The differences between a system of measurement, an abstract mathematical theory of physics and actual reality are described along with some of the consequences. Fundamental constants and laws of nature are, in reality, artefacts of the mathematical theory, not fundamental, not part of reality and not even constant. A Fundamental Law of Nature is best described as a ‘non-negotiable hypothesis within a specific theoretical system‘. The gravitational constant is not a fundamental constant of nature and is not constant.
AI summary
I asked an AI engine “What is the relationship between a ‘system of measurements’ and a mathematical theory of physics?”
A mathematical theory of physics is an abstract framework that uses the rules of mathematics to describe and predict natural phenomena, while a system of measurement provides the concrete, standardized units (like meters or kilograms) to link these abstract mathematical quantities to empirical, real-world observations and experiments.
AI: The relationship can be broken down into several key points:
- Mathematics as the Language of Physics: Mathematical theories express physical laws and principles in a precise, universal, and internally consistent language, independent of any specific unit system. For example, Newton’s second law is universally F=ma, regardless of whether you use SI or Imperial units.
- Measurement for Validation and Quantification: Measurement is how a physical theory is tested against reality. Experimental results, expressed as numerical values combined with specific units, are compared with the predictions of the mathematical model to determine its validity and accuracy.
- Systems of Units Provide Standards: A system of measurement (like the International System of Units, or SI) provides a set of defined, internationally accepted base units (e.g., the meter for length, the second for time) that ensure consistency and reproducibility of experiments across different locations and times.
- Arbitrary vs. Universal: While the underlying mathematical laws are universal, the choice of a specific unit system (SI, Imperial, etc.) is conventional and, to some extent, arbitrary. The mathematical structure of the theory remains the same; only the numerical values and conversion factors change when switching between systems (e.g., an inch is defined in terms of a meter).
In essence, the mathematical theory provides the structure, and the system of measurements provides the “calibration” that allows the theory to make testable, numerical predictions about real-world phenomena. Without a system of measurements, a physics theory remains a set of abstract mathematical rules; without the mathematical theory, a collection of measurements lacks predictive power and underlying explanatory principles
This is a good summary of the relationship but with a few issues and internal inconsistencies that need discussing.
Arbitrary vs. universal
From the AI summary we have:
- While the underlying mathematical laws are universal, the choice of a specific unit system (SI, Imperial, etc.) is conventional and, to some extent, arbitrary.
- For example, Newton’s second law is universally F=ma, regardless of whether you use SI or Imperial units.
What is meant by “While the underlying mathematical laws are universal“? Newton’s F=ma has a strong contender in the form of the Laws of Relativity and so cannot be considered to be either fundamental or universal, but instead just one of several competing abstract theories.
The abstract ‘law’ may be ‘universally F=ma‘ but what is meant by a ‘law’.
We have from the above “Mathematical theories express physical laws and principles in a precise, universal, and internally consistent language, ..”. Ok so now a mathematical theory is not the same as a Law but instead is only expressing a law. This is a contradiction of: “While the underlying mathematical laws are universal“. So again we ask: “What is a ‘law’?”, or maybe “What is a ‘mathematical law’?”.
The language is confused and ambiguous and stems from the desire to claim that reality is somehow mathematical in nature and that mathematics is not only synonymous with reality, but that the current fashionable abstract theory is a reliable description of the Fundamental Nature of Reality.
Physicists seem keen to distinguish themselves from engineers by constantly claiming ‘fundamental’ quantities, forces and relationships. They seem not satisfied with making half-decent predictions concerning laboratory experiments but want to convince themselves that they have somehow described the true nature of reality in terms of a few chosen mathematical equations.
Organising the system
The relationships between Reality, the System of Measurement and the Mathematical theoretical frame work is shown in the diagram below and hopefully brings some clarity to the situation.
- Note the absence of anything definite (let alone fundamental) in the Reality box
- Reality is accessed via measurements only (not mathematics)
- The mathematical framework is two steps removed from Reality
- Mathematical quantities are not measurements
- Mathematical quantities are not Reality
- Mathematical equations describe relationships between the measurements
- Almost all the ‘work’ is within the mathematical domain
- There is a big mismatch between the fundamentals of the measuring system and the fundamentals of the mathematical system
- Measurements are meaningless by themselves and need to be interpreted via the mathematical framework
We need to talk..
The limitations of measurement
What is it we can actually measure directly? The answer is “Not much and most of them are listed above”.
Mass is not directly measurable: Gravity debunked
Note that not even ‘time’ is not directly measurable as the ‘time’ measured by two atomic clocks will vary during an eclipse (Sheldrake’s TED talk), as will that measured by a pendulum. The quantity of ‘time’ is therefore an abstract theoretical quantity that is somehow inferred from the behaviour of such machines.
‘Speed’, similarly is not directly measurable. We need to measure the position of an object at two different ‘times’ and this will enable us to calculate the average speed over the interval. This is therefore not a fundamental, but a derived concept. Similarly, acceleration cannot be measured directly as it involves a change in speed.
Time and again we read, for example: “The gravitational constant has been measured accurately to many decimal places..“, but again, a fundamental constant is part of the mathematical domain and is therefore never measured, but inferred. Fundamental constants such as this are now easily seen to be, not fundamental quantities of Nature, nor even of the measurement system, but to be derived quantities of the mathematical theoretical framework only.
