The Art of Insight in Science and Engineering: Mastering Complexity by Sanjoy Mahajan

• Science and engineering, our modern ways of understanding and altering the world, are said to be about accuracy and precision. Yet we best master the complexity of our world by cultivating insight rather than precision.
• We need insight because our minds are but a small part of the world.
• An insight unifies fragments of knowledge into a compact picture that fits in our minds. But precision can overflow our mental registers, washing away the understanding brought by insight.
• There are two broad ways to master complexity: organize the complexity or discard it.
• To master complexity use:
• divide and conquer
• abstraction
• symmetry and conservation
• proportional reasoining
• dimensional analysis
• lumping
• probabilistic reasoning
• easy cases
• spring models
• The most effective teacher is a skilled tutor. A tutor asks many questions, because questioning, wondering, and discussing promote learning.
• We cannot find much insight staring at a mess. We need to organize it.
• In problem-solving, we organize complexity by using divide-and-conquer reasoning and by making abstractions.
• Reliability comes from intelligent redundancy.
• Divide-and-conquer estimates require reasonable estimates for the leaf quantities.
• The main lesson that you should take away is courage: No problem is too difficult. We just use divide-and-conquer reasoning to dissolve difficult problems into smaller pieces.
• Naming--or, more technically, abstraction--is our other tool for organizing complexity.
• A name or an abstraction gets its power from its reusability. Without reusable ideas, the world would become unmanageable complicated.
• Notations are abstractions, and good abstractions amplify our intelligence.
• Our understanding of the world is built on layers of abstractions.
• The benefit of the abstraction solution, compared to calculating [...] explicitly, is insight.
• Abstraction has a second benefit: giving us a high-level view of a problem or situation. Abstractions then show us structural similarities between seemingly disparate situations.
• The key is to practice effectively.
• Because abstractions are so useful, it is helpful to have methods for making them. One way is to construct an analogy between two systems. Each common feature leads to an abstraction; each abstraction connects our knowledge in one system to our knowledge in the other system.
• Analogies not only reuse work, they help us rewrite expressions in compact, insightful forms.
• A good notation should help thinking, not hinder it by requiring us to remember how the notation works.
• Once you name an idea, you find it everywhere.
• An abstraction connects seemingly random knowledge an insights. By building abstractions, we amplify our intelligence.
• Whenever you reuse an idea, identify the transferable process and name it: make an abstraction. With a name you will recognize and reuse it.
• We use symmetry and conservation whenever we find a quantity that, despite the surrounding complexity, does not change. This conserved quantity is called an invariant. Finding invariants simplifies many problems.
• When there is change, look for what does not change!
• Invariants are powerful partly because they are abstractions.
• Logarithmic scales can make otherwise obscure symbolic calculations intuitive.
• Drag, one of the most difficult subjects in physics, is also one of the most important forces in everyday life.
• High accuracy often requires analyzing and tracking many physical effects. The calculations and bookkeeping can easily obscure the most important effect and its core idea, costing us insight and understanding.
• In the midst of change, find what does not change--the invariant or conserved quantity. Finding these quantities simplifies problems: We focus on the few quantities that do not change rather than on the many ways in which quantities do change. An instance of this idea with wide application is a box model, where what goes in must come out.
• Proportionalities are often called scaling relations.
• Scaling exponents are a powerful abstraction: once you know the scaling exponent, you usually do not care about the mechanism underlying it.
• Scaling relations bootstrap our knowledge.
• Dive and conquer: don't bite off all the complexity at once!
• Proportional reasoning focuses our attention on how one quantity determines another. By guiding us toward what is often the most important characteristic of a problem, the scaling exponent, it helps us discard spurious complexity.
• Make only dimensionless comparisons.
• A quantity with dimensions is, by itself, meaningless. It acquires meaning only when compared with a relevant quantity that has the same dimensions.
• Because dimensionless quantities are the only meaningful quantities, we can understand the world better by describing it in terms of dimensionless quantities.
• Using quantum mechanics, we can predict the properties of atoms in great detail--but the analysis involves complicated mathematics that buries the core ideas. By using dimensional analysis, we can keep the core ideas in sight.
• Dimensional analysis discards complexity without loss of information.
• When the going gets tough, the tough lower their standards: approximate first, and worry later.
