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Preface

  1. Problems with modern mathematics
    1. Emphasizing rigor and rigid specifics, but neglecting guiding principles and flexible ideas.
          [Inc1, p.48, l.18-p.49, l.3] provides the guiding principle describling how to form integral surfaces. [Sne, p.56, l.2-l.14] deemphasizes this guiding principle, but Sneddon presents two examples [Sne, chap. 2, 5 & 6]. As for [Joh], John preserves only one example [Joh, chap. 1, 4 5] and completely ignores the guiding principle.
    2. One ignores important effective algorithms simply because the method of trial and error may provide logical shortcuts. [1]
    3. Due to their poor judgments, the mathematics professors at Harvard University, MIT, and Moscow State University discarded more refined, useful [Inc1, p.80, l.26-l.27], effective, and visualizabe proofs [Inc1, 3.32, 3.4 & 3.41], and preserved only the short proof [Inc1, 3.3 and 3.31] when they wrote textbooks for students. See [Bir, Cod, Pon & Arn1].
          Differentiation is easier than integration, so the proof given in [Inc1, 3.32] is more effective than that in [Inc1, 3.3]. The approach by integrals and series given in [Inc1, 3.3 and 3.31] makes the solution difficult to visualize, while [Inc1, p.76, Fig. 2] gives a clear picture of the solution.
    4. A well-developed theory is less likely to be presented in a textbook because it lacks potential for further research [Inc1, chap. IV].
    5. The modern research is either fragmented or insignificant due to lack of guidance from rich origins.
          If we compare [Inc1, 5.51] with [Coh, vol. I, chap. II, D, 3], we discover that the latter leaves out an important result [Inc1, p.131, l.5-l.9] and has limited resources (i.e., examples and applications) and perspectives. If we compare [Coh, vol. I, chap. II, D, 3] with [Jaco, vol. II, chap. IV, 9] we find that the latter has even more restrictions. For example, its discussion is limited to finite dimensional spaces.
    6. When they fail to consider effectiveness or to focus on their theme, researchers may blow their argument out of proportion.
          [Inc1, 6.11-6.13] give a concise and straightforward solution to linear homogeneous equations with constant coefficients. In contrast, [Pon, p.51, (A); p.52, (B)] digress from the point and blow the argument out of proportion. Pontryagin's unnecessary use of reduction to absurdity in [Pon, p.43, l.-12] makes his proof ineffective. In addition, Pontryagin's use of reduction to absurdity in [Pon, p.54, l.-23] involves a much longer argument to obtain a contradiction than does Ince's in [Inc1, p.136, l.11] .
    7. If one fails to distinguish the nuance of various degrees of precision or fails to understand a notation's limitation in expressing precision, then one's formulation of a theorem, like a pocket with holes, will fail to preserve either the theorem's high precision or its other essential meanings. [1]
    8. Modern mathematics tends to emphasize generalization rather than effectiveness, this approach may mislead readers into ineffectiveness or unnecessary complications [1].
    9. Modern mathematicians often use unnecessarily complicated tools to construct a structure's extension. Thereby, they blur the insight of the extension [1, 2].
    10. Refined methods have almost become endangered species in real analysis simply because mathematicians fail to distinguish a logical existence from a constructive existence [1].
    11. The context of a simple setting may expose and highlight the key point. Improper generalizations or unnecessary complications may make it difficult to find the key points.
          We can see the importance of single-valuedness in [Yos, p.37, l.-1-p.39, l.16] but not in the proof of [Cod, p.100, Theorem 1.1] or the proof of [Har, pp.70-71, Theorem 10.1].
    12. Links {1}.

  2. The philosophy of mechanics serves to
    1. find a more effective method to solve a problem,
    2. reveal the source that inspires the process of synthesizing principles [Mer2, p.268, l.5-l.19].
    3. add understanding of subtle distinctions through comparison.
    4. make sure we fully understand if it is at all possible. Many textbooks say in their prefaces, "There is no statement in this book like 'It can be shown that …'" However, in the end they all contain many unproven results. I will make sure in this web site there are no untied loose ends and no kites with broken lines.

  3. Mechanics versus the Theory of Differential Equations.
    1. Mechanics can be considered an illustrative theory of differential equations because in mechanics we have concrete terms like atoms, molecules, scattering, etc. to describe the subtle point of a scenario [Coh, p.915, (B-25); p.910, (B-9); p.916, (B-33), p.920, (B-47)]. If we fail to describe the subtle points in concrete terms, we may soon get lost in abstractness [Coh, p.916, l.−7; p.920, §b]. Furthermore, the mechanical interpretation of a differential equation often provides a heuristic approach to the solution [Coh, p.919, (B-45); p.921, Fig. 8].
    2. When a problem is formulated in the specific case, the solution tends to be mechanical [Go2, p.442, §10-2]. When a problem is formulated in the general case, the solution tends to be mathematical [Go2 , p.449, §10-4. Separation of Variable]. In physics, the general solution is based on the method of synthesis [Go2 , p.443, (10-17); p.445, (10-25); p.450, (10-32)]. In mathematics, a solution often comes from nowhere.
    3. Only after one finishes studying the theory of fluid mechanics may he write a book on vector calculus. This is because vector calculus is the summary of fluid mechanics in the language of mathematics. If a book on vector calculus is written by someone who does know any fluid mechanics, there is no way that it can provide the physical meanings behind the mathematical formulas. Many mathematicians do not have a solid background in physics, but they attempt to write books in differential geometry or Lie groups, which are sub-topics in classical mechanics. All they can do is to detach the logical frame (mathematical structure) from the soul of mechanics (physics) to hide their ignorance. Their works are doomed to fail because they only make the original physical meanings unrecognizable.


  4. The philosophy of mechanics is simple in the sense that it rarely involves complicated techniques. However, many good scholars have made serious mistakes by ignoring the importance of the philosophy. They detached philosophy from their study because they failed to provide concrete examples to support their philosophical statements, or simply because their focus was narrowly limited to a specific subject.
        Theories of mechanics for various physical objects can only be unified with the theory of differential equations. This is similar to Schöringer did for quantum mechanics. Various topics (eigenvalue problems and perturbation theory [Col , p.260, §18]) of differential equations require mechanics as a guide to relate them naturally [Coh , p.1098, §2]. (We study the same object [energy] in two cases: Eigenvalue problems are designed for the unperturbed case, while perturbation theory is designed for the perturbed case.) Otherwise, their relationships become shallow and artificial.

  5. In this e-book, you will find that every word, every sentence is directed towards consideration of effectiveness for each topic. The topic may be a math term, a theorem or a theory. This effective side has been developed by the centuries-old innovation of mathematicians and still provokes further creativity.