Irving, Gary, Malgosia, list,

Irving, I'm sorry that I gave you the impression that I think that a lemma is 
something helpful but unproven inserted into a proof. I mean a theorem placed 
in among the premisses to help prove the thesis. Its proof may be offered then 
and there, or it may be a theorem from (and already proven in) another branch 
of mathematics, to which the reader is referred. At any rate it is as Peirce 
puts it "a demonstrable proposition about something outside the subject of 

The idea that theorematic reasoning often involves a lemma comes not from me 
but from Peirce. Theorematic reasoning, in Peirce's view, involves 
experimentation on a diagram, which may consist in a geometrical form, an array 
of algebraic expressions, a form such as "All __ is __," etc.  I don't recall 
his saying anything to suggest that theorematic reasoning is particularly 
mechanical.  I summarized Peirce's views in a paragraph in my first post on 
these questions, and I'll reproduce it, this time with the full quotes from 
Peirce. He discusses lemmas in the third quote.
Peirce held that the most important division of kinds of deductive reasoning is 
that between corollarial and theorematic. He argued that, while finally all 
deduction depends in one way or another on mental experimentation on schemata 
or diagrams,[1] still in corollarial deduction "it is only necessary to imagine 
any case in which the premisses are true in order to perceive immediately that 
the conclusion holds in that case," whereas theorematic deduction "is deduction 
in which it is necessary to experiment in the imagination upon the image of the 
premiss in order from the result of such experiment to make corollarial 
deductions to the truth of the conclusion."[2]  He held that corollarial 
deduction matches Aristotle's conception of direct demonstration, which 
Aristotle regarded as the only thoroughly satisfactory demonstration, while 
theorematic deduction (A) is the kind more prized by mathematicians, (B) is 
peculiar to mathematics,[1] and (C) involves in its course the introduction of 
a lemma or at least a definition uncontemplated in the thesis (the proposition 
that is to be proved); in remarkable cases that definition is of an abstraction 
that "ought to be supported by a proper postulate.".[3]

1 a b Peirce, C. S., from section dated 1902 by editors in the "Minute Logic" 
manuscript, Collected Papers v. 4, paragraph 233, quoted in part in 
"Corollarial Reasoning" in the Commens Dictionary of Peirce's Terms, 
2003-present, Mats Bergman and Sami Paavola, editors, University of Helsinki.: 

  How it can be that, although the reasoning is based upon the study of an 
individual schema, it is nevertheless necessary, that is, applicable, to all 
possible cases, is one of the questions we shall have to consider. Just now, I 
wish to point out that after the schema has been constructed according to the 
precept virtually contained in the thesis, the assertion of the theorem is not 
evidently true, even for the individual schema; nor will any amount of hard 
thinking of the philosophers' corollarial kind ever render it evident. Thinking 
in general terms is not enough. It is necessary that something should be DONE. 
In geometry, subsidiary lines are drawn. In algebra permissible transformations 
are made. Thereupon, the faculty of observation is called into play. Some 
relation between the parts of the schema is remarked. But would this relation 
subsist in every possible case? Mere corollarial reasoning will sometimes 
assure us of this. But, generally speaking, it may be necessary to draw 
distinct schemata to represent alternative possibilities. Theorematic reasoning 
invariably depends upon experimentation with individual schemata. We shall find 
that, in the last analysis, the same thing is true of the corollarial 
reasoning, too; even the Aristotelian "demonstration why." Only in this case, 
the very words serve as schemata. Accordingly, we may say that corollarial, or 
"philosophical" reasoning is reasoning with words; while theorematic, or 
mathematical reasoning proper, is reasoning with specially constructed 
schemata. (' Minute Logic', CP 4.233, c. 1902)

2. Peirce, C. S., the 1902 Carnegie Application, published in The New Elements 
of Mathematics, Carolyn Eisele, editor, also transcribed by Joseph M. Ransdell, 
see "From Draft A - MS L75.35-39" in Memoir 19 (once there, scroll down): 

