數學證明

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歐幾里得(Euclid)寫嘅一個數學證明

數學證明粵拼:sou3 hok6 zing3 ming4英文mathematical proof),通常就噉簡稱做「證明」,係數學家研究數學嘅一種工具。喺呢個過程入面,數學家會先諗出一柞公理(axiom)-一啲佢哋認為好明顯係真,唔使證明都可以攞嚟用嘅命題(proposition)-或者係用一啲之前已經證明咗嘅命題(即係所謂嘅定理;theorem)。然後佢哋會靠住用呢啲公理同定理,再用一啲數學證明嘅方法推斷出一啲實啱(always true)嘅新命題,而呢啲新推出嚟嘅命題會俾好多數學家攞嚟詏,如果詏完一輪之後啲數學家覺得個證明冇問題嘅話,條新命題就會變成一條新嘅定理。呢啲新定理又有得攞去用嚟證明新啲嘅定理-於是乎數學知識就係噉增長[1][2]

如果一條命題要俾數學界接受係定理,佢就要有個令人滿意嘅證明,如果佢衹係有人覺得可能係啱但係冇俾人成功證明到嘅話,佢衹會係個猜想(conjecture)[3]

大分類[編輯]

大體上,數學嘅證明有得分做兩種:

  • 非形式化嘅證明(Informal proof)-一種用自然語言-即係好似廣東話呢類日常傾偈用嘅語言-寫出嚟嘅論證,用嚟說服啲讀者某個定理或者論斷係啱。因為呢種證明用咗自然語言,而自然語言好多時冇數學語言噉精確,搞到非形式化嘅證明俾數學家覺得佢寬鬆得滯。非形式化嘅證明通常都係用喺應用嘅場合度,例如係科普講座、口頭辯論、或者初等教育-因為喺呢啲場合度啲讀者嘅水平冇噉高[4]
  • 形式化嘅證明(Formal proof)-一種唔係用自然語言,而係用數學語言寫出嚟嘅證明[5]。數學語言係一連串既定嘅符號,每個符號都有一套標準化而且好明確嘅定義,令到啲數學家唔使憂個證明夠唔夠清楚明確。如果一個數學家係想向成個數學界證明某啲新定理或者理論嘅話,佢一定要用形式化嘅證明,先至可以說服到數學界佢個證明係夠精確嘅。

研究證明嘅形式同方法嘅理論喺學術上俾人嗌做證明論(Proof theory)。

常用嘅證明方法[編輯]

直接證明[編輯]

內文: 直接證明

直接證明(Direct proof)係指直接由啲大家都認為係唔使證明都可以當係啱嘅公理或者係之前俾人證明咗嘅定理嗰度出發,運用推理(Deductive reasoning)去推想證明嗰條定理出嚟,係最簡單直接嘅證明方法[6]。基本上,個思路就係:

  • 有若干數量嘅命題(...),因為佢哋係公理或者之前經已俾人證明咗,所以可以假設佢哋係啱嘅;
  • 由呢柞命題度可以引申到一條新命題
  • 所以命題 都係啱嘅。
例子

要求:證明「任何兩個雙數 加埋一齊,出嘅一定會係一個雙數」。


喺成個證明過程入面,個證明者淨係用咗一條公理,跟手就推咗條新定埋出嚟-係一個直接證明。

數學歸納法[編輯]

內文: 數學歸納法

數學歸納法(Proof by mathematical induction)專係用嚟證明一啲有排序性嘅定理[7][8]。佢成個諗頭係在於要先證明以下兩樣嘢:

  • 嘅時候,命題 係啱嘅;
  • 當命題 係啱嘅時候,可以引伸到 都係啱嘅;

假如以上兩點成立到嘅話,噉就有得話「當 嘅時候, 呢個命題會係啱」,而「當 嘅時候, 呢個命題都會係啱」,如此類推,命題 對應所有嘅自然數(Natural number;平時用嚟數嘢嘅數字)都係啱嘅(For all natural number , is true)。

例子

要求:證明「如果 係非零整數,噉對應所有正整數 係正數。」(想要證明嘅命題


否定證明[編輯]

內文: 否定證明

否定證明(Proof by negation,又或者 Proof by contraposition)係一種利用換質換位(Contraposition)邏輯嚟去證明一啲定理嘅證明方法[3]。「換質換位」指嘅簡單啲講係話「」同「」呢兩句嘢喺邏輯上係有關嘅:如果「 暗示(Imply) 」係啱嘅,噉如果 唔係真,噉 都唔會係真。例如係以下呢個論證:

