Causation and Correlation in Mathematical Discovery and Scientific Progress

7/28/2019 Causation and Correlation in Mathematical Discovery and Scientific Progress

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Mathematics and Science 1

Causation and Correlation in Mathematical

Discovery and Scientific Progress

Eric M. Hodge

February 22, 2010


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Abstract

This paper examines the relationship between mathematical discoveries and progress in

other scientific disciplines. Five specific mathematical developments were examined in

their historical context as representative of breakthroughs in their fields and compared to

their effect on scientific developments. This researcher predicted that mathematical

discoveries would precede scientific progress in accordance with Dr. William Dunhams

assertion that mathematics drives science. This researcher discovered through qualitative

analysis of mathematical and scientific discoveries that the relationship between the two

cannot be described as causative but is still more complex than simple correlation. In

some cases mathematical discovery displayed a causative force on scientific progress,

and in other cases scientific progress displayed a causative force on mathematics. This

suggests that the relationship between mathematical discovery and scientific progress is

not intrinsically causative in either direction but may be correlative due to the specific

paradigm that scientific research is being conducted in.


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Causation and Correlation in Mathematical

Discovery and Scientific Progress

Mathematics and the other sciences have always enjoyed a relationship of mutual

support. Since the Scientific Revolution of the 1600s, scientific disciplines, especially

the natural sciences, have based all their conclusions on the foundations of mathematical

rigor. Physics, biology, chemistry, economics and even linguistics have all benefited, at

one point or another, from improvements in mathematics, and at times mathematical

discovery has even catapulted science in entirely new directions and spawned entirely

new sciences. But what is the actual relationship between mathematics and all the other

scientific disciplines? Is mathematics merely a supportive science that provides

impressive and extremely useful tools to scientists and engineers of other disciplines, or

is it a driving force in science that provides new direction to old scientific ideas and

inspires true innovation? The answer is complex.

Dr. Salomon Bochner, Ph.D., argued that the role of mathematics in the rise of

science is more causative than is generally given credit. The Greeks, specifically, in

Bochners esteem are largely responsible for the Scientific Revolution. Professor William

Dunham likewise argues that mathematics is usually 100 years ahead of science, and that

scientific progress almost always follows naturally from mathematical discovery. It is

true that the Scientific Revolution coincides very closely with the Renaissance

rediscovery of specific mathematical texts from the classical world. Correlation does not

equal causation, though, and Bochners assertions must be examined more closely to see

if they have validity. Philosopher of science Thomas Kuhn suggests more esoterically


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that scientific discoveries occur within a paradigm of scientific research similar to a mode

and that breaking this paradigm is only the result of a multitude of mathematical and

observational inconsistencies that invalidate it. In this case, the relationship between

mathematical discovery and scientific research would be correlative within the specific

paradigm of research. The specific nature of the relationship between mathematical

discovery and scientific progress will help future mathematicians and scientists identify

the applicability of specific mathematical discoveries to the other scientific disciplines

and possibly point to areas where current mathematics are lacking in their ability to

describe observed scientific phenomena.

In the 15th and 16th centuries the scholastic understanding of cosmology was

overturned by the work of three men: Nicolaus Copernicus, Galileo Galilei and Johannes

Kepler. These three men respectively established several important principles, among

them that: (1) The Earth revolves around the Sun, (2) falling bodies fall with uniform

acceleration, and (3) the planets revolve around the sun in elliptical orbits according to

fundamental laws of inertia. While groundbreaking, these three nascent scientists all used

the primarily Eudoxian mathematics of the time. It was not until the 17 th century that new

mathematics was developed to unify these three mens observations into a single,

cohesive set of natural laws.

The Calculus as it is studied today was independently developed by two

mathematicians in the 17th century: Sir Isaac Newton of England and Gottfried Leibniz of

Germany. While Leibniz was attempting to solve the metaphysical tangent problem (in

accordance with his role as a philosopher), it was Newton who was specifically

attempting to solve specific problems that had arisen in his mathematical examinations of


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Keplers work on the movements of celestial bodies. Newton developed his Method of

Fluents and Method of Fluxions as an answer to these problems and, in doing so,

created the first systems for what we today call integral and differential calculus.

