I have been studying mathematics for more than a year now. In general, it's relatively easy for me to solve the exercises and to find proofs for certain propositions myself. In this sense, I would say that I am not too stupid for mathematics. However, I recently noticed that I sometimes perform logical reasoning in a syntactical manner (i.e., in a kind of symbol-pushing) rather than a contentual/intuitive manner. For me, this is unsatisfactory, because I think that mathematics is only fun if it is intuitive.
For example, I have no problems at all finding a proof of the fact that an implication "If $A$, then $B$" is equivalent to its contrapositive "If not $B$, then not $A$" using evident proof methods. Also, it is easy for me to show that, if $n$ isn't even, then $n^2$ isn't even. Then I can syntactically deduce that the proposition "If $n^2$ is even, then $n$ is even." is true – nevertheless I don't understand this contentually, but rather I've inserted the appropriate propositions for $A$ respectively $B$ based on the structure of the previously proven rule of inference that $\neg B\implies \neg A$ implies $A\implies B$.
Now I want to aks you for advice. These are my questions:
Is it normal that one sometimes performs logical reasoning syntactically rather than intuitively (as a beginner)?
What could I do to move against this problem? Is it possible that it just takes some getting used to it? After all, when I was first told that $\forall x(\neg P(x))$ means the same as $\neg\exists x(P(x))$, this wasn't obvious to me at all – but now it is completely obvious for me, after some time of getting used to it.
I've found the following quote by Stanisław Leśniewski (who was the doctoral advisor of Tarski):
Please click here, I can't post images since I do not have 10 reputation.
It seems that he claims that it's sometimes much easier to perform logical reasoning by means of syntactical rules⁺ than to always reason contentually. What do you think of that?
⁺ Of course, the elementary rules of inference should nevertheless be completely obvious
Actually you've a good skill that you can use to your advantage. I always think of mathematical theorems as having two parts, the logical part and the structural part. Your ability to systematically perform symbol-pushing (using valid inference rules of course) means that you'll have no trouble with the logical part, which is often half the problem solved.
But you need to make sure you fully understand the reason behind every single logical inference rule that you use, otherwise you will not have a complete grasp of the logical part of theorems. You mentioned as examples the logical fact that can be expressed prettily as "$\neg \forall \equiv \exists \neg$", as well as the identity "$A \to B \equiv \neg B \to \neg A$". You must know exactly why these are valid, and never merely remember them as black-box rules. Along with the usual ones, you should also be familiar with the disjunctive normal form and the Skolem normal form, because these logical forms are the most useful.
The logical part crops up more frequently than you might be aware of. For a simple example, you ought to know the fact that switching quantifiers in one direction, namely $\exists \forall \to \forall \exists$, yields a valid inference, but not necessarily the other way around. Notice how this is essentially the core of the difference between continuity and uniform continuity, convergence and uniform convergence, and so on. One can in many cases grasp the meaning of the word "uniform" applied to some concept despite never having learnt it, simply by looking at the quantifier structure and seeing where it makes sense to pull an existential quantifier outward past a universal quantifier!
Another simple example is that anything you can say about a structure in terms of its operations and its size is preserved under isomorphism. This is because isomorphism is a relabeling which commutes with the operations and the size of the structure. So one knows without proving anything all kinds of results of this form, such as: (1) The order of an element in a group (including characteristic of a ring) is preserved under isomorphism. (2) A normal subgroup remains a normal subgroup under isomorphism.
Of course, there is still the structural part. In real analysis for example it is highly advantageous to be able to visualize limiting processes such as getting arbitrarily close to a point or going to infinity, because if you just try to push symbols you'll most probably get nowhere in attempting to prove any non-trivial theorem from scratch, such as the extreme value theorem for continuous real functions on a closed bounded set (or even just a closed interval) in Euclidean space.
An interesting case is (Euclidean) geometry, where it's even more obvious how useful a spatial intuition is. The catch is that its intuitiveness is a snare for the logically careless. If you attempt to write down formal proofs of many geometric theorems, you'll quickly find that even simple ones like "angle-at-centre is twice angle-at-circumference" is not so easy, due to issues in handling which side of a line a point is on. Historically it was common to make hand-wavy arguments that are actually invalid. Euclid was not as systematic as he could have been, and his proofs have many gaps, such as for Prop.1.16. And he missed out an axiom as noted by Pasch. Even Hilbert who later attempted a rigorous axiomatization of Euclidean geometry made intuition-driven logical jumps.
Ultimately, different mathematical objects have different mathematical structure, and you may need to study special aspects of their structures independently, even though there may be common aspects that help you to easily visualize new structures using common tools. For instance many mathematical constructions have a nice interpretation when you impose additional structure such as a partial order or a topology. But this also means that a lot of arguments about partial orders or topological spaces carry over without change, and often these arguments have a logical flavour to them (such as a Galois connection).