# The role of visualization and intuition in graduate and postgraduate math and developing it

[I am not an mathematics educator; but because the process of learning is educating yourself, I'm posting it here]

In Visual Complex Analysis's preface, the author gives an analogy with pseudo-deaf musicians and follows the same to mathematics. Mathematics today, he argues, is mostly build on abstract symbolic manipulation rather than much more stronger visual (intuition) method.

Is it a good notion to rely on visualization and intuition solely on graduate and postgraduate mathematics on topics like Analysis, Abstract Algebra and other fields (Topology and differential geometry is pretty obvious - it is probably visual per se) ?

If yes, then how to visualize and develop intuition in graduate and postgraduate mathematics in fields mostly relying on (stupid; ?) symbolic manipulation ?

• Just for clarification, what is the difference between graduate and postgraduate in the context of your question? I suspect that by postgraduate you could mean post-PhD studies, but in certain countries postgraduate studies refer to those undertaken at any level higher than Bachelor's, hence including Master's and PhD. – J W Feb 8 '16 at 16:02
• I don't think relying on "visualization and intuition solely" (emphasis added) is a great idea, but I thought I'd at least point to MESE 1589 in which I linked to Velleman's (2007) The Fundamental Theorem of Algebra: A Visual Approach. (I think you will find it relevant to the notion of using visualization in complex analysis!) – Benjamin Dickman Feb 9 '16 at 1:51
• I am not qualified to give an answer based on research so I will reserve input to comments. I believe this question lies in a broader question about 'form of argument'. I personally find matrix questions easier when they avoid component notation such as $a_{ij}$ and instead are written as $A$. It's not that I cant understand component notation (I can) but it becomes more of a devise for manipulation than something I can concretely visualise as a matrix. I hope this makes some sort of sense, it's hard to put into words exactly what I mean. – Karl Feb 9 '16 at 12:30