Introduction to Theoretical Physics
“I would rather have questions that can’t be answered than answers that can’t be questioned.” Richard Feynman
“Be less curious about people and more curious about ideas.” Marie Curie
Introduction
Imagine standing atop a hill, gazing across a sprawling physics landscape—rivers of motion, mountains of energy, storms of chaos. As a budding theoretical physicist, you can think of yourself as a cartographer, tasked with charting this terrain. Welcome to Lesson 2, where you’ll learn to map theoretical physics. You’ll spot landmarks (phenomena), find signposts (physical laws), and sketch maps (theories) to predict new paths. From observing a falling star to testing predictions using experiments, each step hones your map-making skills.
In the last lesson we learned what learning is. So, what is theoretical physics? I could sit here and write out a definition, then show that it has many terms that you might not actually know. Then I could define each one, point-by-point. I will leave that to you.
Problem 2.1: Define science.
Problem 2.2: Define physics.
Problem 2.3: Define theoretical physics.
The Landscape of Physics
We can imagine physics as a vast landscape that is unexplored. As you explore you will encounter three kinds of objects in this imaginary landscape. The first are physical events (a falling apple, the Sun rising, a thunderstorm in the distance, etc.), we call these phenomena. Along the way you will encounter definite facts. These stand like signposts across the landscape, we can call these statements of fact physical laws, and we will learn more about these later. Every so often you will find a map that illustrates part of the landscape, such maps not only tell us about how the phenomena and laws in the region work, but also allow us to make predictions that we can apply to phenomena we have not even thought of. We call such a map a physical theory. It is the goal of a theoretical physicist to make these maps more accurate, perhaps redraw them, or to chart out unknown regions.
Terms and Definitions
Term/Definition 2.1 Phenomena: Physical events encountered in the world, such as a falling apple, the Sun rising, or a distant thunderstorm.
Term/Definition 2.2 Physical laws: Definite facts or statements of fact that act like signposts in the physics landscape, describing regularities observed in phenomena.
Term/Definition 2.3 Physical theory: A conceptual map that illustrates how phenomena and laws in a region of physics work together, and that allows predictions about new or unobserved phenomena.
Assumptions(the implicit beliefs the author relies on)
Assumption 2.1: Physics can be meaningfully conceptualized as an unexplored landscape, with discoveries analogous to exploration.
Assumption 2.2: The practice of physics naturally divides into observing events (phenomena), identifying regularities (laws), and building predictive frameworks (theories).
Assumption 2.3: Theoretical physics primarily involves improving, refining, or extending these predictive frameworks (“maps”).
Principles (key guiding ideas presented)
Principle 2.1: Physics progresses by observing phenomena, establishing laws from repeated observations, and constructing theories that unify laws and enable predictions.
Principle 2.2: A good physical theory goes beyond describing known phenomena—it must predict new ones.
Principle 2.3: The ultimate role of a theoretical physicist is to refine existing theories, redraw inaccurate ones, or explore and map previously unknown areas of physics.
Principle 2.4: Theories are tools (like maps) for navigating and understanding the physical world, not the world itself.
Exercise 2.1: Begin with Term/Definition 2.1 and copy it into your notebook. Reflect on its meaning for a few minutes. Note any thoughts that come to mind. Then do this for each term/definition, assumption, and principle.
Observation
If we intend to establish something that we can call the cycle of science, the first element of that cycle is the act of locating or even measuring what is going on around us. We call this act observation. At first this act is limited to what information we can acquire through our senses. This is like a cartographer eyeing a river. Eventually we get around to measuring things using technology.
Historically, it took mankind a very long time to realize that you cannot observe something without interacting with it. In fact, that is my personal definition of what we call modern physics. When we take the attitude that you can have an objective point of view, that is what we call classical physics. To explore what happens when you realize that there is no objective point of view, that is modern physics.
Ultimately, observation is the collection of facts. This is tempered by limitations in our senses or measuring technology. Whenever you establish such a fact, then you must also include some estimate of the errors involved.
Terms and Definitions
Term/Definition 2.4 Observation: The act of locating or measuring what is going on around us, initially through our senses and later using technology.
Term/Definition 2.5 Classical physics: The approach to physics that assumes an objective point of view is possible (i.e., observation does not affect the observed system).
Term/Definition 2.6 Modern physics: The approach to physics that recognizes observation necessarily involves interaction with the system, eliminating a fully objective point of view. Some people restrict the definition of modern physics to quantum theory.
Term/Definition 2.7 Facts (in science): Information collected through observation, tempered by the limitations of senses or measuring instruments and always accompanied by an estimate of errors.
Assumptions (the implicit beliefs the author relies on)
Assumption 2.4: All scientific knowledge begins with observation.
Assumption 2.5: Observation is inherently interactive—you cannot observe something without affecting it in some way.
