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Mathematical Philosophy (3 vols)
— A journey to the wonderland of imagination
by Li-Chung Wang

Vol. 1. Chinese Philosophy, Art, Life, and Mathematics.
Vol. 2. Analytic Functions with Mathematical Philosophy  
(more commonly known as Complex Variables with Mathematical Philosophy).   (75% Mathematics, 25% Philosophy)
Vol. 3. Number Theory with Its Mathematical Philosophy.   (50% Mathematics, 50% Philosophy)

Main Ideas:

  1. Why we study mathematical philosophy

    To evaluate a mathematical method, at first we may rely on the personal opinion of an expert, but eventually we have to consider a viewpoint of mathematical philosophy and then let the method speak for itself.
    (vol. 1,  p.1; vol. 2,  pp.80-81,  note 14;  vol. 3,  pp.105-106;  vol. 3,  p.166,  note  8(3)).

  2. Quality

    A mathematical theory of good quality should avoid any unnecessary use of the following methods:

    1. Trial and error   (vol. 3,  p.122,  §2; vol. 3, p.151, note 1(2); vol. 3, p.156, note 10; vol. 3, p.197, note 3(1)).
    2. Reduction to absurdity (vol.3,  p.168,  note  4(4); vol. 3, p.180,  note  11;  vol. 3,  p.192,  note 41(1)).
    3. Mathematical induction (vol. 2,  p.78,  note 6;  vol. 2,   p.94,   note 2(a)).
    4. The axiom of choice (vol. 1,  p.6).

  3. Reduction to absurdity

    1. In a proof we are forced to use reduction to absurdity because the hypothesis is too weak  (vol. 3,  p.107).
    2. How to easily reach a contradiction (vol. 3,  p.154,  note 7; vol. 3, p.163, note 2(1), vol. 3, p.164, note 3(2); vol. 3, p.200, note 7).
    3. Refinements: We refine the method of reduction to absurdity by reducing its scope.
      1. Suppose the method of reduction to absurdity is used to prove the entire statement
        $x (x has the property P),
        we would like to actually find an element x so that reduction to absurdity is only used in proving the portion contained within the parentheses (vol. 3, p.55, Example 5.1).
      2. Suppose we have to use reduction to absurdity to prove AC. We would like to find a middle statement B such that either AB or BC can be proved without using reduction to absurdity. For example, Selberg's elementary proof of The Prime Number Theorem is a refinement of the transcendental proof. (vol. 3, pp.107-108).

  4. Existence

    Any existence is a combination of the following three basic patterns:

    1. Constructive existence.
    2. Logical existence: Assuming {x| x has the property P}= will lead to a contradiction.
    3. Assumptive existence:
      1. The existence is assumed with an axiom.
      2. In a proof we divide into two cases: Case I. x exists; Case II. x does not exist. Via different paths of argument, both cases lead to the same conclusion. In case I the existence is only used in a transition stage. That is, whether or not x really exists is independent of the topic in discussion and can be put on hold temporarily.
      3. We are forced to assume the existence in the hypothesis of a theorem because it is difficult to simulate the effective construction from a specific case to a general case.
      (vol. 2,  pp.97-98,  note  1(d)&(e); vol. 3, pp.109-113).

  5. General theory versus specific case
    1. From practical construction to hypothetical theory.

      In a theory if the construction process of an element with required property relies heavily on a case by case basis, it is expedient to use an axiom to assume the existence of this element in advance.

    2. Application of a general theory to specific cases.

      Suppose the hypothesis of a general theorem involves divisibility. In a complicated example we often fail to provide an effective method to check divisibility because divisibility is well-defined in logic. Thus we create a gap between the theory and the practical example.

    3. Embodiment of a general theorem.

      When we say a theorem is ineffective, it means that when a practical example is given, we can not embody the theorem with this example via the argument of the theorem alone. To embody a theorem demands constructive existence, but the theorem' argument may merely provide logical existences.

    (vol. 2,  p.86, note 6(b); vol. 2, p.92, note 2; vol.2, p.93, note 3(a); vol. 3, pp.114-118; vol.3, pp.159-160, note 3(2); vol. 3, pp.178-179, note 5(3); vol. 3, p.182, note 19(2); vol. 3. p.188, note 32(2)).