  • 如果蘇格拉底係一個人,噉佢嘅壽命會係有限嘅;(
  • 如果蘇格拉底嘅壽命係冇限嘅,噉佢一定唔係一個人。(

如果將 換做某啲數學命題,噉呢種思考方式就有得攞嚟做數學證明。

例子

要求:證明「有個整數 ,如果 係雙數,噉 都一定係雙數。」


反證法[編輯]

內文: 反證法

反證法(Proof by contradiction)係一種古老嘅證明方法,利用咗「如果呢條命題成立,會有個唔合理嘅結果,所以呢條命題冇可能係啱嘅」呢點嚟證明某啲命題係錯嘅。佢條思路係:

  • 先假設想否定嘅命題 係啱嘅;
  • 由命題 嗰度引申一個荒謬嘅結果出嚟;
  • 噉就可以話命題 係錯嘅。
例子

要求:證明「假設 係一個單數,噉 唔會係一個雙數」。


構造法[編輯]

內文: 構造法

構造法(Proof by construction)一般係用嚟證明一啲存在性定理(Existence theorem;一啲話某啲嘢存在嘅定理)-用呢個證明嗰陣,用嗰個人會諗出一件有得用數學描述嘅物件出嚟,列出佢有啲乜嘢特性,再證明一件有呢啲數學物性嘅物件係存在嘅[9]。佢條思路係噉嘅:

  • 揾到有一個情況下,命題 係啱嘅;
  • 證明到「喺至少一個情況下,命題 係啱嘅」。
例子

要求:證明「唔係所有單數都係質數」(即係話「喺至少一個個案入面,有個單數唔係質數」)。


分類證明[編輯]

內文: 分類證明

分類證明(Proof by exhaustion,或者 Proof by cases)係一種用嚟證明啲淨係描述緊數量有限嘅個案度嘅定理嘅一種證明方法。佢嘅過程係要先列出所有個案,再顯示喺所有個案入面,條命題都係成立嘅[10]。條思路如下:

  • 想要證明嘅命題 淨係描述緊數量有限(Finite in number)嘅個案;
  • 將所有命題 描述嘅個案列嗮出嚟;
  • 顯示喺所有個案入面, 嘅預測都係成立嘅;
  • 證明到 呢句命題係啱嘅。
例子

要求:證明「任何整數 係一個單數。」


因為整數一係單數一係雙數,而喺以上兩個個案入面 都係一個單數,所以「任何整數 係一個單數。」呢句命題成立。

相關概念[編輯]

周髀算經(公元前 500 至 200 年)入面用咗圖像方法嚟證明畢氏定理(Pythagorean theorem)。
「證明」畢氏定理嘅動畫

無字證明[編輯]

內文: 無字證明
  • 無字證明(Proof without words),又有叫視覺證明(Visual proof),係指用圖像呢啲直接用眼睇嘅方法嚟到嘗試證明一啲數學定理。呢種「證明」方法好多時會因為條線畫得唔夠直等嘅技術性原因而出錯,所以喺正式嘅數學研究嗰度好少可會有學者接納[11]

證明結尾[編輯]

內文: 證明完畢
  • 有陣時喺一個證明嘅結尾嗰度會寫咗 Q.E.D. 呢三個羅馬字母喺度,呢個係拉丁話「Quod Erat Demonstrandum」嘅縮寫,意思係「本嚟要證明嘅嘢」噉解。
  • 廿一世紀嘅證明完畢符號通常係用「」(實心黑色四方形)。佢個名叫「墓碑」,又或者叫哈爾莫斯符號(Halmos symbol)-因為美國數學家 Paul Halmos 係第一個用呢種做法嘅。個墓碑有時係空心嘅「」。
  • 仲有另外簡單嘅方法係寫「proven」、「shown」、或者「證畢」之類嘅文字[12]

證明唔到嘅嘢[編輯]

  • 有陣時有啲命題係證明唔到嘅,冇方法可以證明佢係真,又冇方法可以證明佢係假,呢啲命題係所謂嘅決定唔到(Undecidable)嘅命題。歐幾里得幾何入面嘅平行公設(Parallel postulate)就係一個例子。
  • 哥德爾不完備定理(Gödel's incompleteness theorem)表示咗,喺數學家會有興趣嘅命題當中至少有一啲係證明唔到嘅[13]

實驗數學[編輯]

內文: 實驗數學
  • 有一啲早期嘅數學家係唔靠證明嘅,但係由幾何學之父歐幾里德(公元前 300 年生)開始,數學證明就一路都係數學發展嘅基礎,直到 19 至 20 世紀都仲係噉[14]。但係自從喺 1960 年代起,電腦嘅運算能力就開始變到愈嚟愈勁,所以開始有數學家研究數學嘢可唔可以用證明以外嘅方法研究-形成咗實驗數學(Experimental mathematics)呢個領域[15]

