Experiments, Computability, and the Existence of Physical Functions

This paper argues that reproducible laboratory experiments function as algorithms computing physical maps from inputs to outputs, demonstrating that the existence of these functions is compatible with finite-precision measurements and distinct from questions of computability or protocol independence.

The Gist

Imagine you are a chef trying to write down a recipe for "the perfect soup." A strict mathematician might ask, "Does 'perfect soup' actually exist as a single, universal object, regardless of who cooks it, what stove they use, or how they taste it?" The mathematician might worry that because every chef's soup is slightly different, there is no single "soup function" to be found.

This paper, written by physicist Isaac Pérez Castillo, argues that this worry is based on a misunderstanding of what an experiment (or a recipe) actually is. The author suggests we stop looking for a magical, invisible "perfect soup" floating in the universe and start looking at the recipe itself.

Here is the paper's argument, broken down into simple concepts and analogies:

1. The Experiment is a Machine, Not a Mystery

The paper starts with a simple definition: An experiment is just a finite list of steps that you follow to get a result.

• The Analogy: Think of a vending machine. You put in a specific code (the input), press a button (the procedure), and after a few seconds, it drops a snack (the output).
• The Point: You don't need to know the deep physics of how the snack was made to know that the machine works. As long as the machine has a clear set of steps, a clear way to start, and a clear way to stop, it is a "procedure." The paper argues that every lab experiment is just like this vending machine. It takes a prepared sample, follows a rule, and spits out a number.

2. The "Bridge" to Math (The Church-Turing Principle)

The author uses a concept called the "Physical Church-Turing bridge principle." This is a fancy way of saying: "If a human can follow a set of rules to get a result, a computer can also follow those rules to get the same result."

• The Analogy: Imagine you are teaching a robot to bake a cake. If you can write down the instructions clearly enough for a human to follow (e.g., "mix for 2 minutes," "bake at 350 degrees"), then a computer can also follow those instructions.
• The Conclusion: Because experiments are just sets of instructions, they are "computable." If a procedure is computable, then the "map" it creates (Input $\to$ Output) exists. The function exists because the machine that runs it exists.

3. The "Finite Precision" Problem (Why We Don't Need Perfect Numbers)

A common objection is: "But experiments aren't perfect! They give us numbers like 3.14 or 3.141, but never the exact infinite number $\pi$. Does the function exist if we can't get the exact answer?"

• The Analogy: Imagine you are trying to measure the length of a room. You use a ruler and get 10 feet. Then you use a tape measure and get 10.1 feet. Then you use a laser and get 10.12 feet. You never get the "infinite" decimal, but you are getting closer and closer.
• The Paper's View: The paper says this is fine. In the world of "computable analysis" (a branch of math), a number is considered "computable" if you can get as close to it as you want, step by step. You don't need to print the whole infinite number in one second. You just need a procedure that says, "If you want more accuracy, here is how to get it."
• The Takeaway: The experiment doesn't need to output a perfect, infinite real number to be valid. It just needs to be able to give you a better approximation whenever you ask for one.

4. The "Solubility" Story (Why Context Matters)

The author tells a story about a chemist friend who was worried about "solubility" (how much sugar dissolves in water). The friend asked, "Does a 'solubility function' exist?" The friend was confused because the answer changes if you change the temperature, the type of water, or how you mix it.

• The Analogy: Imagine asking, "What is the price of a house?" The answer depends entirely on which house, which city, and what time of day you ask. There isn't one single "House Price" for the whole universe.
• The Paper's Solution: The paper says, "Yes, the function exists, but only for the specific recipe you are using."
  • If you fix the temperature, the water type, and the mixing method, you have a specific "Solubility Machine."
  • That machine computes a specific map.
  • The function exists for that machine.
  • If you change the recipe (e.g., use hot water instead of cold), you are building a different machine that computes a different map.

5. What About Randomness? (The Dice Roll)

Some experiments are random. If you run the same test ten times, you might get ten slightly different numbers. Does the function still exist?

• The Analogy: Imagine a slot machine. You pull the lever (input), and it gives you a random number (output). The result isn't the same every time.
• The Paper's View: The function still exists! But instead of a map that gives you one specific number, the function is now a map that gives you a distribution of numbers (a pattern of randomness).
• The experiment computes a "sampler." It doesn't give you a single point; it gives you a reliable pattern of points. The existence claim holds; the object just changes shape from a single dot to a cloud of dots.

Summary: What the Paper Actually Claims

The paper is not saying that everything in physics is computable, or that all experiments will eventually agree on one single "universal truth."

Instead, it makes a much simpler, sharper claim:

1. Stop looking for magic: Don't worry if a "perfect, protocol-independent" function exists in the abstract.
2. Look at the procedure: If you have a fixed recipe (protocol), a fixed set of rules, and a way to report the result, that recipe is a function.
3. It exists because it runs: Because the recipe is a finite set of steps that a computer could follow, the function it computes exists.
4. Context is King: The function belongs to the specific experiment you are running. If you change the experiment, you get a different function. That doesn't mean the first one didn't exist; it just means you changed the machine.

