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opticaltempest
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Is mathematics discovered or created?
HallsofIvy said:Mathematical theorems are created when we choose the axioms for the mathematical system. Of course, what statements are theorems (are provable in that system) is not immediately obvious ("emergent properties" is a good phrase to use here). We "discover" the theorems when we prove them.
opticaltempest said:Is mathematics discovered or created?
That reminds me of a quote I liked.selfAdjoint said:We seem to "find" the theorems in our heads as potential truths and then prove them by creating chains of logically interrelated statements.
(Also spelled Carl Kerenyi.)Karl Kerenyi began his 1976 English language translation of Dionysus with this passage:
"The interdependence of thought and speech makes it clear that languages are not so much a means of expressing truth that has already been established as means of discovering truth that was previously unknown. Their diversity is a diversity not of sounds and signs but of ways of looking at the world."
(http://en.wikipedia.org/wiki/Sapir-Whorf_hypothesis)
There is a debate among mathematicians and philosophers about whether mathematics is a human invention or a discovery of universal truths. Some argue that mathematical concepts are created by humans to describe and understand the world, while others argue that mathematics exists independently of human thought and is discovered through observation and reasoning.
One piece of evidence for the discovery view of mathematics is the existence of mathematical concepts and principles that seem to be universal and unchanging, such as the laws of geometry and arithmetic. These concepts are not dependent on human language or culture, suggesting that they exist independently of human invention.
The concept of mathematical objects, such as numbers, shapes, and equations, is often used to support the idea that mathematics is discovered. These objects are seen as existing independently of human thought, and our understanding of them is a result of discovering their properties and relationships through observation and reasoning.
Some argue that mathematics is both discovered and created. They believe that while the fundamental concepts and principles of mathematics are discovered, humans have the ability to create new mathematical ideas and structures through our imagination and creativity.
The debate about whether mathematics is discovered or created has a significant impact on the study and practice of mathematics. It influences how mathematicians approach their work and the types of questions they seek to answer. It also affects how mathematics is taught, with some educators focusing on the discovery aspect and others emphasizing the creative aspect of mathematics.