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 
Sent: Monday, March 12, 2012 1:14 PM 
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 

Malgosia, list,

Responses interleaved.

----- Original Message ----- 
From: malgosia askanas 
Sent: Monday, March 12, 2012 12:31 PM 
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 

>>[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

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