Wrong Premises That Lead to Correct Answers: The Role and Importance of Paradox and Falsification
The expression, 'There are two types of people in the world: those who can understand my writing and those who cannot,' may seem easy to understand at first glance, but when considered carefully, it is extremely limited and oversimplifies things. This kind of 'dichotomous rhetoric' is used in many situations as a technique to guide arguments.
However, there are also problems with this type of expression. This is because the background—such as what 'understanding' refers to and how its criteria are determined—is ambiguous. In reality, there are 'people who understand completely,' 'people who hardly understand at all,' and everyone in between. In other words, this dichotomy can obscure the essence of the problem and lead to misunderstandings.
Arguments that use such 'wrong premises' can be countered with techniques used in philosophy, logic, and mathematics. These are 'paradox' and 'falsification'.
Paradox: A technique to elicit new perspectives
A paradox is a technique that presents a proposition or situation that seems contradictory at first glance, and by examining it, leads to the realization of new truths or principles. Because many paradoxes are counterintuitive, they have the characteristic of making one think deeply.
Examples of paradoxes
Russell's Paradox:An example in set theory where the concept of a 'set that does not contain itself' leads to a contradiction.
Cantor's Theorem:A paradoxical conclusion regarding the cardinality of infinite sets. These paradoxes shook the foundations of philosophy and mathematics at the time, but as a result, new theories were developed. Similarly, in AI ethics and science, paradoxes are important tools that bring new insights.
Falsification:A technique to promote progress by revealing errors
Falsification is a technique that denies the credibility of a proposition or theory by demonstrating its errors. Through this, incorrect theories are corrected or rejected, and new knowledge is constructed.
Concrete examples of falsification
Falsifiability in science: 'Falsifiability,' proposed by Karl Popper, is a criterion for measuring the soundness of scientific theories. For example, Newtonian mechanics was found to have points that contradicted observational data, leading to its development into the theory of relativity.
Falsification in law: In court, falsification is used to prove the errors in an opponent's argument. For example, there are cases where a lawyer wins an acquittal by pointing out flaws in evidence against the prosecution's claims.
Gödel's Incompleteness Theorems and the Limits of Computational Theory
Gödel's incompleteness theorems led to the profound conclusion that 'no mathematical system can be both complete and self-proving within itself.' This theorem has important implications not only in mathematics but also in computational theory and computer science.
Applications to computer science
Formalization and the limits of computation: This theorem suggests the fundamental principle of computational theory that 'not all algorithms can solve all problems.'
Relation to the halting problem: Linked to Alan Turing's halting problem, it clarified that 'there is no algorithm that can determine whether a given program will always halt.'
Difficulty of program verification: It is also derived from this theorem that it is impossible to completely guarantee that a program will produce correct results for all inputs.
Relevance to Ghost in the Shell:Questions of completeness
Ghost in the Shell depicts a future where human consciousness and identity are digitized. The themes of this work, such as the 'complete understanding of consciousness' and the question 'The net is vast,' have something in common with the 'limits of completeness' presented by Gödel's theorem.
Commonalities
• Gödel's theorem explores completeness from a mathematical perspective. • Ghost in the Shell questions the completeness of consciousness from a philosophical and ethical perspective. Although these are questions from different fields, both share the theme of the 'limits of human understanding.'
The importance of falsification in science
In science, the falsifiability of hypotheses supports the progress of knowledge. New theories and discoveries are born through the falsification and revision of existing theories.
Concrete examples
• Einstein's theory of relativity advanced science by falsifying the limits of Newtonian mechanics. • The hypothesis of evolution is also established as scientific knowledge by having falsifiability.
Summary: The role of paradox and falsification Paradox and falsification are important techniques for eliciting new truths and knowledge from seemingly contradictory situations or incorrect premises. By using these techniques, progress is brought about in the fields of philosophy, mathematics, science, and even AI ethics.
It is important not to fear mistakes, but to have an attitude of learning from them. That is the key to deepening knowledge and opening up the future.
Rintaro Taketake