# How to increase math analytical skills

I am a graduated engineer. I learned math and can do it but in an engineering way (i.e. applying given formulae and bunch of steps to reach a known target) but the problem is that now I realized, math is not about just applying, it is about thinking and analyzing, like what theoretical physicists and mathematicians are doing and I would like to start thinking like them, so how can I change my way of thinking and start analyzing math instead of just applying it?

What kind of book or lectures would you recommend to follow?

• Take basic math books and see how proofs are constructed, what is allowed and what is not and start trying to prove statements on your own before reading the book's proofs. – Yagger Apr 29 '18 at 18:55
• Take some engineering problem that you well know and ask you why the math used to solve it does work. – Emilio Novati Apr 29 '18 at 19:42
• I see your point @EmilioNovati thanks but normally we don't always see behind the 'why' questions if not somebody expert led us to there. – wisdom Apr 29 '18 at 19:47
• I'd recommend number theory. math.stackexchange.com/questions/1714966/… – Ethan Bolker Apr 29 '18 at 20:31

Reading Book 1 of Euclid's Elements is a great place to start on the road to learning pure mathematics. Here are some reasons it may be attractive to you:

• It is self-contained. The text does not assume any previous mathematical knowledge. Every proposition proved in the text can be traced back to the beginning set of principles.
• Book 1 is short enough to be tackled in around two weeks of casual study every evening. It packs a lot of rich material into that short space and culminates in a beautiful proof of the Pythagorean Theorem.
• It provides many chances to practice analysis at any skill level. To practice analysis, draw the diagram step by step as outlined in the "construction" part of the proposition then pause your reading and consider what pieces of information are available to arrive at the conclusion. If you need a hint, read the first few statements of the proof and you may get a feeling for the direction to take.
• Euclid provides excellent examples of synthesis, the counterpart to the skill of analysis. If analysis is about dissecting and extracting information with a view toward what is relevant, then synthesis is about recomposing that same information into a shape that is logically sound, clear, and aesthetically pleasing. You can also practice this in your study of the Elements if you pause your reading, attempt to arrive at your own proof, then clarify your thoughts by writing down each premise in the argument with its supporting data. Compare your argument and its style with that of Euclid's. Students who do this build greater appreciation for the variety of ways to create proofs and a greater aesthetic appreciation for elegance in proofs.
• You will enter into the tradition. The Elements is essentially the prototype of all mathematical texts that have followed over the course of many centuries since its writing. It is the earliest example we have that employs a deductive structure based on axioms and definitions. The Elements is the footpath that very nearly all mathematicians and scientific thinkers in the West have taken as part of their initiation into their subjects. You will be in the company of Newton, Leibniz, Euler, Pascal, Gauss, Einstein, to name only a few. You say that your goal is to start thinking like theoretical physicists and mathematicians. If so it may help to share this core experience with them.

To get the full benefit of the readings, you should have a compass and a straightedge and follow along the steps of the construction on a piece of paper. Mark the diagram with a pencil to show equal lines and angles as you find them. Draw subdiagrams when necessary to break down the information further like in I.5 where there are several overlapping congruent triangles.

As you become more familiar with the style of the propositions there is more you can do. Write your own proofs. Experiment with different ways the "given" objects could be arranged and determine if that would require a different proof.

The more you put pencil to paper while you study, the more you will learn from the process.

• I was going to suggest Introduction To Geometry by Coxeter, or his smaller book Geometry Re-Visited, for approximately the same reasons as yours..... Someone gave you a down-vote. I dk why. – DanielWainfleet Apr 30 '18 at 6:54
• thanks for the suggestion – wisdom Apr 30 '18 at 22:19
• I did not downvote, but I feel that depending on the person Euclid's Elements might be one of the worst resonable recommendations. Traditional geometry is very specific and not representative of modern mathematics. – dtldarek May 1 '18 at 12:48

Let me offer an alternative take. I wish I could provide TL;DR, but I was unable to summarize it well enough, sorry.

Context:

People often say that theorems and proofs are the core concepts of math. I agree that these are frequent topic when discussing and publishing mathematics, but they are not what mathematicians spend the most time on (and I am not talking about sleeping, fixing $\LaTeX$ errors, resolving Python dependencies, etc.). Let me emphasize:

Proofs are just tools that mathematicians use to validate hypotheses. Once a hypothesis is proven, it becomes a theorem and stops being interesting anymore.

We might want to say that the theorem might be still interesting, because we could extend it, but in fact it is this extension (which is not yet proven, so it is a hypothesis, not a theorem) that is interesting.

We might want to say that some particular proof is interesting, because it reveals some deeper structure about the objects we study. Yet, usually it is not the proof itself that is interesting, but, say, some construction in it that forms the structure. However, the whole reasoning could actually be formulated as a lemma claiming the existence of such structure, and the theorem as lemma's immediate corollary.

We might want to say that this structure or the related lemma is interesting, but it is only interesting due to all the hypotheses that stem from it. In a similar vein, proofs are only interesting due to all the other hypotheses that could be reached by applying the same proof technique to other objects.

And so, in my opinion, finding good hypotheses is the basis of mathematician's work. Even when working on a proof, we are still repeatedly (usually even unconsciously) forming and rejecting hypotheses. When my intuition tells me "this approach should work here", my mind forms a hypothesis "given current assumptions, there exists a function that maps..." and so on.

