"The equation e^πi+1 = 0 is true only by virtue of a large number of profound connections across many fields. It is true because of what it means! And it means what it means because of all those metaphors and blends in the conceptual system of a mathematician who understands what it means. To show why such an equation is true for conceptual reasons is to give what we have called an idea analysis of the equation." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)
"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)
"Likewise, complex functions are actually better behaved than real functions, and the subject of complex analysis is known for its regularity and order, while real analysis is known for wildness and pathology A smooth complex function is predictable, in the sense that the values of the function in an arbitrarily small region determine its values everywhere. A smooth real function can be completely unpredictable for example, it can be constantly zero for a long interval, then smoothly change to the value 1." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"The word 'complex' was introduced m a well-meaning attempt to dispel the mystery surrounding 'imaginary' or 'impossible' numbers, and" (presumably) because two dimensions are more complex than one Today, 'complex' no longer seems such a good choice of word. It is usually interpreted as 'complicated', and hence is almost as prejudicial as its predecessors. Why frighten people unnecessarily? If you are not sure what 'analysis' is, you won't want to know about 'complex analysis' - but it is the best part of analysis." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"Consider for example the complex numbers x + iy, where you of course ask what is i = √ −1 when you first encounter this mathematical construction. But that uncomfortable feeling of what this strange imaginary unit really is fades away as you get more experienced and learn that C is a field of numbers that is extremely useful, to say the least. You no longer care what kind of object i is but are satisfied only to know that i^2 = −1, which is how you calculate with i." (Andreas Rosén,"Geometric Multivector Analysis: From Grassmann to Dirac", 2019)
No comments:
Post a Comment