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So, I finished my undergrad with a degree in applied mathematics, but when reading some graduate level texts and/or papers, I often find myself struggling. I eventually get there, but I often feel like I lack the intuition necessary to be able to come up with concepts on my own. I feel like I'm just missing some pivotal step in the journey to mathematical maturity.

Does anyone have any books/references/advice?

PS: If this means anything at all, as I was not a mathematics major, I did not take analysis or abstract algebra.

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  • $\begingroup$ What sort of things have you been trying to read? $\endgroup$
    – davidlowryduda
    Jul 7, 2013 at 18:24
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    $\begingroup$ Someone deleted it, but I mostly interested in machine learning texts. Nonetheless, in general, I'd like to have better intuition and not feel like I'm trying to decipher some esoteric language when I'm looking at advanced texts. I understand that this comes with experience and time, but I feel like I'm just missing some key insight or something, I don't know. Maybe it's just because I'm not used to struggling, but looking at some texts just makes me feel incompetent. $\endgroup$
    – Steve P.
    Jul 7, 2013 at 18:35
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    $\begingroup$ I thought I was hopeless at applied maths when I graduated from my undergrad. But after 3 years of graduate research, I still won't say I am good at it, but at least I can actually read papers in reasonable time now. I think if you keep doing it, you will get better as you acquire more practice and experience. E.g. 1 year ago I was completely hopeless at cooking, but after stubbornly trying to cook over and over again, I can actually produce some tasty dishes. Thinking back, I wonder why I was ever stuck in these two activities. $\endgroup$
    – Legendre
    Jul 7, 2013 at 18:36

5 Answers 5

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I thought I was hopeless at applied maths when I graduated from my undergrad. But after 3 years of graduate research, I still won't say I am good at it, but at least I can actually read papers in reasonable time now.

If you keep doing it, you will get better as you acquire more practice and experience. E.g. 1 year ago I was completely hopeless at cooking, but after stubbornly trying to cook over and over again, I can actually produce some tasty dishes. Thinking back, I wonder why I was ever stuck in these two activities. Same pattern happens for all my other hobbies.

In general, I have three broad advice:

1) Consistency

Keep doing it and do it often. Being good at something is about doing it a lot over a long period of time.

2) Repetition

Keep repeating the same thing. E.g. I used to be really bad at baking scones. Every time I failed, I would try to figure out why I failed and experiment with fixes. After 7 batches of sad looking squashed scones, I fixed all my mistakes and is now able to consistently bake nice looking scones!

3) Don't Delve on Specific Details

In my humble opinion, the biggest thing that has been holding me back from all my activities (mathematics or otherwise) was my inability to move ahead. I tend to get stuck trying to solve specific problems.

I find that it is much better to move on and make as much progress as possible, then come back to the stuck part later to try at it again. If I am unable to resolve it, I would move on again and then come back later. E.g. if you get stuck at a part of the paper, it might help to move on and read the rest of the paper. Or even put this paper aside and read another one.

A mathematics specific advice: it helps a great deal if you have supervision and/or feedback from a professor.

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Not having a foundation in proofs based classes (like analysis and abstract algebra) is a drawback because they teach you the basic tools/arguments/definitions typically used to prove other things in higher math. How important is all of this? It depends in part on what your career goals are. If you want to do mostly research or work in academics, esp at a good institution, then it will probably be important that you improve your background, esp given your comments about your past experiences trying to learn upper-level material. If you work in industry, maybe it isn't important. In my case, where I want to work in statistics in the private sector, I got to know other people already working in the field, and found that many of them did not have nor did they really need many of the upper level math grad courses. Maybe you can similarly survey people in your field. If you do need to remedy your background, consider a Master's program, where you can take some of these courses, en route to your PhD or whatever job you plan to go into. It is not so easy to get a good job with just a BS in math, so this is probably not a bad route for you anyhow. Good Luck

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Since you are interested in machine learning:

For machine learning, what you need is a good understanding of linear algebra, multivariate calculus, probability, algorithms/complexity and perhaps some convex optimization.

Depending on the algorithms you are interested in, convex optimization will probably be the most challenging subject, but you usually won't need deep understanding of this subject to understand the algorithms that use this (e.g. support vector machines). Convex Optimization by Boyd et al. is a great free book on the subject, although it contains much more than you need to understand machine learning.

