I. Core Definitions
A physical model is a method through which human beings use mathematical formulas and symbols to describe natural regularities, rhythms of motion, physical phenomena, and observable signs.
A formula is not the phenomenon itself. Rather, it is an abstract representation of the structures, relationships, and patterns of change identified through observation. A phenomenon does not disappear simply because humanity has not yet understood it. It remains present within nature and spacetime, waiting to be observed, recorded, recognized, and interpreted.
The starting point of scientific inquiry is therefore not the formula, but the phenomenon.
Phenomena Exist Before Formulas
The existence of a phenomenon implies that it already possesses the structure, relationships, conditions, and dynamic order necessary for it to occur.
Whether or not humanity has observed it, understood it, named it, or expressed it mathematically, the phenomenon continues to unfold according to its own natural structure. It does not exist because a formula has been created. Rather, a formula becomes possible because the phenomenon itself contains identifiable regularities.
In this sense, a phenomenon may be understood as a self-consistent and complete natural structure, or as a dynamic process unfolding under specific conditions.
When a phenomenon repeatedly exhibits recognizable rhythms, proportions, exchanges, or directions of change, these regularities can be observed, compared, and recorded. Human beings can then translate the identifiable relationships within the phenomenon into mathematical symbols and construct a formula.
A formula is therefore not an arbitrary structure invented by humanity. It is a mathematical projection of the regularity already present within the phenomenon.
The phenomenon is the primary existence. The formula is an expression of that existence.
The structure of the phenomenon exists first. Observation identifies its regularities. A physical model selects the relationships that must be described. Mathematics then transforms those relationships into a form that can be calculated, derived, tested, and verified.
It is therefore not the formula that creates order in nature. Rather, the inherent coherence and regularity of natural phenomena make the emergence of formulas possible.
However, a formula created by human beings may not immediately or completely represent the phenomenon. The phenomenon itself may be complete, while human observational scales, measurement capabilities, and mathematical models remain limited.
Scientific progress does not alter the phenomenon. It improves the formulas through which humanity describes it, allowing the model to move progressively closer to the structure already present in nature.
“A phenomenon is not the result of a formula. A formula is the projection, in mathematical language, of the structure and dynamic regularity already present within the phenomenon.”
Human beings first observe a recurring change, order, anomaly, or rhythm. They then attempt to express it through mathematical symbols in a form that can be understood, compared, and calculated. Once a physical phenomenon has been translated into a formula, mathematicians can perform calculations, search for solutions, construct derivations, and develop proofs.
The primary function of mathematics is to work with clearly defined relationships. It calculates results, identifies solutions, analyzes conditions, and establishes formal proofs. Mathematical computation alone, however, does not automatically explain why a formula was created, which phenomenon it describes, or where its application should end.
Those questions must still be answered through physical context and observation.
II. Understanding the Spatiotemporal Background of a Problem
Every significant mathematical problem, physical formula, or historical conjecture emerges from a particular period, cultural environment, state of knowledge, and level of observational capability.
To understand a problem, it is therefore insufficient to study only the symbols and statements preserved in their final form. We must also consider the historical circumstances of the person who formulated the problem, what knowledge was available at the time, which observational instruments did not yet exist, and which natural phenomenon or intellectual conflict the problem may originally have addressed.
Important historical conjectures are rarely arbitrary games of numbers. They may instead represent clues left by thinkers who, under the limited observational conditions of their eras, attempted to understand nature, space, time, quantity, and the structure of the cosmos.
A formula may preserve the form of a calculation while gradually losing the historical background, cultural context, and original observational purpose from which it arose.
When later researchers inherit only the symbols but lose the phenomenal background of the original question, scientific inquiry can become an endless process of formal derivation. Computational complexity continues to increase, while the investigation moves further away from the problem that originally required an answer.
This does not mean that calculation or proof has no value. It means that calculation must have a direction.
Before beginning a derivation, scientists should ask:
What phenomenon was this problem originally intended to describe?
Why did it emerge during that particular historical period?
What regularity, contradiction, or anomaly did its originator observe?
Was the formula intended to produce a numerical answer, reveal a structural relationship, or identify the boundary of a natural process?
Only when these questions are reconsidered can mathematical computation become more than a circulation of symbols and reconnect with physical reality.
III. Truth Within Phenomena and the Task of the Scientist
The natural regularities underlying phenomena do not change simply because humanity has not yet discovered them.
Human understanding may change. Models may be revised, formulas may be reinterpreted, and measurements may become increasingly precise. Yet the natural conditions and structures that generate a phenomenon do not change according to human preference.
The task of the scientist is therefore not to create truth from nothing, but to progressively identify relationships that already exist within nature.
Science should not consist solely of endlessly expanding derivations. The essential task is to understand which question a phenomenon is asking humanity to solve.
Some phenomena require the identification of causal relationships. Others require the recognition of critical conditions or the determination of when a system loses stability. Still others do not require an infinitely extendable sequence, but instead require the discovery of the point at which observation and computation should stop.
This point may be called the full stop of observation.
IV. The “Full Stop of Observation” Within a Formula
If a physical formula genuinely corresponds to the real world, it should not function merely as a computational machine that can operate forever without physical purpose.
