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"Wigner argues that mathematical concepts have applicability far beyond the context in which they were originally developed. He writes: "It is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena."[3] He adds that the observation "the laws of nature are written in the language of mathematics," properly made by Galileo three hundred years ago, "is now truer than ever before."Wigner's first example is the law of gravitation formulated by Isaac Newton. Originally used to model freely falling bodies on the surface of the Earth, this law was extended based on what Wigner terms "very scanty observations"[3] to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations."[4] Wigner says that "Newton ... noted that the parabola of the thrown rock's path on the earth and the circle of the moon's path in the sky are particular cases of the same mathematical object of an ellipse, and postulated the universal law of gravitation on the basis of a single, and at that time very approximate, numerical coincidence."Wigner's second example comes from quantum mechanics: Max Born "noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory. However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions." But Wolfgang Pauli found their work accurately described the hydrogen atom: "This application gave results in agreement with experience." The helium atom, with two electrons, is more complex, but "nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshita at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we 'got something out' of the equations that we did not put in." The same is true of the atomic spectra of heavier elements.Wigner's last example comes from quantum electrodynamics: "Whereas Newton's theory of gravitation still had obvious connections with experience, experience entered the formulation of matrix mechanics only in the refined or sublimated form of Heisenberg's prescriptions. The quantum theory of the Lamb shift, as conceived by Bethe and established by Schwinger, is a purely mathematical theory and the only direct contribution of experiment was to show the existence of a measurable effect. The agreement with calculation is better than one part in a thousand."There are examples beyond the ones mentioned by Wigner. Another often cited example is Maxwell's equations, derived to model the elementary electrical and magnetic phenomena known in the mid-19th century. The equations also describe radio waves, discovered by David Edward Hughes in 1879, around the time of James Clerk Maxwell's death."
A scientist named Eugene Wigner talked about something very special: how math, which is made up of numbers, shapes, and patterns, can be used to understand the real world—even in ways people didn’t expect at first.
He said that even when scientists are just guessing or using simple tools, they often find that math gives very accurate answers about how things in nature work. A long time ago, another scientist, Galileo, said that the “laws of nature are written in the language of mathematics,” and Wigner agreed—even more so today!
Later, scientists used the same math to explain more complicated atoms like helium, and they found that their answers matched real experiments—so closely that they were only off by a tiny bit, like one part in ten million! This was amazing, because the math gave them answers they didn’t expect.
In short: Wigner showed that math is like a magic tool. Even when it starts from simple ideas, it can help scientists discover amazing things about the universe—things they didn’t even plan to find!
transform this text into elementary school level
"Wigner argues that mathematical concepts have applicability far beyond the context in which they were originally developed. He writes: "It is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena."[3] He adds that the observation "the laws of nature are written in the language of mathematics," properly made by Galileo three hundred years ago, "is now truer than ever before."Wigner's first example is the law of gravitation formulated by Isaac Newton. Originally used to model freely falling bodies on the surface of the Earth, this law was extended based on what Wigner terms "very scanty observations"[3] to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations."[4] Wigner says that "Newton ... noted that the parabola of the thrown rock's path on the earth and the circle of the moon's path in the sky are particular cases of the same mathematical object of an ellipse, and postulated the universal law of gravitation on the basis of a single, and at that time very approximate, numerical coincidence."Wigner's second example comes from quantum mechanics: Max Born "noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory. However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions." But Wolfgang Pauli found their work accurately described the hydrogen atom: "This application gave results in agreement with experience." The helium atom, with two electrons, is more complex, but "nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshita at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we 'got something out' of the equations that we did not put in." The same is true of the atomic spectra of heavier elements.Wigner's last example comes from quantum electrodynamics: "Whereas Newton's theory of gravitation still had obvious connections with experience, experience entered the formulation of matrix mechanics only in the refined or sublimated form of Heisenberg's prescriptions. The quantum theory of the Lamb shift, as conceived by Bethe and established by Schwinger, is a purely mathematical theory and the only direct contribution of experiment was to show the existence of a measurable effect. The agreement with calculation is better than one part in a thousand."There are examples beyond the ones mentioned by Wigner. Another often cited example is Maxwell's equations, derived to model the elementary electrical and magnetic phenomena known in the mid-19th century. The equations also describe radio waves, discovered by David Edward Hughes in 1879, around the time of James Clerk Maxwell's death."
ChatGPT:
Here’s a version of the text rewritten for elementary school students, using simpler language and clearer examples:A scientist named Eugene Wigner talked about something very special: how math, which is made up of numbers, shapes, and patterns, can be used to understand the real world—even in ways people didn’t expect at first.
He said that even when scientists are just guessing or using simple tools, they often find that math gives very accurate answers about how things in nature work. A long time ago, another scientist, Galileo, said that the “laws of nature are written in the language of mathematics,” and Wigner agreed—even more so today!
Example 1: Gravity
Wigner’s first example is about gravity. That’s the force that pulls things down, like when an apple falls from a tree. A scientist named Isaac Newton used math to explain how gravity works on Earth. Later, he used the same math to show how the planets move in space. Newton realized that when you throw a rock and it follows a curved path, and when the moon goes around the Earth, they are actually following the same kind of shape—a part of an ellipse. He used just a little bit of information to come up with a big idea called the law of gravity, and it turned out to be right!Example 2: Tiny Particles
Wigner’s next example comes from quantum mechanics, which is about tiny particles like electrons that are too small to see. A scientist named Max Born noticed that some of the math used by another scientist, Heisenberg, was just like the math used with something called matrices (which are like special math tables). Born and his team used this math to describe how these tiny particles move. At first, they weren’t sure if it would work in the real world, but another scientist, Wolfgang Pauli, used it to explain how the hydrogen atom works—and it worked really well!Later, scientists used the same math to explain more complicated atoms like helium, and they found that their answers matched real experiments—so closely that they were only off by a tiny bit, like one part in ten million! This was amazing, because the math gave them answers they didn’t expect.
Example 3: Light and Electricity
Wigner’s last example is about something called quantum electrodynamics, which is a fancy way of saying the study of how light and electricity behave. This time, scientists mostly used math to figure things out, and only used experiments to check if they were right. Their math predicted a tiny change in energy called the “Lamb shift,” and when they tested it, the math and the experiment matched very closely—better than one part in a thousand!More Examples
There are other examples, too. For instance, James Clerk Maxwell used math to describe how electricity and magnetism work. His equations also predicted radio waves, which weren’t even discovered until after he died!In short: Wigner showed that math is like a magic tool. Even when it starts from simple ideas, it can help scientists discover amazing things about the universe—things they didn’t even plan to find!
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