This report talks about how people prove things in math. It looks at old ways and new ways with computers.

本報告探討人們在數學中如何進行證明。它分析了傳統的方法以及利用電腦的新方法。

Long ago, mathematicians showed exactly how to find an answer. Later, David Hilbert used a new way. He proved things were true without showing the answer. Some people liked this, but L.E.J. Brouwer did not. They had a big fight about these rules.

很久以前，數學家會詳細地展示如何找到答案。後來，David Hilbert 使用了一種新方法。他證明事物是正確的，而無需展示答案。有些人喜歡這樣，但 L.E.J. Brouwer 並不認同。他們針對這些規則發生了激烈的爭論。

Now, people use computers to check math. Kevin Buzzard uses a computer language called Lean. He wants to put a famous math proof into a computer. This helps people check if the math is correct.

現在，人們使用電腦來檢查數學。Kevin Buzzard 使用一種名為 Lean 的電腦語言。他希望將一個著名的數學證明輸入電腦，這有助於人們檢查數學是否正確。

Today, AI helps write this computer code. AI works very fast. It can write a lot of code in one day. But some people say AI code is messy. Human code is often better and cleaner.

如今，AI 協助編寫這種電腦程式碼。AI 的運作速度非常快，一天之內可以編寫大量程式碼。但有些人認為 AI 的程式碼很凌亂，人類編寫的程式碼通常較佳且更簡潔。

Math is changing. In the future, AI might find answers that are correct, but humans cannot understand them.

數學正在改變。未來，AI 可能會找到正確的答案，但人類可能無法理解。

This report examines how mathematical verification has changed over time, moving from traditional constructive proofs to non-constructive logic and the current use of artificial intelligence to verify complex theorems.

本報告探討數學驗證如何隨時間演變，從傳統的構造性證明轉向非構造性邏輯，以及目前利用人工智慧來驗證複雜定理的情況。

The way mathematicians prove theories changed significantly in the 19th century with the rise of non-constructive proofs. While constructive methods require a mathematician to show a specific example of a mathematical object, non-constructive proofs prove that something exists by showing that the opposite is impossible. For example, the 'pigeonhole principle' asserts that a shared characteristic exists within a group without identifying exactly who has it. David Hilbert promoted this approach, using the 'law of the excluded middle,' which states that a statement must be either true or false.

數學家證明理論的方式在 19 世紀隨著非構造性證明的興起而發生顯著變化。構造性方法要求數學家展示數學對象的具體範例，而非構造性證明則是透過證明相反情況是不可能的，來證明某樣東西存在。例如，「鴿巢原理」主張一個群體中存在共同特徵，而無需確定具體是誰擁有該特徵。大衛·希爾伯特推廣了這種方法，使用了「排中律」，即規定一個陳述必須不是真就是假。

This shift caused a major disagreement between Hilbert's formalism, which treated mathematics as a system of symbols, and L.E.J. Brouwer's intuitionism. Brouwer argued that mathematical objects must be mentally constructed to be valid; consequently, he rejected the law of the excluded middle for infinite sets. This conflict became so intense that it led to a professional split within the journal Mathematische Annalen in 1928. Although Hilbert's view became the standard, Kurt Gödel later challenged it by suggesting that symbolic systems cannot be completely consistent.

這種轉變導致希爾伯特將數學視為符號系統的形式主義，與 L.E.J. 布勞華的直覺主義之間產生重大分歧。布勞華主張數學對象必須經過心智構造才有效；因此，他拒絕將排中律應用於無限集。這場衝突如此激烈，以至於導致 1928 年學術期刊《Mathematische Annalen》內部發生專業分裂。儘管希爾伯特的觀點成為了標準，但庫爾特·哥德爾隨後挑戰了這一點，指出符號系統無法完全一致。

Today, the focus has moved toward using computer languages like Lean to formalize mathematics. For instance, Kevin Buzzard from Imperial College London is leading a project to translate Andrew Wiles's 1993 proof of Fermat's Last Theorem into machine-verifiable code. Furthermore, the use of Large Language Models (LLMs) has greatly increased the speed of this work. However, this introduces a new problem: while humans write efficient code, AI often produces 'verbose' or messy code, which some researchers call 'slop' because it can be unstable during software updates.

如今，焦點已轉向使用如 Lean 等電腦語言將數學形式化。例如，倫敦帝國學院的 Kevin Buzzard 正在領導一個項目，將 Andrew Wiles 在 1993 年對「費馬最後定理」的證明翻譯成機器可驗證的程式碼。此外，大型語言模型 (LLM) 的使用大大提高了這項工作的速度。然而，這也引入了一個新問題：人類編寫的程式碼高效，而 AI 經常產出「冗長」或混亂的程式碼，一些研究人員將其稱為「slop」，因為它在軟體更新期間可能會不穩定。

Mathematics is now moving toward a more industrial process. In the future, AI-driven formalization may create logically correct proofs that are too complex for humans to understand.