Assumption 2.6: The distinction between classical and modern physics hinges on whether one accepts or rejects the possibility of completely objective observation.
Assumption 2.7: Human senses and technological instruments have inherent limitations that affect the accuracy of observations.
Assumption 2.8: Every observational fact must include an estimate of uncertainty or error.
Principles (key guiding ideas presented)
Principle 2.5: The cycle of science begins with observation.
Principle 2.6: Observation progresses from direct sensory experience to technologically assisted measurement.
Principle 2.7: True scientific facts derived from observation are never absolute—they must include estimates of error due to measurement limitations.
Principle 2.8: Recognizing the unavoidable interaction between observer and observed marks the transition from classical to modern physics.
Principle 2.9: Science requires acknowledging the limits of objectivity in observation.
Exercise 2.2: Begin with Term/Definition 2.4 and copy it into your notebook. Reflect on its meaning for a few minutes. Note any thoughts that come to mind. Then do this for each term/definition, assumption, and principle.
Hypothesis Formation
The second element of the science cycle occurs when you begin to ask questions of the facts that you have collected. Use Socratic questioning of the information. When you develop a proposed answer, that is what we call a hypothesis. It is a point for further investigation.
So, if you have a list of facts. how do you go about formulating a hypothesis?
Ask a question of the facts you have accumulated.
Use the techniques from Lesson 1 to develop some background knowledge related to the question.
Based on the facts and knowledge you acquired, make a guess that answers your question. Is any guess a good way to answer your question? No. First, there must be a method to test facts to check if your guess is correct or not. We call a guess with this quality testable. If your guess is not testable, then it cannot be considered as being scientific under any circumstances. The second quality is that your guess must have a way to be proven wrong, note that this is a mathematical/logical process. A guess having this quality is what we call falsifiable. If a guess is not falsifiable, then it cannot be considered as scientific in any way.
When you build a hypothesis make sure to identify all relevant physical quantities. We leave the nature of such quantities until Lesson 13.
Use your developing hypothesis to make predictions by asking questions of the nature, “If this is true, then some thing must happen.” Recall from Lesson 1 that we call this an implication. Such predictions are a process that we call a model.
All guesses require you to make assumptions. While not a formal proof requirement, it is a guideline that the hypothesis requiring the fewest assumptions is probably the best. This is due to the work of the English Franciscan friar, philosopher, and theologian William of Ockham (1287-1347), and is often called Ockham’s razor.
Terms and Definitions
Term/Definition 2.8 Hypothesis: A proposed answer to a question raised about collected facts; a point for further investigation.
Term/Definition 2.9 Testable: A quality of a hypothesis or guess that requires there be a method (using facts/observations) to check whether it is correct or incorrect.
Term/Definition 2.10 Falsifiable: A quality of a hypothesis or guess that requires there be a logical/mathematical/observational way to prove it wrong (i.e., it makes definite predictions that could be contradicted by evidence).
Term/Definition 2.11 Physical quantities: Relevant measurable aspects of a phenomenon that should be identified when building a hypothesis (detailed treatment deferred to Lesson 13).
Term/Definition 2.12 Model (in this context): The process of using a hypothesis to make predictions by asking “If this hypothesis is true, then this must happen” (an implication).
Term/Definition 2.13 Assumptions: Unstated or necessary presuppositions required for a hypothesis to hold.
Term/Definition 2.14 Ockham’s razor: The guideline (attributed to William of Ockham, 1287–1347) that, among competing hypotheses, the one requiring the fewest assumptions is probably the best.
Assumptions (the implicit beliefs the author relies on)
Assumption 2.9: Scientific progress follows a cycle that begins with observation and moves to questioning collected facts.
Assumption 2.10: Good hypotheses must be both testable (verifiable through evidence) and falsifiable (potentially disprovable).
Assumption 2.11: Any guess that lacks testability or falsifiability cannot be considered scientific.
Assumption 2.12: Fewer assumptions in a hypothesis generally indicate a better (more preferable) hypothesis.
Assumption 2.13: Predictions derived from a hypothesis (via implications) are essential for scientific modeling.
Assumption 2.14: Relevant physical quantities are always present and should be explicitly identified in hypothesis formulation.
Principles (key guiding ideas presented)
Principle 2.10: The second step in the scientific cycle is to question observed facts using Socratic questioning.
Principle 2.11: Formulate hypotheses by: asking a clear question of the accumulated facts, building background knowledge (using techniques from Lesson 1), and making an educated guess that answers the question.
Principle 2.12: A scientific hypothesis must be testable (there exists a way to check it against facts) and falsifiable (it can be proven wrong).
Principle 2.13: Identify all relevant physical quantities when constructing a hypothesis.