電腦輔助證明[編輯]

內文: 電腦輔助證明

直至廿世紀為止,學界一般都仲係覺得原則上任何嘅數學證明都可以由能力夠高嘅數學家嚟幫手肯定佢嘅有效性(Validity)[16],但係家吓數學界成日都會用電腦輔助證明(Computer-assisted proof)-即係用電腦幫手證明一啲定理,又或者用電腦做一啲長得滯搞到用人手計唔到嘅運算,例如四色定理(Four-color theorem)嘅第一個證明就係由電腦輔助做嘅。初頭有唔少數學家擔心用電腦幫手會搞到啲證明有可能出錯,質疑電腦輔助證明嘅有效性。現時嘅數學家會用好多方法令到電腦輔助證明令人更加有信心,好似係重複噉檢查同埋用多個唔同程式做證明呀噉,而事實係,就算用人手做數學證明都係有可能出錯嘅。所以喺廿一世紀,數學界一般都唔抗拒用電腦幫手做數學證明[16]

睇埋[編輯]

[編輯]

  1. Clapham, C. & Nicholson, JN. The Concise Oxford Dictionary of Mathematics, Fourth edition. "A statement whose truth is either to be taken as self-evident or to be assumed. Certain areas of mathematics involve choosing a set of axioms and discovering what results can be derived from them, providing proofs for the theorems that are obtained."
  2. Gossett, E. (2009). Discrete Mathematics with Proof. Definition 3.1, p. 86. John Wiley and Sons. ISBN 0-470-45793-7
  3. 3.0 3.1 Cupillari, Antonella. The Nuts and Bolts of Proofs. Academic Press, 2001. Page 3.
  4. Buss, Samuel R. (1998), "An introduction to proof theory", in Buss, Samuel R., Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, 137, Elsevier, pp. 1–78, ISBN 9780080533186. See in particular p. 3: "The study of Proof Theory is traditionally motivated by the problem of formalizing mathematical proofs; the original formulation of first-order logic by Frege [1879] was the first successful step in this direction."
  5. Bogomolny, Alexander. "Mathematics Is a Language". www.cut-the-knot.org. Retrieved 2017-05-19.
  6. Cupillari, page 20.
  7. Cupillari, page 46.
  8. Proof by induction, University of Warwick Glossary of Mathematical Terminology.
  9. 余紅兵; 嚴鎮軍. 《構造法解題》. 中國科學技術大學出版社. 2009.
  10. Reid, D. A. & Knipping, C. (2010). Proof in Mathematics Education: Research, Learning, and Teaching. Sense Publishers, p. 133.
  11. Weisstein, Eric W. "Proof without Words". MathWorld.
  12. Paul R. Halmos, I Want to Be a Mathematician: An Automathography, 1985, p. 403.
  13. What is Gödel's proof?. Scientific American.
  14. "What to do with the pictures? Two thoughts surfaced: the first was that they were unpublishable in the standard way, there were no theorems only very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, but theorems were coins of the realm ant the conventions of that day dictated that journals only published theorems", David Mumford, Caroline Series and David Wright, Indra's Pearls, 2002.
  15. "Mandelbrot, working at the IBM Research Laboratory, did some computer simulations for these sets on the reasonable assumption that, if you wanted to prove something, it might be helpful to know the answer ahead of time."A Note on the History of Fractals Archived 2009-02-15 at the Wayback Machine.
  16. 16.0 16.1 The History and Concept of Mathematical Proof, (2007). Steven G. Krantz.

參考[編輯]

  • Alibert, D., & Thomas, M. (2002). Research on mathematical proof. In Advanced mathematical thinking (pp. 215-230). Springer Netherlands.
  • Fallis, Don (2002), "What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians", Logique et Analyse, 45: 373–388 .
  • Franklin, J.; Daoud, A. (2011), Proof in Mathematics: An Introduction, Kew Books, ISBN 0-646-54509-4 .
  • Hanna, G., & Jahnke, H. N. (1996). Proof and proving. In International handbook of mathematics education (pp. 877-908). Springer Netherlands.
  • Hardy, G. H. (1929). Mathematical proof. Mind, 38(149), 1-25.
  • Pólya, G. (1954), Mathematics and Plausible Reasoning, Princeton University Press .
  • Solow, D. (2004), How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, Wiley Publishing, ISBN 0-471-68058-3 .
  • Velleman, D. (2006), How to Prove It: A Structured Approach, Cambridge University Press, ISBN 0-521-67599-5 .