What is different about math:

The above paragraphs might sound like a truism, because this is the basis of scientific approach and not even specific to math. However, I think it is important to realize that math can take different forms from what stereotypes tell. What separates math from everything else is the way mathematicians validate or reject hypotheses, i.e., what constitutes a proof. In math, we validate or reject hypotheses based on the assumptions we have and inference rules we agreed (often implicitly) to use, while in sciences we do so (usually) based on results of experiments. That is, I guess, why many people claim that the proof (i.e., the path using inference rules, from assumptions to the claim in question) is the core concept of mathematics. I agree that the nature of mathematical proof is what sets math apart from the rest of STEM. Nevertheless, the hypotheses are what mathematicians are concerned about.

In other words, if you want to increase your math skills, increase your ability to imagine/find/recognize good hypotheses.

What is a good hypothesis:

This is a very hard question, which is similar to "what is good art" or "what is good design". Certainly a good hypothesis is one that is verifiable, but usually it is impossible to tell at a glance if the hypothesis is a theorem (otherwise it would not be interesting). Thus, the best one can do is to look for things that seem plausible, e.g. we have an intuition which explains why it could be true. However, a big part of forming good hypotheses is about how well it represents the object in question and how well it interacts with other theorems and hypotheses (not all of which might have in your mind, some even not conceived yet). There is a bit of art to it, and it is something that is usually picked up with experience, but you can speed up the process slightly by observing how others do that (e.g., by reading textbooks or well-written papers).

How to increase math skills?

There are obvious avenues like reading math-related things, going to math classes and doing exercises. Some basic skills you will acquire in this way are necessary to being a good mathematician (e.g., it is hard to succeed in verifying a complex hypothesis if one has problems with standard related mathematical objects).

Nevertheless, the most important thing you could try to do is finding hypotheses on a daily basis at your work, whatever you do. Try to understand what are the assumptions of your hypothesis, what are conclusions, how does it interact with objects around it, what happens if you change it a little, what happens if you add/remove/modify some assumptions, is there anything particular about the assumptions you have that seems to make this hypothesis true?

Try to start with concrete stuff, and move to more abstract things later. The more abstract you go, the more math-y the thing will seem, but the tower of abstraction in math is quite high, and on upper levels you get so removed from reality, that some people question usefulness of such research (while the term abstract nonsense was not intended as derogatory, it exists for a reason).

There are some famous real-life examples of such idle curiosity: Seven Bridges of Königsberg, and while I don't expect to found a new theory out of my on-the-bus musings, it can be an amusing exercise to try searching for math problems and their solutions in your everyday life. Here is one situation I had recently:

I accidentally overheard someone saying that he got a free one-pass ticket from a friend, but was unsure how could he maximize its potential. At that point I observed that indeed, while some web-services allow us to search for cheapest connections, we can't really ask for the cheapest route assuming we can nullify the cost of one leg of the trip (e.g., the cheapest option in such setting might be the one that was previously very expensive due to one particular leg costing very much).

I decided that cannot be a hard problem, because given a transportation-network-graph $G = (V, E)$, there are only $|E|$ possibilities for which edge we might want to nullify, and running standard shortest paths algorithm $|E|$ times would constitute a crude brute-force solution. Then, I noticed we could just run the Dijstra's algorithm twice, once from the source vertex and once backwards from the destination, and later go over all the edges to pick the cheapest connection.

Finally, I managed to generalize that solution for the case of having $k$ free tickets, at which point I had to disembark the bus and occupy my mind with other matters ;-)

I hope this helps $\ddot\smile$

• very nice :) ...that is kind of what am now looking for, making hypothesis, assumptions and build on that (that is incredibly amazing). Always great scientists in math and physics puzzle and astonish me when I read about their hypothesis (later on theories) and how it is working in giving us deeper understanding to the universe around us!! Therefore more often I look for "how and why scientist X came up with this new assumption, how did he/she reached that deep understanding and conclusion"......>>> – wisdom May 5 '18 at 14:57
• >>> So that is why now I am looking for some texts that can help me in founding a good basis for analytical skills "how to look to everything around me and every problem - I leart before- from different perspective, different view to be able to critisize, analyze and deeply understand it" – wisdom May 5 '18 at 14:57

The "engineering way" (of doing mathematics) is different from the "mathematician way" because their purposes are not the same.

So, in order to start thinking like a methematician, the first step is to discover what is the purpose of a mathematician. For this, I suggest you reading the Section Proof, the mathematician's travelogue in Chapter 2 of Sautoy's book. This section (as well as the entire book) will not teach you any mathematics. But it will explain what is the most important thing to a mathematician: the proof.

After read that, you will have a good notion of the mathematician's point of view on the concept of proof and its importance. Thus, the next step is to start learning proofs and doing proofs. For this, you have to study mathematics in books which are intended to teach math to mathematicians (basically, the aim of this kind of book is to present definitions and prove theorems; and the exercises will ask you to prove things that were not proved in the text).

What book to choose depends on what subject you want to learn and what is your background. You could try Analysis I by Terence Tao. In Chapter 2, for example, it is proved that $4$ is different from $0$. Yes, this is an obvious thing for an engineer. But a mathematician does not accept it until prove it.

• yeah knowing the mathematicians' perspective is a vital point! thanks for the references – wisdom Apr 30 '18 at 22:22