Principle 2.14: Use the hypothesis to generate predictions (implications of the form “If hypothesis, then this must occur”)—this is modeling.
Principle 2.15: Explicitly recognize the assumptions underlying any hypothesis.
Principle 2.16: Prefer hypotheses with the fewest assumptions (Ockham’s razor).
Principle 2.17: Non-testable or non-falsifiable guesses are outside the realm of science.
Exercise 2.3: Begin with Term/Definition 2.8 and copy it into your notebook. Reflect on its meaning for a few minutes. Note any thoughts that come to mind. Then do this for each term/definition, assumption, and principle.
Experimentation
The only way to be completely confident that you have successfully tested a hypothesis is to develop a systematic method of verification by duplicating the guess under controlled conditions. We call this an experiment. Experiment is the only method of determining scientific truth in any way. Observations are not sufficient since we have no way of controlling them—this allows unknown factors to enter in—so we can never be sure of what we are observing. Here are the elements of making an experiment scientific:
You must establish conditions where you can manipulate one or more well-defined physical quantities to observe how the change effects some other well-defined physical quantity. This is called controlled testing.
Use statistical methods to design experiments to test a hypothesis against evidence obtained by direct and measured observations. Such observations are called empirical. The process is called hypothesis testing.
Key aspects of the experiment should be able to be reproduced given similar circumstances. Reasonably enough, this property is called reproducibility.
The result of any experiment is a collection of facts acquired through direct observation and measurement.
Many experiments have one or more runs where variables are not changed at the same time that other runs experience changes. We call such unchanged runs a control group. We can then match the changing runs to the unchanged runs.
We try to reduce the effect of any quantities we are not changing, this allows greater confidence that observed changes are due to the controlled quantities. We can call this isolating the variables.
If an experiment is careful to include all of these aspects, then we consider it good and valid. The fewer aspects from this list that are present, then the less reliable the results are.
Terms and Definitions
Term/Definition 2.15 Experiment: A systematic method of verification that duplicates a hypothesis under controlled conditions; the only reliable way to determine scientific truth.
Term/Definition 2.16 Controlled testing: Establishing conditions where one or more well-defined physical quantities are manipulated to observe their effect on another well-defined physical quantity.
Term/Definition 2.17 Empirical (observations): Direct and measured observations obtained during an experiment.
Term/Definition 2.18 Hypothesis testing: Using statistical methods to design experiments that test a hypothesis against empirical evidence.
Term/Definition 2.19 Reproducibility: The property that key aspects of an experiment can be duplicated under similar circumstances to yield the same results.
Term/Definition 2.20 Control group: One or more experimental runs where variables are deliberately not changed, allowing comparison with runs where changes are made.
Term/Definition 2.21 Isolating the variables: Reducing the influence of quantities that are not being intentionally changed, to increase confidence that observed effects are due to the manipulated variables.
Assumptions (the implicit beliefs the author relies on)
Assumption 2.15: Experiments are the only fully reliable method for establishing scientific truth, because passive observations cannot eliminate unknown influencing factors.
Assumption 2.16: Scientific experiments must involve active manipulation and control to be trustworthy.
Assumption 2.17: The quality and validity of an experiment depend directly on how many key aspects (controlled testing, statistical design, reproducibility, control groups, variable isolation) it incorporates.
Assumption 2.18: Unknown or uncontrolled factors in observations can always potentially confound results.
Assumption 2.19: Reproducibility under similar conditions is a necessary hallmark of valid science.
Assumption 2.20: Statistical methods are essential for rigorously testing hypotheses against evidence.
Principles
Principle 2.18: Scientific truth requires active verification through controlled experiments, not merely passive observation.
Principle 2.19: Design experiments with controlled manipulation of specific physical quantities to clearly link cause and effect.
Principle 2.20: Always incorporate statistical methods for hypothesis testing against empirical data.
Principle 2.21: Ensure experiments are reproducible by others under similar conditions.
Principle 2.22: Include control groups (unchanged runs) for comparison with experimental (changed) runs.
Principle 2.23: Isolate variables by minimizing the influence of unchanged quantities.
Principle 2.24: The reliability of experimental results increases with the number of these key aspects included; missing aspects reduce confidence in the conclusions.
Principle 2.25: A good and valid experiment carefully incorporates all (or as many as possible) of these elements.
Exercise 2.4: Begin with Term/Definition 2.15 and copy it into your notebook. Reflect on its meaning for a few minutes. Note any thoughts that come to mind. Then do this for each term/definition, assumption, and principle.
Exercise 2.5: Study a famous physics experiment. Write a one page report on it.
Data Analysis
Raw and unprocessed facts and experimental results are what we call data. Data can be words, descriptions, observations, measurements, and numbers. Measurements and numbers are the most useful kinds of data, since we can analyze those kinds of data most effectively.
