Hi,

Can it be that continuum is the complementary union of continuity and
discontinuity, in the same sense that light can be viewed as the
complementary union of wave and particle ?  Are there any Peircean texts
that would either support or deny this possibility ?

With all the best.

Sung
__________________________________________________
Sungchul Ji, Ph.D.
Associate Professor of Pharmacology and Toxicology
Department of Pharmacology and Toxicology
Ernest Mario School of Pharmacy
Rutgers University
Piscataway, N.J. 08855
732-445-4701

www.conformon.net







> For the deeply skeptical, and finicky, who see only “continuous” and
> “continuum” but not “continuity,” one need only back up a few lines in the
> same text to find that third integrally related term thrown into the mix,
> where CSP contrasts the Dedekind/Cantor notion of continuity [of the line]
> with his own, and asks “whether I am right in contending, solus, that
> though
> such a series of points [omega-1] no doubt has what is called continuity
> in
> the calculus and theory of functions, it has not the continuity of the
> line”
> [which for Peirce is the true continuum].
>
>
>
> A [truly/imperfect] continuous object is by definition a [true/imperfect]
> continuum, which by definition exhibits [true/imperfect] continuity.
>
>
>
> This can be beaten up quite a bit more, I’m certain, but I haven’t yet
> seen
> a step away from this obvious truism.
>
>
>
> From: Michael DeLaurentis [mailto:[email protected]]
> Sent: Tuesday, November 18, 2014 10:14 AM
> To: 'Jeffrey Brian Downard'; 'Peirce-L'
> Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism,
> etc.
>
>
>
> Just to give one more example of the integral – inseparable – relation
> between continuity [and the continuous] – and continua, here’s CSP from a
> 1908 Monist supp [after laying out Cantor’s aleph-1 notion]: “I proceed to
> define what I think it is that constitutes a true continuum [CSP
> emphasis],
> or continuous object [my emphasis].”
>
>
>
> Again I ask: where is the language IN PEIRCE that separates these patently
> integrated concepts?
>
>
>
> From: Michael DeLaurentis [mailto:[email protected]]
> Sent: Tuesday, November 18, 2014 9:33 AM
> To: 'Jeffrey Brian Downard'; 'Peirce-L'
> Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism,
> etc.
>
>
>
> Jeff -- If I may briefly intercede, a few too-brief comments since my time
> is very limited.  (1) This is indeed a very different discussion from the
> one I was trying to have with Jerry, with a quite different set of
> questions.  (2) I think I made pretty clear through implication, if not
> expressly (though I think I was sufficiently express), that, as you state
> in
> very similar terms below, “the new term [continuum – is used] to refer the
> thing that has the property of being continuous.” That’s an elementary
> point
> of diction, whatever else it is.  It’s not just that “the function of the
> monadic predicate:  ‘-is continuous’ …  tend[s] to [function] like an
> adjective”:  “continuous” IS an adjective.  So these are at the first
> level
> merely usage points. (3)  One of my points in the prior exchanges was that
> Peirce was reconsidering his conception of continuity and the continuum
> [always used in his sense quite distinctly from the common usage of those
> terms by mathematicians, though he often used them as starting points for
> contrast] in very fundamental ways, reflected in conceptions like the
> “imperfect continuum” and “incipient cohesion” [the former obviously
> reflecting the latter in specific instantiations, thus, again,
> illustrating
> the difference in usage but integral connection in concept].  I would say
> that what you are addressing in late CSP, and what CSP was late
> addressing,
> is grades of continuity, again reflected in corresponding grades of
> continua.  But the key point which does carry over from the prior
> discussion
> is this pairing: interleaving more and less perfect/complete
> examples/degrees of continuity necessarily involves a parallel refinement
> in
> the continua instantiated. That merely illustrates that the conceptions
> are
> integrally – I stress integrally – and necessarily related. The purported
> separation of these conceptions, other than in usage and diction, is
> simply
> not to be found – in CSP anyway. [I’m not sure I even know what such a
> separation would mean: as with differentiation and continuity, I assume
> one
> would have to illustrate this purported separation with an example of a
> grade of continuity that reflects a connectivity different from that
> reflected in what is an independently correlated continuum – which seems
> to
> me utter nonsense.]  And your exposition below does not indicate, or
> purport
> to indicate, otherwise. So, yes, I indeed think “there is little” – I
> would
> emphatically say ‘NO” – “difference between the adjectival use of
> continuous, and the conception of continuity that we arrive at by
> hypostatic
> abstraction” other than their diction. (4)  So to answer your
> “simple-minded
> question [--]…is an imperfect continuum continuous, or is it not
> continuous?” – I say, with intentionally boring obviousness, “An imperfect
> continuum [necessarily] exhibits imperfect continuity, just as a true
> continuum exhibits genuine continuity; and refinement into finer grades of
> continuity will be reflected in corresponding grades of continua – and in
> each such case, this is necessarily so. It’s all just usage  -- the
> conceptions of continuous, continuity, and continuum at each grade of
> however many grades necessarily go hand in hand.”  I cannot imagine Peirce
> taking exception to these, indeed, simple-minded statements, all masking
> the
> genuinely substantive point that his late reconsideration of continuity
> and
> related continua reflect a new uncertainty about the nature of continuity
> and continua themselves, even in the perfect sense he had propounded in
> prior years. I’m not sure I see quite where CSP imagined he was heading
> with
> these hypothetic refinements, which, to me, seem only to classify and
> label
> stages of crowding, degrees of connectedness, en route to his genuine
> continuum.
