There are three fundamental ways science uses mathematics and observations to understand the reality of our world.
The first involves developing a mathematical description by directly observing how its components interact.
For example, Isaac Newton developed his law of gravity by observing the movement of planets and realizing that they could be understood by assuming they exerted a force on their neighbors that was directly proportional to their mass. He then derived a mathematical equation which allowed one to quantify their future movements based on those observations. These equations could also be applied to the movement of planets that had not been observed. For example the position of the planet Neptune was predicted before it was observed by the use of his equations.
In other words he first conceptually defined or understood their movements through observations and then derived a mathematical expression that quantified the reality of their world from that understanding.
The second involves extrapolating a conceptual understanding of the properties of our observable environment to an unobservable one.
For example Schrödinger developed an equation that defined the quantum mechanical properties of energy/mass by extrapolating an understanding of waves gained from observations of our three-dimensional environment to the unobservable one of subatomic particles.
In other words he mathematically defined the reality of a quantum mechanical world in terms of the reality of his observable properties of waves.
The third method involves developing a mathematical expression based not on a conceptual understanding of how the components of our observable universe interact as Newton and Schrödinger did but on analyzing the quantified result of those interactions.
For example string theorists analyze the quantified results of particle interactions and then developed mathematical expression that predicts them directly from those results. They then define the reality of a “string” world based on a conceptual understanding of the equations that quantify them.
In other words string theorists define the “reality” of a “strings” environment based only on the mathematical structure of the equations that they use to define that reality.
For the last century researchers have favored the approach taken by string theorists in that they use the quantified result of observations to mathematically define how the components of an environment interact to generate those results.
Why?
Because many feel they must rely totally on their imagination, intuition, and mathematics to guide them on the road to understanding because they cannot directly observe the worlds they are analyzing.
Unfortunately it is possible to use intuition and mathematics to create completely self contained worlds that can predict observations which may or may not be connected to their reality they are attempting to define. Therefore the validity of the worlds they create can only be verified if they are anchored in a real non abstract world provided through direct observation of one’s environment.
For example the Copenhagen interpretation of Quantum Mechanics does not view the equations that define its theoretical concepts in terms of the observable properties of waves as Schrödinger had done but in only in terms of a mathematical probability based on them.
However history has shown the most powerful way to understand the physical properties of our environment are through observations. Before Isaac Newton developed his law of gravitation scientists could still make accurate quantifiable predictions of planetary movements even though no one understood why they moved that way.
Some scientists of that period felt there was no need to look any further because they could still make accurate predictions of their motion without understanding why. However, Newton by observing his environment developed an understanding that enabled one of what many feel is the greatest conceptual leaps in humankind understanding of the universe.
Presently we seem to be in a situation similar to that which occurred before Newton developed his gravitational theory.
Quantum mechanics can make extremely accurate predictions of the quantum mechanical properties of energy/mass based on a self contained mathematical environment called a wave function. Yet no one can explain its physicality in terms of observable properties of our environment.
However, similar to the scientists who came before Newton many believe that the self contained abstract mathematical environment of quantum mechanics does not need explaining because it can make accurate a predictions of the properties of energy/mass.
Yet Newton demonstrated how important connecting a mathematical environment to an observational one is to our understanding of its reality.
Mathematics is a very powerful tool for helping us understand the world we live in but because of its abstract nature it can be used to create self contained environments which can predict our physical world and still not be connected to its reality. The only way to make sure the reality they define is connected to our world is by anchoring it in physical observations of our environment.
Later Jeff
Copyright 2011 Jeffrey O’Callaghan