists as an artifact of the real world; it makes John plan for taking courses, reading books, buying a telescope, etc. It is "subjective" (ie, non-objective) in the sense that it resides exclusively in John's volition; should he change his mind and decide to study pharmacology instead, the purpose ceases to exist.
If you intend it like "it is objective because it exists in the material world", then virmaior's answer applies. If you intend it like "it is objective because it is not subjective", then no; there is no purpose, no intention, without somebody's volition, and so, there is no such thing as an "objective purpose".
In other words, there is no such thing as a "final cause" outside of subjectivity. If we say that "the purpose of rain is to avoid the death of plants", we are either attributing a mind to natural phenomena (clouds, water, air, wind) or, perhaps inadvertently, creating a fictitious mind ex-nihilo (gods, devils, angels).
From an older Alchemical view recaptured by one common theory of modality and reiterated in pyschological terms by Lacan, the answer has to be no.
A purpose is 'cardinal' or 'symbolic' information that relies upon either empathy or authority for its content. It is tied intrinsically to the person (or other entity like a tradition that we treat as a person) that the purpose serves, so it cannot be objective to some degree. It can have meaning only 'deontically' or 'cardinally', with reference to a goal, and therefore to the desire of a personality.
For Lacan, the mind lives in three worlds, which he labels Real, Ideal and Symbolic which correspond to the Astrological aspects of the Fixed, the Mutable and the Cardinal.
Real or Fixed statements are about physical objects that we experience as outside our minds. They are objective in the normal sense and take an indicative mood or an alethic modality.
At the other extreme, Ideal or Mutable statements are ideas, inside our minds. Since we can describe things to one another, and once we describe them they become independent of their source, Ideal things are also objective, in a different way. Their extreme and complete subjectivity lends them a form of objectivity. They transfer between subjects and everyone has their own view of them so no one controls them. Some of them are personal wishes, but my wish exists for you only as my wish, and not as a fact about other things. So it retains objectivity by being necessarily relativized. These take a subjunctive or optative mood or an optative modality.
But a Symbolic or Cardinal statement is a link between these two, and we are confronted right off with the fact that individuals and cultures generally map the two apparently shared realms back and forth idiosyncratically. Its contents are not objective and cannot be rendered objective unless they are explicitly relativized. They take a jussive or imperative mood or a deontic modality. (Purpose is morality for the inanimate.)
A hammer has a purpose only in a context of human usage. Without someone around who wants to beat on something with a certain level of precision, a hammer does not have a purpose. When we humans are all dead, the purpose will vanish. So it is not an objective fact, only a contexual one. A hammer may be intended for a given purpose by its creation, or its identification, but a given real hammer does not have an objective destiny to fulfill. It might hang on a hook in a store until it becomes too rusty to use, and then go to a landfill. Then in what way was there ever an objective purpose?
So it may be objective that virmiaor has a purpose for his statement, but the idea that it really has that purpose is entirely dependent upon him as a subject. This purpose can only be understood through his or someone else's eyes, by projection of oneself into someone's mind: i.e. subjectively and not objectively.