>
>
>
> So, yes, an imperfect continuum is imperfectly continuous; and no, an
> imperfect continuum is not perfectly continuous; and we all knew this all
> the time. And, to return to your first comment below, these elementary,
> even
> simple-minded, considerations contribute only a first, but critically
> important, step in the full understanding and implications of synechism.
>
>
>
> -----Original Message-----
> From: Jeffrey Brian Downard [mailto:[email protected]]
> Sent: Tuesday, November 18, 2014 2:29 AM
> To: Peirce-L
> Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism,
> etc.
>
>
>
> Gary R., List,
>
>
>
> My aim in raising a different sort of question was to move the recent
> discussion about the conceptions of continuity and the continuum in a
> different direction.  The passage Michael copied into the email is one
> that
> I've spent some time trying to make out--and I find that it is not a
> simple
> point that Peirce is making.  As such, I was trying to return to the
> starting point in our recent discussion of continuity--which began with a
> question about the relationship between generality and continuity and the
> motivations behind Peirce's synechism.
>
>
>
> Gary R., would you clarify your last remark, where you say:  "I don't see
> that it offers any particular difficulties in answering (Peirce is quite
> explicit), and Michael in passing did offer a remark clearly articulating
> this distinction (I didn't quickly find it)."  My initial hunch reading
> this
> line was that you are referring in both clauses to Michael's response to
> Jerry.  Is this hunch on the right track?  Or, are you responding to me by
> saying that Peirce is being quite explicit about what he means in CP
> 4.642--and that it shouldn't take much work on our part to clarify what is
> being said?  If you are saying that latter, then I disagree.
>
>
>
> At the risk of defeating my own aim in trying to move the discussion in a
> different--and hopefully more productive--direction, let me consider the
> question:  "How are the conceptions of continuity, continuous and
> continuum
> connected to each other?"
>
>
>
> Let me say that I am working on the assumption that the root conception we
> use in talking about things that appear to be connected in the way that
> our
> experience of something like time is connected is to say, "time is
> continuous."  What is the function of the monadic predicate:  "-is
> continuous"?  I tend to think that it often functions like an adjective.
> That is, we say about our common experience of moving from what we were
> thinking a few seconds ago to what we are thinking now is "of a continuous
> series of changes."
>
>
>
> The term "continuity" is the result of taking this adjective and turning
> it
> into an abstract noun.  As such, it is the result of hypostatic
> abstraction.
> How is the conception of the continuum connected to the conception of
> continuity?  One place where we use the conception of "the continuum" is
> in
> mathematics.  If we move from one area of mathematical inquiry to another,
> we come across a number of different examples of things that seem--at
> least
> to some mathematicians--to be cases of continua.  For example:  there is
> the
> line in geometry; and the field of real numbers in analysis; and the power
> set of the real numbers in set theory.  After considering a number of such
> examples, the mathematicians ask:  do these different cases that appear to
> involve continuity all have something in common?  That is, can we arrive
> at
> a conception of "The continuum" that is at the root of the plurality of
> examples of things that appear to be continuous?  That, I take it, is the
> kind of thing Gödel's was talking about when he posed the question:  “What
> is Cantor’s continuum problem?” (1947)
>
>
>
> Peirce also uses the term 'continuum' in philosophy to talk about things
> that appear to be continuous.  He is following the lead of other
> philosophers who, in the mid-1700's, appear to have coined the term by
> taking the Latin "continuus" and the English "continuous" and then used
> the
> new term to refer the thing that has the property of being continuous.  In
> this way, Peirce asks questions about the nature of the continuum of
> feelings, and the continuum of space, etc.  We might, if we were thinking
> like mathematicians, ask if these different examples all have something in
> common.  Is there a philosophical conception of "The continuum" that is at
> the root of these different continua?
