Mathematics Supports Science
Formula of James Clerck Maxwell derived from the research of Faraday (Source: Campbell, N. What Is Science?.1921)
Mathematics manipulates the data of science without adulteration
Test of mathematics
Validity is the test of mathematics (Russell, B. Human Knowledge: Its Scope and Limits. 1947). The bare bones of mathematics are concepts and rules that are not contradictory.
We can start with one set of concepts and non-contradictory rules. Take the game of chess. It consists of concepts (king, queen, bishop, horse, rook, and pawn). The rules on how pieces and pawns move correspond to relationships or maxims in the game.
Maybe it will be better for this discussion to start with some familiarity with chess. Better yet with an ability to play it. We are not after proficiency or expertise as to be a national master, or international master, or grandmaster, which are ranks in mastery of the game.
We focus on the concepts and rules, absence of contradiction among rules that result in validity.
Now, as to the question: are the concepts like pieces and pawns real? That is, do they have extension? Do they represent some physical entity? They are derived. For example, the king was derived from kings; the queen was derived from queens; the rook was derived from towers in a castle; the bishops were derived from bishops of the religious; the horse was derived from horses; the pawn was derived from soldiers. We say 'derived' not exact representation of physical reality. We are not talking of physical reality that these pieces and pawns are made of wood or plastic or marble.The chess king does not represent a king of Thailand who now reigns over that country.
Do the rules represent relationships among physical entities? They don’t. The rules are relationships, or maxims that are pure inventions. They are a priori, according to Emmanuel Kant, a philosopher.
When the pieces and pawns are moved according to rules, the game of chess is alright. A game of chess is amenable to arbitration and a win or a draw is enforceable even by laws or statutes that is why we have world chess championships (Anand of India is the reigning world champion). The game is logical that is amenable to mathematical calculation. One move (n) has a value of n3. That is, one move has a ramification of 3 that makes for variations, difficulty and attractiveness of the game. To calculate three moves ahead is mind boggling: 33 = 27 or 27 variations.
Chess as a game passes the test of validity because the concepts are well defined and maxims are not contradicting each other and are consistent. A move in chess is valid; a chess game is valid; a draw is valid; a win is valid. That is like mathematics.
Mathematics is more than arithmetic. It starts with logic, according to Bertrand Russell and Alfred North Whitehead in their book Principia Mathematica. North Whitehead and Russell were philosophers and mathematicians; Russell was a Nobel Prize winner in Literature (Philosophy).
For example: "One is a class with a solitary member," “Two is a class of couples;” “three is a class of triplets.” “Zero is a class without a member.” This last sentence shows that mathematics has no physical reality but an invention. The reason is that there are more classes than there are things, according to Russell. There is ‘zero’ but no such a reality exists, in layman’s language.
It is easy to see that these concepts are inventions and maxims are inventions. However, they pass the test of validity. That is, no contradiction is found. One feature that they possess is that no matter how you group them their identities as defined and their relationships as defined do not change. Two times two will always be equal to four. Two plus two will always be equal to four.
Mathematics helps science
Now a commonsense question arises: how does mathematics help science?
Quantities of physical entities can be defined into mathematics, like arithmetic that is used in counting (2, a couple of pupils). Relationships of physical entities or psychological phenomenon can be defined into mathematical formulas such as c2 (speed of light squared). Such definitions are fixed as to be amenable to symbols (a, alpha) or terms (energy). Then these symbols or quantities can be manipulated into relationships such that the physical realities they represent are unadulterated. This unadulterated manipulation is done by mathematics. Thus physical realities or psychological phenomena are translated into mathematics.
For example: Suppose you are asked how much you like roses. You start on a red rose. You place your attitude in a scale of five: I dislike it a lot = 1; I dislike it a bit = 2; I neither like it nor dislike it = 3; I like it a bit = 4; I like it a lot = 5. You check the number that corresponds to how much you like the rose. Suppose you check 5. Therefore, you like the rose a lot. Now you have translated into mathematics a psychological phenomenon. You may do the same for a white rose; for a purple rose; for a green rose. Add the scores you placed for each color then get the average. When you add the scores for each color and take the average, you make unadulterated manipulations in mathematics. The above procedure is a method in semantic differential used in poll surveys, a fast one in political campaigns to get the attitudes of voters to a speech, to an issue, to a candidate. It is being used in: "How do you like Obama;" in "How do you like Romney;"
Physical realities and relationships with which science deals can be translated into symbols or formulas. This was started by James Clerck Maxwell, a British physicist and mathematician. He played around with mathematical inventions or constructs (please see Maxwell's formula at the start of Hub):
Symbols on the left (i = dy = dB; dz dx; j = dα= dy; dz dx; k = dB= dα; dx dy) indicate electrical laws, Ampere’s law and Faraday’s law (Campbell, N. What is Science? 1921:154). Maxwell assigned mathematical symbols to Faraday’s physical ideas. That is to say, these symbols indicate some facts in the world. Faraday “... was the first to produce an electric current from a magnetic field, invented the first electric motor and dynamo, demonstrated the relation between electricity and chemical bonding....” (Encyclopedia Britannica 2008).
