Gary F, List
GF: Gary, thank you for reminding us of “The Simplest Mathematics”, i.e. “of
these very simple branches of mathematics which lie at the root of formal
logic.” When we juxtapose ["The Simplest Mathematics"] with some of
Peirce’s other writings on mathematics, logic and phaneroscopy. . . it
raises some questions which are not simple at all.
GR: I agree, and I believe you pointed to one of the most important of the
"other writings" in this context, viz., “The Logic of Mathematics: An
Attempt to Develop My Categories From Within.”
GF: I admire Bellucci’s attempt to untangle some of the paradoxes involved
in phaneroscopic analysis, but I don’t regard it as definitive. I think
some of these questions simply have to remain open, as long as we are
determined to find answers fully consistent with *Peirce’s* writings and
diagrams.
GR: I agree that not only Bellucci's but virtually every "attempt to
untangle some of the paradoxes involved in phaneroscopic analysis," at
least, those I know of, cannot be seen as definitive, that many questions
remain open, and that in a strong sense that the work of explicating and
developing "phaneroscopic analysis" is still in its infancy, the the
science-egg has a long way to go before it's hatched and the (potentially)
beautiful bird emerges.
GF: To give just one example: is “formal logic” *formal* in the same sense
that the “formal elements of the phaneron” are *formal*?
GR: I have tended to think of it rather in the reverse order that you frame
the question; so that the question as I see it is: are the formal elements
of the phaneron formal in the same sense as “formal logic”is formal? That
is, since phaneroscopy derives at least some of its principles from formal
logic, it seems sensible to me to answer your question (albeit reversed),
yes! The formal elements are deeply related in so far as valency theory and
the reduction thesis in mathematics leads to the Three Universal Categories
of phaneroscopy.
As you wrote, Peire, in “The Logic of Mathematics” holds that “mathematics
performs its reasonings by a *logica utens* which it develops for itself,
and has no need of any appeal to a *logica docens;* for no disputes about
reasoning arise in mathematics which need to be submitted to the principles
of the philosophy of thought for decision” (CP 1.417).
And as you added: "This explains the priority of mathematics over
[normative] logic as “the philosophy of thought” — and over phenomenology,
which comes between mathematics and (normative) logic in Peirce’s later
classification of sciences. . ."
GR: So far we are, I think, in agreement.
However, I have trouble with your next comment, that "it seems highly
unlikely that “formal logic” can be considered a *logica utens* rather than
a *logica docens*." Why is it unlikely, at least, and perhaps especially in
consideration of "the simplest mathematics"? I can't say that I follow
here. What supports your contention, hardly Peirce's, that a *logica docens*
is a requirement of formal logic, mathematical logic?
You continue: "And if the mathematics that phaneroscopy depends on is based
on a *logica utens*, then it would seem that the phaneroscopist does not
need any specialist or *formal* training in mathematics."
GR: The phaneroscopist needs only enough "formal training in mathematics"
to grasp the simplest mathematics and valency theory to the extent that the
reduction thesis makes good sense. That much seems absolutely a
requirement. I believe that it is highly doubtful that even Kant
understood these simplest principles (which are only simple once you grasp
them), principles which Peirce explicates if, indeed, he doesn't introduce
them.
GF: What she would need instead, as De Tienne suggests in his slides, is
“training” and practice in sensory (and imaginative) *observation* of the
constituents of the phaneron. The *mathematics* involved, on the other
hand, is as simple as one, two, three. The closest Peirce comes to
*formulating* it is his diagram showing the “valency” of a *rhema* or a
“spot” in EGs.
GR: I do not agree that the mathematics involved are "as simple as one,two,
three" the "diagram showing the 'valency' of a *rhema.*" At least one needs
to add to this the reduction thesis which is not "as simple as one, two,
three."
In short, what Peirce analyses in "The Simplest Mathematics" as regards
trichotomic relations, and how all relations can be reduced to them, is
hardly obvious until it is made so, first by him. That seems to me to be
the way of many a fundamental mathematical idea from Pythagoras through
Euclid to Peirce and beyond: Eureka! once you're shown the (now) obvious.
GF: But what I’ve just written is no more definitive than Bellucci’s 2015
paper on the question of what constitutes phaneroscopic analysis. On the
contrary, I’m just giving an example intended to keep such questions open.
GR: I agree that it is important "to keep such questions open," but I
believe Bellucci's 2015 paper does a pretty good job analyzing some of the
most important points -- and you do as well in *Turning Signs*.
Nonetheless, as I see it, for both the simplest mathematics in formal
mathematics and for phanersocopy, "The most fundamental fact about the
number three is its generative potency."
GF: As for the *practice* of phaneroscopy, *solvitur ambulando*, as they
say in Latin.
GR: I would agree, and not only for the practice of phaneroscopy. Peirce
was certainly not alone in suggesting that getting up from one's desk and
walking around tended to enliven a person's thinking. And as a (now)
occasional composer of music, I know that thinking/creating on your feet is
much to be recommended.
Best,
Gary R
“Let everything happen to you
Beauty and terror
Just keep going
No feeling is final”
― Rainer Maria Rilke
*Gary Richmond*
*Philosophy and Critical Thinking*
*Communication Studies*
*LaGuardia College of the City University of New York*
On Fri, Sep 10, 2021 at 8:50 AM <g...@gnusystems.ca> wrote:
> Gary, thank you for reminding us of “The Simplest Mathematics”, i.e. of “of
> these very simple branches of mathematics which lie at the root of formal
> logic.” When we juxtapose it with some of Peirce’s other writings on
> mathematics, logic and phaneroscopy, and the relations of “dependence”
> between them, it raises some questions which are not simple at all. I
> admire Bellucci’s attempt to untangle some of the paradoxes involved in
> phaneroscopic analysis, but I don’t regard it as definitive. I think some
> of these questions simply have to remain open, as long as we are determined
> to find answers fully consistent with *Peirce’s* writings and diagrams.
