Does 1. extra-, 2. intra-polation fall under abduction, induction or deduction?

Question

1. How does extrapolation relate to abduction, deduction, and/or induction? Scilicet, does abduction, deduction, and/or induction fully encompass Extrapolation?

2. Same question for Interpolation.

I screenshot from this Youtube video that distinguishes abduction, deduction, vs. induction.

Abduction as an Aspect of Retroduction | Chiasson, Phyllis | Commens

Definition of Terms

Despite the risk of temporarily violating Peirce’s “terminological ethics,” let us begin by examining the terminological basis for identifying abduction and retroduction by distinct meanings. This “terminological” violation is only temporary, since Peirce, by selecting the terms “abduction” and “retroduction,” provided us with a terminological justification for applying these terms to distinguish between two levels of concepts. Since he was so precise in his use of definitive language, the rationale for separating the meanings in this way should begin with an examination of the root meanings of the words “retroduction,” “abduction” (and, while we are at it, for “deduction,” and “induction” as well).

Retroduction:
The prefix “retro,” occurs in loanwords from Latin having to do with going backward. Yet, the prefix “retro” provides an implication of deliberateness–of deliberately “choosing” to go backward for a purpose. Thus “retroactive” means choosing to go back to an earlier date and make something operative as of that date. “Retrofit” means choosing to go back and modify an earlier model of something with an improvement of some sort. The combination of the prefix “retro” (as deliberately “going backward”) with the suffix “ductive” from the Latin ducere (to lead) places the meaning of retroduction as “deliberately leading backward.” This implies that retroduction is intended to be a deliberate and recursive process involving more than the making of an abductive inference. Its Latin roots indicate that “retroduction” refers, not only to the apprehension of a “surprising fact,” and an ensuing hunch, but also that the hunch, once formed, is deliberately and recursively taken “backward” for analysis and adjustment (requiring deduction and induction), before it is engendered into a hypothesis worthy of extensive testing.

Abduction:
The prefix “ab” appears in loanwords from Latin where it meant “away from.” Thus we have words like “abdicate” and “abolition”–going “away from” the throne and from slavery, respectively. Thus, when the prefix “ab” (away from) is combined with the suffix “ductive” (from the Latin ducere, meaning to lead) we have the meaning of abduction as “leading away from.” The term “abduction” fits well with the concept of abduction as moving “away from” a particular course or topic, as one would when responding to an anomaly, or a “surprising fact.” The Latin root for “abduction” does not fit with the idea of going backward to explicate and evaluate an idea. Rather, this root indicates that the outward movement of an abductive inference allows the result of such an inference to be left as a completion, or used as the sole means for further exploration of possibilities–as in the arts.

Deduction:
The prefix “de” from Latin loanwords refers to separation, removal, and negation. When we combine the prefix “de” (to separate) with the suffix “ductive” (to lead), we have the meaning of deduction as “leading to separation, removal, or negation,” which are the goals and consequences of deductive reasoning.

Induction:
The prefix “in,” also from the Latin has to do with inclusion. Thus, the prefix “in” (to include) combined with the suffix “ductive” means “leading into” (or including), as one would do when reaching a conclusion by estimating from a sample, or generalizing from a number of instances.

Therefore, based upon their Latin derivations (to which Peirce was partial, as he was for Greek roots) our four terms have the following meanings:

• Retroduction = deliberately leading backward.

• Abduction = leading away from

• Deduction = leading to separation, removal, or negation.

• Induction = “leading into” (or including).

Answer 1

Short Answer

Both mathematical interpolation and extrapolation are forms of estimation. Any form of methodological estimation leads to an uncertain conclusion about the object of the mathematical model by definition. This means that both practices are inductive forms of reasoning, even if they rely on deductive techniques of calculation to provide an answer.

Long Answer

Interpolation provides additional data based on inference of an existing set. Extrapolation provides additional data based on inference outside of the existing set. The former technique might be thought to "fill holes" in the data set, whereas the latter, is about "predicting" future values. A simple way to understand this is to have a data set of points on a plane that approximates a continuous curve. For any set of points, it's possible to come up with multiple polynomials that contain all domain/range pairs. Once a continuous curve is selected, one then can ask what a point would be further in the domain by calculating.

In both cases, there is only relative certainty about the conclusions drawn about the interpolated and extrapolated points. The uncertainty in interpolation exists because of the indeterminacy of a finite set leading to the construction of a function over R. Extrapolation is indeterminant in this example because the point generated, though done so deductively through calculation, ultimately rests on the indeterminacy of the curve selected from the available family of curves. Note that the example cited from curve fitting is but one example, and there are several classes of inductive methods used in mathematics.

Where the intersection is particularly relevant is the use of mathematical models in science to model complex phenomena. In pure mathematics, which might be considered largely concerned with constructing provable theorems within consistent systems axiomatically, scientists who use applied mathematics are frequently concerned with how well their models fit the physical phenomena. For instance, in biological population modeling, one certainly doesn't measure every organism, but rather applies statistical and algebraic models to make a prediction about an ecosystem, for example. Science has long been the playground of the inductivist because physical reality is most accurately modeled by complex systems. Any physicist knows for instance that the moment one goes from modeling a two-body to a three-body system, mathematical models quickly lose their efficacy because of the stochiastic nature of nature. Charles Sanders Pierece, in fact, held as one of his core tenets that the random was built into the physical universe. From his page on WP:

Peirce held that science achieves statistical probabilities, not certainties, and that spontaneity (absolute chance) is real.

His tychism presaged the conclusion of quantum mechanics who eventually settled on the idea that "God does play dice" with the universe, in defiance of Einstein's claims during the debate on the philosophical implications of quantum physics. From 'tychism', WP:

Tychism (Greek: τύχη, lit. 'chance') is a thesis proposed by the American philosopher Charles Sanders Peirce that holds that absolute chance, or indeterminism, is a real factor operative in the universe. This doctrine forms a central part of Peirce's comprehensive evolutionary cosmology. It may be considered both the direct opposite of Albert Einstein's oft-quoted dictum that: "God does not play dice with the universe" and an early philosophical anticipation of Werner Heisenberg's uncertainty principle.