The Bottom Line:
The paper tells us to stop asking, "Does the true solubility exist?" and start asking, "What does this specific experiment compute?" Once you define the experiment clearly, the answer is always "Yes, it computes a function." The function exists right there in the machine's output.

Technical Summary

Technical Summary: Experiments, Computability, and the Existence of Physical Functions

Problem Statement
The paper addresses a conceptual difficulty in experimental science regarding the "existence" of physical functions. Using the example of a solubility function, the author notes that while laboratories routinely measure quantities, the reported values depend heavily on specific protocols (solvent, temperature, equilibration criteria, sample preparation) and reporting conventions. This dependence raises the question of whether a well-defined, protocol-independent mathematical function exists. Traditional mathematical views often require functions to be continuous maps between well-defined spaces with protocol-independent domains and codomains, leading to skepticism about the existence of such functions in messy experimental contexts. The paper seeks to resolve this by reframing the existence question through the lens of computability theory.

Methodology
The author employs a framework combining the philosophy of measurement, metrology (specifically the International Vocabulary of Metrology and the Guide to the Expression of Uncertainty in Measurement), and computable analysis. The core methodological move is to treat a reproducible laboratory experiment not as a physical process evolving in time, but as a finite, effective procedure (an algorithm) that maps admissible inputs to finite reports.

The argument proceeds in three logical stages:

1. Structural Analysis: Defining an experiment as a sequence of steps involving a protocol ($P$), background conditions ($C$), and a reporting rule ($R$) that terminates with a finite output.
2. Bridge Principle: Introducing a restricted Physical Church–Turing bridge principle. This thesis posits that for every fixed, finitely specified, reproducible laboratory protocol, there exists a Turing-computable procedure that reproduces the protocol's input-output behavior.
3. Computable Analysis: Applying the tools of computable analysis to handle finite-precision measurements and stochasticity, distinguishing between the direct computation of finite reports and the approximation of real-valued quantities.

Key Contributions and Results

• Existence of the Computed Map (Proposition 1):
  The paper establishes that a fixed experimental protocol defines a computable function $F: D \to \text{Rep}_{\text{fin}}$, where $D$ is the set of admissible finite input descriptions and $\text{Rep}_{\text{fin}}$ is the set of finite report objects (e.g., rational numbers, intervals, strings with uncertainty). Under the Physical Church–Turing bridge principle, the experiment is an effective procedure that halts with a report; therefore, the map it computes exists in the minimal mathematical sense. The existence of the function is not a precondition but a consequence of the experiment being a halting effective procedure.

• Finite Precision and Real-Valued Idealizations (Proposition 2):
  The paper clarifies that experiments do not output exact real numbers in a single step. Instead, using computable analysis, a real-valued quantity is considered computable if there is a procedure that produces rational approximations to any requested accuracy ($|q_n - y| \le 2^{-n}$). The paper argues that finite-precision measurement is not an obstacle to the existence of real-valued functions but is the very mechanism by which they are computed. A real-valued function exists in this sense only if the finite reports can be organized into a coherent refinement scheme (e.g., increasing integration time or replicate counts) that converges to a target value.

• Protocol Dependence and Operational Measurands:
  The paper explicitly rejects the idea that a single protocol-independent function is required for existence. Instead, it defines an operational measurand ($F_{P,C,R}$) which is specific to the protocol, conditions, and reporting rule. Changing these parameters changes the computed map. The question of whether different protocols measure the "same" underlying quantity is a separate, higher-level scientific achievement requiring calibration and robustness arguments, not a prerequisite for the existence of the function computed by a specific protocol.

• Stochasticity as a Computable Object:
  For stochastic experiments where repeated runs yield different reports, the paper argues that the computed object is not a failure of existence but a shift in the type of function. In stochastic cases, the experiment computes a sampling procedure for a probability distribution over reports ($\mu_{x,k}$). The existence claim holds: the experiment defines a computable map from inputs to report distributions (or a sampler for them), rather than a deterministic point-valued map.

Significance and Scope
The paper's significance lies in separating three distinct questions often conflated in the philosophy of science:

1. Whether the function exists (answered affirmatively for fixed protocols via the bridge principle).
2. Whether the function is computable (answered affirmatively under the bridge principle).
3. Whether results from different protocols measure the same quantity (a separate scientific problem requiring convergence and calibration).

The author explicitly limits the scope of the claims:

• The paper does not prove a universal Physical Church–Turing thesis for all physical processes or arbitrary physical evolution.
• It does not claim that all protocols converge to a single protocol-independent real-valued quantity.
• It does not address computational complexity or the practical feasibility of evaluating these functions.

The conclusion is that the "existence" of a physical function is secured once an experiment is specified as a finite halting procedure. The scientific work then shifts from proving existence to explaining the structure of the computed map, comparing it across protocols, and determining when unification into a protocol-independent quantity is warranted. This reframes the problem from a metaphysical worry about the reality of unmeasured quantities to a methodological analysis of what specific experimental procedures actually compute.