Projection and Invariance in Scientific Explanation

This paper argues that the persistence of superseded theories, the coexistence of incompatible frameworks, and the productivity of multiple descriptions in science are best explained by a structural account of "projection," which maps underlying complexity to structured descriptive spaces to reveal invariants, thereby reconciling scientific realism with explanatory pluralism through a distinction between vertical refinement and horizontal irreducibility.

The Gist

The Big Idea: Science is Like Taking a Photo

Imagine you are trying to describe a busy city street to a friend.

• Option A: You describe every single atom in every car, every pedestrian, and every leaf on every tree. This is the "complete truth," but it's so overwhelming that your friend learns nothing about how the city works. It's useless.
• Option B: You take a photo from a helicopter. You see the flow of traffic, the red lights, and the grid of streets. You ignore the individual cars and people. You see the pattern (the traffic jam) that you couldn't see if you were standing on the sidewalk.

Sticker argues that all scientific theories are like Option B. They are "projections." They take a messy, complex reality and intentionally blur out (suppress) the tiny details so we can see the big, stable patterns (invariants) that actually matter.

The paper tries to solve three big mysteries about how science works using this idea.

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Mystery 1: Why do "Wrong" Theories Keep Working?

The Puzzle: We were taught that Newton's physics was "wrong" because Einstein came along and fixed it. So why do NASA engineers still use Newton's math to send rockets to the moon? If it's wrong, why does it work so perfectly?

The Explanation:
Think of Newton's physics as a low-resolution map. It's not the "true" map of the universe (Einstein's is), but for driving a car or flying a plane, the low-resolution map is perfect.

• The Analogy: Imagine a map of a country that shows cities and highways but ignores the potholes. If you are driving a truck, you don't care about the potholes; you care about the highway. The map is "incomplete," but it is invariant—the highway stays a highway whether you zoom in or out.
• The Takeaway: Old theories aren't just "mistakes." They are limiting cases. They are the "zoomed-out" version of the new theory. They keep working because they are capturing a real pattern that still exists, even if we now understand the deeper details.

Mystery 2: Why Do Scientists Argue About the Same Thing?

The Puzzle: In biology, there are over 20 different definitions of what a "species" is. In personality psychology, there are different models (Big Five vs. HEXACO). If science is finding the "one true answer," why are there so many competing answers that all seem useful?

The Explanation:
This is because different questions require different lenses.

• The Analogy: Imagine a forest.
  • A botanist looks at the trees and sees "species" based on leaf shape and DNA.
  • A firefighter looks at the same forest and sees "fuel loads" based on how dry the wood is.
  • A tourist sees "scenic spots."
    None of them are lying. They are all looking at the same forest, but they are using different projections to answer different questions.
• The Takeaway: Science doesn't always converge on one single definition. Sometimes, the world is rich enough to support multiple valid ways of looking at it. The "Big Five" personality model and the "HEXACO" model aren't fighting to be the only truth; they are just highlighting different patterns in human behavior that are useful for different purposes.

Mystery 3: Why Can't We Always "Zoom In" to Explain Things?

The Puzzle: We often think that if we explain something using tiny particles (like atoms or neurons), we have the "real" explanation. But Sticker says sometimes, looking at the tiny parts makes the big picture disappear.

The Explanation:
Some patterns only exist when you look at the group, not the individual.

• The Analogy: Traffic Jams.
  Imagine a traffic jam moving backward on a highway.
  • If you look at one car, you see a driver pressing the gas pedal. You cannot find the "traffic jam wave" inside that one car.
  • The "wave" is a pattern that only exists when you look at hundreds of cars together.
  • If you tried to explain the traffic jam by studying the physics of a single engine, you would fail. The "jam" is a horizontal invariant—a truth that exists at the group level and cannot be found by just zooming in on the parts.
• The Takeaway: Some laws (like Gresham's Law in economics: "Bad money drives out good") only work at the level of human behavior. You can't find them by studying the atoms in a dollar bill. The "explanation" requires ignoring the tiny details to see the group pattern.

