The German mathematician Klaus Janich has a wonderful response to this question in his book on topology, which is *intentionally* very non-rigorous and intuitive:

*It is often said against intuitive, spatial argumentation that it is not really argumentation,but just so much gesticulation-just 'handwaving'.Shall we then abandon all intuitive arguments?Certainly not.As long as it is backed by the gold standard of rigorous proofs,the paper money of gestures is an invaluable aid for quick communication and fast circulation of ideas.Long live handwaving!* (Janich, *Topology*,page 49,translation by Silvio Levy)

It was later said by Levy that Janich told him that this particular passage was inspired by Janich's concerns that German mathematical academia (and textbooks in particular) were beginning to become far too axiomatic and anti-visual and that this was hurting the clarity of presentations to students.

There is indeed much that is wise in this quote and it really gives what I think is an excellent "rule of thumb" for determining when a "proof" in mathematics has crossed the line and really become non-rigorously vague by 21st century mathematical standards to the point is really proves nothing: If the intuitive argument **cannot** be rewritten in a completely axiomatic and pedantic manner where there is a completely logical progression from premises to a conclusion, then we should seriously reconsider whether or not our intuitive argument is a valuable one. I think you'll notice most of Hatcher's arguments would pass this test,even if it would probably take a considerable amount of spade work to make them completely rigorous in the same sense as a real analysis or algebra proof.But since algebraic topology is so closely related to classical geometry, completely abstract reasoning would probably strip away much understanding of the sources of most of the central concepts,which I believe was Hatcher's reason for writing the text in this manner.

(Sadly, there are too many algebraic topology texts that take the uncompromisingly rigorous and non-visual,categorical/functorial perspective-such as the old classics by Spanier and Dold and more recently, the beautiful texts by May and tom Dieck. The down side of this approach is that it completely disconnects the subject from it's geometric roots and it becomes simply another branch of algebra whose roots are utterly mysterious.No one quite seems to have figured out yet how to effectively interpolate between the 2 approaches in a textbook. The closest anyone's ever come to pulling it off to me is Rotman. )

Of course, as it's stated, this isn't an exact science. The question of what constitutes a proof has constantly been questioned and revised since the beginnings of mathematics in the Ancient World. I have no doubt it will continue to undergo scrutiny in future ages. But I think Janich has given some quite good advice to the novice here.