Utility of mathematics
The utility of mathematics encompasses its practical applications across various fields, as well as philosophical debates regarding its nature and origins. Mathematics is often viewed as both a construct of the human mind and a discovery of universal truths. This duality raises questions about whether mathematics serves purely as an intellectual pursuit or has inherent utility in understanding the world. Historical perspectives, such as those from ancient philosophers like Plato, have explored these themes, suggesting that mathematics can be appreciated for its beauty in addition to its practical use.
Over time, mathematics has been applied to diverse areas such as physics, architecture, and computer science, often leading to groundbreaking advancements from theoretical concepts that initially seemed abstract. Notable examples include the applications of algebraic topology and number theory in modern technology and science. The ongoing discussion about how best to teach mathematics—balancing pure knowledge with its practical relevance—reflects a long-standing inquiry into its purpose. This inquiry highlights a fundamental tension: while some mathematicians celebrate the beauty of mathematics for its own sake, others emphasize its critical role in solving real-world problems and advancing human knowledge.
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Utility of mathematics
SUMMARY: Although performing a mathematics operation does not necessarily require utility as an outcome, there are many examples of applications in various fields.
To discuss the utility of mathematics, there must be some agreement on the definition of the term “mathematics.” Many may agree that mathematics is a pure creation of the human mind. It is a body of knowledge at which one arrives by pure reason and does not rely upon any observations of the phenomenal world. This characteristic makes it free from the limitations imposed by the particular way that human minds create experience from their understanding of the underlying phenomena. The argument comes down to the following: is mathematics the complete construction of the human mind or is it universally inherent, only being discovered/uncovered by mathematicians? Many books have been written to discuss this question, and no decision has been (or will be) made on one side or the other. Ancient philosophers such as Plato put forth much of this contemplation. Mathematical platonism suggests that humans did not develop or invent mathematics but rather discovered it.
There are examples of people in the mathematics community, such as G. H. Hardy in A Mathematician’s Apology, who see a difference between pure and applied mathematics based solely on utility and revel in the fact that nothing that they will do will be useful to humanity. This statement was in part a response to the work of Andrew Littlewood and a group of mathematicians who worked strenuously for the British War Department during World War I.
There are ample examples in the historical record of mathematics, done for its own sake, that were later discovered to be applicable to real-world problems. The theory of tensors by Giovanni Ricci-Curbastro and Tullio Levi-Civita proved to be a cornerstone for Albert Einstein’s work on relativity. The purely algebraic area of twistor theory in physics, which predicted the existence of certain subatomic particles in the 1980s, started in the area of finite algebraic geometry. Areas that in the 1980s and 1990s were considered pure mathematics now find themselves at the forefront of application: algebraic topology used to study distribution of sensors; hyperbolic geometry used to study the extent and reach of the Internet; number theory used in architecture and cryptography; and category theory used in studying social behavior. Where will the applicable mathematics of the mid-twenty-first century come from?
What sort of mathematics should be taught in schools? This question dates back to at least the early 1800s in the United States and is discussed in the work of Charles Davies. E. R. Hedrick again raised the same question in his address to the New York Section of the Mathematical Association of America in 1933. The question arises with each new generation. Should only the mathematics that is currently known to be applicable be taught to those who will only use the tool, or should they be exposed to the whole of mathematics?
The purpose of asking this question lies in the purpose of mathematics. Should mathematics, or is mathematics, done only for its own sake? If that is the case, then why has mathematics been so useful to science? This is the question raised by Eugene Wigner in his 1960 work The Unreasonable Effectiveness of Mathematics in the Natural Sciences. The question and his answer have again brought to the fore this long standing argument.
Historical Context
From the earliest recordings in Babylonian and Egyptian mathematics, historians and archaeologists found books with mathematical exercises. These were created to train mathematical neophytes—possibly young priests—in the algorithms that were used in disciplines such as building and surveying. The excercises were not all applied problems but included examples of mathematics being done for its own sake. This was not the norm, however. Most mathematics of these earlier eras seemed to have been for inherently practical purposes.
From the Western perspective, it was the Greeks under the Pythagorean School, which took the idea of mathematics and made it deified. The Platonic school did not avoid the question of the “utility of mathematics.” There is a quote, ascribed to Euclid in Stobaeus’ Extracts “A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, inquired, ‘What do I get by learning these things?’ So Euclid called a slave and said ‘Give him three pence, since he must make a gain out of what he learns.’” Already the teacher has to answer the long-asked question, “What is this good for?” The Platonic school may have been one of the first in which mathematics was studied for its own beauty and internal structure—not being required to have any other purpose. Archimedes saw the utility of mathematics; whether he held the same philosophical beliefs as did the Platonists, we cannot be certain.
The Romans were extremely interested in the utility of mathematics to warfare, navigation, and architecture. It was the Greeks and the Alexandrians, though, that kept mathematics moving forward until it was rescued from the fate of much of the ancient world’s science by the Islamic mathematicians. Not only did they need mathematics for navigation and geometry, but they also imbued into the geometry the need to glorify Allah with the perfectness of the geometric form.
In the Renaissance in the late thirteenth century, the early scientist, Roger Bacon, made statements about the utility of mathematics, “Mathematics is the door and key to the sciences,” and “. . . mathematics is absolutely necessary and useful to the other sciences.”
Perhaps the best summary can be found in various quotes:
The Universe is a grand book which cannot be read until one first learns to comprehend the language and become familiar with the characters in which it is composed. It is written in the language of mathematics.…
— Galileo Galilei (1564–1642)
Mathematics is a game played according to certain rules with meaningless marks on paper.
— David Hilbert (1862–1943)
(Cantor’s work on set theory)… the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.”
— David Hilbert (1862–1943)
From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.
— James Hopwood Jeans (1877–1946)
I have never done anything “useful.” No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.
— G. H. Hardy (1877–1947)
…enigma that researchers of all times have worried so much about. How is it possible that mathematics, a product of human thinking independent of any experience, so excellently fits the objects of physical reality?
— Albert Einstein (1879–1955)
As far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, the do not refer to reality.
— Albert Einstein (1879–1955)
The unreasonable effectiveness of mathematics in the natural sciences… that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.
— Eugene Wigner (1902–1995)
…[enumerating cases where structures needed in physics have already been found and developed by mathematicians]… long before any thought of physical application arose. It is positively spooky how the physicist finds the mathematician has been there before him or her.
— Steven Weinberg (1933–)
This universality of application [of mathematics] can be traced back to the fact that all aspects of Nature and areas of life are governed by the same principles of order and intelligence that have been discovered subjectively by mathematicians by referring back to the principles of intelligence in their own consciousness.
— Maharishi Mahesh Yogi (1914–2008)
Was it not the Pisan scientist who maintained that God wrote the book of nature in the language of mathematics? Yet the human mind invented mathematics in order to understand creation; but if nature is really structured with a mathematical language and mathematics invented by man can manage to understand it, this demonstrates something extraordinary.
— Benedict XVI (1927–)
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