Here, then, we have an example of scientific discovery, through observation,

driving mathematical inquiry in direct contradiction to the assertions of Bochner and

Dunham. If not for the baffling observations of Copernicus, Galileo and Kepner, Newton

might never have developed his methods. That being said, the contributions of Leibniz

cannot be ignored. Leibniz developed his calculus while investigating a purely

mathematical and metaphysical problem of tangents. It is possible that modern calculus

could have arisen completely independently of scientific inquiry and observation.

Differential and integral calculus have had an enormous impact on science since

their discovery. Calculus is used in every branch of the physical sciences, actuarial

science, computer science, statistics, engineering, economics, business, medicine,

demography, and in other fields wherever a problem can be mathematically modeled and

an optimal solution is desired. The applications of differential calculus to physics alone

could fill several pages, but it is enough to say that all calculations of velocity,

acceleration and time derivative rest solely on the foundation of differentials. Therefore,

the entirety of rocketry and ballistics, for example, are completely dependent on

differential calculus, as are all other arenas of classical mechanics. Applications of

integral calculus arise whenever the problem is to compute a number that is

approximately equal in principle to the sum of the solutions of many smaller problems.

For example, if the pollution density along a river is known in relation to the position of


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measurement, then the integral of that density can be used to determine how much

pollution there is in the whole length of the river.

Another example of scientific observation driving mathematical discovery comes

to us from economics. In the late 17 th century an Italian mathematician named Jacob

Bernoulli was examining the problem of compound interest. Bernoulli noticed that if

interest is calculated more often throughout a year then more money would be owed. For

example, if 100% interest is calculated for $1.00 at the end of a year then the debtor will

owe $2.00. If interest is calculated once every six months then the debtor will owe $2.25.

If the interest is calculated quarterly then the debtor will owe $2.44 (with additional

remainders). Bernoulli also noticed that the amount of increase reaches a limit that we

now call the force of interest. He calculated that, Using n as the number of

compounding intervals, with interest of 1/n in each interval, the limit for large n is a

constant. This number came to be known as e. Therefore, with continuous compounding,

the account value will reach $2.71 (with remainders). More generally, an account that

starts at $1, and yields (1+R) dollars at simple interest, will yield eR dollars with

continuous compounding.

This constant came to be known as Eulers constant, named after a later

mathematician named Leonhard Euler who calculated it out to 18 digits. e was later

introduced to calculus in order to perform integrals and differentials with exponential

functions and logarithms such as those necessary in aviation, meteorology and

astrophysics. Regardless of the later mathematical applications of the constant, it started

out firmly rooted in the science of economics and came about as a solution to a specific

observation. Based on these examples it might be tempting to write off Bochner and


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Dunhams assertions as baseless, but there are also numerous examples of purely

mathematical discoveries with no starting foundation in scientific observation that lead

inexorably to scientific progress.

Carl Friedrich Gauss, the Prince of Mathematics, is generally credited with

doing the first groundbreaking work in what is now known as modular arithmetic. The

discovery of Congruence Modulo is one of the foundations of number theory, touching

on almost every aspect of its study, and provides key examples for group theory, ring

theory and abstract algebra. It is sometimes called clock arithmetic because of its

relationship to numbers that wrap around back to one another arithmetically. Though it

began largely as an investigation into numerical relationships, which forms the

predecessor of what is now called Number Theory, modular arithmetic has numerous

and profound impacts in all arenas of science. In cryptography, for example, modular

arithmetic directly underpins public key systems such as RSA and Diffie-Hellman, as

well as providing finite fields which underlie elliptic curves, and is used in a variety of

symmetric key algorithms including AES, IDEA, and RC4. In computer science, modular

arithmetic is often applied in bitwise operations and other operations involving fixed-

width, cyclic data structures. What is interesting to note is that Gauss discovered

congruence modulo in 1801, but it wasnt until the early 1900s that this concept saw

practical application in the sciences. This correlates very closely to Dunhams 100

years assertion.