What do we mean by analyzing data? Here are a number of ideas that go into data analysis:
The process of removing or otherwise correcting inaccuracy, coping with missing data, and making sure the process of collecting and storing data maintains the validity of the data is what we can collectively call data cleaning.
When we look at data to list its main characteristics by using statistics, visualization, or the strength and direction of the relationship between data quantities (what we can call correlation) is collectively what we call data exploration.
There are numerous techniques to transform data into more useful forms.
We can predict patterns and relationships in data using statistics, mathematical analysis, or even machine learning. We can call this modeling the data.
The ability to identify patterns in the data is vitally important. This forms the basis of what we might call scientific laws. We will cover specific techniques of data analysis as we develop mathematical techniques.
Terms and Definitions
Term/Definition 2.22 Data: Raw and unprocessed facts and experimental results, which can include words, descriptions, observations, measurements, and numbers (measurements and numbers are highlighted as the most useful for analysis).
Term/Definition 2.23 Data cleaning: The collective process of removing or correcting inaccuracies, handling missing data, and ensuring the collection and storage process maintains data validity.
Term/Definition 2.24 Data exploration: The process of examining data to list its main characteristics using statistics, visualization, or assessing the strength and direction of relationships between quantities (correlation).
Term/Definition 2.25 Correlation: The strength and direction of the relationship between data quantities.
Term/Definition 2.26 Modeling the data: Using statistics, mathematical analysis, or machine learning to predict patterns and relationships in data.
Assumptions (the implicit beliefs the author relies on)
Assumption 2.21: All scientific progress relies on data as the starting point (raw facts/experimental results).
Assumption 2.22: Quantitative data (measurements and numbers) are superior to qualitative data (words/descriptions) for effective analysis.
Assumption 2.23: Data is often imperfect (inaccurate, incomplete, or corrupted during collection/storage) and requires preprocessing to be useful.
Assumption 2.24: Patterns and relationships in data are discoverable and form the foundation of scientific laws.
Assumption 2.25: Advanced techniques (statistics, math, machine learning) are necessary for deep data analysis, but will be introduced gradually as mathematical tools become available.
Principles
Principle 2.26: Start scientific inquiry with raw data—prioritize measurements and numbers for best analyzability.
Principle 2.27: Always clean data first: correct inaccuracies, handle missing values, and preserve validity during collection/storage.
Principle 2.28: Explore data thoroughly before deeper analysis: use statistics, visualization, and correlation to understand main characteristics.
Principle 2.29: Transform data into more useful forms when needed (various techniques exist).
Principle 2.30: Model data to predict patterns and relationships (using statistical, mathematical, or machine learning methods).
Principle 2.31: Identifying patterns in data is essential, as it directly leads to formulating scientific laws.
Principle 2.32: Data analysis techniques should be learned progressively alongside mathematical tools.
Exercise 2.6: Begin with Term/Definition 2.22 and copy it into your notebook. Reflect on its meaning for a few minutes. Note any thoughts that come to mind. Then do this for each term/definition, assumption, and principle.
The Nature of Physical Laws
Say that we have analyzed a collection of data (what are might call a set of data) relating to some specific question or phenomena. This analysis leads us to conclude that a pattern has emerged from the data. Such a perceived pattern is not considered a fact, only a potential fact. We examine other, related, data sets to see if that pattern exists there, too. If the pattern exists in every case we examine, we then gain a level of confidence in it. We then, and only then, consider it to be a fact. That confident pattern is often called a law of nature.
Laws of nature are considered to be true everywhere. This is sometimes called universality.
Laws of nature are almost always expressed as a mathematical relationship between physical quantities. This lends itself to allow us to make predictions based on the relationships.
Laws of nature are always provisional. It is possible that tomorrow someone will discover, in data, that a law of nature is wrong, or incomplete. Since such laws are open to testing, we can say that they are falsifiable.
Terms and Definitions
Term/Definition 2.27 Law of nature: A confident pattern that emerges consistently from analyzed data across multiple related data sets; considered a fact only after repeated confirmation, often expressed as a mathematical relationship between physical quantities.
Term/Definition 2.28 Universality: The property that laws of nature are considered true everywhere (in all relevant contexts).
Term/Definition 2.29 Provisional (nature of laws): Laws are always subject to potential revision or falsification based on new data.
Term/Definition 2.30 Falsifiable (laws): Laws are open to testing and can potentially be proven wrong or incomplete by future evidence.
Assumptions (the implicit beliefs the author relies on)
Assumption 2.26: Patterns in data are not automatically facts; they require repeated confirmation across multiple data sets to gain confidence.
Assumption 2.27: Scientific laws must be derived from empirical data analysis and extensive verification.
Assumption 2.28: Laws of nature apply universally (everywhere and always, within their domain).
