數(shù)學(xué)基礎(chǔ)概論

時(shí)間：2024-09-24 03:09:36 數(shù)學(xué)畢業(yè)論文我要投稿

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數(shù)學(xué)基礎(chǔ)概論

摘要：對(duì)于數(shù)學(xué)這門學(xué)問(wèn)，當(dāng)我們從它的最熟悉的部分出發(fā)時(shí)，可以在兩個(gè)相反的方向進(jìn)行研究。比較熟悉的方向是構(gòu)造性的，逐步增強(qiáng)復(fù)雜程度：從整數(shù)到分?jǐn)?shù)實(shí)數(shù)復(fù)數(shù)；從加法到乘法到微分和積分，以至更高等的數(shù)學(xué)。另1較為生疏的方向，是通過(guò)分析走向抽象性和邏輯簡(jiǎn)單性；我們不討論從假定的起點(diǎn)開始可以定義什么或可以演繹什么，而去探討能夠找到什么可以由之定義或演繹出我們的出發(fā)點(diǎn)的1般思想。羅素在《數(shù)學(xué)哲學(xué)導(dǎo)論》中所說(shuō)的后1方向既數(shù)學(xué)哲學(xué)或數(shù)學(xué)基礎(chǔ)。我們難以確切地說(shuō)出數(shù)學(xué)基礎(chǔ)所研究的具體對(duì)象，但我們可以這樣粗略地理解數(shù)學(xué)基礎(chǔ)的內(nèi)容，即：給現(xiàn)存數(shù)學(xué)1個(gè)穩(wěn)固的基礎(chǔ)，1個(gè)合理的解釋，1種完美統(tǒng)1的形式，通過(guò)這種形式可以演繹全部現(xiàn)存數(shù)學(xué)。也許這只是1個(gè)理想，但仍然可以作為數(shù)學(xué)基礎(chǔ)研究的1個(gè)終極目標(biāo)。在這篇文章里，先討論數(shù)學(xué)基礎(chǔ)的1些重要問(wèn)題；然后簡(jiǎn)述3個(gè)學(xué)派的起因，成果以及他們所面臨的困境；接著介紹結(jié)構(gòu)主義的基本觀點(diǎn)；最后試著給出關(guān)于數(shù)學(xué)基礎(chǔ)的1些基本看法，這也是本文所要完成的最后1個(gè)目標(biāo)。

關(guān)鍵詞：數(shù)學(xué)基礎(chǔ)；無(wú)限；存在；真理；相容性；邏輯主義；直覺(jué)主義；形式主義；公理化；結(jié)構(gòu)；集合論

Introduction to Foundations of Mathematics

Abstract: As to the science of mathematics, we can make our research toward two opposite directions when we begin from the most familiar part. The comparative familiar diction is about structure. Increasing the degree of complexity gradually, which is from integer to fraction, real number and plural, from addition to multiplication, differentiation and integration, even get to more advanced mathematics. The other comparative strange direction is to reach abstraction and logical simplicity through analysis. However, we do not discuss what we can definite deduct when beginning with presumption, but to explore the general ideas of our point of departure we find out to definite and deduct. In Introduction to Mathematical Philosophy, the later view as Russel refer to is about mathematical philosophy and foundations of mathematics. Although we can not exact tell the concrete object the foundations of mathematics research into, we can understand its content in general. That is to say, we endow present mathematics with a stable basis, a reasonable explanation and a perfect and united form. And through this form, we can deduct the whole present mathematics. Maybe this only to be a dream, but we can still take it as a terminal goal in foundations of mathematics studies. In this passage, well first discuss several important issues about the foundations of mathematics. Then, well state in brief the origin and outcomes of three schools, and the hardship they are facing. Next, well introduce the elemental views of structuralism. At last, well try to put forward a few basic opinions about the foundations of mathematics. And this is also the final aim we want to attain.

Keywords: foundations of mathematics; unlimitation; existece; truth; compatibility; logicism; intuitionalism; formalism; axio matization structure; set theory

前言

關(guān)于數(shù)學(xué)基礎(chǔ)的討論在柏拉圖時(shí)代就開始了，由柏拉圖發(fā)展起來(lái)的唯理論在現(xiàn)代數(shù)學(xué)中仍有重要的地位，之后數(shù)學(xué)基礎(chǔ)問(wèn)題也不間斷地被討論到，而數(shù)學(xué)基礎(chǔ)發(fā)展的輝煌時(shí)期卻是19世紀(jì)末期及20世紀(jì)前310年。
當(dāng)實(shí)數(shù)的理論建立后，分析的基礎(chǔ)也正式確立了，而代數(shù)學(xué)也從當(dāng)時(shí)的紊亂中走了出來(lái)，并且非歐幾何的相容性問(wèn)題也由歐氏幾何得出，這樣，到1900年為止，數(shù)學(xué)的主要分支—算術(shù)，代數(shù)，分析及幾何已經(jīng)被嚴(yán)密化，而且邏輯學(xué)在當(dāng)時(shí)也得到了極大的豐富和發(fā)展。面對(duì)這種情景，數(shù)學(xué)家們欣喜不已，就像彭加勒在第2次國(guó)際數(shù)學(xué)家大會(huì)上所說(shuō)的：“今天我們可以宣稱絕對(duì)的嚴(yán)密已經(jīng)實(shí)現(xiàn)了！”。
然而情況真的如此嗎？數(shù)學(xué)已經(jīng)有了1個(gè)可靠的基礎(chǔ)了嗎？就在這時(shí)，集合論的幽靈已經(jīng)出現(xiàn)，數(shù)學(xué)史上的第3次數(shù)學(xué)危機(jī)正在醞釀，而羅素打開了潘多拉的盒子，它們將再1次考驗(yàn)數(shù)學(xué)家們。1903年，羅素發(fā)表了羅素悖論，即：設(shè)N是由所有不屬于自身的集合組成的集合，那么N有屬于誰(shuí)呢？若N屬于N，按照定義不應(yīng)如此；若N不屬于N，則依定義其應(yīng)屬于N。其實(shí)在羅素悖論發(fā)表之前，已經(jīng)在集合論的基數(shù)理論和序數(shù)理論中發(fā)現(xiàn)了悖論，之后也構(gòu)造了好些悖論，然而以羅素悖論最為有名，這1悖論動(dòng)搖了元素的類這1在數(shù)學(xué)中廣泛應(yīng)用的概念，希爾伯特稱這個(gè)悖論對(duì)數(shù)學(xué)界有著災(zāi)難性的后果。
為了解決悖論，以及由此而引起的數(shù)學(xué)基礎(chǔ)問(wèn)題，特別是相容性問(wèn)題及數(shù)學(xué)本性問(wèn)題，數(shù)學(xué)家們做出了各種努力，嘗試為數(shù)學(xué)提供1個(gè)可靠的基礎(chǔ)。我們就從數(shù)學(xué)基礎(chǔ)中最為關(guān)心的重要數(shù)學(xué)基礎(chǔ)問(wèn)題開始我們的敘述，接著討論基礎(chǔ)學(xué)派所做的各種努力，然后著重分析數(shù)學(xué)的困境和數(shù)學(xué)的本性，最后給出自己的1個(gè)初步結(jié)果，試圖從困境中走出來(lái)，達(dá)到對(duì)數(shù)學(xué)的1種整體的理解，給數(shù)學(xué)1個(gè)合理的解釋，這就是自己要達(dá)到的目標(biāo)。

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