數學現在正走向一個更工業化的過程。未來，AI 驅動的形式化可能會創造出邏輯正確，但複雜到人類無法理解的證明。

This report examines the transition in mathematical verification from traditional constructive proofs to non-constructive logic and the contemporary integration of artificial intelligence in formalizing complex theorems.

本報告探討了數學驗證從傳統構造性證明向非構造性邏輯的轉型，以及當代如何將人工智慧整合於複雜定理的形式化過程中。

The historical trajectory of mathematical proof underwent a significant shift in the 19th century with the emergence of non-constructive proofs. Unlike constructive methods, which require the explicit exhibition of a mathematical object, non-constructive proofs establish existence through logical contradiction. A primary example is the 'pigeonhole principle,' which asserts a shared attribute within a set without identifying the specific subjects. David Hilbert championed this approach, notably in 1888 when he demonstrated the existence of finite generating sets for invariants without specifying their composition. This methodology relied upon the 'law of the excluded middle,' a logical axiom stating that a proposition is either true or its negation is true.

數學證明的歷史軌跡在 19 世紀隨著非構造性證明的出現而發生重大轉向。與需要明確展示數學對象的構造性方法不同，非構造性證明透過邏輯矛盾來確立存在性。一個主要例子是「鴿巢原理」，它主張一個集合中存在共同屬性，而無需識別具體對象。大衛·希爾伯特支持這種方法，尤其是在 1888 年，他證明了不變量具有有限生成集，而未指定其組成。此方法依賴於「排中律」，即一個邏輯公理，主張一個命題要麼為真，要麼其否定為真。

This shift precipitated a philosophical schism between Hilbert's formalism—which viewed mathematics as the manipulation of symbols—and L.E.J. Brouwer's intuitionism. Brouwer contended that mathematical objects must be mentally constructible to be valid, thereby rejecting the application of the law of the excluded middle to infinite sets. This intellectual conflict manifested institutionally within the journal Mathematische Annalen, culminating in the 1928 dismissal of the editorial board by Hilbert. While Hilbert's approach became the prevailing standard, Kurt Gödel later challenged formalism via his incompleteness theorem, suggesting that symbolic manipulation cannot achieve total consistency.

這一轉向導致了希爾伯特的形式主義（將數學視為符號操作）與 L.E.J. 布勞威爾的直覺主義之間的哲學分歧。布勞威爾主張數學對象必須在心智上可構造才有效，因此拒絕將排中律應用於無限集。這場知識衝突體現於期刊《數學年誌》中，最終導致希爾伯特在 1928 年解散編輯委員會。雖然希爾伯特的方法成為了主流標準，但庫爾特·哥德爾隨後透過其不完備定理挑戰了形式主義，指出符號操作無法實現完全的一致性。

In the contemporary era, the focus has shifted toward the formalization of mathematics through computational languages such as Lean. Kevin Buzzard of Imperial College London is currently leading an initiative to formalize Andrew Wiles's 1993 proof of Fermat's Last Theorem. This process involves translating human-readable proofs into machine-verifiable code, contributing to the Mathlib repository. The integration of Large Language Models (LLMs) has accelerated this process; for instance, a recent workshop saw the volume of project code double in a single day. However, the use of AI introduces a dichotomy between 'efficient' human-authored code and 'verbose' AI-generated code, the latter of which some researchers characterize as 'slop' due to its potential for instability during software updates.

在當代，焦點已轉向透過 Lean 等計算語言將數學形式化。倫敦帝國學院的 Kevin Buzzard 目前正領導一項計劃，將安德魯·懷爾斯 1993 年關於費馬最後定理的證明形式化。此過程涉及將人類可讀的證明翻譯成機器可驗證的代碼，並貢獻至 Mathlib 儲存庫。大型語言模型 (LLM) 的整合加速了這一過程；例如，最近的一場工作坊見證了項目代碼量在單日內翻倍。然而，AI 的使用在「高效」的人類編寫代碼與「冗長」的 AI 生成代碼之間產生了對立，後者被部分研究人員稱為「廢料」(slop)，因其在軟體更新時具有潛在的不穩定性。

Mathematics is currently transitioning toward an industrialization of the intellectual process, where AI-driven formalization may eventually produce logically valid proofs that exceed human cognitive comprehension.

數學目前正轉向知識過程的工業化，AI 驅動的形式化最終可能會產生邏輯有效但超出人類認知理解能力的證明。