I'm near the end of Velleman's How to Prove It, self-studying and learning a lot about proofs. This book teaches you how to express ideas rigorously in logic notation, prove the theorem logically, and then "translate" it back to English for the written proof. I've noticed that because of the way it was taught I have a really hard time even approaching a proof without first expressing everything rigorously in logic statements. Is that a problem? I feel like I should be able to manipulate the concepts correctly enough without having to literally encode everything. Is logic a crutch? Or is it normal to have to do that?
Logic – Is Logical Notation a Crutch When Writing Proofs?
formal-proofslogicsoft-question
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My advice: Do not get too distracted by your own shortcomings. Study mathematics with passion, curiosity, and enthusiasm (whether it's mathematics or any other discipline, these are three characteristics that will take you a very long way in life). I will say, however, that the pursuit of mathematics does require a large degree of passion.
If you strive to "master" mathematics, then I would suggest teaching yourself from the classical works of those who laid the foundations of the various branches of your interest. For instance, if you are interested in "Number Theory", then study the works of Gauss (see, "Disquisitiones Arithmeticae"). If you told me you are interested in "Set Theory" then I'd tell you to study the original works of Georg Cantor that laid the foundations of set theory. You can easily search the internet to find various PDF's of original papers. However, a good book is "God Created the Integers" by Stephen Hawking. If you wish (and my advice), pick a random "mathematician" from this book and read his works. From there, supplement it with modern literature and practice until you fully understand it. You will be well versed if you proceed in this manner.
Despite the branch(es) you may be interested in (and I strong recommend against the idea of "specializing"), your primary focus should be to understand the classical works while supplementing this objective with modern texts. This has the benefit of not only allowing you to recognize the intuition that served as the basis of these theories but you will also come to realize first-hand that contemporary proofs are largely simplified. By that I mean, do not become discouraged by the degree to which a proof appears to be "trivial" in a modern text; the originator's version was much longer and more complicated, I assure you.
Now, regardless of what branch of mathematics you happen to be studying at a given time, I've always found that it is supremely beneficial to find a geometric interpretation for things that are purely algebraic and similarly, when working with geometric notions, I try to find the algebraic interpretation of those notions. This is not always an easy thing to do, but IF/ WHEN you do it, the solution of many complicated problems will become more intuitive.
Generally speaking, you want to find a practical application for every mathematical concept you come across. Newton's genius was his ability to not only develop the calculus but to translate it to various properties of the universe. My point though, is that calculus seems to be more "obvious" once you find this application. Most of mathematics works this way.
None of what I'm saying here is "absolute". This is mere opinion. However, I hope it helps you in your future endeavors. The most important aspect though, is your own degree of passion and curiosity. None of the above is relevant without that.
The traditional reading of "if $A$, then $B$", i.e. $A \to B$, is :
"$A$ is a sufficient condition for $B$" and "$B$ is a necessary condition for $A$".
What happens with :
"$A$ if and only if $B$" ?
It is only a "syntactical" issue. The abbreviation is made of : "$A$ if $B$" and "$A$ only if $B$".
In turn, "$A$ if $B$" is $B \to A$, while "$A$ only if $B$" is $A \to B$.
Thus, from the point of view of natural language, "$A$ if and only if $B$" is :
"$B \to A$ and $A \to B$".
But when "if $B$, then $A$", we say that "$A$ is a necessary condition for $B$", and when "if $A$, then $B$" we also say that "$A$ is a sufficient condition for $B$".
Thus, cooking them together, we read :
"$A$ if and only if $B$"
as :
"$A$ is a necessary and sufficient condition for $B$".
Best Answer
It's perfectly normal. In fact, I think that's how a mathematitian's mind grows.
At least, that's how it's worked for me. Mind you, the intuition in the final step is very different than the original intuition. The original one is the brash "d'ah, how can you ever doubt that?!" kind of thing that is embarasingly bad at doing math.
The more developed intuition in step 3 is much different. It has a lot more thought put into it. It's more "yeah, this is so, and I know approximately how I can prove this using strict logic, but since it would take 3 pages, I won't use strict logic."
Sure, the second intuition can also be wrong. And every once in a while, it is. But its performance is way way way way way better than in the beginning, and it also has a fall-back. If all else fails, your final answer is no longer "well, but, but how could it not be true?", the final answer is "oh alright fine, I'll spell it out rigorously!"