  Deduction is only of value in tracing out the consequences of hypotheses, 
which it regards as pure, or unfounded, hypotheses. Deduction is divisible into 
sub-classes in various ways, of which the most important is into corollarial 
and theorematic. Corollarial deduction is where it is only necessary to imagine 
any case in which the premisses are true in order to perceive immediately that 
the conclusion holds in that case. Ordinary syllogisms and some deductions in 
the logic of relatives belong to this class. Theorematic deduction is deduction 
in which it is necessary to experiment in the imagination upon the image of the 
premiss in order from the result of such experiment to make corollarial 
deductions to the truth of the conclusion. The subdivisions of theorematic 
deduction are of very high theoretical importance. But I cannot go into them in 
this statement. (Carnegie Institute Application, from Draft A - MS L75.35-39, 

3. Peirce, C. S., 1901 manuscript "On the Logic of Drawing History from Ancient 
Documents, Especially from Testimonies', The Essential Peirce v. 2, see p. 96. 
See quote in "Corollarial Reasoning" in the Commens Dictionary of Peirce's 

  "This appears to be in harmony with Kant's view of deduction, namely, that it 
merely explicates what is implicitly asserted in the premisses. This is what is 
called a half-truth. Deductions are of two kinds, which I call corollarial and 
theorematic. The corollarial are those reasonings by which all corollaries and 
the majority of what are called theorems are deduced; the theorematic are those 
by which the major theorems are deduced. If you take the thesis of a corollary, 
-- i.e. the proposition to be proved, and carefully analyze its meaning, by 
substituting for each term its definition, you will find that its truth 
follows, in a straightforward manner, from previous propositions similarly 
analyzed. But when it comes to proving a major theorem, you will very often 
find you have need of a _lemma_, which is a demonstrable proposition about 
something outside the subject of inquiry; and even if a lemma does not have to 
be demonstrated, it is necessary to introduce the definition of something which 
the _thesis_ of the theorem does not contemplate. In the most remarkable cases, 
this is some abstraction; that is to say, a subject whose existence _consists_ 
in some fact about other things. Such, for example, are operations considered 
as in themselves subject to operation; lines, which are nothing but 
descriptions of the motion of a particle, considered as being themselves 
movable; _collections_; _numbers_; and the like. When the reform of 
mathematical reasoning now going on is complete, it will be seen that every 
such supposition ought to be supported by a proper postulate. At any rate Kant 
himself ought to admit, and would admit if he were alive today, that the 
conclusion of reasoning of this kind, although it is strictly deductive, does 
not flow from definitions alone, but that postulates are requisite for it. ('On 
the Logic of Drawing History from Ancient Documents, Especially from 
Testimonies', EP 2:96, 1901)

Best, Ben

----- Original Message ----- 
From: Irving 
Sent: Tuesday, March 13, 2012 4:33 PM 
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 

Ben, Gary, Malgosia, list

It would appear from the various responses that. whereas there is a consensus 
that Peirce's theorematic/corollarial distinction has relatively little, if 
anything, to do with my theoretical/computational distinction or Pratt's 
"creator" and "consumer" distinction.

As you might recall, in my initial discussion, I indicated that I found 
Pratt's distinction to be somewhat preferable to the theoretical/computational, 
since, as we have seen in the responses, "computational" has several 
connotations, only one of which I initially had specifically in mind, of hack 
grinding out of [usually numerical] solutions to particular problems, the other 
generally thought of as those parts of mathematics taught in catch-all 
undergrad courses that frequently go by the name of "Finite Mathematics" and 
include bits and pieces of such areas as probability theory, matrix theory and 
linear algebra, Venn diagrams, and the like). Pratt's creator/consumer is 
closer to what I had in mind, and aligns better, and I think, more accurately, 
with the older pure (or abstract or theoretical) vs. applied distinction.

