This latest post caught my attention. Since my first degree was a B.S. in "computational mathematics," I thought that I would weigh-in.
One can make the distinctions as follows, beginning with pure vs. applied mathematics. I will give a negative definition, since I am not so skilled with the Peircean terminology used so far; applied mathematics is the use of mathematics as a formal, ideal system to specific problems of existence. For instance, consider the use of statistical confidence intervals to solve problems in manufactoring relating to the rate of production of defective vs. non-defective goods. Pure mathematics is not bound by existent conditions, but "pure" becomes "applied" when used in that context. Hence, I am treated applied mathematics as an informal, existential constraint that alters the purpose and use of pure mathematics. Computational mathematics is for the most part a subset of applied mathematics, which focuses on how to adapt computational formulas so that they may be run or run more efficiently on a given computation system, e.g., a binary computer. Computational mathematics, then, is primarily focused on formulas and computation of said formulas, which is to be more specific about the limits that make it an applied mathematic. I offer this as a different viewpoint, one coming from where the distinction has practical effects. Jason H. On Mon, Mar 12, 2012 at 12:47 PM, Benjamin Udell <bud...@nyc.rr.com> wrote: > ** > > Malgosia, Irving, Gary, list, > > I should add that this whole line of discussion began because I put the > cart in front of the horse. The adjectives bothered me. "Theoretical math" > vs. "computational math" - the latter sounds like of math about > computation. And "creative math" vs. what - "consumptive math"? > "consumptorial math"? Then I thought of theorematic vs. corollarial, > thought it was an interesting idea and gave it a try. The comparison is > interesting and there is some likeness between the distinctions. However I > now think that trying to align it to Irving's and Pratt's distinctions just > stretches it too far. And it's occurred to me that I'd be happy with the > adjective "computative" - hence, theoretical math versus computative math. > > However, I don't think that we've thoroughly replaced the terms "pure" and > "applied" as affirmed of math areas until we find some way to justly > distinguish between so-called 'pure' maths as opposed to so-called > 'applied' yet often (if not absolutely always) mathematically nontrivial > areas such as maths of optimization (linear and nonlinear programming), > probability theory, the maths of information (with laws of information > corresponding to group-theoretical principles), etc. > > Best, Ben > > ----- Original Message ----- > *From:* Benjamin Udell <bud...@nyc.rr.com> * > **To:* PEIRCE-L@LISTSERV.IUPUI.EDU * > * > *Sent:* Monday, March 12, 2012 1:14 PM > *Subject:* Re: [peirce-l] Mathematical terminology, was, review of > Moore's Peirce edition > > Malgosia, list, > > Responses interleaved. > > ----- Original Message ----- > From: malgosia askanas > To: PEIRCE-L@LISTSERV.IUPUI.EDU > Sent: Monday, March 12, 2012 12:31 PM > Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's > Peirce edition > > >>[BU] Yes, the theorematic-vs.-corollarial distinction does not appear in > the Peirce quote to depend on whether the premisses - _up until some lemma_ > - already warrant presumption. > >>BUT, but, but, the theorematic deduction does involve the introdution of > that lemma, and the lemma needs to be proven (in terms of some postulate > system), or at least include a definition (in remarkable cases supported by > a "proper postulate") in order to stand as a premiss, and that is what > Irving is referring to. > > >[MA] OK, but how does this connect to the corollarial/theorematic > distinction? On the basis purely of the quote from Peirce that Irving was > discussing, the theorem, again, could follow from the lemma either > corollarially (by virtue purely of "logical form") or theorematically > (requiring additional work with the actual mathematical objects of which > the theorem speaks). > > [BU] So far, so good. > > >[MA] And the lemma, too, could have been obtained either corollarially (a > rather needless lemma, in that case) > > [BU] Only if it comes from another area of math, otherwise it is > corollarially drawn from what's already on the table and isn't a lemma. > > >[MA] or theorematically. Doesn't this particular distinction, in either > case, refer to the nature of the _deduction_ that is required in order to > pass from the premisses to the conclusion, rather than referring to the > warrant (or lack of it) of presuming the premisses? > > [BU] It's both, to the extent that the nature of that deduction depends on > whether the premisses require a lemma, a lemma that either gets something > from elsewhere (i.e., the lemma must refer to where its content is > established elsewhere), or needs to be proven on the spot. But - in some > cases there's no lemma but merely a definition that is uncontemplated in > the thesis, and is not demanded by the premisses or postulates but is still > consistent with them, and so Irving and I, as it seems to me now, are wrong > to say that it's _*always*_ a matter of whether some premiss requires > special proof. Not always, then, but merely often. In some cases said > definition needs to be supported by a new postulate, so there the > proof-need revives but is solved by recognizing the need and "conceding" a > new postulate to its account. > > >[MA] If the premisses are presumed without warrant, that - it seems to me > - does not make the deduction more corollarial or more theorematic; it just > makes it uncompleted, and perhaps uncompletable. > > [BU] That sounds right. > > Best, Ben > > --------------------------------------------------------------------------------- > You are receiving this message because you are subscribed to the PEIRCE-L > listserv. To remove yourself from this list, send a message to > lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in the body > of the message. 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