Understanding of the maths you're doing

[I_laff]

Whilst I'm solving a maths problem I normally try and understand the problem visually. For example, in calculus I try to justify what I'm doing by seeing how certain expressions relate with the geometry of the problem. However, sometimes when the step are quite lengthy, it feels like I'm just rearranging symbols without a real intuition as what I am doing. Will the lack of computational intuition damage my understanding of mathematical concepts, and if so what can I do to fix it?

 

[member 587159]

Whilst I'm solving a maths problem I normally try and understand the problem visually. For example, in calculus I try to justify what I'm doing by seeing how certain expressions relate with the geometry of the problem. However, sometimes when the step are quite lengthy, it feels like I'm just rearranging symbols without a real intuition as what I am doing. Will the lack of computational intuition damage my understanding of mathematical concepts, and if so what can I do to fix it?

Not necessarily. In more abstract settings, like group theory or topology, geometric intuition becomes less important. You will discover that you develop another kind of intuition for these subjects.

 

[fresh_42]

Whilst I'm solving a maths problem I normally try and understand the problem visually. For example, in calculus I try to justify what I'm doing by seeing how certain expressions relate with the geometry of the problem. However, sometimes when the step are quite lengthy, it feels like I'm just rearranging symbols without a real intuition as what I am doing. Will the lack of computational intuition damage my understanding of mathematical concepts, and if so what can I do to fix it?

Intuition is a two-sided coin. It often will help to guide you along a path, but sometimes, it'll be the cause of problems in understanding. I think of examples like Peano curves. As @Math_QED has already mentioned, topology has many such examples. In addition different people have very different ways to imagine or not what they are doing.

There is a famous anecdote about John von Neumann. At a party he has been asked: "Imagine two locomotives being 100 km apart and running at the same speed of 100 km/h on the same track heading towards a collision. Now a fly is traveling at twice the speed of the locomotives back and forth between them. When will the fly be smashed?" Von Neumann answered without any hesitation: "After half an hour." - "Oh,"said the questioner "you already knew this one." - "No, I didn't. Why should I?" - "Because the trick is that usually mathematicians set up a series and calculate its value." Von Neumann's answer: "But I did calculate the limit of the series!"

This short story tells us that you don't necessarily need an intuition. The question with the fly has a simple solution if you think about it a bit. But of course you can also calculate the length of the way the fly has to take before the crash. For most of us the second method will be a bit troublesome and might be a source of errors. John von Neumann has been famous to do calculations in mind very fast, so he didn't need intuition, just techniques.

Intuition is very helpful for the majority of people, but it can also be an obstacle or - as in many cases - difficult to achieve. If you want to improve it, try to draw as much as you can, and for longer proofs, outline the main arguments in advance. Drawings don't need to be accurate or detailed, far more important is that they contain the clue of a problem. Nobody can draw e.g. the manifold $Sp(4),$ but to get an intuition, any curved surface will likely do. However, if you'll occasionally lack a good intuition, don't worry. Start to worry if you have neither an intuition nor the techniques.