The attempt to determine whether, and, if so, how well, Peirce's 
theorematic/corollarial distinction correlates to the theoretical/computational 
or creator/consumer distinction(s) was not initially an issue for me. It was 
raised by Ben Udell when he asked me: 
"Do you think that your "theoretical - computational" distinction and likewise 
Pratt's "creator - consumer" distinction between kinds of mathematics could be 
expressed in terms of Peirce's "theorematic - corollarial" distinction?"

I attempted to reply, based upon a particular quote from Peirce. What I gather 
from the responses to that second round is that the primary issue with my 
attempted reply was that Peirce's distinction was bound up, not with the truth 
of the premises, but rather with the method in which theorems are arrived at. 
If I now understand what most of the responses have attempted to convey, the 
theorematic has to do with the mechanical processing of proofs, where a simple 
inspection of the argument (or proof) allows us to determine which inference 
rules to apply (and when and where) and whether doing so suffices to 
demonstrate that the theorem indeed follows from the premises; whereas the 
corollarial has to do with intuiting how, or even if, one might get from the 
premises to the desired conclusion. In that case, I would suggest that another 
way to express the theorematic/corollarial distinction is that they concern the 
two stages of creating mathematics; that the mathematician begins by examining 
the already established mathematics and asks what new mathematics might be

Ben Udell also introduces the issue of the presence of a lemma in a proof as 
part of the distinction between theorematic and corollarial. His assumption 
seems to be that a lemma is inserted into a proof to help carry it forward, but 
is itself not proven. But, as Malgosia has already noted, the lemma could 
itself have been obtained either theorematically or corollarially. In fact, 
most of us think of a lemma as a minor theorem, proven along the way and 
subsequently used in the proof of the theorem that we're after.

I do not think that any of this obviates the main point of the initial answer 
that I gave to Ben's question, that neither my theoretical/computational 
distinction nor Pratt's "creator" and "consumer" distinction have anything to 
do with Peirce's theorematic/corollarial distinction.

In closing, I would like to present two sets of exchanges; one very recent 
(actually today, on FOM, with due apologies to the protagonists, if I am 
violating any copyrights) between probability theorist William Taylor 
(indicated by '>') and set theorist Martin Dowd (indicated by '>>'), as follows:

>> More seriously, any freshman philosopher encounters the fact that there are 
>> fundamental differences between physical reality and mathematical reality.

> Quite so.  And one of these is noted by Hilbert (or maybe Hardy, > anyone 
> help?) >-

> "The chief difference between scientists and mathematicians is that 
> mathematicians have a much more direct connection to reality."

>> This does not entitle philosophers to characterize mathematical reality as 
>> fictional.

> Quite so; but philosophers tend to have a powerful sense of entitlement.

the other, in Gauss's famous letter November 1, 1844 to astronomer Heinrich 
Schumacher regarding Kant's philosophy of mathematics, that: 
"you see the same sort of [mathematical incompetence] in the contemporary 
philosophers.... Don't they make your hair stand on end with their definitions? 
...Even with Kant himself it is often not much better; in my opinion his 
distinction between analytic and synthetic propositions is one of those things 
that either run out in a triviality or are false."

----- Message from ---------
    Date: Mon, 12 Mar 2012 13:47:10 -0400
    From: Benjamin Udell <>
Reply-To: Benjamin Udell <>
Subject: Re: [peirce-l] Mathematical terminology, was, review of 
Moore's Peirce edition

> 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 edition
> 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 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 with the line "SIGNOFF PEIRCE-L" in
> the body of the message.  To post a message to the list, send it to

----- End message from -----

Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159

You are receiving this message because you are subscribed to the PEIRCE-L 
listserv.  To remove yourself from this list, send a message to with the line "SIGNOFF PEIRCE-L" in the body of the 
message.  To post a message to the list, send it to PEIRCE-L@LISTSERV.IUPUI.EDU

Reply via email to