One of the most powerful examples, if not the most powerful example, of

mathematics driving scientific progress is the non-Euclidian geometry of Jnos Bolyai

and Nikolai Lobachevsky. In classical Greece, a mathematician named Euclid wrote the


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seminal work of geometry: The Elements. For nearly 2000 years it stood as the single

most important work in the field of geometry next to Archimedes Of the Sphere and the

Cylinder. Examining the 5 postulates that form the necessary assumptions of Euclidian

geometry in the early 19th century, Bolyai and Lobachevsky were both confounded by

what they felt should be a proof rather than an axiom. Euclids Fifth Postulate, known as

the Parallel Postulate, defines parallel lines as those that do not meet when taken to the

infinite and was noted as being conspicuously more complex in nature than the other four

postulates. Independently both Bolyai and Lobachevsky discovered that if geometric

operations are performed without Euclids Fifth Postulate they continue to yield a

thoroughly consistent geometry. Bolyai performed these operations on an elliptic while

Lobachevsky performed them on a hyperbolus, but the results were the same and form

the backbone of what is today studied as non-Euclidian geometry.

Bolyais and Lobachevskys non-Euclidian geometries were used by Bernhard

Riemann in the mid 19th century to form Riemannian geometry, which acts as Euclidian

geometry in some instances and as non-Euclidian geometry in others. Riemanns work

was groundbreaking for mathematics, but in the early 20th century it would be used by

perhaps the most famous physicist of all time to describe peculiar properties of light and

energy. In 1915, Albert Einstein published a paper on a geometric theory of gravity now

known as General Relativity Theory. In it Einstein describes space as a generally flat

plane excepting regions near where energy is present, whereupon space becomes

elliptical. Since, according to an earlier theory of his called Special Relativity Theory,

energy and mass are equivalent (i.e. matter is simply frozen energy), the appearance of

masses such as planets forces space into an elliptical framework that distorts light and


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energy to form gravitation. This revolutionary theory unified special relativity and

Newtonian mechanics and serves as the description of gravitation to this day.

As if revolutionizing physics as we know it isnt enough, this wouldnt be the last

time Riemanns mathematics would be used to revolutionize a field of science. In 1896,

Riemann published a paper providing a functional equation hypothesizing a relationship

between the distribution of zeroes and the emergence of prime numbers. This equation,

known as the Riemann Zeta Function, is the backbone of the as-yet-unsolved Riemann

hypothesis, which is widely considered the most important unsolved mathematical

problem today. While the exact meaning of the Riemann hypothesis is esoteric to all but

the most advanced mathematicians, its mathematical elegance is widely respected. In

1935, linguist George Kingsley Zipf proposed an empirical law of linguistics, now known

as Zipfs Law, that accorded closely to Riemanns Zeta Function. Zipf's law states that

given a body of natural language spoken words or text, the frequency of any word is

inversely proportional to its rank in the frequency table. In other words, the most frequent

word will occur approximately twice as often as the second most frequent word, which

occurs twice as often as the fourth most frequent word, and so on. Zipfs Law has proven

true for all known languages, and also has important implications for cryptography.

It is obvious to any researcher that mathematical discovery and scientific progress

go hand in hand. What is not so obvious is the nature of that relationship. In many cases

scientific observations have demanded new mathematics. In these cases mathematics is

not the instigator of change but the result. In other cases discoveries in mathematics

predate practical scientific progress by as much as a century. For this reason it is clear

that the relationship between mathematical discovery and scientific progress cannot be


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described as strictly causative. Instead, after analyzing several of the most important

discoveries in mathematics and their application to other scientific disciplines, the data

suggests that the relationship between mathematical discovery and scientific progress is

merely correlative in accordance with the current paradigm of scientific and

mathematical research.


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References

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Hawking, S. (2003). On the Shoulders of Giants. Philadelphia: Running Press.

King, J. (2009).Mathematics in 10 Lessons: The Grand Tour. Amherst, NY: Prometheus

Books.

Kuhn, T. (1962/1996). The Structure of Scientific Revolutions. Chicago: University of

Chicago Press.

Pickover, C. A. (2008).Archimedes to Hawking: Laws of Science and the Great Minds

Behind Them. New York City: Oxford University Publishing USA.

Pickover, C. A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250

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Trudeau, R. (1986). The Non-Euclidian Revolution. Boston: Birkhauser.

Wigner, E. P. (1980). The Unreasonable Effectiveness of mathematics in the Natural

Sciences. Comunications on Pure and Applied Mathematics, 13, 1-14.

Causation and Correlation in Mathematical Discovery and Scientific Progress

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