Mismatch between ‘fundamentals’
We can only measure very basic quantities such as ‘length’ and ‘position’ and even have to do some theoretical calculations to measure ‘time’. The ‘measurables’ of the system are very limited but we must constrain ourselves to using these if we are to obtain an objective characterisation of Reality.
However, the ‘fundamentals’ of the mathematical model are in abundance. We have mass, time, forces and all manner of fundamental constants to use as a basis for a complex theoretical framework.
Note that these are precisely the quantities that physicists claim as the fundamentals of actual reality, but also note that they derive immediately from the fundamentals of the measuring system, not actual reality. Note also that none of these quantities themselves are directly measurable! There is a therefore major mismatch between what we measure and what we theorise as fundamental to the theoretical model.
We need, therefore, some sort of interpretive system interposed between the measurement system and the theoretical system to make sense of the measurements. When does a set of measurements of length etc. translate to a ‘mass’ and when does it translate to a measurement of ‘time’ etc.?
Such interpretation is necessarily part of the theoretical model and is dependent upon it. A different theoretical model requires a different interpretive procedure to translate measurements to fundamentals. The ‘measurable’ quantities are not going to change and so the so called fundamentals of reality are really just artefacts of the abstract mathematical theoretical framework. A new framework will necessarily result in a new set of fundamentals.
What is a Law of Nature?
Newtons 2nd Law of Motion is invariably described as Fundamental Law of Nature:
Force = mass x acceleration
We can now easily see what this means. None of these quantities are part of our measurement system, being not directly measurable, and nor are they part of Reality as far as we know, but instead they are all part of the abstract theoretical framework.
The quantities of the measurement system are just point measurements and have no real meaning by themselves but when somehow translated into the theoretical framework we expect to see meaningful relationships between the quantities which will:
- Allow the formulation of quantitative ‘predictions’ concerning further measurements to validate the model
- Enable some sort of ‘understanding’, i.e. some representation of reality which sits comfortably within the human cognitive system
Note that only a finite number of measurements have ever been made of reality and even then they were made with limited techniques and primitive equipment. Note also that the theoretical framework ultimately derives from such limited measurements and is therefore necessarily constrained by such limitations.
Note also that if the theoretical framework itself changes then so do the relationships; relativistic mass is different from Newtonian mass for example.
Definition: A Fundamental Law of Nature can now be described as a ‘non-negotiable hypothesis within a specific theoretical system‘ and that is all.
The relationship between a Law of Nature and Actual Nature is always via a limited system of measurements which never directly measures any of the Fundamental Properties of the theory itself.
Different ways of measuring?
We hear often that there are now better (different) ways of measuring various quantities such as mass, gravitational constant, permittivity of space, speed of light etc. but this is deceptive language aimed at making it seem that things are improving all the time.
The idea that there are two different ways of measuring mass, for example, is just nonsense. Mass is not a measurable quantity and isn’t even part of the system of measurement. Mass is a theoretical interpreted quantity and therefore not susceptible to measurement.
The way that scientists use language gives the impression that mass is part of Reality, that there is something there to be be measured and that successive ‘refinements’ to measurement techniques are all that is needed to get an improved value for its magnitude.
The idea that there is something absolute to be measured if only we knew how, lends respectability to what might be termed ‘goal oriented’ refinements to experimental techniques. If you really think that there is something there to be measured then both the attainment of greater agreement between experiments and the increasing ‘accuracy’ of such results leads inevitably to the conclusion that you are getting nearer the ‘real’ answer.
In particular, if successive refinements seem to be converging towards a specific value, then this value will assumed to be the true goal regardless of the fact that it is never actually attained.
So what?
The calculated values of the gravitational constant vary considerably over time and according to a repeated pattern. It s clearly not constant and so there is something wrong with the theory and the theory should be ditched.
Why do measurements of the gravitational constant vary so much? – Lisa Zyga
https://phys.org/news/2015-04-gravitational-constant-vary.html
However, the idea that Big G is a fundamental of Nature itself as opposed to a derived value in an abstract theoretical system, means that physicists are loathe to admit that it is the theory that is wrong and instead blame the measurement system:
Now scientists have found that the measured G values oscillate over time like a sine wave with a period of 5.9 years. It’s not G itself that is varying by this much, they propose, but more likely something else is affecting the measurements. – Zyga
This is odd since the measurements themselves are dependent upon technology and it is the technology that has been improving steadily over the years. We are being asked to believe that the technology is not only wrong but is wrong according to regular pattern.
The idea that the equations of Newton are not theory but immutable features of physical reality fact is impeding scientific progress.
π
Pi is of course not a fundamental constant of nature but a fundamental constant of geometry; the two are not the same.
Special and general relativity
The theories of relativity are clear examples of confusion between theory and reality. A simple abstract mathematical idea, that of a coordinate system soon becomes conflated with actual reality and ends up being regarded as synonymous with physical space-time, even in the absence of sufficient support from the measurement system itself: Einstein’s relativity vs. actual reality
Summary
The tendency to conflate an abstract mathematical theory with an actual physical process seems to be irresistible, but nevertheless leads to severe interpretive problems. The fashion of the day is framed as a fundamental law or equally fundamental constant and scientific progress is immediately impeded by the requirement to express any new idea within the self-imposed constraints. Nothing that is declared ‘fundamental’ can be challenged without cries of ‘pseudo-science’!
Again, the ‘fundamentals’ of nature are nothing of the sort but are instead the bases of an abstract mathematical system which tries to give meaning to a finite number of measurements. These measurements in turn derive from severely limited measurement techniques with the hope that they somehow represent the entire of Reality.