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Mathematics is the language of physics
Natural philosophy [i.e., physics] is written in this grand book – I mean the universe – which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. [The universe] cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
- Galileo, Opere Il Saggiatore p. 171.
Mathematics is the language of physics. Physical principles and laws, which would take two or even three pages to write in words, can be expressed in a single line using mathematical equations. Such equations, in turn, make physical laws more transparent, interpretation of physical laws easier, and further predictions based on the laws straightforward.
- Mesfin Woldeyohannes, Assistant Professor, Western Carolina University
ἀεὶ ὁ θεὸς γεωμετρεῖ – Aei ho theos geōmetreî. God always geometrizes.
- Plato, 400 BCE, classical Greece, as quoted by Plutarch in his The Moralia, Quaestiones convivales. (circa 100 CE)
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
Wigner begins his paper with the belief, common among those familiar with mathematics, that mathematical concepts have applicability far beyond the context in which they were originally developed. Based on his experience, he says “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.”
He then invokes the fundamental law of gravitation as an example. Originally used to model freely falling bodies on the surface of the earth, this law was extended on the basis of what Wigner terms “very scanty observations” to describe the motion of the planets, where it “has proved accurate beyond all reasonable expectations”.
Another oft-cited example is Maxwell’s equations, derived to model the elementary electrical and magnetic phenomena known as of the mid 19th century. These equations also describe radio waves, discovered by David Edward Hughes in 1879, around the time of James Clerk Maxwell’s death. Wigner sums up his argument by saying 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”. He concludes his paper with the same question with which he began:
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
- The Unreasonable Effectiveness of Mathematics in the Natural Sciences. (2016, September 11). In Wikipedia, The Free Encyclopedia
Math is different from physics
Mathematics does not need to bother itself with real-world observations. It exists independently of any and all real-world measurements. It exists in a mental space of axioms, operators and rules.
Physics depends on real-world observations. Any physics theory could be overturned by a real-world measurement.
None of maths can be overturned by a real-world measurement. None of geometry can be.
Physics starts from what could be described as a romantic or optimistic notion: that the universe can be usefully described in mathematical terms; and that humans have the mental ability to assemble, and even interpret, that mathematical description.
Maths need not concern itself with how the universe actually works. Perhaps there are no real numbers, one might think it is likely that there is only a countable number of possible measurements in this universe, and nothing can form a perfect triangle or point.
Maths, including geometry, is a perfect abstraction that need bear no relation to the universe as it is.
Physics, to have any meaning, must bear some sort of correspondence to the universe as it is.
Why-is-geometry-mathematics-and-not-physics? Physics StackExchange, by EnergyNumbers