Advanced Technology Replacing
Mathematicians?
How Modernized Technology Helps to Solve Millennium Issues
Written by: Tommy Li | Edited by: Mariano Frare | Photo by MART PRODUCTION
Many current groundbreaking technological advancements were all based on the foundational subject of mathematics. How do these stunning inventions help out mathematics in return?
As the advancement in computational modeling is growing rapidly, it broadens our imagination and turns what was previously considered impossible into reality. Many significant breakthroughs in mathematical proofs occur with the help of computer computations, producing models that are otherwise impossible for human beings to create.
One of the most prominent mathematical proofs in recent decades is Fermat’s Last Theorem, which was proven by a British mathematician named Andrew Wiles in 1993. Before 1993, Fermat’s Last Theorem puzzled human civilization for more than 300 years and was labeled an unsolvable problem. This theorem states that xn+yn=zn, where x, y, and z cannot be any positive whole number if n is greater than 2. Many mathematicians took different paths to prove this relatively straightforward equation, but all failed. However, Andrew Wiles took a different direction in his proof, utilizing the Taniyama-Shimura Conjecture. This theorem connects two unrelated mathematical terms, the elliptic curve and modular forms. An elliptic curve is a donut shape function, as this image suggests.
A modular form is a highly symmetrical function that takes in a complex number with a fundamental part and outputs another complex number. Based on the work of previous mathematicians, if the Taniyama-Shimura Conjecture were accurate, Fermat’s Theorem must also be true. With the help of computer computations that verified his calculation, Wiles could derive concrete proof that shocked the whole world. As Wiles said in his documentary: “This is a 20th-century proof, and Fermat would never be able to derive it during his time.” Technology, particularly computer computations, played a crucial role in Wiles' proof, helping him check and visualize many of his processes and making the final result accessible.
Besides assisting with the proof, the newly developed computers completely changed the verification process; their incredible computational ability allowed the scientists to find some counterexamples to some conjectures. One famous example is Euler’s conjecture, a conjecture derived from Fermat’s Last Theorem, which states:
Unfortunately, while this equation seemed straightforward, it was disproved under repetitive computer verification that checks every possible number. For example, one counterexample under the condition when k=4 was found, which is: 2682440^4 + 15365639^4 + 18796760^4 = 20615673^4. This counterexample, found with the help of the computer, was a significant breakthrough in mathematical research. It demonstrated the power of technology in dealing with repetitive computation problems that would only be possible for humans to accomplish solely due to the large numbers involved.
As technology continues to grow, mathematicians have an expanding array of tools to tackle seemingly impossible puzzles and eliminate much redundant work for mathematicians. While some may doubt these machines, many mathematicians embrace their potential, integrating them into their work to handle the more mundane calculations. Yet, these machines are far from replacing or becoming the major contributor to solving these millennium puzzles as technologies are incapable of creating generalized proofs and can only verify the proof numerically. By combining the wisdom and creativity of human beings, along with computational skills, we would soon be able to conquer many impossible obstacles.
These articles are not intended to serve as medical advice. If you have specific medical concerns, please reach out to your provider.