Correctness in Applied Mathematics

Wandering Significance, p. 534:

As observed before, an important purpose of establishing a semantic picture of this type is that it allows us to check various inference rules $\mathcal{R}$ (e.g. Heaviside's well-known "shift" procedure) for their generic soundness: that is, within every mathematical structure $\mathcal{S}$ compatible with the Laplace + distributions reading, $\mathcal{R}$ carries true claims to true claims. Assigning a fresh picture to a term can therefore serve as a novel source of corrective directivities upon its usage.

Wandering Significance, p. 215:

applied mathematicians have learned that they must investigate, to the best of their ability, the validity of their reasoning principles from a generic and correlational point of view. That is, they must first model mathematically the range of physical circumstances $\mathcal{S}$ in which they expect to apply the rule and then verify whether the sentences progressively ground out by the method will unfold in proper alignment with every $\mathcal{s}$ in $\mathcal{S}$.

Wandering Significance, p. 220:

Indeed, the modus operandi of most correctness proofs validity [sic] with which I am familiar proceed by first characterizing the (often abstract) nature of this correlated data and then showing that each step within the routine handles such information appropriately under generic conditions.

Wandering Significance, p. 528:

An examination of the sort sketched in the fine print, where we study the unfolding steps of a formalism to insure that they always lead to proper results with respect to the settings for which they are intended, is called a soundness proof in a logical context. We have already witnessed another verification of the same type in 4,x, in the guise of probing Euler's method for its correctness (the more usual term than soundness in such contexts). Through those means we discovered that a naive employment of Euler's rule is not correct and supplementation by a Lipschitz proviso is required… I consider both cases to illustrate studies of predicate behavior in relation to proposed pictures of their intended settings.

Wandering Significance, p. 552:

a chief manner in which semantic pictures assist us lies in their ability to collaborate or reject established inferential rules $\mathcal{R}$ through generic investigations of soundness in the mode of 4,xi. That is, we use our picture $\mathcal{P}$ to carve out a generic range of mathematical settings which we hope will accurately model all of the potential settings in which $\mathcal{R}$ may be applied. $\mathcal{P}$ then certifies $\mathcal{R}$ as sound if $\mathcal{R}$ always carries true assertions to true assertions over very setting tolerated by $\mathcal{P}$.

Wandering Significance, pp. 626-7:

Both parties to this dispute have misunderstood the real, but limited, value that soundness proof evaluations play within the unfolding histories of predicative vocabulary. This oversight is the consequence of considering logical specimens in isolation from their cousin projects in rule ratification, such as scrutinizing Euler's rule to learn its proper range of applications (4,xi) or propping up the operational calculus through Laplace transforms and distributions (8,ix). As I have continuously stressed, such studies also represent examinations of how inferential rules $\mathcal{R}$ behave with respect to generic associated models of their word/world correlations. Nobody would fancy that such probes are trivial or uninformative, yet they utilize "true in $\mathcal{S}$" determinations in exactly the same manner as our logical examination.

'models of word/world correlations' = different 'pictures' of how Heaviside's manipulations work. e.g. Schwartz picture vs Mikukinski vs Laplace transform.
Of course the deflationist could always replace "true in __" with some longer description.

Wandering Significance, p. 627:

It is in this context that the proper salience of our soundness proof for reasoning by cases should be addressed: our examination lays down the prerequisite conditions to be sought within the more detailed picture that we provide for our more powerful reasoning rules. If we can't align "D" and "t" with simple sets and objects inside those models, we must be wary of following logical impulse willy-nilly across this language… our soundness proof sketches the minimal structure that we must locate within a broader scale picture to be able to announce, "Whew! At least we won't have to worry about failures of logical reasoning here."

Wandering Significance, p. 264:

we merely need to substitute "Euler's method" for "practices" into Johnson's [sic] "the only real legitimation of these practices consists in showing their worthiness to survive on the testing ground of everyday life" to generate a palpable falsehood. Indeed, the better form of "legitimation" we desire for Euler's method is a proof of its correctness (as in 4,x: a result squarely based upon the correlational studies that Johnston abjures).

"What Can Contemporary Philosophy Learn from Our 'Scientific Philosophy' Heritage?", p. 169:

we are considering a non-logical equivalent to what logicians call a soundness question with respect to a logical reasoning rule R: beginning with premises $\Gamma$ ($\approx$ our data $\mathbf{D}$), will the conclusions reached by $\mathbf{R}$ remain true in every model $\mathbf{M} \in M$ ($\approx \mathbf{C} \in C$) satisfying $\Gamma$? R is said to be semantically justified if it proves 'sound' or 'correct' under such a generic correlational study. A chief purpose of a 'semantic portrait' of descriptive language is to provide the underpinnings for 'soundness' evaluations such as this.

"What Can Contemporary Philosophy Learn from Our 'Scientific Philosophy' Heritage?", p. 174:

semantics-for-the-sake-of-soundness-proofs should not be viewed as completely capturing in a traditionalist manner the hypothetical 'established contents' of the words under examination. Instead, the correlational studies embodied in successful soundness proofs should be primarily regarded as important vehicles for reorienting established usage towards higher standards of performance, exactly in the 'get me the right numbers' vein that the 'scientific philosophy' tradition has always emphasized.

page revision: 12, last edited: 30 Sep 2013 13:21