An explanation connects stated assumptions to an outcome in a way that constrains what else can happen. Agreement with observations is necessary for an empirical account, but a successful fit alone does not identify a unique mechanism or guarantee a reliable prediction.
A fit answers only part of the question
A formula may reproduce every measurement in a small table. We can then ask two further questions: why does that formula apply, and what should happen in a case that did not help select it? The first asks about structure or mechanism; the second probes the reach of the account.
Explanations need not all have the same form. Some show how an interaction produces an effect. Others derive a constraint from a symmetry or establish a result under explicit mathematical assumptions. Their strength depends on the connection they demonstrate, including which assumptions do the explanatory work.
Three points, many matching models
Take three exact illustrative observations: (0, 0), (1, 1) and (2, 4). The quadratic y = x² passes through all three. So does every member of this family, for any coefficient λ:
yλ(x) = x² + λx(x − 1)(x − 2).
The added term vanishes at each observed input. At x = 3, however, the prediction is 9 + 6λ. Choosing λ = −1, zero or one gives 3, 9 or 15. Perfect agreement at the three calibration points leaves a substantial predictive difference.
Explore a shared fit
Same observations, different predictions. The gold points are fixed illustrative data. Change the alternative model and choose a new input at which to compare its prediction with the quadratic.
Solid green: quadratic x². Dashed purple: selected alternative. Gold circles: the three calibration points. Burgundy square: the alternative's prediction at the selected input. All quantities are dimensionless.
The selected input is a question posed to the model, not an additional measured point. No probability distribution over the coefficient has been assumed. The slider range is chosen for illustration, rather than inferred as an uncertainty interval from the three data.
What would additional evidence establish?
An independently obtained measurement at x = 3 could discriminate among these alternatives, subject to its uncertainty. If the result were 9 exactly, it would select λ = 0 within the displayed family. It would not prove that no other functional family could pass through all four points.
For example, adding a multiple of x(x − 1)(x − 2)(x − 3) preserves all four values. This does not make evidence powerless. It shows why finite data are interpreted together with physical knowledge, restricted model classes and checks on their assumptions.
Choosing the simpler quadratic may be reasonable when an independent argument supports that form. Simplicity alone is not a guarantee that nature has selected it. NIST's modelling guidance distinguishes model selection, fitting and validation; the activities reinforce each other without being interchangeable.
From postulate to prediction
For SMM/DTT, the broad proposal of a unified generative source gains explanatory force when shared assumptions restrict several outputs. The book's account of an auditable model asks for explicit inputs and predictions tested beyond the cases used to choose those inputs. A mathematical result can be sound while its physical identification remains a separate task.
This distinction helps readers assess progress fairly. A conditional theorem establishes what follows from its premises. A fit establishes agreement within the fitted setting. A successful new prediction supports the model over alternatives that disagree there. None should silently stand in for all the others.
Give the explanation a testable extension
When evaluating an account, identify one outcome it excludes, one assumption on which that exclusion depends and one new comparison it predicts. An explanation becomes more informative as its independently justified commitments leave less room to adjust every result after it is known.
Sources and further reading
- NIST/SEMATECH, Basic steps for developing an effective process model.
- Mohamed Haj Yousef, Tests with explicit alternatives and A model that can be audited.
- Continue with Matching claims with evidence and Recognizing circular derivations. The polynomial family is an original illustration.