Assumption 2.29: Mathematical formulation is the standard (and most useful) way to express laws of nature.
Assumption 2.30: All scientific laws are inherently tentative and open to revision—no law is absolutely final.
Assumption 2.31: Falsifiability is a core requirement for something to qualify as a scientific law.
Principles
Principle 2.33: Scientific laws emerge from rigorous data analysis: identify a pattern in one data set, verify it consistently in others, and only then elevate it to “law” status.
Principle 2.34: Confidence in a pattern (and thus its status as a law) increases with broader and repeated confirmation.
Principle 2.35: Laws of nature possess universality—they are assumed to hold everywhere.
Principle 2.36: Express laws mathematically to enable precise predictions based on relationships between physical quantities.
Principle 2.37: Treat all laws as provisional: they are always subject to potential falsification by new data.
Principle 2.38: True scientific laws must be falsifiable—they make testable predictions that could, in principle, be contradicted.
Exercise 2.7: Begin with Term/Definition 2.27 and copy it into your notebook. Reflect on its meaning for a few minutes. Note any thoughts that come to mind. Then do this for each term/definition, assumption, and principle.
Problem 2.8: Choose a well-known physical law (for example, Newton’s First Law of motion). Research its history: Who discovered it? What observations led to its formulation? How has it been modified or expanded upon over time? Write a short essay or presentation on this journey, focusing on the interplay between observation and theory.
Problem 2.9: Research a physical law and explore its limitations. For example, consider Newton’s Second Law of Motion in the context of very high speeds (where relativity takes over) or very small scales (where quantum mechanics applies). Hypothesize or research what happens when these laws are pushed beyond their typical domains.
The Problem of Induction
How do we know that an observed pattern in nature, represented as a physical law, is universal? Physical laws are often derived through inductive reasoning. Many observations and experiments lead us to find the underlying pattern. This leads to the problem of uncertainty of universality. This is an assumption. The assumption is based on fact, but there is no way to be sure as we cannot see every part of the universe or every time.
In fact, the Scottish philosopher David Hume (1711-1776) questioned how we can justify that we expect the future to resemble the past. He argued that induction leads to a fallacy where we assume what we are trying to prove. The Austrian/British philosopher Karl Popper (1902-1994) responded to Hume’s argument by stating that science does not attempt to prove a law by induction, rather it tries to falsify the law. If the law can’t be falsified, then it is not scientific. Of course, this solves nothing, it just changes the focus.
For the practicing scientist, since the process of induction seems to work, we will continue to use it until it doesn’t—even if it leads to a level of uncertainty. Some scientists adopt the notion that physical laws represent reality, they can be termed realists. Others see the laws as tools for making predictions, and do not correspond to reality—they are called instrumentalists.
What is the implication of the problem of induction? All laws are provisional. There is always the possibility that a result will break them. This emphasizes the need for continuous experimentation and observation, how else can laws be tested? Note that this point of view avoids the problem of choosing between realism and instrumentality.
Terms and Definitions
Term/Definition 2.31 Inductive reasoning: The process of deriving physical laws from many observations and experiments, generalizing a pattern from specific instances.
Term/Definition 2.32 Problem of uncertainty of universality (or problem of induction): The philosophical issue that there is no logical guarantee that a pattern observed in some parts of the universe/time will hold everywhere/always, since we cannot observe everything.
Term/Definition 2.33 Realists: Scientists who believe physical laws represent actual reality.
Term/Definition 2.34 Instrumentalists: Scientists who view physical laws as useful tools for making predictions, without claiming they correspond directly to reality.
Assumptions (the implicit beliefs the author relies on)
Assumption 2.32: Physical laws are derived primarily through inductive reasoning from observations and experiments.
Assumption 2.33: Universality of physical laws is an assumption, not a provable fact, because we cannot observe every part of the universe or every moment in time.
Assumption 2.34: Induction, while philosophically problematic (as highlighted by Hume), practically works well enough to be used in science until it fails.
Assumption 2.35: Science progresses through ongoing testing (falsification attempts) rather than absolute proof.
Assumption 2.36: All physical laws are inherently provisional and could be overturned by future evidence.
Assumption 2.37: Continuous experimentation and observation are necessary to test and refine laws.
Principles
Principle 2.39: Physical laws’ claimed universality rests on induction— patterns observed repeatedly are assumed to hold everywhere/always.
Principle 2.40: Induction cannot logically justify expecting the future/past/unobserved to resemble the observed (Hume's critique)—it involves circular reasoning.
Principle 2.41: Science avoids proving laws via induction; instead, it focuses on attempting to falsify them (Popper's response)—unfalsifiable claims are not scientific.
Principle 2.42: Despite philosophical uncertainties, practicing scientists continue using induction because it has been empirically successful.