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The Two Types of Scientific Progress

Sticker says science moves in two directions:

1. Vertical Progress (Refining the Lens):

   • This is when we get a sharper, more detailed picture that includes the old picture.
   • Example: Einstein didn't throw away Newton; he showed that Newton's laws are just what happens when things move slowly. It's like upgrading from a standard definition photo to a 4K photo. The old photo is still there, just inside the new one.

2. Horizontal Progress (Changing the Lens):

   • This is when we discover a completely new way of looking at the world that the old way couldn't see.
   • Example: Realizing that "money" isn't about metal or paper (physics), but about trust and value (economics). You can't get to this truth by studying physics harder; you have to switch to a different "projection."

The Bottom Line: "Blindness" is a Feature, Not a Bug

The paper concludes with a surprising thought: Science works because it leaves things out.

If a theory tried to describe everything (every atom, every thought, every coin flip), it would be useless.

• The Coin Flip: Before you flip a coin, the chance of heads is 50%. This is a real fact about the situation, not just your ignorance. Once the coin lands, the physics takes over, and the probability becomes 0% or 100%. The "50%" didn't disappear because it was fake; it disappeared because you changed the question (from "what will happen?" to "what happened?").

In short:
Scientific knowledge isn't a ladder leading to a single "God's Eye View" of the universe. It's a toolbox. Sometimes you need a hammer (physics), sometimes a screwdriver (biology), and sometimes a wrench (economics). They all describe the same world, but they highlight different patterns by ignoring different details. And that's exactly how we understand the world.

Technical Summary

1. Problem Statement

The paper addresses three persistent structural features of scientific knowledge that the standard "progressive replacement" model (where better theories simply displace wrong ones) fails to explain:

1. Persistence of Superseded Theories: Why do "wrong" or incomplete theories (e.g., Newtonian mechanics, phlogiston theory) continue to work with high precision within their original domains after being superseded?
2. Principled Pluralism: Why do mature sciences sustain multiple, incompatible frameworks simultaneously (e.g., competing definitions of "species" in biology, Big Five vs. HEXACO in psychology) without this constituting a failure of knowledge?
3. Blindness to Omission: Why do representational systems (including scientific theories) fail to register their own limits? (Illustrated by Mariotte's blind spot in vision, where the gap is filled seamlessly rather than appearing as an absence).

The author argues that the standard view treats probability and theory change as binary (real vs. subjective, correct vs. wrong), whereas these phenomena suggest a third possibility: that scientific descriptions are projections that reveal genuine invariants by suppressing specific underlying details.

2. Methodology and Core Framework

The paper proposes a structural framework based on the concept of Scientific Projection. This is not merely idealization or abstraction, but a specific representational operation defined by four linked components:

• Projection as Mapping: A principled mapping from an underlying complex state space to a structured descriptive space.
• Compatibility Classes: The projection partitions the underlying states into equivalence classes. States within a class are treated as interchangeable for the projection's purposes.
• Descriptive Variables: Variables are defined over the "quotient space" created by these classes.
• Invariants: Patterns, laws, or regularities that remain stable across the variation suppressed within a compatibility class.

Key Distinctions:

• Representational Structure vs. Substrate Interpretation: Every framework has a structure (variables, classes) and a substrate story (what causes the invariants). The structure can survive even if the substrate interpretation is replaced (e.g., Darwin's selection theory surviving before the discovery of DNA).
• Legitimacy Criteria: A projection is adopted if it satisfies:
  1. Empirical Adequacy: Accounts for known facts without excessive distortion.
  2. Ontological Consonance: Consistent with the best current understanding of what entities/relations exist (a rational, revisable constraint, not just tradition).
• Explanatory Success Criterion: A projection is successful when the variation it suppresses within a compatibility class is irrelevant to the invariant it seeks to capture.