>
>
>
> It might appear that there is little difference between the adjectival use
> of continuous, and the conception of continuity that we arrive at by
> hypostatic abstraction, and the thing we refer to by calling it a
> continuum.
> But, we should keep in mind that Peirce labored over the differences
> between
> the conceptions of relative, relation and relationship.  I think we could
> probably draw on the points he makes about the conceptions of relative,
> relation and relationship in order to get clearer about the conceptions of
> continuous, continuity and continuum.  But that is just my hunch.  I’ll
> leave that task to another time.
>
>
>
> With this much having been said, let me ask a simple-minded question about
> what Peirce seems to be saying in CP 4.642.  In this passage, he makes a
> distinction between a perfect continuum and an imperfect continuum.
> Elsewhere, he offers the following definition about the construction of
> Cantor's set:  “but now I define a pseudo-continuum as that which modern
> writers on the theory of functions call a continuum. But this is fully
> represented by, and according to G. Cantor stands in one-to-one
> correspondence with, the totality of real values, rational and irrational;
> and these are iconized, in their turn, according to these writers [by the]
> entire body of decimal expressions carried out to the right to all finite
> powers of 1/10 without going on to Cantor's {ö}th place of decimals.” (CP
> 6.176)  Peirce uses this definition in an argument that the
> pseudo-continuum
> is not really continuous.  So, here is my simple-minded question:  is an
> imperfect continuum continuous, or is it not continuous?
>
>
>
> --Jeff
>
>
>
>
>
> Jeff Downard
>
> Associate Professor
>
> Department of Philosophy
>
> NAU
>
> (o) 523-8354
>
> ________________________________________
>
> From: Gary Richmond [[email protected]]
>
> Sent: Monday, November 17, 2014 10:39 PM
>
> To: Peirce-L
>
> Subject: Re: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism,
> etc.
>
>
>
> Michael, Jeff D., Jerry, list,
>
>
>
> Michael wrote: I part with this very late gem [ca. 1908], where I think
> the
> continuity-continuum seamlessness is quite evident – continuous, if you
> will.
>
>
>
> I quite agree, and if one could say of a quotation offered to prove a
> point
> that it did, one might say of this one: Q.E.D.
>
>
>
> Jeff asked: What might we do to arrive at greater clarity about what the
> distinction between the conceptions of a perfect and an imperfect
> continuum
> consists in and what the basis of the distinction rests on?
>
>
>
> As Michael noted earlier, this is a very different question than the one
> he
> was arguing with Jerry. I don't see that it offers any particular
> difficulties in answering (Peirce is quite explicit), and Michael in
> passing
> did offer a remark clearly articulating this distinction (I didn't quickly
> find it).
>
>
>
> Best,
>
>
>
> Gary R
>
>
>
>
>
>
>
> [Gary Richmond]
>
>
>
> Gary Richmond
>
> Philosophy and Critical Thinking
>
> Communication Studies
>
> LaGuardia College of the City University of New York C 745
>
> 718 482-5690< <tel:718%20482-5690> tel:718%20482-5690>
>
>
>
> On Mon, Nov 17, 2014 at 11:53 PM, Michael DeLaurentis <
> <mailto:[email protected]%3cmailto:[email protected]>
> [email protected]<mailto:[email protected]>> wrote:
>
> Lest you lost sight of my sample citations -- and for your further
> musement
> -- I part with this very late gem [ca. 1908], where I think the
> continuity-continuum seamlessness is quite evident – continuous, if you
> will.