Principle 2.43: Scientists may adopt realism (laws describe reality) or instrumentalism (laws are predictive tools), but neither resolves the induction problem.
Principle 2.44: The implication of the problem of induction is that all laws are provisional—always open to potential falsification by new evidence.
Principle 2.45: Ongoing experimentation and observation are essential to test laws and maintain scientific progress.
Exercise 2.10: Begin with Term/Definition 2.31 and copy it into your notebook. Reflect on its meaning for a few minutes. Note any thoughts that come to mind. Then do this for each term/definition, assumption, and principle.
Exercise 2.11: For a week, note instances where you make predictions based on past experiences. Reflect on how these predictions could fail if the future doesn’t resemble the past.
The Meaning of a Physical Theory
So, a physical theory can be thought of as the specific presentation of a set of physical situations, along with their consequences. Theories are almost always formulated by mathematics and represent a huge body of work. So, if someone says, “It is only a theory,” it is important to understand that this means many—if not most—scientists accept it as a potential fact. This, of course, recognizes the provisional nature of such potential facts. How are theories developed? There are many ways, I will present several, in no particular order.
The first method is that of estimation—a collection of techniques that allow you to guess, usually based on solid physical principles, the value of important physical quantities to within a power of ten. These are often called Fermi problems, from the famous physicist Enrico Fermi, who was always making such estimates.
Elementary abstraction is where we remove all but the simplest, most abstract, aspects of the problem. We begin by making the assumption that the simplification can be accomplished. Once this is done, we then attempt to work out the consequences so that we can test the simplification against actual data. We will get into the details of this kind of work as we go.
At the other extreme to induction is specification. Here you begin with a general principle or a mathematical formulation and you apply it to a specific situation. In this way, you can derive the necessary mathematical or computational framework tailored to a specific situation. If you use this method to make specific predictions for your situation, then you are creating a mathematical or computational model.
Another method of deriving a formulation is to propose a relationship among the variables where the structures are imposed by the requirements of the units of the physical quantities involved (we will get into the details of this in Lesson 13). This is called dimensional analysis. It is used a lot for deriving required formulas accurate to within an arbitrary constant.
Of course, one way of doing theoretical physics is to imagine a situation and then work out its consequences using your physical intuition. Such an activity is called a thought experiment. The success of such experiments is dependent upon your depth of physical intuition. Understand that calculations will likely need to be made, but the physics is the important part.
Another way of doing theoretical physics is to frame your physical phenomena according to some mathematical or computational formulation of the theory. You then make predictions based on this analogy of the real situation. As we have already noted, this is called a model, and this forms a large part of theoretical physics.
If you have a mathematical formulation that is too complicated, you might be able to simplify the situation by deriving a new quantity, and then reformulating your theory with respect to that quantity. This type of theory is called constructive, because you are constructing a new formulation.
Another way of writing a formulation is to identify if there are situations where the answer is the same no matter how you look at it. This property is called symmetry. As an example, if we can say that the answer looks the same no matter what direction you look at it, we call that spherical symmetry and it reduces the problem from three directions down to only one.
There is another way of writing theories, it rests on the assumption that different phenomena are part of a single theory that encompasses many phenomena. We call such a larger theory a unification. It was thus that Sir Isaac Newton unified the phenomena of gravity on the earth with the orbits of planets and moons.
Still, another way of doing theoretical physics is to just play around with the ideas. See if you can make something work without any formal structure.
A final aspect, allowing you to choose between competing theoretical systems is simplicity. I tend to discount simplicity arguments, unless one idea is obviously simpler than others. There is no real criteria for simplicity other than a vague sense of aesthetics.
Terms and Definitions
Term/Definition 2.35 Physical theory: A specific presentation of a set of physical situations along with their consequences; almost always formulated mathematically and representing a large body of accepted work (provisionally treated as potential fact by most scientists).
Term/Definition 2.36 Estimation (or Fermi problems): Techniques to guess the value of important physical quantities to within a power of ten, based on solid physical principles (named after the Italian/Naturalized American Physicist Enrico Fermi (1901-1954) who was widely known for his estimates).
Term/Definition 2.37 Elementary abstraction: Removing all but the simplest, most abstract aspects of a problem, assuming the simplification is valid, and then testing consequences against data.
Term/Definition 2.38 Specification: Starting with a general principle or mathematical formulation and applying it to a specific situation to derive a tailored framework or model.
Term/Definition 2.39 Mathematical/computational model: Application of a specific formulation derived by applying general principles to specific situations, used to make predictions.
Term/Definition 2.40 Dimensional analysis: Proposing relationships among variables constrained by the units of physical quantities (accurate up to an arbitrary constant; details in Lesson 13).
Term/Definition 2.41 Thought experiment: Imagining a situation and working out its consequences using physical intuition (may involve calculations, but physics/intuition is primary). The distinction between a thought experiment and a model can become murky.