• Asking calculators to do simple arithmetic dulls our ability to navigate the quantitative world. The antidote is to do the computations ourselves, but approximately--by placing quantities on a logarithmic scale and rounding them to the nearest convenient value.
• The simplest method of round is to round every number to the nearest power of ten. That simplification turns most calculations into adding and subtracting integer exponents.
• "Nearest" is judged on a logarithmic scale, where distance is measured not with differences but with ratios or factors.
• Rounding to the nearest power of ten gives a quick, preliminary estimate. When it is too approximate, we just round more precisely. The next increase in accuracy is to round to the nearest half power of ten.
• Lumping not only simplifies numbers, where it is called rounding, it also simplifies complex quantities by creating an abstraction: the typical or characteristic value.
• Using typical or characteristic values allows us to reason out seemingly impossible questions while sitting in our armchairs.
• Proportional reasoning reduces complexity by showing us a notation for ignoring quantities that do not vary.
• Lumping rescues us by replacing changing values with a single, constant, typical value--making the relations amenable to proportional reasoning.
• A powerful form of lumping is to replace complex shapes by a comparably sized cube.
• In everyday life, an important feature of fluid flow is drag.
• The essential physical idea is that the viscous force, a force from a neighboring region of fluid, slows down fast pieces of fluid and speeds up slow pieces.
• Lumping smooths out variation.
• Lumping replaces a complex, changing process with a simpler, constant process.
• The antidote to complicated integrals is lumping.
• By using lumping to introduce quantum mechanics, we will gain a physical intuition for the effect of quantum mechanics.
• In mechanics, the simplest useful model is motion in a straight line at constant acceleration. In quantum mechanics, the simplest useful model is a particle confined to a box.
• Lumping is our first tool for discarding complexity with loss of information. By doing so, it simplifies complicated problems where our previous set of tools could not. Curves become straight lines, calculus becomes algebra, and even quantum mechanics becomes comprehensible.
• Probabilistic reasoning helps us when our information is already incomplete--when we've discarded even the chance or the wish to collect the mission information.
• The essential concept in using probability to simplify the world is that probability is a degree of belief. Therefore, a probability is based on our knowledge, and it changes when our knowledge changes.
• The Bayesian interpretation is based on one simple idea: a probability reflects our degree of belief in a hypothesis. Probabilities are therefore subjective: someone with different knowledge will have different probabilities. Thus, by collecting evidence, our degrees of belief change. Evidence changes probabilities.
• Random walks are everywhere.
• In large complex systems, the information is either overwhelming or not available. Then we have to reason with incomplete information. The tool for this is probabilistic reasoning--in particular, Bayesian probability.
• Probabilistic reasoning helps us manage incomplete information.
• A correct analysis works in all cases--including the simplest. This principle is the basis of our next tool for discarding complexity: the method of easy cases.
• Easy-cases reasoning is a way of introducing physical knowledge.
• Special relativity is Einstein's theory of motion. It unifies classical mechanics and classical electrodynamics (the theory of radiation), giving a special role to the speed of light c.
• The speed of light is the universe's speed limit, and special relativity obeys it.
• The Heisenberg uncertainty principle restricts how small we can make these uncertainties, and therefore how accurately we can determine the position and momentum [of a particle].
• Look at the easy cases first. Often, we can completely solve a problem simply by understanding the easy cases.
• Our final tool for mastering complexity is making spring models. The essential characteristics of an ideal spring, the transferable abstractions, are that it produces a restoring force proportional to the displacement from equilibrium and stores an energy proportional to the displacement squared. These exceedingly specific requirements are met far more widely than we might expect.
• Sunlight is an oscillating electric field.
• Many physical processes contain a minimum-energy state where small deviations from the minimum require an energy proportional to the square of the deviation. This behavior is the essential characteristic of a spring. A spring is therefore not only a physical object but a transferable abstraction.
• For long-lasting learning, the pieces of knowledge should support each other through their connections. For when we remember a fact or use an idea, we activate connected facts and ideas and solidify them in our minds.
• So, for long-lasting learning and understanding, make bonds: connect each new fact and idea to what you already know. This way of thinking will help you learn in one year what took me two or twenty.
• Use your reasoning tools to weave a richly connected, durable tapestry of knowledge.