
How do you prove the evidence? Sometimes we don’t
Lucidio Studio Inc./Getty Images
A mathematician opens her office door to find a small fire starting. Instead of panicking, she spots a fire extinguisher and exclaims, “Oh, I have a solution!” before closing the door and resuming her tasks. This comedic anecdote illustrates a clever aspect of modern mathematics: the concept of unconstructive proofs.
Consider this non-mathematical example: Imagine there are 367 people in a room. What is the likelihood that at least two of them share the same birthday? The answer is 100%. With only 366 possible birthdays (accounting for leap years) and 367 birthdays to fill, at least two individuals must share a birthday. This illustrates the “pigeonhole principle,” a classic method in non-constructive proofs, showing that even without identifying specific individuals, a birthday match is guaranteed.
Traditionally, proofs involved displaying a tangible mathematical object for verification. This approach shifted significantly in the 19th century when mathematicians began to embrace nonconstructive proofs as powerful tools. Leading this transformation was David Hilbert, a prominent figure in mathematics known for his controversial ideas.
Hilbert investigated complex problems regarding algebraic objects, aiming to uncover the essential invariants. For example, when considering a square, rotating it by 90 degrees results in the same shape. This phenomenon is termed “rotational symmetry,” conveying that a square remains unchanged despite rotation.
Hilbert sought to understand how many invariants were necessary to construct algebraic objects, building upon the foundation laid by mathematician Paul Gordan. Gordan identified finite generating sets for certain objects, but his proofs were often complicated. In 1888, Hilbert surprisingly proved this for a wider class of algebraic objects without detailing the composition of the generating set. By assuming the existence of invariants not produced by the generating set, he demonstrated a contradiction, ultimately concluding that a generating set must exist.
Initially, Gordan reacted negatively to this unconstructive evidence, labeling it as “theology” rather than mathematics. However, he later acknowledged Hilbert’s valid perspective, recognizing the potential advantages of such an approach.
The conflict over formalism and intuitionism further complicated Hilbert’s endeavors. While Hilbert championed formalism, viewing mathematics as symbol manipulation, intuitionist L.E.J. Brouwer critiqued this stance, arguing that mathematics is a product of human thought and should involve tangible construction of objects.
This clash of philosophies engaged the law of excluded middle—asserting that every proposition must be either true or false. For instance, if it’s stated that “Hilbert is a cat,” the proposition is either true or false (in this case, false).
Human Mathematician David Hilbert
From Ulstein Newspaper, Getty Images
Hilbert’s 1888 proof claimed, “not all invariants can be generated by a finite generating set,” which he refuted by demonstrating a contradiction, affirming that every invariant must indeed be generated by a finite set. However, Brouwer contested the application of the law of excluded middle in infinite sets, acknowledging its validity for finite ones since all elements can be checked for specific properties.
Hilbert mocked this viewpoint, equating the constraints of the law to “banning a boxer from using his fists.” In stark contrast, Brouwer labeled Hilbert as “my enemy.” Compounding their rivalry was their involvement in Mathematics Annalen, a leading mathematical journal. Hilbert’s influence led to Brouwer’s expulsion from the editorial board in 1928, prompting Albert Einstein’s resignation, who questioned the absurdity of their conflict.
From a pragmatic standpoint, Einstein’s perspective held weight. Today, many mathematicians utilize non-constructive proofs as valuable methodologies, suggesting Hilbert’s theory prevailed. Conversely, Brouwer’s relevance waned as he became increasingly isolated after leaving his editorial role in Mathematics Annalen. Nonetheless, Hilbert’s formalism faced a severe setback from Kurt Gödel, whose incompleteness theorem revealed that symbol manipulation games lack complete consistency. Notably, Gödel’s work, while not aligned with Brouwer’s intuitionism, drew inspiration from it in his challenge against Hilbert’s views.
The lasting influence of Brouwer and Gödel has permeated contemporary computer science, impacting figures like Alan Turing and addressing the computability of problems. As mathematicians increasingly explore AI and formal proof verification, the dialogue around unconstructive proofs remains relevant. Machines may generate proofs in ways that transcend human understanding, possibly leading to scenarios where Brouwer may ultimately find vindication.
Topic:
Source: www.newscientist.com