Term/Definition 2.42 Model (repeated context): Framing physical phenomena according to a mathematical or computational analogy to make predictions.
Term/Definition 2.43 Constructive (theory): Simplifying a complicated mathematical formulation by deriving a new quantity and reformulating the theory around it.
Term/Definition 2.44 Symmetry: The property that the answer remains the same regardless of how the situation is viewed (e.g., spherical symmetry reduces a 3D problem to 1D).
Term/Definition 2.45 Unification: Combining different phenomena into a single overarching theory (e.g., The English polymath Sir Isaac Newton (1643-1727) unifying terrestrial gravity and celestial orbits).
Term/Definition 2.46 Simplicity (in theory choice): A vague aesthetic criterion for preferring one theory over competing ones (often discounted unless one is obviously simpler).
Assumptions (the implicit beliefs the author relies on)
Assumption 2.38: Physical theories are highly developed, mathematically formulated bodies of work that most scientists provisionally accept as potential facts.
Assumption 2.39: Theories are always provisional and open to revision.
Assumption 2.40: Multiple valid methods exist for developing theories; no single method is superior.
Assumption 2.41: Simplifications (e.g., elementary abstraction) are legitimate starting points, provided they are tested against data.
Assumption 2.42: Physical intuition is a reliable guide in thought experiments and theoretical development.
Assumption 2.43: Symmetry, dimensional constraints, and unification are powerful tools for formulating theories.
Assumption 2.44: “Playing around” informally with ideas can lead to valid theoretical insights.
Assumption 2.45: Simplicity is subjective and often overrated as a decisive criterion for choosing theories.
Principles
Principle 2.46: Dismiss casual dismissal of theories (“it’s only a theory”) — theories represent extensive, widely accepted work and are treated as potential facts.
Principle 2.47: Theories can be developed through diverse methods, including: estimation/Fermi problems, elementary abstraction (idealize then test), specification (general → specific), dimensional analysis, thought experiments (intuition-driven), analogical modeling, constructive reformulation (new derived quantities), exploiting symmetry, unification of phenomena, and even informal exploration (“play around”).
Principle 2.48: Always test theoretical consequences against data, especially after simplification or abstraction.
Principle 2.49: Use mathematical/computational models to make specific predictions.
Principle 2.50: Symmetry reduces complexity and guides formulation.
Principle 2.51: Unification is a powerful goal—seek broader theories encompassing multiple phenomena. This, of course, assumes such phenomena are linked.
Principle 2.52: Simplicity may guide theory choice but should not be over-relied upon due to its subjective nature.
Principle 2.53: Theoretical physics is creative and multifaceted—no single rigid path exists.
Exercise 2.12: Begin with Term/Definition 2.35 and copy it into your notebook. Reflect on its meaning for a few minutes. Note any thoughts that come to mind. Then do this for each term/definition, assumption, and principle.
Problem 2.13: Research a physical theory (e.g., atomic theory) and create a timeline showing its development from ancient ideas to modern understanding. Reflect on what changes in observation or technology prompted these evolutions.
Theoretical Models
If a theory is a map of a region of physics, then a model can be seen as a detailed map of a specific phenomenon, or class of phenomena, within that landscape. It is an application of a theory to the phenomena so that we can predict the behavior of that phenomenon. Most of the work we do as theorists is to invent and tinker with models of specific phenomenon. These models can be mathematical or computer-based, what we call computational.
Formulations
The specific mathematical or computational methods that we use to express a theory, or a model, is what we call a formulation of the theory or model. Getting back to the analogy of the landscape of physics, a formulation is the language that we use to make our map. The theory might show you the forest of, say, classical mechanics. The formulation gives you the trail through the forest to get to the harmonic oscillator, as an example.
Particles
The idea of a particle is an abstraction. Let’s say you want to study the motion of a car. There are the components of the engine moving and exchanging fluids, the wheels rotating and bending, people inside the car moving and breathing, the car itself moving on the road and creating turbulence in the air as it passes. This is a very complicated system.
Imagine that you can zoom out so that you can avoid most of this complexity. You no longer see the internal workings of the car. You have the same system, but it’s a simpler picture.
Now imagine that you zoom out till all you see is a speck in the distance. This speck still has all of the properties of the car that we started with. All of the internal systems are still there. We just don’t need to worry about them.
The next part requires an act of faith. We have to believe that we can treat an object as if we were zoomed out till we can treat it as if it were a speck. Think about what we are losing by treating the object as a speck. The first thing we lose is all of the internal complexities of the object. We also lose the size and shape of the object. Whenever we have an object where we choose not to worry about its size, shape, or its internal workings we can look at it as if it were a speck. Such a simplified object is called a particle. The highest level of abstraction for an object of any kind is to treat it as if it were a particle, even though we know that no object is really a particle.
