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Physics LibreTexts

1.15 Mathematics

The language of science is mathematics, a rigorous and deductive system for replacing concrete objects by numbers and symbols that represent them. Most of the physical principles in astronomy can be expressed simply in mathematics. Whenever you see such a principle, it can be expressed in words, or abstractly, in symbols. This ability to relate a mental image or a physical symbol to something concrete is at the root of mathematics.

Random mathematical formula illustrating the field of pure mathematics. This is an example of the symbols and abstract thought that we use to try and understand the universe. Click here for original source URL.

Mathematics is the wondrous system developed on Earth — we believe uniquely — by humans that makes full use of symbols and abstract thought. Until a baby is a couple of months old, an object like a toy or a rattle that is removed from direct view is forgotten. Each time it reappears it is new to the baby's world. Then, as the baby's brain grows, something remarkable happens. Babies become able to hold the idea of an object in their heads, so that the abstract idea of it is retained even after the object is removed from view. In that simple transition, the infant's universe expands immeasurably. A practical example of abstraction is money. Money used to be made of gold or silver, where its value was based on the scarcity of the metal or the difficulty in extracting it from the ground. In the United States, the dollar bill has become an abstract symbol for a certain amount of value or labor, backed only by the confidence that we all agree to use it and that the federal government will honor it. If you seek an example of something abstract, look no further than your pocket.

?Logarithmic spiral (dashed blue curve) plotted on top of this shell. For this spiral, the radius grows by a factor of 1.1923 per 1 radian of angle. It is not an exact fit to the Nautilus, being off if different directions in different parts, but is a good approximate fit. Click here for original source URL.

Abstract reasoning isn't unique to humans; parrots can count small numbers of objects just like people do. The ability to assign words and symbols to numbers can also be learned by several different animals. What makes us special is our origination of written numbers and the derivation of the rules of math (up through calculus) that can describe everything from the ebb and flow of water to the complex geometry of sea shells. It is in the application of mathematics to the natural world that science found its origins. Humans make mental models of the world they live in all the time, but only scientists create mental models based on mathematics.

Section of a woodcut showing Pythagoras with bells in Pythagorean tuning. From Theorica musicae by Franchino Gaffurio, 1492. Click here for original source URL.

The practice of the scientific method raises a serious philosophical issue. Cultures dating back to the ancient Greeks have proposed that the universe is ruled by numerical relations. The Greek word cosmos means more than just the universe and everything in it. It also means order and harmony, reflecting the ancient Greek ideal of a universe governed by understandable laws. For example, Pythagoras made the amazing discovery that to sound a musical note on a vibrating string that is just one octave below another note, you need a string exactly twice as long! The human brain can hear that these two notes are somehow related; but it took Greek science to discover that the relation was a precise numerical relation in the lengths of the string, not 2.7 or 1.94 or some other number, but precisely 2. In the past hundred years Einstein and other great thinkers have posed the question: why is the universe so well described by mathematics? Nobody knows why - perhaps is part of the fundamental design of the universe - but the application of mathematics to the natural world gives science its great explanatory and predictive power.'