Is it usual to have no intuition to certain proofs and simply do them mechanically?

Question

At least two books I have read, claim that proofs are not supposed to be intuitive and are meant to be terse and mechanical. I find this confusing as mathematicians are humans too. Differential geometry once upon a time was seen as simply a theoretical mathematical concept which had no real world applications. Einstein proved this wrong by using differential geometry in his theory which actually implies that Euclidean geometry is an approximation and differential geometry is more accurate for our world. Doesn't this say that Einstein could just 'feel' the way the world was non - Euclidean?

Differential geometry may seem hostile at times but if one is to translate those mysterious proofs and mechanisms to real world situations, surely Einstein could just speak mathematics as if it were German to him? When we are proving theorems on rings, fields and number systems, the proofs all materialise mysteriously and we are at times satisfied with the way they are non-intuitive. Satisfied simply because we can recall the entire proof and have a logical flow as well. Logical flow will do when we are studying the proof. But didn't someone actually come up with the proof? That person too must have had his 'Eureka' moment when he suddenly notices something that immediately gives him an idea how to prove something.

Answer 1

According to my knowledge and experience, sometimes proofs are more intuitive or clear and less technical, sometimes conversely.

To balance the opinions from books your read I present two others:

Nicolas Bourbaki: “the mathematician does not work like a machine, nor as the workingman on a moving belt; we can not over-emphasize the fundamental role played in his research by a special intuition, which is not the popular sense-intuition, but rather a kind of direct divination (ahead of all reasoning) of the normal behavior, which he seems to have the right to expect of mathematical beings, with whom a long acquaintance has made him as familiar as with the beings of the real world”

Henri Poincaré wrote a paper “Mathematical Creation”. You may also look at his paper “Intuition and Logic in Mathematics” (for instance, in this book).

PS. A few of my old answers to similar MSE questions:

• Question about long proofs?
• How are long proofs "planned"?
• Big list of "guided discovery" books

Answer 2

To quote Spivak from his preface to 'Calculus':

...precision and rigour are neither deterrents to intuition nor ends in themselves, but the natural medium in which to formulate and think of mathematical questions.

But the question may arise why take one author's words and disregard another's. Quite simply because saying things like proofs are 'supposed' to be non-intuitive is not true in any way. Not only is it untrue but it also confuses and discourages amateur mathematicians.

To an uninformed person, Einstein's field equations are nothing more than a bunch of Greek letters put together. If that person asks what those symbols actually mean, he will get explanations about how the world really is non-Euclidean and about the 4th dimension and whatnot. It may seem terse, unapproachable and even false. Even a trained physicist may accept the equation but he or she may not agree that it is intuitive.

But when one hears of the way Einstein himself got his idea. The way he was staring at everyday objects and thinking of them in different situations; thought experiments as they were called. The way he just let his imagination loose like a child, but a mathematically inclined one. The little ideas were so simple that when one hears of them he will immediately connect with them. (Although the mathematics behind them will seem intuitive only to a handful)

Every proof you see, I guarantee you, has intuition behind it. If it seems like a bunch of symbols strewn here and there it is one's incompetency to understand the language clearly.

Whether what is intuitive to one is intuitive to another or not is a different story. But one thing is certain; the more you explore, the more you think, the more thought experiments you conduct, the more mistakes you make...you have a higher chance to connect with the most impenetrable proofs

Answer 3

To extend the comments above, I think this also very much is a reflection of the idiosyncratic manner in which people learn. I personally find that 'intuition' for me almost exclusively takes the form of visual insight of some sort into a problem. I know many people who operate precisely in the opposite way though and who find algebraic/mechanical manipulations the most insightful. When I read their proofs, it seems like black magic of some sort at first pass. What may appear terse and mechanical to you in these cases may be very intuitive to someone with the familiarity with the relevant tools, especially if you're at all like me.

Alternatively though, the kernel of the idea behind many proofs is often geometric or visual in nature, and a toolbox with only the more mechanical methods may take longer to arrive at the same place as someone more geared toward visual thinking.

Answer 4

I guess I'd make the following points:

1. While it may be usual to have no intuition, it's definitely not ideal! I find that proofs are often mysterious or inexplicable to begin with. Intuition comes with time and thought. Intuition is extremely valuable because it suggests a way forward, a direction to check. Simply understanding the truth of individual steps of a proof is unsatisfying, and doesn't allow you to see the totality of the proof, the overarching story of why something is true. I don't consider myself to have understood a proof until I have some kind of intuition. But getting this kind of intuition is hard and takes a lot of time. However it's time well spent because you will see things that others can't.
2. Maybe this isn't the case everywhere, but I have always been able to eventually find an intuition for proofs. Sometimes an idea in mathematics is quite bizarre and outside the real of ideas that you would encounter in ordinary life. Intuitions might resemble something like "Loosely, more X seems to generate more Y in this system" or "This kind of structure A seems to frequently appear frequently in situations B". Terry Tao for instance always tries to introduce his intuition on a subject in lectures, even with something as abstract as real analysis. His lectures were the ones that leveraged intuition the most out of all subjects I've ever taken, and I think there are good reasons for that.
3. Sometimes when a subject is new, the proof and results are extremely unintuitive. People are more concentrated on just proving the result than trying to make how they proved it understandable to someone else (there's also the issue of page limits). But over time, the newer presentations of the same subject matter make more and more sense and are baffling no longer.

In summary, having a low-level understanding of proofs is not only unsatisfying, but makes it extremely difficult to advance your research.

Answer 5

First,no proofs on any topic are done mechanically without any motivation

WHY:Mathematical Proofs are nothing but paintings on canvas.Ever seen an artist just gazing on the blank paper on his canvas to find out where to start his drawing from.

Just like that! Without any motivation it is hard to come up with a proof of any mathematical phenomenon example theorems,lemmas,etc.

Maybe when we see the whole proof of the theorem,it makes us feel how does one on earth come up with such a beautiful proof.

Secondly theorems do not come out from the hat just like that.You are feeling they are not intuitive because in a book it is just not possible to give every detail of how the proof came up.A book is just organized to include the theorems and its proofs otherwise you would end with a book with $1000-1200$ pages which does not make sense.

Just imagine it took many years to ultimately come to the definition of a GROUP in Abstract Algebra but when we read it in any book of Abstract Algebra,we find everything is so organized and properly given.If it were to include all the details it would never end.

You can just google in to find out more about GROUPS.

And Lastly if you see a complete picture on the canvas can you just figure out by seeing it that from where did the artist start painting it from?