# Hard time delivering other than algebraic proofs

I'm enrolled in some different math (related) courses thought by the CS department at a university (although I'm unable to attend most of the classes due to work). Even though the focus of these courses doesn't really rely on mathematical rigor per se, there's no way to avoid it. Every once in a while you're asked to prove some propositions. When we first started logic, we could do this using truth tables, later on we got to use standard equivalences to rewrite propositions. Following that we got into natural deduction (using Fitch/flag proofs in our case).

This all went pretty smooth, even though I'm sadly by no standards blessed with any mathematical talents.

But then we got to the point where proving propositions using flag proofs or stand equivalances was so tedious and long winded we got introduced by written proofs. And from that point onward I just didn't get it anymore.

The same is happening now. Having to proof combinatorial identities using a combinatorial proof is proving head breaking. Algebraically I can manage, albeit with some help here in there.

Now this question might be too vague or specific to my own problems, but what can I do to remedy this? It seems like I'm missing a certain amount of creativity (or perhaps just intellect as a whole) to manage these problems...

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It sounds like you may use some good course in remedial algebra, meaning: refreshing all the algbera taught at high school level and, perhaps, a little above. – DonAntonio Feb 12 '13 at 20:39
Perhaps looking at well-presented examples of other proof-kinds in books would help explain how to prove something. There are a variery of techniques (by contradiction, induction, direct, etc), and in my case it helped a lot to go through lots of standard examples with practice... – gt6989b Feb 12 '13 at 20:39
Sometimes there is a relation between difficulty in the course and unable to attend most of the classes. But not always. – GEdgar Feb 23 '13 at 17:27

Firstly I'll mention the difference between proofs using natural deduction systems or formal proofs and "standard" mathematical proofs.

The former is an actual deduction system, you're proving statements in a formal way by using certain inference rules.

The latter is completly different! A more correct term for the latter would be plausibility argument. It's just an informal reasoning to get from the hypothesys to your goal, that's what people usually mean by proof (though an actual proof must be formal).

Secondly, as I have little information about what exactly you're having trouble with, I can only guess what the problem is, but here's my guess anyway: you're having a hard time following standard mathematical proofs because your only training is in formal proofs, you're too used to the clarity that comes with them. Also you probably are really unused to what I named above as plausibility arguments or informal proofs. From here on when I mention the term proof I mean a proof in the latter sense.

Thirdly, I'll address how to solve this. I'm assuming that what you mean by an algebraic proof is actually an arithmetical proof, i.e., proofs in which you're basically doing some calculations.

Mathematics is built on blocks. Some blocks are tougher than others, but all of them can be divided into atoms. In an atomic state everything is simple (or at least as simple as it can get). If you're having trouble with a proof, break the block into smaller blocks, if you still can't handle it, break the smaller blocks into even smaller blocks and keep doing that until you understand the proof.

This is easier said than done, but nothing is more important than you trying it yourself, I believe. Also if you can't understand a proof, there's a good chance you can't identify which block to break or how to break it. To handle this issue I suggest that every time you infer something (or something is infered in the proof), you ask yourself "why is this true?", when you can't answer that question, you've found a block you want to break. Obviously it also helps that you have someone you can ask what you don't understand and break blocks with you.

I also suggest that you try going through some simpler proofs. For instance like the ones one would find in an elementary number theory course. It's a nice bridge between arithmetical proofs and the other proofs, while still mantaining a lot of logical simplicity. Finally, don't think for a minute that my next (and last) suggestion is any less worthy than the previous. If I was forced to give you an answer in ten words or less, this would be it: How to Prove It: A Structured Approach, by D.J. Velleman. Go through it. I hope I understood your issues correctly and wish you luck.

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You understand correctly, I will be putting more time aside to focus on delivering proofs. The book you recommended is on its way as we speak :) Apart from that one I also ordered amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/…, which is the recommended book from a uni course. So'' Ill work trhough both of them :) – DiligoDiligentia Feb 23 '13 at 10:28
@DiligoDiligentia Glad I could be of assistance. – Git Gud Feb 23 '13 at 10:29
+1 for noting that typo. Thanks again. :-) – Babak S. Feb 24 '13 at 16:26
I like the atom analogy. I sometimes like to think of proofs as going from nail in the wall A, to nail in the wall B. Reading a proof involves investigating a very thin (terse) string connecting the two nails. To read a proof, you have to dig your fingernails into the thin thread and expand it, perhaps with a few fat parts so you can back up and sort of see how the proof went. – Brady Trainor Jun 23 '13 at 8:58