>
>
>
> In going over the proofs of this paper, written nearly a year ago [1907],
> I
> can announce that I have, in the interval, taken a considerable stride
> toward the solution of the question of continuity, having at length
> clearly
> and minutely analyzed my own conception of a perfect continuum as well as
> that of an imperfect continuum, that is, a continuum having topical
> singularities, or places of lower dimensionality where it is interrupted
> or
> divides… . If in an otherwise unoccupied continuum a figure of lower
> dimensionality be constructed __ such as an oval line on a spheroidal or
> anchor ring surface __ either that figure is a part of the continuum or it
> is not. If it is, it is a topical singularity, and according to my concept
> of continuity, is a breach of continuity. If it is not, it constitutes no
> objection to my view that all the parts of a perfect continuum have the
> same
> dimensionality as the whole. (Strictly, all the material, or actual parts,
> but I cannot now take the space that minute accuracy would require, which
> would be many pages.) That being the case, my notion of the essential
> character of a perfect continuum is the absolute generality with which two
> rules hold good, first, that every part has parts; and second, that every
> sufficiently small part has the same mode of immediate connection with
> others as every other has. (CP 4.642)
>
>
>
> With that, I will rest my case.
>
>
>
> From: Michael DeLaurentis [ <mailto:[email protected]>
> mailto:[email protected]]
>
> Sent: Monday, November 17, 2014 7:56 PM
>
> To: 'Jerry LR Chandler'; 'Peirce List'
>
> Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism,
> etc.
>
>
>
> Jerry – A lot of words, but no explication whatsoever of any distinction
> CSP
> makes between continuity and its instantiation in continua. After some
> irrelevancy re the Continuum Hypothesis, you make some statements about
> continuity and the philosophy of the natural sciences, continuity and
> chemistry,  and “continuity in term of units, individuals and collections”
> -- three sets of comments about continuity, none involving the continuum.
> Where’s the purported distinction between continuity and the continuum –
> i.e., between the continuity exhibited in continua and any other alleged
> continuity?  Where are CSP’s words that indicate such a distinction?  What
> has this claim – “One notion of continuity was constructed by CSP from
> units, individuals and collections” -- contrasted with this claim – “A
> second notion of  usage of continuity emerges from Cantor's view of the
> number line as a closed interval that could be separated into two notions
> of
> distance” – got to do with a purported distinction PEIRCE makes between
> continuity and the continuum, other than that the former is exhibited in,
> and only in, continua? And how do these meandering musings confirm that
> “Kirsti’s intuition was spot on”?  Nothing you say even begins to
> addresses
> this.
>
>
>
> I cited the late articles and passages where, to the contrary, CPS, as
> even
> Kirsti has now acknowledged, moves seamlessly between the two, in just the
> manner I have described. You have splashed a bunch of disconnected
> comments
> around, but have cited nothing in Peirce to the contrary.
>
>
>
> I don’t see the point of continuing this thread if you’re just going to
> toss
> a hodge-podge of unrelated statements around. And with this post, I will
> therefore close end my responses to these aimless meanderings.
>
>
>
>
>
> From: Jerry LR Chandler [ <mailto:[email protected]>
> mailto:[email protected]]
>
> Sent: Monday, November 17, 2014 7:26 PM
>
> To: Peirce List
>
> Cc: Michael DeLaurentis
>
> Subject: Re: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism,
> etc.
>
>
>
> List, Michael, John, Kirsti:
>
>
>
> On Nov 14, 2014, at 11:41 AM, Michael DeLaurentis wrote:
>
>
>
> Jerry – All due respect, but my post concerned the distinction Kirsti
> claimed to find, not anything in your post. So I don’t see the relevance..
>
>
>
> The immediate relevance of my post is that this is a listserve for CSP
> writings and the recent correspondence relate to his writings, although I
> do
> not think that is what was of concern to you.
>
>
>
> Your post leave me puzzled about your penultimate post which in turn was
> puzzling, so I reviewed the tread from its beginning and read more widely.
>
>
>
> The immediate motivations for my contributions and, as I understand it,
> yours also, was the issue raised by  Kirsti with respect to the possible
> distinction the continuum and continuity in philosophy, mathematics and
> CSP's writings.
>
>
>
> Your post (Nov 12)  expresses this perspective:
>
> Continuity is simply the unique quality which continua, and  only
> continua,
> exhibit.
>
>
>
> Does this assertion close the philosophical issue that Kirsti raised?