Algorithms and complexity can also be complicated if you are not used to dealing with these things. There are several good books on algorithms, Introduction to Algorithms by Cormen et al. is probably the most famous.

Finally, depending on the subject, some of the probability material can be quite advanced, for example in probabilistic graphical models. Again there are many great books on the subject (e.g. Probabilistic Graphical Models by Koller et al.).

BTW for all of these you can find free online video courses.

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I regard that mathematical maturity is not universal mathematical intuition. The intuition of a mature mathematician may not work properly even in branch of mathematics very closed for him. So the lack of intuition is natural, and the intuition is formed by experience. But it is not routine, but a creative process, because you are creating your intuition. :-)

The master’s help is specific.

I don’t know are there pivotal steps in the journey to mathematical maturity.

I described my journey here.

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I attended engineering school after I three years active military service. Although I received good grades I felt that something was missing. Over the years since I graduated and then retired I have read many algebra, geometry and precalculus texts to gain insight but my goal still eluded me.

Since I retired I had the time needed to better focus on my lack of mathematical insight. At present I have identified three issues impeding my developing mathematical maturity.

  1. Language. It should be obvious that just writing and simplifying formulas is not sufficient to understand the math involved. Mathematical proofs are important for those folks who are sufficiently mathematically mature but are premature for those of us who have not developed that maturity. Developing the skill to write clearly and concisely about mathematical solutions is essential. The best place to start is to tell the story of how a particular type of problem was solved. Describe how the result was obtained in terms that an intelligent individual who graduated 8th grade could understand.

Consider the story of the child prodigy Carl Gauss whose class was told to solve problem of adding all the integers from 1 to 100. Gauss immediately answered that the sum of these integers is 5050. When asked how he was able to answer correctly so quickly he explained that he added to each element of the set of consecutive integers of 1 through 100 the elements of this set but in decreasing order. The young Gauss noted that each of the 100 elements of the resulting set of integers had a value of 101. So the sum of a 100 integers with the same value being 101 is twice the sum of the consecutive integers of 1 through 100. Gauss then answered that 5050 was equal to the sum of 50 integers which had the same value of 101. Note that Carl did not provide his answer in the form of the proof.

  1. Learn basic mathematical problem solving skills that are assumed by many mathematical textbook authors to be so basic that they too often state that "It can be easily seen that Blah Blah  Blah is Blah Blah Blah". Oh yah! Those who math can't teach math because they assume someone else has taught the basic math learning skills such as focus on detail, which in turn requires motivation, and understanding that periods of intense concentration should be short and followed by other activities such as walking or even a short nap.

Take a look at No Bullshit Guide to Mathematics by Ivan Solov. ISBN 9780992001032 and

  1. Again look at a mathematical dicription that you found difficult to comprehend. Ask yourself what is information do I need? For example, as I have tried to improve my understanding  of mathematical proofs I realized that the proof that the square root of 2 is irrational was the wrong place to start.

So then reviewed my Geometry texts are considered the proof for the Pythagorean theorem. They classic proof based upon Euclid's axioms is not sufficient algebra had not existed. Then I found a tiling proof that resonated for me at Wikipedia.

As I scrolled down to the rearrangement proof I instinctively understood. To improve my understanding of how this proof was developed I sketched congruent right triangles with sides of lengths a, b and c.  To the triangle on the left I sketched a square which had one of its sides coincident with the hypotenuse of the right triangle. On the right I sketched the congruent shape for the sum of the squares of sides a and b of the right triangle.

I noted that on both figures the smallest congruent squares bounding these shapes, that sides on both bounding  measured a plus b. Since both areas would measure the square of the sum of, a plus b. On the right  shape  this is equal to area of the small square a plus the combined areas of the four congruent right triangles plus the area of the large b.

On the left the combined areas is equal to the area of the largest square whose side equal the hypotenuse of the right triangle, plus the combined areas of the four congruent right triangles. Now when the combined areas of the four congruent right triangle are removed from the shapes on left and  right, what remains is that square of the hypotenuse is equal to the sum of the squares of the sides that are adjacent to the right angle of the right triangle.

This description may seem a bit wordy but it provided to me a better understanding of the process by which this proof was developed.

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