The first precept states
I undertake the training rule to abstain from killing.
So does this mean that punching someone in the nose isn't actually against the first precept and the five precepts generally. It really feels like it should be.
The Pāḷi word for "killing living beings" is usually pāṇātipātā. It is composed by pāṇa + atipātā:
Technically, it will depend on how one translates it. Either as killing\destroying or injuring\attacking. However, if one takes into account the entire Buddha's teaching it is evident, at least for me, that it means injuring\attacking, not only killing\destroying.
So I would say, yes, assault is against the first precept. However, one should always keep in mind that intention is key. One can injure a baby, for example, by extracting from his throat a swallowed toy. In such cases, I wouldn't consider it as against the first precept.
The following five conditions must be satisfied to break the first precept.
So punching someone in the nose does not break the 1st precept. But it weakens it. Since the question is if it's against(not if it breaks) the precept, the answer is yes.
Weakening here means the merits you acquire by keeping to first precept are reduced or weakened. Also, when it comes to the consequences of Kamma, Killing results in one having a short life span if born as a human again. Just injuring or torturing results in having bad health if born as a human again. killing usually involves inflicting pain or torturing as well. So it can result in bad health too.
The precepts are usually expressed in purely negative terms, but they are intended to express much broader positive principles. In other words, keeping the precepts is just as much about keeping the spirit of the law as it is the letter. For example, in the case of the first precept, the Buddha gives this description:
Assault is clearly against the positive principle in the first precept regardless of whether or not it fits within the strict wording of the precept. Hitting someone might not be classified as pāṇātipātā but it certainly is incompatible with having mercy and compassion.