Shelley cannot make his grand claims about poetry without first establishing that in his mind poetry is a broad category. Poetry is not limited to lines of verse. Instead the term refers to any product of the imagination, such as music and visual arts. It seems natural enough to group the arts together as contrasting the sciences, but once the reader allows for this genre-blurring, Shelley seizes the opportunity to push his argument further. If poetry is anything requiring abstract, creative thought then philosophy and law are poetry too.
The idea that philosophy and law qualify as forms of poetry does not entirely protect poetry from Peacock's allegations of irrelevance and outdatedness. Shelley's ingenious strategy on this point is to insist on poetry's currency not by renouncing poetry of the past but by applauding it. Shelley argues that true poetry is timeless. This would explain why literary scholars continue to read Shakespeare centuries after the playwright's death while scientists discard publications that are only a few years old. In Shelley's estimation, this fact shows that poets possess a superior genius. He goes so far as to compare them to prophets.
A skeptic might say that Shakespeare's authority persists because art is subjective and therefore lenient. Scientists have to prove their work; poets do not. Shelley addresses this issue by comparing reason and imagination. He explains that you use reason to see differences and distinctions, and you use imagination to see relationships. An exclusively reasonable person would open a book and just see groups of letters. This would be accurate but meaningless. You need imagination to recognize relationships -- to recognize groups of letters as words and, more broadly, to give data value.
Shelley admits that poetry is not materialistic. Products of the imagination are usually abstract, and it follows that poetry will not make it easier to harvest a crop or sail a ship. Shelley rejects the idea that this means poetry is useless. This argument parallels his argument about reason and imagination. You can only benefit from information if you can interpret it, and a society can only benefit from technological advances and wealth if it can give them a purpose. According to Shelley, economists and scientists have only managed to widen the gap between rich and poor. Poetry makes people more compassionate, and therefore more likely to put material progress to noble use.