Mathematics and the physical sciences

One of the primary purposes of the physical sciences is help us understand how the different components of our universe physically interact to create its observable properties.  However many scientist’s use the abstract properties of mathematics to extract, by quantification the underlying rules governing them.  In other words they define those rules not in terms of the physical objects that they are studying but in terms of non physical or abstract properties of mathematical equations.

Granted many of our most successful theories began as a mathematical study of "real world" problems. In other words scientists attempt to use mathematics to quantify "real world" environments and to establish the underlying rules that govern them.

However the fact that one can mathematically quantify an environment does not mean that they accurately defined the "reality" of the rules that govern it.

For example, Isaac Newton made qualitative observations of how objects in a "real world" environment interacted with the earth’s gravitational field. He then used the understanding develop form those observations and his knowledge of mathematics to derive a theoretical model that could not only quantity them but also explain why they interacted the way they did in terms of those observations.

In other words he was able to provide a direct physical connection between the abstract properties of his mathematics and a "real world" perception of those interactions based on observable components of his environment.

However, with the advent of higher mathematics and advance computing technology physicists now feel they have to ability to define the "reality" of what we observe in purely abstract mathematical terms

For example, String Theory is based purely on mathematically analyzing the quantitative observation of the "real world" and then, using only that information defines its reality. In others words they not only define the observable properties of the environment but also the rules governing the interaction of its components in terms of abstract mathematics without physically connecting them to the "real world" perception of how those components interact.

Therefore, String Theory does not and cannot provide a physical connection to the observable universe because its description is based purely on abstract properties of mathematics and not on the physical observations as Isaacs Newton’s were, of how its components interacted to form the environment they are describing.

These two different approaches to theoretical philosophies are called Empiricism and Realism.

On the surface they both to be appear to equally viable methods for defining the rules that governing our observable environment even though their methodologies are very different.

Empiricists say that our theoretical models should only be concerned with the quantifiable properties of observations while the Realist tell us that our theories should not only make accurate quantitative predictions of an environment but also allow us to understand why nature behaves the way it does based on the observable properties of the environments they are describing.

For example empiricists feel that, as mentioned earlier science should only be concerned with quantifying observations and that they should not be tested against the observable properties of the "real world".  In other words they are not interested in or feel that it is important to integrate the observations of how objects interact in the "real world" to create our observable environment. This is the attitude most string theorist take because they attempt to define not only observations but how the "real world" behaves the way is does in terms of the abstract properties of mathematics.

Realists, on the other hand believe that science should not only be concerned with quantifying experiences but also explaining why the real world behaves the way it does based on observations. In other words they feel that mathematics should not only be used to quantify an environment but also should explain why objects in the "real world" interact the way we do in terms of the observable properties of that environment. This, they feel would give the underlying essence of a physical environment developed by mathematics a stronger tie to its reality.

For example, Einstein who some would call a realist first developed a conceptual understanding of space-time, based, in part on the assumption that the speed of light was constant in all reference frames. However unlike the Empiricists he then developed the theoretical structure of Special Relativity by forming a physical image of what it would be like to chase after a beam of light based on observable properties of the "real world" and then translated or transposed that understanding to define how and why matter and energy in motion would interact in a space-time environment. Later he developed the equations that quantified and verified the accuracy of his conceptual model based on observations of speed of light in the "real world".

However, the proponents of Empiricism take the opposite approach to science. They observe the quantitative results of observations and then, through trial and error define a series of abstract equations, which can accurately predict them. They then use those equations to define a theoretical structure which then predicts the reality or rules governing the underlying essence of that environment.

For example, Quantum Theories, which espouses the empiricist approach because defines the observations of the quantum mechanical environment of energy/mass based solely on mathematical probability functions or equations. They then use those abstract equations to not only quantify those observations but to define the rules which govern the environment they occupy.

However this circular method of predicting both observations and the operating environments based on only on mathematics does not allow one to determine the physical reality of the environments they define because those mathematically created environments are by definition abstract and therefore are independent of the physical world they are defining.

But is there a way science can verify when a mathematical created environment which defines the underlying essence of the "real world" does not have a direct "physical connection" to it.

The realist answer to this is that it is possible connect them, as Newton and Einstein did to the physical environment they are defining though observations.

For example Quantum theory makes predictions based on the abstract mathematical environment of probability functions. However because its abstract properties are not connected to any physical images of the "real world" all observations, no matter how inconsistent or bazaar they are can be incorporated into it.

This is in sharp contrast to the space-time environment defined by Einstein because he, as mentioned earlier developed the theoretical structure of a space-time environment based on a physical image of what it would be like to chase a beam of light in the "real world". This not only gives the abstract properties of his mathematics a physical connection to the "real world" it also give science a way of checking its conceptual validity.

For example Einstein’s theory would be invalidated if it was found that something could travel faster than the speed of light because that would contradict the physical model he define.

If however if some observation happened to contradict principals of quantum mechanics such as simultaneously observing of both the particle and wave properties of mass it could easily explained by the fact that its probability functions tells us that anything that can happen will eventually happen. Therefore it is impossible to find any observation that would contradict its basics assumption in a "real world" that is based on probabilities because it tells us that anything can, will and must happen at some time in the future even if it is direct contraction to its basic theoretical concepts.

Yet this can only happen in an abstract environment which is not bound by the physicality our observational world because in the "real world" we observe that some things just do not happen.

But why should science put in the effort to understand the observational reality of our world when both the abstract mathematical foundation of quantum mechanics and the physical imagery of Einstein’s theories make very accurate predictions of future events based on the past.

Because the only way to determine if the rules provided by abstract mathematical equations apply to our observable world is through observations.

For example with powerful enough computers one could still use the geocentric or earth centered model of planetary motion to accurately quantify their relative motions.  However it could not explain the observation that objects such as the orbits of moon of Jupiter are centered on a different objects other that the earth.   If we were unable to make those observations then we may still incorrectly think that the earth is the center of planetary motion.

We as educators must make sure that our students are aware of the difference between defining and describing our observable universe and the importance of having those definitions correlate with the observations they are defining.

Later Jeff

Copyright Jeffrey O’Callaghan 2019

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