### Experiments in learning Math- II

#### by ayushkhaitan3437

**Disclaimer**: I am a graduate student about to begin his PhD. I don’t claim that the following method is the “best” way to learn Math with any credible authority. I am only sharing some thoughts that I think have improved my Math, and hope that they can help others in my position too.

I spend an inordinate amount of time thinking about how best to learn Math. It might seem a waste of time to most people, as they might think that I should spend more time doing actual Math than waste time on such pedagogical questions. However, I stuck to this process of introspection for very many years, and it is precisely because of such analyses that I believe I am doing the best Math of my life these days.

**Method**: Whenever I learn a new concept, I attach a **geometric example **to it. When studying Algebra, for instance, most of my examples come from Algebraic Geometry. Hence, a prime ideal becomes the set of polynomials disappearing at a point. The Snake’s Lemma becomes the set of polynomials on a line, mapping to the set of polynomials on a point of that line.

**Reason for coming up with this method**: While thinking about how to best learn new Math a couple of weeks back in the library, I realized that I think much better with pictures. For instance, I would be much more comfortable thinking of a continuous function as a “curve without breaks” as compared to a map between two sets such that the inverse of an open set is an open set.

One may argue that the “line without a break” definition does not capture all the implications and consequences of a function being continuous. A continuous function is a much more general concept that may be used as a map between spaces that the human mind cannot even imagine. One must not lose sight of the beauty and the power of this concept that much smarter mathematicians have contrived over a century.

That person would be absolutely right. However, let me offer a bad analogy: imagine a programmer trying to program. The programmer speaks a certain language that comes more naturally to him, and the computer understands a different language. The compiler translates the language that the programmer is more comfortable with, into the language that the computer can understand. The programmer may not be comfortable using the language that the computer uses.

Similarly, think of formal Math as the language that the computer uses. It is the most powerful language that the human mind has ever come up with, and is the closest approximation to “truth” that we have. However, despite years of trying, I’ve never really been very good at this language. Now I’ve stopped trying. And it is definitely for the best. I have started thinking in elementary visual examples, sacrificing generality for visual intuition. And the process of translating such intuition into formal Math proofs is not very difficult.

**Anecdotal evidence**: To give you an idea of how much easier this approach has made life for me, let me just say that I am now comfortable with more Algebra, Algebraic Geometry, Algebraic Topology, Differential Geometry and Complex Analysis than I have ever been in my life.

The way Math is generally taught pushes people away from the discipline. I really hope that this post helps people learn Math a little better.

EDIT: New mathematical blog posts coming soon.

I have had a similar experience. Mathematics is not conceptually difficult. The difficulty lies in the formalism. Intuition, for better or worse, is not a proof; a proof must follow some rules of logic. To proceed further and further in mathematics, one must become expert at exploiting these rules to meet some particular objective. At some point, intuition diverges from reality (what we can verify empirically) altogether and the proof would seem to be entirely self-contained within some logical structure. Perhaps this is what Hardy meant when he said mathematics forms its own reality. The opposite is true in physics; there is almost always some physical model to reference. It is a paradox in some sense; the conclusions drawn from physics, based on the interactions of four fundamental forces, describe the Universe completely. Thus, physics is the reality we experience empirically or otherwise. Yet it makes great use of a language (mathematics) that makes no claim to describing the Universe, given that its axiomatic structure is entirely self-contained.

While I think there is a sense in which mathematics is simply the study of structure, which can be broken down in terms of the basic objects of your axiom system, that is not how it works in practice. Mathematics is social; fundamentally, mathematics is about human understanding.

As it turns out, almost everyone understands mathematics by examples, simplifications, examples, and familiarity, not dry, logical, abstract symbols. Similarly, progress in mathematics usually begins with examples. People usually develop the theory to match the examples, not the other way around. John Conway has some wisdom here:

“To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples…”

Halmos also emphasized the importance of examples. The following quote, which is probably so well known in part thanks to the blog Annoying Precision, demonstrates this fact:

“A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one.”

The mathematics educational system, at least in the United States, does a poor job of emphasizing these features of mathematics, especially at the advanced undergraduate and beginning graduate level. The goal is absolutely rigorous theorem proving, no more, no less. In undergraduate mathematics, you need to learn how to prove theorems and deal with abstract, formal mathematics. In beginning graduate school—where lectures are so often too fast, with too few examples, and with too few deviations from the definition-theorem-proof format Vladimir Arnold so hated—you simply practice applying this new skill to new, more sophisticated example.

We often fail to emphasize adequately the roles of intuition, example, time, and play. Students maximize the number of theorems they learn and prove in a small period of time, but they do so at great cost.