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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?

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Logic is a crutch only because our intuitive brains are permanently broken legs. Look no further than the monte hall problem. – corsiKa Jan 21 at 20:09
@corsiKa Although the problem is unintuitive...the (modern) statement of the problem hides the trick in a synthetic way. No one ever says if you have three doors one already open and guaranteed to be a goat, but you have a door picked before that door was opened... – Zach466920 Jan 22 at 21:59
@Zach466920 I'm pretty sure the monte hall problem confuses most people even when totally and properly explained. But that's just an example, there are countless mathematical properties that most people can not intuitively grasp. Look at the calc 1 failure rates. Look at the almost anything to do with infinity. Or imaginary numbers. – corsiKa Jan 22 at 22:31
@corsiKa I agree, however, according to a Google search, intuition is the ability to reason without conscious effort. So having rules like the Op describes fits the definition. In this case, manipulation of logical symbols/statements without having to "think" about what these conjured results "really mean". – Zach466920 Jan 22 at 22:56
up vote 53 down vote accepted

It's perfectly normal. In fact, I think that's how a mathematitian's mind grows.

  • First, you are naive and "intuistic", and you do a lot of "well, of course this is so!" like statements that are not well founded.
  • After you are repeatedly hit over the head with examples where your intuition fails, you take a huge step back. You realize that even simple statements may be wrong, and that you need a rigorous way of proving them. You use strict logic notation to avoid any and all confusion.
  • When you are more and more practiced, you begin to transition back a little. Yes, you still know that every statement has a strict logical form, but you don't write it down anymore. You begin, again, to rely on intuition.

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!"

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I would say my process is still in bullet two mode in my head, where I maintain rigorous logical thinking internally, and then when I write a proof I mostly use words instead of symbols to make it more readable. Perhaps I haven't transcended the bullet two style of thinking because of being less advanced, but I do feel like maintaining rigorous internal logic is valuable. I find it easier to think "in symbols" and more readable to write out the words. +1 – Todd Wilcox Jan 21 at 17:47
See also Terence Tao's well-known blog post on this subject, which mentions a similar process. – Marc Paul Jan 21 at 20:51
I know for me, what bullet I'm in depends on how familiar I am with a certain branch of math (although I usually try to skip bullet 1 if I can). – PyRulez Jan 22 at 6:06
@MarcPaul Oh wow. Now I'm kinda proud of myself for writing almost the same thing as the great Terence Tao :D – 5xum Jan 22 at 6:54
This looks like another formulation of the four stages of competence: 1. Unconscious incompetence, 2. Conscious incompetence, 3. Conscious competence and 4. Unconscious competence. – skyking Jan 22 at 11:53

Logic is not a crutch. It is a way to check that you haven't made a mistake. It might not be the best first step to make a proof, but it is definitely a necessary step.

As you get more experienced, you tend to start by sketching the proof in very informal terms. This is like navigating by large landmarks: "I am headed towards that mountain over there, hm, that ridge seems like a good approach. OK, lets go."

However, to actually get to that mountain you still put one foot in front of another, one step at a time for hours. Logic is like walking one step at a time, while looking at your feet and the nearby ground. It is necessary, but sometimes you need to look up to see where you should be going.

Now, if you are going to write the proof in English eventually, you don't have to actually write down all the logic in symbols. I find that I can write English (or Norwegian) while keeping the formal symbols in my mind.

Some published proofs aren't very formal. The reason for this is that formal proofs can be very long. Instead they give arguments that convince the reader that it is possible to work out the proof in all its details, even if neither the writer or the reader have actually done so.

Professional mathematicians become good at telling whether such an informal argument is valid or not, but they occasionally slip up, even peer-reviewed articles can be wrong. So, take care.

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