Exploring the Works of an Imaginary Mathematician: Discovering New Insights in Mathematics

One of the most influential figures in modern mathematics, Nicolas Bourbaki, has reportedly been researching for almost a century, producing numerous books and publications that guide the entire field. Interestingly, Bourbaki is a pseudonymous figure who does not exist as an actual individual.

Bourbaki represents a secretive collective of mathematicians, initially formed in France in 1934. Their primary objective was to modernize mathematics textbooks, transforming them to meet contemporary reader needs. However, this endeavor culminated in the creation of an innovative approach to mathematical writing, impacting the field for decades.

The group initially anticipated that their study would comprise about 1,000 pages and be completed in six months. By 1935, Bourbaki had expanded its vision to include six interconnected volumes, aiming to “provide a comprehensive foundation for modern mathematics,” as expressed in an explanatory preface. While they were correct about the length, they were notably mistaken regarding the timeline.

Though these volumes (which eventually comprised several physical books) were intended to be read sequentially, Bourbaki’s first published text in 1939 turned out to be the concluding chapter of the first book on set theory. The group later published different sections intermittently before returning to finish set theory in 1954, finally completing the entire project in 1970. Collectively labeled as elements of mathematics, this singular title underscores the cohesion of the mathematicians’ work. The completion of this monumental collection extended into the 1980s, reaching nearly 4,000 pages. Even after that, Bourbaki continued to release new works as the original scope broadened.

This unorthodox publishing approach stemmed from Bourbaki’s distinctive methodology. The original group comprised six young mathematics professors, including Andre Weil, a prominent figure in number theory and algebraic geometry. Most members were former students of the École Normale Supérieure in Paris, and the group’s name emerged from a prank revolving around the notoriously obscure Bourbaki theorem.

This playful spirit fostered a strong sense of camaraderie. Meetings were lively, often involving shouting matches and humorous banter. One member crafted the proposed text and presented it line by line for critique and discussion, leading to a revised draft before reaching consensus. Given that chapters took an average of ten years to produce, the protracted timeline is understandable. This mathematical endeavor spanned generations, as Bourbaki members were required to retire at 50, making way for new recruits.

Eternal Challenges in Mathematics

So, what was Bourbaki’s actual contribution? Despite its unorthodox methods, the group’s work was notably serious and thoroughly detailed. The cornerstone of their research, set theory, aimed to tackle the perennial challenge in mathematics: the idea that mathematical objects are fundamentally independent of human language and symbols.

To illustrate this, consider the word “addition” or the symbol “+”. These terms have an arbitrary connection to the underlying mathematical concepts. As long as there’s an agreement on the meaning of “addition,” any string of symbols could be utilized to indicate it. Conversely, addition has a definitive relationship with subtraction; one operation is the inverse of the other, independent of their nomenclature.

In practical terms, labeling mathematical concepts does not present a significant challenge, as mathematicians adhere to standardized mappings between ideas and symbols. However, in principle, contradictions and inconsistencies may emerge.

Bourbaki was not the inaugural attempt at formalization (as mentioned in my previous writings), but his approach was perhaps the most meticulous. For instance, he took care to define the number 1 in a footnote on page 158 of set theory. Bourbaki clarified that “the symbol ‘1’ should not be confused with the common language interpretation ‘one'”; instead, it should be understood through a rigorous definition:

τZ ((∃u)(∃U)(u = (U, {∅}, Z) and U ⊂ {∅} × Z and (∀x)((x ∈ {∅}) ⇒ (∃y)((x, y) ∈ U)) and (∀x)(∀y)(∀y’)(((x, y) ∈ U and (x, y’) ∈ U) ⇒ (y = y’)) and (∀y)((y ∈ Z) ⇒ (∃x)((x, y) ∈ U))))

Don’t worry if this seems daunting; a simplified explanation is that ∅ represents a set devoid of elements, referred to as the “empty set.” Consequently, 1 is defined as {∅}, indicating a set containing only one item (which, in this case, is the empty set). More details on this concept can be found in a previous column.

Astonishingly, embedded within this sea of symbols is a broader and more complex formal definition. Each symbol is elaborately defined based on earlier texts using only designated symbols. Bourbaki never elaborated these entirely; the footnote mentions that completing this definition would require tens of thousands of symbols — an estimation soon revealed to be vastly understated. Later mathematicians calculated that articulating the full formula for the number 1 would necessitate over 4.5 billion symbols, or more precisely, 2,409,875,496,393,137,472,149,767,527,877,436,912,979,508,338,752,092,897 symbols, depending on your definition of precision.

Clearly, mathematicians would need to occasionally abandon such stringent formalism if they wished to accomplish their objectives. Bourbaki acknowledges this necessity, while maintaining that utilizing shorthand terms like “1” is an “abuse of language.” By establishing foundational rules, Bourbaki granted mathematicians the flexibility to deviate as needed.

Emerging Mathematical Challenges

So, what achievements stemmed from all this labor? One significant outcome was Bourbaki’s aspiration to unite mathematics as a cohesive discipline. In theory, if terms and concepts from various mathematical domains could be expressed using a common set of symbols, it would yield a rigorous framework for transitions between fields. Although few actually practice this, it positions mathematics on a more solid philosophical foundation.

In the decades that followed, Bourbaki’s influence has proven unexpectedly significant, particularly as mathematicians increasingly explore computer-assisted formalization to verify proofs generated by artificial intelligence. The collective also introduced numerous concepts and symbols, many of which remain integral to contemporary mathematics (for instance, ∅ representing the empty set). On a broader scale, the Bourbakian writing style continues to shape modern mathematical textbooks.

However, Bourbaki was not without critique. Following the publication of elements of mathematics, some mathematicians expressed discontent with the group’s claims of excessive rigor. Oddly enough, Bourbaki inadvertently incited a misguided initiative to reform mathematics education in schools. Emerging in France during the late 1950s, this movement, dubbed New Mathematics, sought to replace traditional educational methods with rigorous set-theoretic approaches based on Bourbaki’s teachings. The intention was to grasp the general principle of multiplication rather than memorizing specific multipliers, such as 3 × 4 = 12.

The “New Math” movement faced extensive criticism and was largely deemed a failure. Parents and teachers alike struggled to understand the curriculum. Bestselling critiques like Why Can’t Johnny Add? emerged, and by the late 1970s, the initiative had largely dissipated. Additionally, this decade brought challenges for Bourbaki, including legal disputes with publishers over copyright and royalties.

Despite these hurdles, Bourbaki remains relevant today. New chapters will be released this year alone. However, the identity of the author remains a well-guarded secret. This anonymity allows mathematicians to regard Bourbaki as a quirky, eccentric relative: appreciated for essential contributions, yet sparing themselves from the discomfort of personal association.