# 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

## 1 Answer

First, I am a great fan of Visual Complex Analysis.

Nevertheless, I disagree with the statement that

Mathematics today [...] is mostly built on abstract symbolic manipulation rather than on [...] visual (intuition)

Even the areas of mathematics that do not easily lend themselves to visualization—say, abstract algebra, or mathematical logic—still rely on intuition, just not visual intuition. Someone adept at group theory develops strong intuitions about group structures, strong enough to sense whether a claim or conjecture might be true, whether a proof is sound, etc.

I am convinced by Needham that visual thinking can greatly enhance understanding of complex analysis. I am not convinced the same is true when "complex analysis" is replaced by other areas of mathematics.

• @ArkaKarmakar: By constant, deep immersion in the topic. – Joseph O'Rourke Feb 9 '16 at 12:58