More specifically in mathematics, the interplay between formalism and intuition, like a mental danse or gymnastics, is a powerful process in furthering our understanding of mathematical truths:
From a formal perspective, mathematical objects can be created in so many ways, some constructions being more intuitive and beautiful than others (axioms, groups, rings, fields). The formalism itself let us see what intuition can't.
From an intuitive perspective, it's useful to latch on whatever concept one have to learn the next level of abstraction, while acknowledging that the intuition might not be 100% correct. Like using addition to intuitively understand multiplication, or addition and multiplication to intuitively understand fields. The intuition let us familiarize with otherwise novel ideas.
Ironically, this article wants to be very normative about which mathematical intuition is better (which there isn't, I'm sure many don't think of "multiplicand" as something special), while disregarding any cues from any formalism.