3. Key Contributions and Results

The paper distinguishes two types of scientific progress/cases, resolving the tension between reductionism and pluralism:

A. Vertical Cases (Refinement and Embedding)

• Definition: Successive projections that refine the same fundamental structure. Earlier projections survive as limiting cases of more general ones.
• Mechanism: The successor projection expands the ontological space or mathematical generality, embedding the predecessor.
• Examples:
  • Astronomy: Ptolemaic circles $\to$ Copernican circles $\to$ Keplerian ellipses. The circle is the degenerate case (eccentricity = 0) of the ellipse.
  • Physics: Newtonian mechanics is a limiting case of Einsteinian relativity at low velocities.
  • Biology: Darwin's theory of natural selection (1859) identified the correct variables and invariants before the substrate (genetics) was known. The representational structure survived the arrival of the causal substrate.
• Result: Superseded theories work because they track genuine invariants within a specific domain; they are not "errors" but limiting cases of a broader truth.

B. Horizontal Cases (Level-Specific Irreducibility)

• Definition: Projections that reveal invariants constitutively specific to a level of description. These invariants are not recoverable from lower-level descriptions as simple refinements because the lower-level descriptions define different compatibility classes.
• Mechanism: Multiple Realizability. The same higher-level property (e.g., "money," "traffic wave") is instantiated by physically heterogeneous states. The projection suppresses physical differences to reveal a functional invariant.
• Examples:
  • Gresham's Law: Holds across coins, paper, and digital currency. A neural or physical description of the coins cannot reveal the law because the law depends on the functional projection (monetary role), not the physical composition.
  • Traffic Flow: Traffic jams propagate as waves with specific speeds. This invariant is invisible at the level of individual vehicle mechanics; it only emerges under an aggregate projection.
  • Universality Classes (Physics): Different physical systems (water, magnets) with different microscopic constituents share identical critical exponents near phase transitions. The Renormalization Group projection suppresses microscopic details to reveal shared structural invariants.
• Result: Lower-level descriptions do not "explain away" higher-level laws; they render them explanatorily inaccessible. Explanation requires suppression.

C. Reinterpretation of Theory Change (Kuhn Revisited)

• The paper revises Thomas Kuhn's account of scientific revolutions:
  • Trigger: Change is not driven solely by anomaly accumulation (crisis) but by the constructive arrival of a new projection that resolves anomalies while preserving the predecessor's successes.
  • Incommensurability: The paper rejects "full" incommensurability. While substrate interpretations (worldviews) may change sharply, representational structures often carry forward via embedding. The "gestalt shift" is the discovery that a new geometric/ontological space was always available but previously forbidden.

D. Application to Probability

• The opening puzzle of the coin flip is resolved by viewing probability as an invariant of a specific projection.
• Before the flip: Probability (1/2) is a real invariant of the coarse-grained projection (suppressing trajectory details).
• After the flip: The projection changes to a finer-grained one (deterministic trajectory). The 1/2 probability does not vanish because it was "wrong"; it becomes inapplicable because the question (and the projection) changed.
• Interpretations: Frequentist, Bayesian, and Propensity theories are viewed as different legitimate projections of the same underlying phenomena, each revealing different invariants.

4. Significance and Implications

• Reconciliation of Realism and Pluralism: The framework allows for scientific realism (invariants are real) while validating explanatory pluralism (multiple frameworks can be simultaneously legitimate).
• Normative Criterion for Progress: It provides a standard to distinguish genuine theoretical progress (embedding/refinement) from mere replacement. Progress is not just "more detail" but the discovery of new levels of description where new invariants exist.
• Defense of Special Sciences: It validates the autonomy of fields like economics, biology, and psychology. Their laws are not "approximations" of physics but genuine invariants that require specific projections (suppression of physical detail) to be visible.
• Nature of Explanation: The paper argues that explanation requires suppression. A description that suppresses nothing is identical to the phenomenon and explains nothing. The "incompleteness" of scientific description is not a defect but the constitutive condition of explanatory success.
• Ontological Consonance: It reframes ontological commitments not as absolute truths but as rational constraints that are revisable based on evidence, explaining why scientists feel their frameworks are "necessary" (blindness to omission) even when they are provisional.

In conclusion, the paper posits that scientific knowledge is not a single converging line toward a "view from nowhere," but a landscape of projections. Each projection partitions reality to make specific invariants visible, and the persistence of old theories and the coexistence of new ones are structural features of how we access and explain a world that is inexhaustible by any single framework.