There is a drawback to this! By removing the complexities you also remove levels of reality. So what is the point of the abstraction? It makes the problem simple enough to start analyzing it. Once you understand the simplified problem, you can begin to put the complexity back to make it more realistic. This process of abstraction is the heart of theoretical physics.
Thought Experiments
One aspect of theoretical phsyics that is almost as fundamental as models is the thought experiment. In the expansive, ever-changing landscape of physics, where theories and formulations map out the known and unknown, thought experiments are like the trails we blaze in our minds. They’re the imagined journeys we take through this landscape, exploring paths that haven’t been physically traveled, yet revealing new insights and possibilities.
If you picture yourself as an explorer, standing at the edge of a vast, uncharted terrain. A thought experiment is like sitting down with your map (theory) and imagining a journey through this landscape, testing routes and outcomes without ever taking a physical step. Just as you might imagine climbing a mountain or crossing a river to see what lies beyond, thought experiments allow physicists to explore the implications of theories or laws in scenarios that might be impractical, impossible, or too costly to test in reality.
Such thought experiments rely heavily on your imagination, guided by the principles of physics. They require a deep understanding of the “map” (theory) to predict what might happen if you followed a particular path. They often involve “what if” questions. What if you could travel at the speed of light? What if a cat were both alive and dead? These scenarios are like choosing different trails on your map to see where they lead. Thought experiments push the boundaries of current understanding, much like an explorer testing the edges of their map to see if it holds true or if there's more to discover.
One famous example of such a thought experiment was when Albert Einstein imagined riding alongside a beam of light, which led him to question the nature of time and space, eventually shaping his special theory of relativity. It was like imagining a path through spacetime itself.
Another example was when Erwin Schrödinger proposed a cat in a box with a poison vial triggered by a quantum event, both alive and dead until observed. This thought experiment, like a paradoxical trail, highlighted the bizarre nature of quantum superposition.
Our final example occurred when James Clerk Maxwell envisioned a tiny creature, a demon, sorting fast and slow particles to defy the second law of thermodynamics. This mental journey explored the limits of entropy and information, like finding a hidden shortcut in the landscape of thermodynamics.
How do you build a thought experiment? You begin with a solid understanding of the current scientific theory. What does the map say? Where are its edges or uncertainties? Then you propose a “what if” that challenges or extends the theory. It’s like looking at your map and wondering, “What if this mountain were twice as high?” You then use the principles of physics as your compass to navigate this imagined scenario logically. What would happen? What contradictions or insights arise? Such thought experiments often reveal gaps or inconsistencies in the map, prompting new theories, formulations, or real experiments to explore further.
In essence, thought experiments are the adventurous daydreams of science, allowing us to traverse the scientific landscape in our minds, discovering new paths, challenging old ones, and sometimes, rewriting the map entirely. They’re the art of exploration without leaving the comfort of your study, yet they have led to some of the most profound shifts in our understanding of the universe.
Problem 2.14: Study famous thought experiments like Schrödinger’s Cat, Einstein’s Elevator, or the Trolley Problem. Write an essay or discussion piece analyzing what these thought experiments reveal about physics, ethics, or reality. How do they push the boundaries of our understanding?
Problem 2.15: Speculate on what happens when someone who does not have a deep physical intuition or knowledge begins to make thought experiments.
Only Experiments Confirm Theories
We speak of thought experiments and computer experiments. These are no substitute for laboratory experiments. They do not constitute reality. Only direct physical experiment can verify any theoretical construct—whether a theory, a model, or a thought experiment. The most beautiful theory can be beautifully wrong.
Summary
Make a summary of this lesson as an exercise.
Connect the ideas of Lesson 1 to this lesson.
For Further Study
Joseph J. Carr, (1992), The Art of Science. HighText Publications, Inc. A very elementary, but still good, overview of experimental and observational science.
E. Bright Wilson, (1952), An Introduction to Scientific Research, McGraw-Hill Book Company (republished in 1990 by Dover Publications, Inc.) This is a wonderful book on scientific research, it is a little dated, but still good.
W. I. B. Beveridge, (1957), The Art of Scientific Investigation, W. W. Norton &Company, Inc. A detailed discussion of most areas of scientific research.
David Hume, (1740), An Enquiry Concerning Human Understanding, (The version I have is published in 2007 by Oxford University Press). The part on the induction problem is section IV, “Sceptical Doubts concerning the Operations of the Understanding.”
Karl Popper, (1959), The Logic of Scientific Discovery. Hutchinson & Co (republished in 2002 by Routledge).
Grok 2 helped me write some of these sections. This is a very nice Large Language Model (LLM). You can find it here: https://x.com/i/grok .