>
> To give just one example: is “formal logic” *formal* in the same sense
> that the “formal elements of the phaneron” are *formal*?
>
> Among the many Peirce texts that should be considered in trying to answer
> such a question would be Peirce’s c. 1896 essay on “The Logic of
> Mathematics” (subtitled “An Attempt to Develop My Categories From Within”)
> where he says that “mathematics performs its reasonings by a *logica
> utens* which it develops for itself, and has no need of any appeal to a
> *logica
> docens;* for no disputes about reasoning arise in mathematics which need
> to be submitted to the principles of the philosophy of thought for
> decision” (CP 1.417). This explains the priority of mathematics over logic
> as “the philosophy of thought” — and over phenomenology, which comes
> between mathematics and (normative) logic in Peirce’s later classification
> of sciences — but it seems highly unlikely that “formal logic” can be
> considered a *logica utens* rather than a *logica docens*. And if the
> mathematics that phaneroscopy depends on is based on a *logica utens*,
> then it would seem that the phaneroscopist does not need any specialist or
> *formal* training in mathematics. What she would need instead, as De
> Tienne suggests in his slides, is “training” and practice in sensory (and
> imaginative) *observation* of the constituents of the phaneron. The
> *mathematics* involved, on the other hand, is as simple as one, two,
> three. The closest Peirce comes to *formulating* it is his diagram
> showing the “valency” of a *rhema* or a “spot” in EGs.
>
> But what I’ve just written is no more definitive than Bellucci’s 2015
> paper on the question of what constitutes phaneroscopic analysis. On the
> contrary, I’m just giving an example intended to keep such questions open.
> As for the *practice* of phaneroscopy, *solvitur ambulando*, as they say
> in Latin.
>
>
>
> Gary f.
>
>
>
> } Her untitled mamafesta memorialising the Mosthighest has gone by many
> names at disjointed times. [*Finnegans Wake* 104] {
>
> https://gnusystems.ca/wp/ }{ living the time
>
>
>
> *From:* peirce-l-requ...@list.iupui.edu <peirce-l-requ...@list.iupui.edu> *On
> Behalf Of *Gary Richmond
> *Sent:* 9-Sep-21 16:15
>
>
>
> List, Gary F,
>
>
>
> A while back I took another look at "The Simplest Mathematics" and began a
> draft intended as a response to something you'd written, Gary, but which I
> wasn't able to complete at the time. I decided to post the draft today as
> the quotations are of considerable interest. [All the quotations below are
> from "The Simplest Mathematics" (1902)]
>
>
>
> Peirce begins his discussion "with a little a prior chemistry."
>
>
>
> *The most fundamental fact about the number three is its generative
> potency.* This is a great philosophical truth having its origin and
> rationale in mathematics. It will be convenient to begin with a little a
> priori chemistry. An atom of helion, neon, argon, xenon, crypton, appears
> to be a medad (if I may be allowed to form a patronymic from méden). Argon
> gives us, with its zero valency, the one single type A (CP 4.309, emphasis
> added).
>
>
>
> He continues in this vein taking up other valencies which eventually leads
> to a discussion of triads.
>
>
>
> *Triads, on the other hand, will give every possible variety of type. *Thus,
> we may imagine the atom of argon to be really formed of four triads, thus
> [. . .] We may imagine the monadic atom to be composed of seven triads; [.
> . . ] A dyad will be obtained by breaking any bond of A; while higher
> valencies may be produced, either simply [. . . ] or in an intricate manner
> (emphasis added, CP 4.309 [note: each bracket above is a diagram in the
> text])
>
>
>
> He arrives at what might be seen as the crux of the matter:
>
> . . . .
>
> *It would scarcely be an exaggeration to say that the whole of mathematics
> is enwrapped in these trichotomic graphs*; and they will be found
> extremely pertinent to logic. *So prolific is the triad in forms that one
> may easily conceive that all the variety and multiplicity of the universe
> springs from it* [. . . ] All that springs from -[this symbol] [in the
> text, a diagram of the valental triad] -- an emblem of fertility in
> comparison with which the holy phallus of religion's youth is a poor stick
> indeed (CP 4.310, emphasis added).
>
>
>
> Finally, and somewhat abruptly, he arrives at the conclusion of this
> section of the paper.
>
>
>
> Other points concerning trichotomic mathematics are more of logical than
> of mathematical interest, and are so woven with logic in my mind that I
> will not attempt to set them forth from a purely mathematical point of
> view. Here, then, I conclude what I have to say of these very simple
> branches of mathematics which lie at the root of formal logic (CP 4.323,
> emphasis added).
>
> I haven't much to say about these excerpts at the moment, so I'll simply
> conclude by emphasizing the very first point: "*The most fundamental fact
> about the number three is its generative potency."*
>
>
>
> Best,
>
>
>
> Gary R
>
>
> “Let everything happen to you
> Beauty and terror
> Just keep going
> No feeling is final”
> ― Rainer Maria Rilke
>
>
>
> *Gary Richmond*
>
> *Philosophy and Critical Thinking*
>
> *Communication Studies*
>
> *LaGuardia College of the City University of New York*
>
>
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