>
>
>
> In response to John Deely's questions (post 18 in the listing) "Deely,
> John
> N." [[email protected]< <mailto:[email protected]>
> mailto:[email protected]>] kirjoitti:
>
> Kirsti, would you mind clarifying for me, if possible (but not
> necessarily)
> with some specific ref. to a Peirce text(s), your remark re the difference
> between "continuity" and "continuum":
>
>
>
> I believe that it is relevant to cite the specific texts 4.172-176 from
> CSP
> in part of the answer to John's and Kristi's questions.
>
> These paragraphs brought to mind the Cantor's famous "Continuum
> Hypothesis"
> .  A couple of citation from the web place the concept of the Continuum is
> a
> completely different context that that of mere continuity.
>
>
>
> [Introduction. Arguably the most famous formally unsolvable problem of
> mathematics is Hilbert's first problem: Cantor's Continuum Hypothesis:
>
> {The proposal originally made by Georg Cantor that there is no infinite
> set
> with a cardinal number< <http://mathworld.wolfram.com/CardinalNumber.html>
> http://mathworld.wolfram.com/CardinalNumber.html> between that of the
> "small" infinite set of integers<
> <http://mathworld.wolfram.com/Integer.html>
> http://mathworld.wolfram.com/Integer.html> [aleph_0]  and the "large"
> infinite set of real numbers<
> <http://mathworld.wolfram.com/RealNumber.html>
> http://mathworld.wolfram.com/RealNumber.html>  (the "continuum<
> <http://mathworld.wolfram.com/Continuum.html>
> http://mathworld.wolfram.com/Continuum.html>"). Symbolically, the
> continuum
> hypothesis is that [aleph_1=c] . Problem 1a of Hilbert's problems<
> <http://mathworld.wolfram.com/HilbertsProblems.html>
> http://mathworld.wolfram.com/HilbertsProblems.html> asks if the continuum
> hypothesis is true.]
>
>
>
> Another aspect of this issue was how did CSP relate his views on
> continuity
> to the philosophy of the natural sciences?
>
> And these to synechism?
>
> 4.584 (1906) It is that synthesis of tychism and of pragmatism for which I
> long ago proposed the name, Synechism
>
>
>
> Yet, with respect to chemistry and continuity, he writes CP1.62 (1896?)
> Now
> it enters into every fundamental and exact law of physics or of psychics
> that is known. The few laws of chemistry which do not involve continuity
> seem for the most part to be very roughly true. It seems not unlikely that
> if the veritable laws were known continuity would be found to be involved
> in
> them…
>
>
>
> This is to be contrasted with his statements in 4. 173 where he justifies
> the origin of continuity in term of units, individuals and collections,
> strongly implies a consistency with the legisigns of chemistry with atoms
> as
> units, individuals as proper names of elements and collections becoming
> continuous.
>
>
>
> Thus, my conclusion from these readings is that Kristi's intuition was
> spot
> on.
>
> One notion of continuity was constructed by CSP from units, individuals
> and
> collections.
>
> A second notion of  usage of continuity emerges from Cantor's view of the
> number line as a closed interval that could be separated into two notions
> of
> distance, as shown in his well know "removal of the middle third" argument
> to construct infinite numbers of continuous closed intervals from a line
> of
> UNIT length.  The "Continuum Hypothesis" is a proposition about Cantor's
> mathematical philosophy.  It is not an extension of CSP's notion of
> continuity.
>
>
>
> On another topic, I think it is important to support Stefan's quote of CP
> 5.131:
>
> "Man makes the word, and the word means nothing which the man has not made
> it mean, and that only to some man. But since man can think only by means
> of
> words or other external symbols, these might turn round and say: ”You mean
> nothing which we have not taught you, and then only so far as you address
> some word as the interpretant of your thought.“ In fact, therefore, men
> and
> words reciprocally educate each other; each increase of a man‘s
> information
> involves and is involved by, a corresponding increase of a word’s
> information."
>
>
>
> I was not aware of this quote, but have had a similar thought in mind for
> decades from my sensory experiences in the world. The observation that
> meaning is individualized is true for all individuals as a consequence of
> their antecedent sensory experiences.  It is also true of language usage
> among disciplines. It is particularly important for those who love
> knowledge.
>
>
>
> Cheers
>
>
>
> Jerry
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> From: Jerry LR Chandler [ <mailto:[email protected]>
> mailto:[email protected]]
>
> Sent: Friday, November 14, 2014 12:33 PM
>
> To: Peirce List
>
> Cc: Michael DeLaurentis; John N. Deely; Määttänen Kirsti
>
> Subject: Re: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism,
> etc.