  6. Axiomatic approaches

    Solving a mathematical problem is like assembling a jigsaw puzzle. It is easier and wiser to assemble from the perimeters than from the center because we have more clues from the environment once the perimeter is established. By the same token a footnote should not be placed at the beginning of a written passage,
    (vol. 2, pp.84-85, note 5(a); vol. 3,  p.132; vol. 3, p.167, note 2; vol. 3, pp.175-176, note 3).

  7. Applications

    Weakening a theorem's hypothesis makes its application more difficult.
    (vol. 2, p.87, note 10; vol. 3, pp.123-124).

  8. Effectiveness

    1. Effective construction.

      A more generalized theorem reveals less features. A less generalized theorem about construction can be more easily visualized because it has more resources available and its construction can be more specific and effective.

    2. When we construct a mathematical element in finite steps, we should bring trial and error to a level as basic as possible so that the construction can be more specific and more relationship can be revealed.

    (vol. 3, chap.7, §2, §3.1 and their notes, pp.84-90 & pp.174-186; vol. 3, pp.121-122).

  9. Mathematical induction

    The well-ordering principle contains too many variables either in its hypothesis or in its conclusion to be a simple axiom. (vol. 3, pp.131-133; vol. 3, p.197, note 2(2)).

  10. Comparison

    Sometimes mathematical philosophy is so subtle that its truth is not clear unless we give examples to compare and criticize them (vol. 3, p.3).

    Simply displaying two different approaches without comparing them is nothing more than leading to two dead ends.

    Whenever we compare two proofs of a theorem, we must make their sameness as strong as possible and their difference as crucible as possible (vol. 3, pp.125-127).

  11. Contradictions

    In mathematics, we use parallelism or compatibility to deal with the problem of contradiction (vol. 3, pp.119-120; vol. 3, p.197, note 3(1)).

  12. Origins and originality

    If we tasted a delicious dish, we would ask the chef how he made it. In this way we could best capture the original flavor. Similarly, the first-hand works of a great mathematician are a rich and irreplaceable resource to understand how this mathematician developed his ideas. (vol. 2, p.85, note 6(a); vol. 3, p.128-130; vol. 3, p.149, note 7(1); vol. 3, pp.174-175, note 2; vol. 3, p.191, note 38).

  13. Definitions

    A solution often means a way to retrace the definition of problem. If we enter the problem through concise definition, we may solve it easily.

    In a mathematical theory, we often define a term by logic without an effective algorithm to check the definition. Actually, the algorithm can be the very backbone of the theory. Lack of the backbone will make the theory essentially empty. (vol. 2, p.75, note 1; vol. 3, pp.138-144).

  14. Typical examples and counterexamples

    A typical example of a definition should properly show both the latitudes and the restriction of the definition.

    The best counterexample should be perfect in almost every aspect; more importantly, it should be based on practical research experiences. (vol. 2, pp.86-87, note 9; vol. 3, pp.134-135).

  15. Timing, order, and arrangement
    1. Timing.

      A formula should be developed to the stage in which it is free from complicated theory. Otherwise, whenever a new factor is added into the assumption, we have to trace its influence all the way back to the beginning of the formula's development, even though such trouble is unnecessary. (vol. 3, p.185, note 24(5); vol. 3, p.193, note 41(2); vol. 3, p.198, note 4).

    2. Order and arrangement (vol. 1, p.16; vol. 1, p.22; vol. 2, p.94, note 1; vol. 3, pp.129-130, IV).

  16. Isolation from complexity

    There is a famous Taoist story in Chaung Chou  (around 300 B.C.). Once upon a time there was a butcher who did not understand the anatomy of a cow. He often broke his knife by cutting too deeply and striking the bones. His meat was always mixed with crushed bones and his bared bones were always left with a good amount of meat on them.

    In a theorem if a variable is intuitively independent of the conclusion, it is better to show that the variable affects the theorem's proof in a simple manner even though the entire proof can be long and complicated.

    (vol. 3, pp.154-155, note 8(1); vol. 3, pp.196-197, note 2(1); vol. 3, p.198, note 4).

  17. Right choices

    Making a right choice will help avoid unnecessary adjustment and complications (vol. 3, p.151, note 2; vol. 3, p.155, note 9(1); vol. 3, p.190, note 35(3) note 36).