"Algebra is what most people associate with mathematics. In a sense, this is justified. Mathematics is the study of abstract objects, numerical, logical, or geometrical, that follow a set of several carefully chosen axioms. And basic algebra is about the simplest meaningful thing that can satisfy the above definition of mathematics. There are only a dozen or so postulates, but that is enough to make the system beautifully symmetric." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)
"Analysis is also a heavily explored subject, and it is just as general as algebra: essentially, analysis is the study of functions and their properties. The more complicated the properties, the higher the analysis." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)
"Data is there to be used, so one should pick up the data and play with it. Can it produce more meaningful data?" (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)
"It is also a good idea to not apply any given technique or
method blindly, but to think ahead and see where one could hope such a
technique to take one; this can allow one to save enormous amounts of time by
eliminating unprofitable directions of inquiry before sinking lots of effort
into them, and conversely to give the most promising directions priority."
"Like and unlike the proverb above, the solution to a problem begins (and continues, and ends) with simple, logical steps. But as long as one steps in a firm, clear direction, with long strides and sharp vision, one would need far, far less than the millions of steps needed to journey a thousand miles. And mathematics, being abstract, has no physical constraints; one can always restart from scratch, try new avenues of attack, or backtrack at an instant’s notice. One does not always have these luxuries in other forms of problem-solving (e.g. trying to go home if you are lost)." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)
"Mathematics is a multifaceted subject, and our experience and appreciation of it changes with time and experience." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)
"Mathematics is sometimes thought of as great entity, like a tree,
branching off into several large chunks of mathematics, which themselves branch
off into specialized fields, until you reach the very ends of the tree, where
you find the blossoms and the fruit. But it is not easy to classify all of
mathematics into such neat compartments: there are always fuzzy regions in
between branches and also extra bits outside all the classical branches."
"Mathematical problems, or puzzles, are important to real mathematics (like solving real-life problems), just as fables, stories, and anecdotes are important to the young in understanding real life. Mathematical problems are ‘sanitized’ mathematics, where an elegant solution has already been found (by someone else, of course), the question is stripped of all superfluousness and posed in an interesting and (hopefully) thought-provoking way. If mathematics is likened to prospecting for gold, solving a good mathematical problem is akin to a ‘hide-and-seek’ course in gold-prospecting: you are given a nugget to find, and you know what it looks like, that it is out there somewhere, that it is not too hard to reach, that it is unearthing within your capabilities, and you have conveniently been given the right equipment (i.e. data) to get it. It may be hidden in a cunning place, but it will require ingenuity rather than digging to reach it." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)
"Number theory may not necessarily be divine, but it still has an aura of mystique about it. Unlike algebra, which has as its backbone the laws of manipulating equations, number theory seems to derive its results from a source unknown." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)
"Understand the data. What is given in the problem? Usually, a question talks about a number of objects which satisfy some special requirements. To understand the data, one needs to see how the objects and requirements react to each other. This is important in focusing attention on the proper techniques and notation to handle the problem." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)
"Understand the objective. What do we want? One may need to find an object, prove a statement, determine the existence of an object with special properties, or whatever. Like the flip side of this strategy, ‘understand the data’, knowing the objective helps focus attention on the best weapons to use. Knowing the objective also helps in creating tactical goals which we know will bring us closer to solving the question." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)
"Write down what you know in the notation selected; draw a
diagram. Putting everything down on paper helps in three ways: (a) you have an
easy reference later on; (b) the paper is a good thing to stare at when you are
stuck; (c) the physical act of writing down of what you know can trigger new inspirations
and connections."
"At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate." (Terence Tao, "Topics in Random Matrix Theory", 2012)
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