“The symbols i, j, k represent in those laws an electric current” (Campbell, pages 155-156). The substitution of DX dY dZ for these dt dt dt symbols, defines a fact about the world: electromagnetic field.
"An electric current produced in one place is duplicated in another place through space."
This was a hypothesis or scientific guess made by Maxwell in 1870 that arose from his substitution. But Maxwell did not have the device to verify it. That device, coherer, was invented by Heinrich Hertz.
“Maxwell's theory suggested that electromagnetic waves could be generated in a laboratory, a possibility first demonstrated by Heinrich Hertz in 1887, eight years after Maxwell's death. The resulting radio industry with its many applications thus has its origin in Maxwell's publications” (Encyclopedia Britannica 2008).
A valid mathematical formula suggested a physical reality that is electromagnetic waves. These waves could be propagated through space. It was never thought that such propagation would lead to the abandonment of ether as carrier of light which is also a form of electromagnetic wave. Such propagation does not involve a carrier as Einstein later on saw.
The existence of physical entities (particles and waves) and relationships (propagation) was rendered probable because the symbols of concepts and relationships were based on physical reality. The manipulation of such concepts and relationships pointed to the existence of concepts and relationships hitherto unknown.
We now know that electromagnetic waves in the electromagnetic spectrum consist of (1) galactic cosmic waves, (2) gamma rays, (3) X-rays, (4) ultraviolet light, (5) visible light, (6) infrared light, (7) microwaves and radar, (8) TV and FM radio, (9) shortwave radio, (10) AM radio, (11) aircraft and shipping bands (Goldsmith, M., Dr. Guglielmo Marconi. 2003:5). These are arranged from the highest energy to the lowest.
Guglielmo Marconi took over Hertz and invented the radio, radar, precursors of cellphones, microwave oven and more.
Einstein showed another way how mathematics helps science. His special theory of relativity and generalized theory of relativity were couched in mathematics. He used Lorentz transformation and the speed of light as fundamental concepts. The speed of light is a constant, that is, it was measured by Michelson and Morley, refined further and is now unchanging no matter what relationship you place it into. That is, as if transformed into a concept in mathematics. Even the Lorentz transformation was a mathematical formula invented by Hendrik Antoon Lorentz. Einstein used it as an assumption.
Einstein’s formulation of the special theory in the language of mathematics was valid.
Again, Einstein’s generalized hypothesis of relativity was couched in mathematical language published in 1915. To pave the way for its verification, Einstein derived several statements of fact from it. Some such statements were: “the universe is expanding;” “light bends when it passes in the vicinity of great body of mass.” The second statement was verified by Sir Arthur Eddington in 1929; the first was verified by Edwin Hubble. These graduated the general hypothesis of relativity into the general theory of relativity.
The special theory of relativity deals with two coordinating systems in uniform motion; the general theory of relativity deals with two coordinating systems in relative motion. The former does not involve gravity; the latter involves gravity.
Newton invented his calculus, used it together with Euclidean geometry to come up with his laws of motion. (Leibniz also invented calculus independently of Newton). These were inadequate for use in the problems that confronted Einstein. Einstein adopted the Riemannian geometry, refined by Bernard Riemann, that is non-Euclidian. The Riemannian geometry enabled Einstein to include the concept of time as a dimension to the Euclidian dimensions and came up with a theory of gravity. Gravity is a curvature of space-time in the general theory of relativity. It is not a force. In Newton’s physical theory, gravity is a force.
So, that’s how mathematics helps science.
Some scientists take the relationship between mathematics and science for granted. They make enormous amounts of computations with computers confident that they will come up with results unadulterated by the mathematics. That confidence is guaranteed.
Mathematics manipulates (calculation) classes not properties of things. That makes for convenience. It also ensures that the properties are not changed by the manipulation.
Mathematics manipulates relationship among classes not relationships of molecules. This makes for convenience especially in statistical or small and numerous objects like electrons. For example, orbital that is the path of several electrons around the nucleus. In oxygen, the first orbital is occupied by two electrons, the second orbital is occupied by four paired electrons and two unpaired electrons. You cannot, or it is extremely difficult to follow each of the two electrons in the first orbital, It is not necessary to follow each, if that is not your specific purpose. The relationship can be derived from experiment that in 95% you can catch one electron in the first orbital. But you don't have to say that the electron is comprised of muons or quarks. That is the field of nuclear physics. If we cannot group electrons, or census into classes, doing science would be cumbersome and confusing. Making groups or classes is the business of mathematics.
You may not meet the same difficulty if you were dealing with massive determinate objects like moons, planets, and comets. Tycho Brahe and Johannes Kepler followed and plotted the orbits of planets; they did not meet inconvenience because there are only a few planets.
Relationships in mathematics still have to be interpreted. The test of mathematics is validity. The test of a statement is truth. We make a generalization: the test of science is truth. In common language the language of mathematics must be interpreted in the language of science. For example, in a census of cancer victims who got well from taking noni juice. Suppose out of 100 who took noni juice, 95 got well of cancer.
Mathematics shows that 95% got well from taking noni juice. Now we make the interpretation: noni juice treats cancer. Such simple interpretation is sometimes taken for granted. But once the mathematical formulas become more complicated, there is a need to master the mathematics and their interpretations in reality.