Processes in nature usually have conditions, ranges, stages, and boundaries. Certain exchanges cease. Certain structures collapse. Some motions reach a steady state. Some measurable signals fall below instrumental resolution. Other systems cross a critical point and transform into a different state.
A physical model must therefore describe not only how a system changes, but also:
under which conditions the process begins;
within which range the formula remains valid;
at which time or state the process ends;
and whether the original formula retains physical meaning after that endpoint.
The full stop of observation is the effective endpoint of a physical process within a particular model and observational scale.
It is not an arbitrary truncation of calculation, nor is it a point chosen merely because the researcher is unable to continue. It is a boundary determined by physical reality.
When dynamic exchange has ceased, when the system has reached a steady state, when observable variables no longer undergo meaningful change, or when the assumptions of the model are no longer satisfied, the original formula should reach its full stop.
This full stop reminds us that mathematics may be extended indefinitely, but a physical model cannot be extended indefinitely beyond the conditions under which it remains valid.
A recurrence relation, for example, may continue forever in pure mathematics. However, if it originally represents a physical process occurring over a finite period, then once the actual exchange has ended, the energy has been dissipated, or the system has undergone a phase transition, further calculation may no longer correspond to the original phenomenon.
The full stop within a formula may therefore represent more than a temporal ending. It may correspond to:
a spatial boundary;
a lower energy limit;
a stable state;
a critical value;
a measurement limit;
a transition of scale;
the end of a causal relationship;
or the endpoint of a model’s validity.
The purpose of the observational full stop is to establish a clear boundary between physical reality and pure mathematical continuation.
V. Formulas Must Not Be Altered Arbitrarily, but Models Must Remain Testable
When a formula accurately describes a particular phenomenon, researchers should not alter its structure merely to obtain an expected result.
The symbols, coefficients, relationships, and boundary conditions within a formula should have identifiable physical origins. Any modification must be supported by new observations, measurements, or a more complete model. It should not be introduced simply to keep a calculation functioning.
However, the principle that a formula should not be altered arbitrarily does not mean that formulas can never be revised.
Scientific models must remain open to observation, experiment, and falsification. When new evidence demonstrates that an existing model has a limited domain of validity, a higher-order model may be developed, but researchers must clearly explain:
the conditions under which the earlier formula remains valid;
which limitations have been revealed by new observations;
which variables or boundaries have been added;
and how the old and new models are connected.
We should respect the original regularity preserved by a formula without transforming the formula into an unquestionable belief.
VI. Avoiding Purposeless Cycles of Proof
Some longstanding problems in mathematics and science may remain unresolved not only because computational power is insufficient, but also because researchers continue to derive results within the same formal framework without reexamining the physical background, original definitions, or observational boundaries of the problem.
When an investigation enters a prolonged cycle, scientists may reconsider three levels of the problem:
Has the problem itself been correctly understood?
Is the mathematical language being used appropriate for the original phenomenon?
Has the research overlooked an observational full stop that should be present?
Some problems that appear to require an infinite proof may instead require a redefinition of observational scale. Some calculations that appear to have no endpoint may lack a physically meaningful stopping condition. Some unresolved conjectures may preserve information about nature that could not be clearly expressed using the knowledge and instruments available when they were first proposed.
These possibilities cannot replace rigorous proof, nor can they directly establish whether any conjecture is true or false. They can, however, provide new directions for research and prevent investigators from remaining indefinitely trapped within a single framework.
VII. Guidance for Future Scientists
When future scientists confront formulas, conjectures, and complex models, they should not ask only, “How much further can this be calculated?” They should also ask, “Why does this calculation exist?”
They must cultivate three interconnected abilities:
the ability to observe phenomena;
the ability to construct physical models;
and the ability to perform mathematical calculations and proofs.
Mathematics without observation may lose its connection to physical reality.
Observation without mathematics may prevent phenomena from being precisely compared or reproduced.
A model without verification may become an account that cannot be tested.
A complete scientific method should therefore proceed as follows:
First, observe the phenomenon and identify the problem.
Second, construct a model that expresses the relevant regularity.
Third, perform calculations to obtain solutions and testable consequences.
Finally, return to physical reality and determine where the model is valid, where it fails, and where observation should reach its full stop.
When future scientists understand the spatiotemporal background, physical meaning, and observational endpoint behind a formula, they will no longer see formulas merely as sequences of symbols waiting to be extended. They will recognize the structure, rhythm, and vitality preserved within them.
Scientific research can then reduce purposeless cycles of derivation and direct finite human lives and computational resources toward questions that possess genuine physical meaning, can be observed, can be tested, and may ultimately be solved.
Conclusion
Phenomena exist before formulas.
Formulas allow phenomena to be described. Mathematics allows formulas to be calculated. Observation determines the relationship between a formula and reality.
Mathematics may extend toward infinity, but every physical model has its own conditions, scale, and boundary.
The full stop of observation is not an end to the exploration of knowledge. It is a reminder that when a model reaches the endpoint of its physical validity, purposeless continuation should cease, and inquiry should return to the phenomenon itself.
The purpose of science is not merely to calculate a formula further. It is to understand what is being calculated, why it is being calculated, and where the calculation should stop.
Only then can humanity move beyond the surface of symbols and gradually approach the core question that the phenomenon itself requires us to understand.