# 數學證明

(由證明跳轉過來)

## 大分類

• 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。

## 常用嘅證明方法

### 直接證明

• 有若干數量嘅命題（${\displaystyle P}$ ${\displaystyle Q}$ ...），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 ${\displaystyle R}$
• 所以命題 ${\displaystyle R}$  都係啱嘅。

### 數學歸納法

• ${\displaystyle n=1}$  嘅時候，命題 ${\displaystyle P_{n}}$  係啱嘅；
• 當命題 ${\displaystyle P_{k}}$  係啱嘅時候，可以引伸到 ${\displaystyle P_{k+1}}$  都係啱嘅；

### 否定證明

• 如果蘇格拉底係一個人，噉佢嘅壽命會係有限嘅；（${\displaystyle P\rightarrow Q}$
• 如果蘇格拉底嘅壽命係冇限嘅，噉佢一定唔係一個人。（${\displaystyle \neg Q\rightarrow \neg P}$

### 反證法

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

### 構造法

• 揾到有一個情況下，命題 ${\displaystyle P}$  係啱嘅；
• 證明到「喺至少一個情況下，命題 ${\displaystyle P}$  係啱嘅」。

### 分類證明

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

## 相關概念

「證明」畢氏定理嘅動畫

### 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

• 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

• 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

• 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]

## 攷

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. 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 互聯網檔案館歸檔，歸檔日期2017年11月14號，., 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. The History and Concept of Mathematical Proof, (2007). Steven G. Krantz.

## 參考

• 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.