> Some things are less intuitive because of the dense equations. You'd have to be slightly insane to say that "x^2 = y^2 + z^2" is a more intuitive way to explain what a circle is than a picture of a circle.
It's the difference between qualitative and quantitative descriptions. Which one captures the full depth of the concept? Neither on its own; you need to contemplate both at the same time to capture that depth ("range" may be a better word). There is information hiding in the interplay between the two that is missed when considering them in isolation.
I don't believe formalism is all we need, but it is necessary if the problem at hand requires more than just intuition. Conversely, formalism without qualitative understanding is opaque and sterile. Together, they combine operational ability and simple intuition into a higher form of intuition (operational intuition), the most desirable kind.
It's the difference between qualitative and quantitative descriptions. Which one captures the full depth of the concept? Neither on its own; you need to contemplate both at the same time to capture that depth ("range" may be a better word). There is information hiding in the interplay between the two that is missed when considering them in isolation.
I don't believe formalism is all we need, but it is necessary if the problem at hand requires more than just intuition. Conversely, formalism without qualitative understanding is opaque and sterile. Together, they combine operational ability and simple intuition into a higher form of intuition (operational intuition), the most desirable kind.