The question about whether all mathematical theorems can be deduced from axioms is the cornerstone question in the current debate about whether AI can simulate human intelligence. But the original intent to base all of mathematics on a finite set of axioms began much earlier. Roughly speaking, mathematics is a collection of theorems derived by deduction from a set of basic a priori assumptions, called axioms. The sequence of deductions that lead from the axioms to the statement of a theorem is known as a proof of that theorem. So, mathematics may be regarded as a collection of proofs–links from a priori “assumptions” to inevitable conclusions.
When mathematical exploration was resuscitated in the Renaissance, the axiomatic structure of mathematics introduced by Euclid continued to be the basis of certainty and “truth” in geometry. However, the intuitive nature of the axioms that caused problems in geometry were also causing difficulties in the other branches of mathematics. Algebra and analysis, like geometry had also relied on intuitive notions that were not well defined and proofs were not rigorous in the modern sense. Newton’s development of calculus using “fluxions” had been based on intuitive notions of motion and change rather than precisely defined concepts. As mathematics began to venture into the realm of infinite quantities, convergence, and limits, the formal manipulation of symbols often led to contradictions. Mathematicians became increasingly aware of the importance of examining all axioms for hidden assumptions that might later yield contradictions. Even number theory, known as “the higher arithmetic,” came under scrutiny. In 1889, Giuseppe Peano published a set of nine axioms, precisely formulated in the language of set theory. These axioms were designed to put algebra on a firm footing by replacing all intuitive notions of whole numbers with unambiguously stated properties.
In 1899, David Hilbert revised Euclid’s axioms in a similar way, replacing intuitive notions with precisely-stated properties relating points, planes, and lines. His revision of Euclid’s axioms was part of what became known as the Hilbert program. In 1920, Hilbert proposed that a new research project be launched with a two-fold purpose:
1. To underpin all of mathematics with a finite set of axioms.
2. To develop a “metalanguage” that could be used to prove those axioms consistent.
The mathematicians who followed the Hilbert approach, subscribed to the idea that mathematics can be reduced to rules for manipulating formulas without any reference to their meaning. Members of this so-called formalist school believed that the mathematical symbols, and the inferential rules that govern their relationships constitute the totality of mathematical thought.
While Hilbert was attempting to achieve rigor by showing that all of mathematics could be deduced from a set of basic axioms and simple rules of inference, without using the concept of number or set, Gottlob Frege and Bertrand Russell were attempting to use set theoretic language and formal symbolic logic to achieve absolute rigor.
In 1931, Austrian mathematician, Kurt Gödel, at age 25 published his doctoral thesis, On Formally Undecidable Propositions of Principia Mathematica and Related Systems. In this epoch-making paper, Gödel enunciated a theorem and corollary that registered a magnitude ten on the seismometer of mathematical quakes. The programs of both the formalists and the logicists collapsed like giant skyscrapers whose foundations had crumbled under a tectonic shift. Though it took many years for the mathematics community at large to appreciate what Gödel had achieved, those who were working in the foundations of mathematics soon recognized the far-reaching implications of his powerful theorem and corollary. Gödel proved that any sufficiently comprehensive mathematical system (such as number theory) cannot prove its own consistency. Furthermore, there will be some theorems that cannot be deduced from its axioms.
The importance of Gödel’s theorem has come to the fore in recent discussions about artificial intelligence. Since a computer operates on algorithms that are based on the axioms that are hard-wired into its logic circuits, many mathematicians and computer scientists, including Roger Penrose argued that Gödel’s Theorem revealed that computers can never replicate human cognition. Others, including Douglas Hofstadter author of Gödel, Escher, Bach: An Eternal Golden Braid argued that computers could go beyond algorithmic thinking. He stated, “Human thinking in all its flexible and fallible glory can, in principle be modeled by a ‘fixed set of directives,’ provided one is liberated from the preconception that computers can do nothing but slavishly produce the truth.” He further argues that human intuition can be simulated by computer technology. The debate continues as we probe more deeply into human cognition and the processes by which we access “truth.”