>
>
>
> List, Michael, Kirsti, John:
>
> On Nov 12, 2014, at 11:47 AM, Michael DeLaurentis wrote:
>
> I don’t find any such distinction, implicit or explicit, in Peirce’s late
> writings.
>
> Motivated by your assertions, I re-read 4.172 and later paragraphs,
> searching for distinctions between CSP logic and set theory logic.
>
> In contrast to your assertion, I certainly find numerous critical
> philosophic distinctions between CSP logic and Cantorian/Russellian logic
> with respect to inquiry into the mathematics/logic of the continuum.
>
> Although a large number of texts could be cited, availability of time and
> energy restrict my rhetoric principally to 4.172 to 4.176.
>
> 1. 4.173 introduces with the notion of a collection.
>
>   A collection is a consequence of "bring or gather together", parts of a
> whole.
>
> CSP bases his notion of relation on collections as parts of a whole.  It
> requires activity to bring together a collection.
>
> Thus, CSP is grounding his argument, among other mathematical concepts, on
> the theory of numbers, the collectability of numbers, and the antecedent
> parts being brought together to construct a whole.
>
> This is clearly distinct from Cantor / Russell views which pre-supposes a
> geometric line.
>
> 2. 4.174 (and 4.172) introduces the notion of a relative of a part versus
> the relative of a whole, drawing on the statistical example in 4.172 and
> the
> concept of a unit of a partition of a role of a pair of dice.  Each role
> of
> the pair of die generates a relative value among all possible roles of the
> pair of six-sided die, exactly 36.
>
> This is clearly distinct from Cantor / Russell views.
>
> 3. 4.175   "But when the units lose there individual identity because the
> collection exceeds every positive existence of the universe, the word
> multitude ceases to be applicable.  I will take the word multiplicity to
> mean the greatness of any collection discrete or continuous."
>
> I infer from this, in light of 4.172-175, that individual identity is
> related to parts of a whole such that parts, as units, can be collected
> into
> whole, generating the NOUN, collection.  The "bringing together" of a
> collection is of the nature of a sublation. The quality of the collection,
> is, presumable for CSP, a matter of sensory experience, as one perceives
> from the usage of the term "because" in this sentence, inferring
> causality.
>
> (And qualities are an aspect of sensory experiences, are they not?)  This
> is
> clearly distinct from Cantor / Russell views of memberships and classes.
>
> Yes, set theory, as a dominant force in modern mathematics, has ignored
> the
> logical basis of CSP notion of multitude and his terminology for
> distinguishing between parts and wholes, points and lines, and sensory
> experiences.
>
> But, CSP’s philosophy expressed in 4.172-4.175 is consistent with many
> aspects of chemical logic; modern mathematics is not consistent with
> chemical logic for very specific reasons of the non-transitivity of the
> mathematics of chemical sublations of individual identities.
> Non-transitivity is illustrated, for example, by the handedness of
> chemical
> isomers.)  I conclude that although many many aspects of CSP logic and set
> theory logic are consistent with one another, the distinction between them
> (modes of constructions) at the rhetorical and semantic levels differ in
> mathematically profound ways.
>
> The basic conundrum of the nature of distinction between discrete and
> continuous mathematics remains alive and open.  Indeed, a very active
> subfield of mathematics is the Brouwer School of intuitionism.
>
> (   <http://en.wikipedia.org/wiki/Intuitionistic_logic>
> http://en.wikipedia.org/wiki/Intuitionistic_logic )  Parenthetically (or
> perhaps metaphorically) I conclude that studying CSP texts without an
> in-depth knowledge of the state of the science in the 2nd half of the 19
> Th
> Century is like attempting to solve a crossword puzzle with only the
> superficial "across" clues.  The depth of his thought corresponds with
> knowledge of mathematics and the natural sciences and the natural
> propositions in his time, that is, the "down" clues.
>
> Extending the metaphor, the sensory experiences of the American cultural
> milieu of the late 19 Th Century are interwoven into the very fabric of
> CSP's text.
>
> Cheers, Jerry
>
> (BTW, Thanks to Gary F. for suggesting a puzzle analogy for hermeneutics.)
>
>
>
>
>
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