History of Formalisms Leading to Advance in Knowledge
History of Formalisms Leading to Advance in Knowledge A short history lesson on how this has occurred and the historic Role of Formalisms, e.g. Mathematics:
Bear in mind that philosophy and science are community sports. Or to put it more seriously, life forms societies to develop them. Many minds have to work together. In philosophy it requires sharing ideas. But often, shared ideas are not completely the same ideas. Thus philosophy can develop many divergent schools of thought on any given subject.
Mathematics functions quite differently. Mathematical concepts are spelled out in detail. To draw a circle will never produce a circle, and an actual circle cannot exist. This is the mystery of mathematics. Nothing in mathematics is manifest, yet it provides the most powerful way of thinking and understanding reality. The secret is that mathematics does not describe; it provides thought recipes. Everyone in the society of inquiry can now think the same way. When that happens the inquiry becomes scientific rather than philosophical.
Science is based on the discovery of organizing principles. I think not many realize that for physics mathematics had to come first. Breakthroughs in physics require new math. The following history lesson gives examples of this.
- Galileo (1564-1650) experimented with motion. He established the relationship that distance traveled equals speed multiplied by time. He also expressed the hope that everything could be reduced to calculation. Then, instead of arguing, philosophers could get out paper and pencils and calculate.
- Descartes (1596-1650) developed a coordinate system for expressing locations in space. Today it is called cartesian coordinates. (It is considered a rare honor that cartesian is spelled with a small c.) He also introduced analytic geometry relating algebra and geometry. This ultimately enabled the development of Newton’s calculus.
- Tycho Brahe (1546-1601) observed planetary motion producing a catalogue of planetary orbits.
- Newton (1643-1727) inherited all this. With Descartes’ contribution force and speed would become vector quantities. That means some numeric quantity pointing in some direction. Speed became velocity. Now the problem became relating force and velocity. Many philosophers of the time got involved, including Descartes. Obviously, the stronger the force, the faster the speed or velocity. None of them could solve the problem until Newton. He recognized that force changed the rate of change in velocity, i.e., force produced acceleration. To express his discovery Newton invented the calculus, which is a mathematics of change.
- Concomitantly, Leibniz also invented the calculus but giving it easier notation. Today we use the Leibniz notation. The equation of motion became F = d(mV)/dt where mV is momentum. (Capital letters are vectors. Small letters are scalars.) The expression d(mV)/dt is the derivative, the rate of change. This is known as a differential equation. The calculus process of integration solves it.
Having produced the equation of motion Newton turned to Tycho Brahe’s planetary orbits. The orbits have a peculiar property known as the equal area law.
To picture this, imagine that the orbit is an ellipse around the sun. Draw a line from the sun to a point on the orbit where the planet is. Next, draw a second line from the sun to where the planet will be after some time interval. The two lines and the orbit enclose an area.
Next, sample twice. First draw lines to the orbit where it is close to the sun. Then draw the lines to the orbit far from the sun. Use the same time interval for both. The two cases will enclose the same areas. This is the equal areas law. Far from the sun the planet is moving slowly. Near the sun the planet is racing along.
Newton wanted to find the force that would produce this effect. He found it as an inverse square law:
F = kMm/r2
Where M is the mass of the earth, m is the mass of the planet, and r is the distance between them, while k is just a constant. (Forget about the apple, it played no role. There is some speculation that Newton started the apple story in answering a question.)
Now in mathematics it can be proven that any orbit in an inverse square law field has to be a conic section. The orbit of Mars became a problem. Its orbit is almost an ellipse, but it is rotating in space. The Newtonian theory is false. But we use it anyway because for most practical purposes it does fine. But something else is going on! Something else has been found to be relativity.
- Now I want to move on to another example that can be stated briefly. From Euclid on, mathematicians worried about his fifth postulate, also called the parallel postulate. If two lines are parallel they will never meet even if extended to infinity. From the very beginning there seemed to be problems with the postulate. I think most practical people would think it too obvious to waste time thinking about it.
In 1840 mathematicians realized that they could test the postulate by denying it and proceeding to develop geometry. The test was to see if they would ultimately derive an inconsistency. They proceeded and derived new geometries; curved geometries. They were convinced that the real geometry of the world was Euclidean, but the curved geometries were interesting so they continued to develop them. Of course in curved geometry what is locally parallel can converge at a distance.
- Early in the 20th century Einstein was able to develop relativity because there was Riemannian geometry: a curved geometry.
- Maxwell, in 1860, reduced what was known about the interactions of electricity and magnetism to four differential equations. He noticed that they were wave equations. Could it be that there are electromagnetic waves? There was no experiential evidence but Maxwell went looking for them; and found them. There are octaves of electromagnetic waves accounting for radio waves, x-rays, cell phones, light waves and photons carrying energy from the sun to us.
- In 1928, P.A.M. Dirac made a change to Schrödinger’s quantum wave equations. The result led to the discovery of zero point energy, which fills all space. This is a very exciting area of research. It promises to change all our notions of reality in ways that are very important to understanding life itself.
- But my favorite example is the square root of –1. I am not sure when this first appeared. It was in the far distant past. It is my favorite since it results from mathematical esthetics. Given a system of algebraic equations including X2-1 = 0, the solution is obvious. X = 1 or X= -1. However change it to X2+1 = 0 and we have a problem. Now X is equal to the square root of –1. There was no known number to satisfy that condition. However, mathematicians made up a number. They called it i. This led to numbers with two parts, X + iY. X was called the real part; iY was called the imaginary part. Such numbers were called complex numbers. In the past some people objected to the teaching of imaginary numbers. It was alleged to be a waste of time. Now we know that such imaginary numbers play a very important role in some very real and practical disciplines, such as electronic circuit theory. More, this imaginary number plays a starring role in quantum equations. Quantum equations typically employ complex numbers.