I have trouble understanding the difference between these two words, especially when people use them while talking about aims and goals. Any help please?
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This question is bordering on philosophy, but it does extend to developments in the English language, so I will try to give an answer.
Just as in English grammar, a subject in philosophy is someone who acts, and an object is that which is acted upon.
At some point, the term subject came to be used in English (but also other languages) primarily for the person who perceives or thinks about something, and object for the thing that is perceived or thought about by this person. This is in itself mainly a linguistic development, not a philosophical one.
Thus subjective in English came to mean "pertaining to the percipient or thinker", objective "pertaining to something existing outside and independently of the percipient or thinker".
Now, when we consider the nature of cats, one might say they are "ferocious killers", another "adorable pets", yet another "small feline carnivores". The first two descriptions can be said to depend very much on the person describing them: to a cat owner, they seem like adorable pets, but to a mouse, they seem like ferocious killers. Especially the words ferocious and adorable suggest that these descriptions depend very much on whoever is describing cats, i.e. on the subject. We would hence call these descriptions subjective.
Conversely, small feline carnivores depends very little on the person describing cats; it is more like an independent fact, a description based mostly on cats as they are and exist in the outside world. As it can be said to centre on the cat itself and not on any perceiver, this is generally called an objective description.
You probably already knew this; I've just tried to explain how these words acquired the sense they now usually have.
Goals and aims are inherently subjective—that is, unless you're talking about a goal that is not yours / the subject's, but rather the object's. E.g., when we, as cat owners, are talking about cats in the attic ('we' are the subject), their goal as objects might be to feed themselves on juicy mice, whereas our subjective goal to have them present there is to kill off the pests that poop on our stored furniture.
Note that all of this applies mostly to other languages as well, as this part of English has commonly been part of international discussions and communications.