數學之美

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曼德勃羅集合的邊界

大多數的數學家會由他們的工作及一般數學裡得出美學的喜悅。他們形容數學是美麗的來表示這種喜悅。有時，數學家會形容數學是一種藝術的形式，或至少是一個創造性的活動。通常拿來和音樂和詩歌相比較。

伯特蘭·羅素以下列文字來形容他對數學之美的感覺：

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry. (The Study of Mathematics, in Mysticism and Logic, and Other Essays, ch. 4, London: Longmans, Green, 1918.)

保羅·埃爾德什形容他對數學不可言說的觀點，而說：「為何數字美麗呢？這就像是在問貝多芬第九號交響曲為什麼會美麗一般。若你不知道為什麼，其他人也沒辦法告訴你為什麼。我知道數字是美麗的。且若它們不是美麗的話，世上也沒有事物會是美麗的了。」

[编辑] 解法之美

數學家形容一特別愉悅的證明方法為優美的。依據其內容，這可能是指：

用了少量的額外假設或之前結論的證明。
極短的證明。
由意外的方式導出的證明(即由一表面上無關的定理或一群定理)。
基於新的及原本洞悉的證明。
可輕易推至相似問題之解題的證明方法。

為了尋找一個優美的證明，數學家常會尋求不同證明的方法，而第一個被找到的證明可能不是最好的。有著最多被找到的不同證明方法的定理可能為勾股定理，已有上百的證明被發表了出來。另一個被以許多不同方法證明出來的定理為二次互反律的定理－光高斯一人便給出了此定理八種不同的證明。

相反地，若結論是邏輯上正確的，但包含有費工的計算、過度複雜的方法、極普通的處理方法或需依靠大量有力的公理或不被認為優美的之前結論，則稱此為醜陋的或笨拙的。這和奧卡姆剃刀的概念有關。

[编辑] 結論之美

數學家看數學結論的美於其在第一眼看似完全無關的兩個數學領域間建立著關連性。這種結論通常被形容為深奧的。

因為很難得到一結論是否為深奧的共識，某些例子便常被引用來說明。其中一個為歐拉恆等式 e^iπ + 1 = 0，它被費曼稱為「數學內最著名的公式」。現代的例子則包含有建立起橢圓曲線與模型式之間關連性的谷山-志村定理(此結論使安德魯·懷爾斯和羅伯特·郎蘭茲得到了沃爾夫獎)，和以弦理論接連了怪獸群與模函數的怪獸月光(理查·波傑蒂斯因此得到了菲爾茲獎)。

和深奧的相對的為當然的。一當然的定理會是個可以由一已知結論經一明顯及簡單的方法導出的結論，或者是只應用在如空集合等特定集合的結論。但有時一定理的敘述亦可因夠原始而被認為是深奧的，即使其證明是很簡易的。

[编辑] 體驗之美

些許對於操縱數字和符號的喜好是從事任何數學相關的必須要件。在科學及工程內的數學工具，似乎都會在其技術社會裡主動地培育出美學來，尤其是在其自身的科學哲學裡。

對於大多數的數學家而言，數學之美最強烈的體驗來自於積極地從事數學研究。以純粹被動的方式是很難享受及欣賞數學的－在數學裡，是沒有觀眾及聽眾的。

伯特蘭·羅素指這是數學的樸素之美。

[编辑] 美麗與哲學

Some mathematicians are of the opinion that the doing of mathematics is closer to discovery than invention. These mathematicians believe that the detailed and precise results of mathematics may be reasonably taken to be true without any dependence on the universe in which we live. For example, they would argue that the theory of the natural numbers is fundamentally valid, in a way that does not require any specific context. Some mathematicians have extrapolated this viewpoint that mathematical beauty is truth further, in some cases becoming mysticism.

Pythagoras (and his entire philosophical school of the Pythagoreans) believed in the literal reality of numbers. The discovery of the existence of irrational numbers was a shock to them - they considered the existence of numbers not expressible as the ratio of two natural numbers to be a flaw in nature. From the modern perspective Pythagoras' mystical treatment of numbers was that of a numerologist rather than a mathematician. In Plato's philosophy there were two worlds, the physical one in which we live and another abstract world which contained unchanging truth, including mathematics. He believed that the physical world was a mere reflection of the more perfect abstract world.

Galileo Galilei is reported to have said "Mathematics is the language with which God wrote the universe", a statement which (apart from the implicit deism) is consistent with the mathematical basis of all modern physics.

Hungarian mathematician Paul Erdős, although an atheist, spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would exclaim "This one's from the Book!". This viewpoint expresses the idea that mathematics, as the intrinsically true foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God by different religious mystics.

Twentieth-century French philosopher Alain Badiou claims that ontology is mathematics. Badiou also believes in deep connections between math, poetry and philosophy.

In some cases, natural philosophers and other scientists who have made extensive use of mathematics have made leaps of inference between beauty and physical truth in ways that turned out not to be confirmed. For example, at one stage in his life, Johannes Kepler believed that the proportions of the orbits of the then-known planets in the Solar System had been arranged by God to correspond to a concentric arrangement of the five Platonic solids, each orbit lying on the circumsphere of one polyhedron and the insphere of another. As there are exactly five Platonic solids, Kepler's theory could only accommodate six planetary orbits, and was disproved by the subsequent discovery of Uranus. James Watson made a similar error when he originally postulated that each of the four bases of DNA connected to a base of the same type in the opposite strand (thymine linking to thymine, etc.) based on the belief that "it is so beautiful it must be true." ^{[來源請求]}

[编辑] 參考文獻

Aigner, Martin, and Ziegler, Gunter M. (2003), Proofs from THE BOOK, 3rd edition, Springer-Verlag.

蘇布拉馬尼揚·錢德拉塞卡 (1987), Truth and Beauty. Aesthetics and Motivations in Science, University of Chicago Press, Chicago, IL.

雅克·阿達馬 (1949), The Psychology of Invention in the Mathematical Field, 1st edition, Princeton University Press, Princeton, NJ. 2nd edition, 1949. Reprinted, Dover Publications, New York, NY, 1954.

高德菲·哈羅德·哈代 (1940), A Mathematician's Apology, 1st published, 1940. Reprinted, C.P. Snow (foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992.

Hoffman, Paul (1992), The Man Who Loved Only Numbers, Hyperion.

Huntley, H.E. (1970), The Divine Proportion: A Study in Mathematical Beauty, Dover Publications, New York, NY.

Loomis, Elisha Scott (1968), The Pythagorean Proposition, The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem.