This is called experimentation.

## What Is Quantum Mechanics Good for? - Scientific American

When assumptions become universally accepted as an established rule, they are called axioms and sometimes universal laws. The most ancient written records from every culture show forms of measurement and basic arithmetic. Mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events.

The history of mathematics ranges from Babylonian mathematics with its sexagesimal base 60 system dating back 4, years, to Euclidean Geometry which dates to B. The development of physics has always been intertwined with mathematics and measurements.

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Mathematics and measurement are the basic testing ground for determining the usefulness of a theory. Mathematics can be broadly subdivided into the study of quantity, structure, space, and change i. Each of these broad fields of mathematics has branched out into more abstract and complex forms due to the nature of the complex systems they must explain. Mathematical shorthand has been invented to create simple notations to denote not only simple values, but entire formulas, systems, or algorithms.

The test of theory in theoretical physics relies strongly upon mathematics. The necessary prerequisite mathematics include algebra, geometry, algebra 2, trigonometry, calculus single variable and multivariable , analytic geometry, linear algebra, ordinary differential equations, partial differential equations, complex numbers, the complex plane and probability theory.

Today much of theoretical physics relies on vector calculus. However, the mathematics itself is meaningless without the measurements that it represents, therefore, the international system of units must be understood as well which involve such measurements as the kilogram, second, metre, ampere, Kelvin, and mole.

### General Physics (so even mathematicians can understand it!)

In theoretical physics, if the mathematics of a theory is found to be inaccurate, the theory collapses. Therefore, the importance of mathematical rigor in theoretical physics cannot be overstated. The Greeks also had theories on light, its properties and its speed. They thought light did not have movement or speed, because if it did, the speed would be too great to contemplate. They concluded that light must simply either be there or not be there. Aristotle, the famous Greek natural philosopher, said that " It is certainly not a body, Empedocles and with him all others who used the same forms of expression was wrong in speaking of light as 'traveling' or being at a given moment between the earth and its envelope, its movement being unobservable by us; that view is contrary both to the clear evidence of argument and to the observed facts; if the distance traversed were short, the movement might have been unobservable, but where the distance is from extreme East to extreme West [i.

Note: comments in brackets added for clarification. This indicates that the Greeks believed the speed of light to be infinite i. This idea was not overturned even by Galileo who attempted to measure the speed of light in the 17th century and it continued to hold in Newton's day. Euclidean geometry mentioned earlier as a form of mathematics originating with the Greeks in about B.

It can be described by the Cartesian system of coordinates.

## The Quantum Physicists: And an Introduction to Their Physics

In Euclid's Elements , Euclidean space is "flat". This flatness is obvious from Euclid's definition 23 which states: 'Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. Classical mechanics is generally classified as preth century mechanics. Aristotle's views of motion held sway for 2, years until an Italian scientist, Galileo Galilei developed such ingenious and conclusive experimentation that it not only began the destruction of Aristotelian physics but demonstrated the absolute necessity of experimentation in science.

Galileo studied the physics of motion by performing experiments in which he was able to calculate the speed and acceleration of falling bodies. He developed new theories to explain motion based upon his experiments. Galileo made advancements in theoretical physics by postulating a theory of inertia contrary to any previous assumptions. Galileo asserted his theory of motion which is stated as his Principle of Inertia: "A body moving on a level surface will continue in the same direction at constant speed unless disturbed. Isaac Newton capitalized upon Galileo's observations and created theories based upon this notion that objects tend to maintain momentum.

Newton was an English scientist who systemized and generalized the assumptions made by Galileo, extended the predictions of the theory, and put them in a way that could be described by mathematical formulae that could be tested. A direct translation from the Latin of Newton's first law of motion with Newton's own explanation is:. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air.

The greater bodies of the planets and comets, meeting with less resistance in more free spaces, continue ever their motions both progressive and circular for a much longer time. Newton begins his Laws of Motion with Galileo's idea of inertia. Inertia in Newton's first law of motion is a concept and an assumption and therefore cannot be proven by experimentation or mathematics.

## The physicists’ library

Inertia has two meanings today, and later, under the heading "inertia", the term inertia will be considered in its second definition which is measurable. But the idea of "inertia" that is present in Newton's first law is an imagined concept of Galileo that once you get something into momentum, it stays that way until a force affects it.

In Newton's first law, rest is the state of zero momentum or zero movement, so that inertia can be simply stated as the tendency to maintain movement. The idea comes before the law and creates the law.

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Without the idea of inertia there are no Newtonian laws of motion. Working from Galileo's principle, Newton's reasoning was that Aristotle postulated continual circular movement in the heavens and that laws on earth were different for Aristotle because Aristotle believed the earth was at rest, a fixed motionless reference frame, and at the center of the universe.

Newton knew the earth was not at rest since by Newton's time the Copernican system with the sun at the center was the generally accepted theory. Therefore Newton realized that the same laws in the heavenly bodies had to apply for earthly bodies as well. Therefore the same maintenance of movement, inertia, that had been postulated to exist in celestial bodies had to exist on earth.

It was this reasoning that made Newton believe that something on earth was making things come to rest. What Galileo called a disturbance, Newton called a force. Inertia was Newton's way of describing the "default state" of matter that we now consider a law of the universe. Aristotle had described the default state of earthly objects as "rest" natural place. Newton contradicted or enlarged on the Aristotle default state and said that to the contrary the default state is "inertia" and upon that seminal idea built the laws of motion. Without the "idea", the mental abstraction of "inertia", Newton could have never formulated the Laws of Motion.

Newton could then form his first law extrapolate how much force it would take to change the momentum of an object. Newton therefore stated his Second Law of Motion thus:. And this motion being always directed the same way with the generating force , if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

Newton here is basically saying that the change in the momentum of an object is proportional to the amount of force exerted upon the object.

He also states that the change in direction of momentum is determined by the angle from which the force is applied. Interestly too, Newton is restating in his further explanation another prior idea of Galileo being what we call today the Galilean transformation or the addition of velocities. An interesting fact when studying Newton's Laws of Motion from the Principia is that Newton himself does not explicitly write formulae for his laws which was common in scientific writings of that time period.

In fact, it is today commonly added when stating Newton's second law that Newton has said, "and proportional to the mass of the object. In fact, the idea of mass is not introduced until the third law. This is actually a combination of laws two and three of Newton expressed in a very useful form.

This formula in this form did not even begin to be used until the 18th century after Newton's death, but it is implicit in his laws. Newton's Third Law of Motion states: "LAW III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse if I may so say will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other.

If a body impinge upon another, and by its force change the motion of the other, that body also because of the equality of the mutual pressure will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies.