[PEIRCE-L] Conflict between deduction and discovery in mathematics

[Matias]

List,

I am new to Peirce's work, but I am curious to know if this is a valid
interpretation of his work. I would also like to thank you for your
answers to my previous post.
In various places, Peirce states that all deduction is diagrammatic.
For example, in a letter to William James (1909, L224), he affirms
that he "first found, and subsequently 'proved', that every deduction
involves the observation of a diagram (...) and having drawn the
diagram, one finds the conclusion to be represented by it" (NEM 3:869,
his emphasis). I believe there is no risk in taking this thesis to
mean that diagrammaticity is a necessary feature of deductive
reasoning, even if it is not always explicit. In other words,
diagrammatic reasoning is not a product of our cognitive limitations,
nor is it merely another type of valid reasoning. What this statement
does not say is why “every deduction involves the observation of a
diagram.”
However, in PAP (1906), Peirce offers an argument that seems to ask
this question. First, he states that “necessary reasoning makes its
conclusion evident” (NEM 4:317). He then states that this evidence
"consists in the fact that the truth of the conclusion is perceived in
its full generality, and in the generality the how and why of the
truth is perceived" (Ibid.). He then considers the evidence that can
be furnished by the three kinds of signs, symbols, indices and
diagrams. Finally, he concludes that diagrams are the only signs that
"literally show, as a percept shows the perceptual judgment to be
true, that a consequence does follow, and more marvellous yet, that it
would follow under all varieties of circumstances accompanying the
premisses" (NEM 4:318).
The argument seems to rely on a distinction between what Peirce calls
"following a rule of thumb" and "reasoning properly called".
Previously, to present the argument in PAP, Peirce makes this
distinction as follows:

"[I]t is necessary to distinguish reasoning, properly so called, where
the acceptance of the conclusion in the sense in which it is drawn, is
seen evidently to be justified, from cases in which a rule of
inference is followed because it has been found to work well, which I
call following a rule of thumb, and accepting a conclusion without
seeing why further than that the impulse to do so seems irresistible.
In both cases, there might be a sound argument to defend the
acceptance of the conclusion; but to accept the conclusion without any
criticism or supporting argument is not what I call reasoning." (NEM
4:314)

The citation is in accordance with other citations from Peirce, such
as this one, in which he makes a similar distinction between "reason"
and the work of a machine.

"[I]t has been shown that all possible general conclusions can be
arranged in a serial order and as soon as anybody wished to defray the
not extravagant cost, the specifications will be ready for a machine
that will actually turn new theorems from a given set of premises, one
after another, as long as they continue to have any interest. But
though a machine could do all that, and thus accomplish all that many
an eminent mathematician accomplishes, it still cannot properly be
called a reasoning machine, any more than the sort of a man I have in
view can be called a reasoner. It does not reason; it only proceeds by
a rule of thumb." (NEM 3:115, n. d., Our Senses as Reasoning Machines)

Based on this argument, we can conclude that diagrams are necessary
for deduction according to Peirce, because they are the only signs
that furnish the evidence that properly called reasoning requires.
This does not mean that it is impossible to draw conclusions from
premises without the use of diagrams. A logical machine could do this,
but such a process would not be considered proper reasoning. It is a
blind process that is analogous to following a rule of thumb.
Furthermore, this is not psychologism, although it involves a subject
who sees that the inference is justified.
In PAP Peirce also mention that “there are at least two other entirely
different lines of argumentation each very nearly, and perhaps quite,
as conclusive as the above, though less instructive, to prove that all
necessary reasoning is by diagrams. One of these shows that every step
of such an argumentation can be represented, but usually much more
analytically, by Existential Graphs. Now to say that the graphical
procedure is more analytical than another is to say that it
demonstrates what the other virtually assumes without proof. Hence,
the Graphical method, which is diagrammatic, is the sounder form of
the same argumentation. The other proof consists in taking up, one by
one, each form of necessary reasoning, and showing that the
diagrammatic exhibition of it does it perfect justice" (NEM 4:319).
However, he does not follow these other two lines of argumentation in
the mentioned text.
I am not sure if I am correct about this interpretation. I apreciate
your feedbak. Thank you very much in advance for your 

Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Ben Udell]

Matias, Jon, list,

Jon is quite right.  Don't get hung up on Peirce's remark on the 
ampliative-explicative distinction as applied to deduction.


SIDE NOTE: I'm getting a bit rusty.  In a previous post I said Peirce 
never discussed nontriviality or depth in their later sense, but that 
was my bad editing; he discussed them by discussing theorematic 
deductions; I meant that he didn't call them by such words as 
"nontrivial" and "deep". I should add that it's kind of interesting that 
Gilman discussed some deductive categorical-syllogistic conclusions (all 
corollarial) as being, in effect, _/less/_corollarial than others.

END OF SIDE NOTE.

The following, with various links to texts, is about *Peirce's 
terminology*, maybe helpfully to Matias and lurkers.


By "ampliative" Peirce USUALLY meant "non-deductive" and there several 
passages of his across the years that show it.


Peirce usually made a distinction between "ampliative" and "deductive" 
rather than between "ampliative" and "explicative", since some 
deductions are too vacuous to be called "explicative".


The English words "explicative" and "ampliative" were used by writers 
before "analytic" and "synthetic" to translate Kant's division of 
conclusions. Peirce regarded mathematical reasoning as based on 
hypotheses and did not hold with Kant's idea of the synthetic a priori, 
an idea strongly associated with the use of the English words "analytic" 
and "synthetic" in logic.


The following are gathered from my notes. I place underlines and 
asterisks around italicized and bolded text respectively because the IU 
peirce-l archive's list-serv program is designed to sadistically refuse, 
among other things, to render visually bold tags, italic tags, and other 
simple HTML markup. I checked the online archive texts' markup and the 
italic tags, bold tags, etc., are there, but they don't change the 
appearance of the text. Or maybe the owner (IUPUI) has to pay extra for 
such "amenities".


**Peirce *1883**:*

   We are thus led to divide all probable reasoning into deductive and
   ampliative, and further to divide ampliative reasoning into
   induction and hypothesis. [….]

   —Page 143
   https://books.google.com/books?id=xq8RYAAJ=RA3-PA143
    in “A
   Theory of Probable Inference”
   
https://www.google.com/books/edition/Studies_in_Logic/xq8RYAAJ?gbpv=1=PA126
   

   in_Studies in Logic_.

**Peirce 1889–91**:

***ampliative*** (am*/′/* pli-ạ̄-tiv) […]   Enlarging; increasing; 
synthetic. Applied — (/a/) In _/logic/,_ to a modal expression causing 
an ampliation (see _/ampliation/,_ 3); thus, the word _/may/_ in “Some 
man may be Antichrist” is an _/ampliative/_ term. (/b/) In the _/Kantian 
philosophy/,_ to a judgment whose predicate is not contained in the 
definition of the subject: more commonly termed by Kant a _/synthetic/_ 
judgment. [“Ampliative judgment” in this sense is Archbishop Thomson’s 
translation of Kant’s word /Erweiterungsurtheil/, translated by Prof. 
Max Müller “expanding judgment.”]


   No subject, perhaps, in modern speculation has excited an intenser
   interest or more vehement controversy than Kant’s famous distinction
   of analytic and synthetic judgments, or, as I think they might with
   far less of ambiguity be denominated, explicative and _/ampliative/_
   judgments. _/Sir W. Hamilton./_//

— _Century Dictionary_, p. 187 
https://archive.org/stream/centurydictipt100whituoft#page/187/mode/1up , 
in Part 1: A – Appet., 1889–91, of Volume 1 of 6, and identically nd 
identically on p. 187 
https://books.google.com/books?id=dHsxAQAAMAAJ=PA187=ampliation 
 
in Volume 1 of 12, 1911 edition . The brackets around the sentence 
mentioning Archbishop Thomson are in the originals.


Peirce also opposes “explicative inference” and “ampliative inference” 
to each other in his long definition of inference 
https://archive.org/details/centurydictipt1100whituoft/page/3081/mode/1up 
in CD.  Note that _/explicative inference/_ there amounts to non-vacuous 
deductive inference.


   [….] **Explicative inference,** an inference which consists in the
   observation of new relations between the parts of a mental diagram
   (see above) constructed without addition to the facts contained in
   the premises. It infers no more than is strictly involved in the
   facts contained in the premises, which it thus unfolds or
   explicates. This is the opposite of _/ampliative inference,/_ in
   which, in endeavoring to frame a representation, not merely of the
   facts contained in the premises, but also of the way in which they
   have come to present themselves, we are led to add to the facts
   directly observed. [….]

**Peirce1892**: Peirce treats “non-deductive” and “ampliative” as 
alternate labels for the same kinds of 

Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Jon Alan Schmidt]

Matias, List:

In the quoted passage, Peirce suggests "that there are two kinds
of deductive reasoning, which *might, perhaps*, be called explicatory
and ampliative" (emphases mine). However, he immediately adds that "no
mathematical reasoning is what would be commonly understood by ampliative,"
and goes on to say that "Kant's characterization of all deductive
reasoning" as *strictly *explicative is also incorrect. He ultimately
proposes instead calling "the two kinds of deduction *corollarial *and
*theorematic*" (emphases in original).

MAS: Why did Peirce claim that his own studies on the logic of polyadic
relations did not yet fully explain mathematical reasoning?


He states in the quoted passage, "These studies [of the logic of polyadic
relations] threw a great deal of light upon logic; but still they did not
really explain mathematical reasoning, until I opened up the subject of
abstraction." He elaborates elsewhere, as follows.

CSP: Another characteristic of mathematical thought is the extraordinary
use it makes of abstractions. ...
Look through the modern logical treatises, and you will find that they
almost all fall into one or other of two errors, as I hold them to be; that
of setting aside the doctrine of abstraction (in the sense in which an
abstract noun marks an abstraction) as a grammatical topic with which the
logician need not particularly concern himself; and that of confounding
abstraction, in this sense, with that operation of the mind by which we pay
attention to one feature of a percept to the disregard of others. The two
things are entirely disconnected. The most ordinary fact of perception,
such as "it is light," involves *precisive *abstraction, or *prescission*. But
*hypostatic *abstraction, the abstraction which transforms "it is light"
into "there is light here," which is the sense which I shall commonly
attach to the word abstraction (since *prescission *will do for precisive
abstraction) is a very special mode of thought. It consists in taking a
feature of a percept or percepts (after it has already been prescinded from
the other elements of the percept), so as to take propositional form in a
judgment (indeed, it may operate upon any judgment whatsoever), and in
conceiving this fact to consist in the relation between the subject of that
judgment and another subject, which has a mode of being that merely
consists in the truth of propositions of which the corresponding concrete
term is the predicate. ... Abstractions are particularly congenial to
mathematics. Everyday life first, for example, found the need of that class
of abstractions which we call *collections*. Instead of saying that some
human beings are males and all the rest females, it was found convenient to
say that mankind consists of the male *part *and the female *part*. The
same thought makes classes of collections, such as pairs, leashes,
quatrains, hands, weeks, dozens, baker's dozens, sonnets, scores, quires,
hundreds, long hundreds, gross, reams, thousands, myriads, lacs, millions,
milliards, milliasses, etc. These have suggested a great branch of
mathematics. Again, a point moves: it is by abstraction that the geometer
says that it "describes a line." This line, though an abstraction, itself
moves; and this is regarded as generating a surface; and so on. So
likewise, when the analyst treats operations as themselves subjects of
operations, a method whose utility will not be denied, this is another
instance of abstraction. Maxwell's notion of a tension exercised upon lines
of electrical force, transverse to them, is somewhat similar. These
examples exhibit the great rolling billows of abstraction in the ocean of
mathematical thought; but when we come to a minute examination of it, we
shall find, in every department, incessant ripples of the same form of
thought, of which the examples I have mentioned give no hint. (CP
4.234-235, 1902)


He also says later in the manuscript that you quoted, "Theorematic
reasoning, at least the most efficient of it, works by abstraction; and
derives its power from abstraction" (NEM 4:11, 1901). In an alternate
version of the same text, he says that "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" (EP 2:96, 1901).

MAS: Additionally, I do not fully understand the relation between the
notion of theorematic deduction and Peirce's thesis about the diagrammatic
character of all deduction.


I refer you again to CP 4.233 (1902) and CP 4.612-616 (1908), the first of
which includes the following explanation.

CSP: Just now, I wish to point out that 

[PEIRCE-L] Conflict between deduction and discovery in mathematics

[John F Sowa]

The Bourbaki were a group of brilliant mathematicians, who developed a totally 
unusable system of mathematics.  That example below shows how hopelessly 
misguided they were.  Sesame Street's method of teaching math is far and away 
superior to anything that the Bourbaki attempted to do.  Sesame street 
introduces the number 1 as the starting point of counting.  That is also 
Peirce's method.

Furthermore, the Bourbaki banished all diagrams from their system, and thereby 
violated every one of Peirce's principles of diagrammatic reasoning.  Sesame 
Street emphasizes diagrams and imagery.  Mathematics without diagrams and 
imagery is blind.

The so-called "new math"  disaster of the late 1960s was a hopelessly misguided 
attempt to inculcate innocent students with set theory as the universal 
foundation for everything.  Another violation of Peirce's methods.

Finally, there is no conflict whatever between deduction and discovery.  As 
Peirce insisted, all discovery is based on diagrams (or images mapped to 
diagrams).  Deduction is just an exploration of the content of some diagram or 
system of diagrams.  There are, of course, many challenges in discovering all 
the provable implications.  But once again, those implications are determined 
by elaboration and analysis of the starting diagrams.

There is much more to say, and it is closely related to my previous note about 
problems with AI.  I'm currently writing an article that shows how Peirce's 
diagrammatic reasoning is far and away superior to the currently popular 
methods of Large Language Models.  The LLMs do have some important features, 
but the LLMs are just one special case of one certain kind of diagram (tensor 
calculus).  The human brain (even a fruit fly brain) can process many more 
kinds.

There is, of course, much more to say about this issue, but it will take a bit 
more time to gather the references.

John


From: "Evgenii Rudnyi" 
Sent: 8/22/23 11:13 AM

Recently I have seen a paper below that could be of interest to this
discussion as it shows that to work deductively even with the number 1
is not that easy.

Best wishes, Evgenii

Mathias, Adrian RD. "A Term of Length 4 523 659 424 929." Synthese 133,
no. 1 (2002): 75-86

"Bourbaki suggest that their definition of the number 1 runs to some
tens of thousands of symbols. We show that that is a considerable
under-estimate, the true number of symbols being 4 523 659 424 929, not
counting 1 179 618 517 981 disambiguatory links."
_ _ _ _ _ _ _ _ _ _
► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON 
PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . 
► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu 
with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the 
body.  More at https://list.iupui.edu/sympa/help/user-signoff.html .
► PEIRCE-L is owned by THE PEIRCE GROUP;  moderated by Gary Richmond;  and 
co-managed by him and Ben Udell.

Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Matias]

Jon, list,

I thank you very much for your answer.

As you suggest, I believe that Peirce's answer to the problem lies in
his notion of theorematic deduction. However, I'm having trouble
understanding what he means by that.

For example, I am confounded by the meaning of this citation.

"It was because those logicians who were mathematicians saw that the
notion that mathematical reasoning was as rudimentary as that was
quite at war with its producing such a world of novel theorems from a
few relatively simple premisses, as for example it does in the theory
of numbers, that they were led,-first Boole and DeMorgan, afterwards
others of us, -to new studies of deductive logic, with the aid of
algebras and graphs. The non-relative logic having soon been
exhausted, we went into the study of the logic of relatives, first the
dyadic, and subsequently I, almost alone, into polyadic relations.
These studies threw a great deal of light upon logic; but still they
did not really explain mathematical reasoning, until I opened up the
subject of abstraction. It now appears that there are two kinds of
deductive reasoning, which might, perhaps, be called explicatory and
ampliative. However, the latter term might be misunderstood; for no
mathematical reasoning is what would be commonly understood by
amp/iative, although much of it is not what is commonly understood as
explicative. It is better to resort to new words to express new ideas.
All readers of mathematics must have felt the great difference between
corollaries and major theorems, although these words are not sharply
distinguished. It is needless to say that the words come to us, not
from Euclid, but from the editions of Euclid's elements. The great
body of the propositions called corollaries (all but 27 in the whole
13 books) are due to commentators, and are of an obvious kind. Kant's
characterization of all deductive reasoning is true of them: they are
mere explications of what is implied in previous results. The same is
true of a good many of Euclid's own theorems; probably the numerical
majority of the whole 369 of them are of this character. But many of
them are of a different nature. We may call the two kinds of deduction
corol/arial and theorematic." (NEM 4:1, 1901)

Here, Peirce first gives some hints about the history of the problem.
He then puts his own contribution in this context, acknowledging the
limits of his studies of polyadic logic. Finally, he affirms that the
problem arises when deduction is reduced to Kant's characterization.
Nevertheless, he conjectures that there are in fact two kinds of
deductions, which are explicative and "ampliative". This can
eventually throw light on the problem by explaining how deductions can
be both certain and novel.

However, within what framework should Peirce's reference to Kant's
characterization of deductive reasoning be interpreted: the new logic
of relations or syllogistic logic? Why did Peirce claim that his own
studies on the logic of polyadic relations did not yet fully explain
mathematical reasoning? Is it because the logic of relatives cannot
explain some inferential steps, for example, the introduction of
abstractions or the construction in Euclid's propositions? Or is it
because we cannot find premises that can transform every proof into a
corollarial or explicative proof? Or is there another reason?

Additionally, I do not fully understand the relation between the
notion of theorematic deduction and Peirce's thesis about the
diagrammatic character of all deduction. Here, I suspect there is some
important clue to understanding Peirce's argument.

Thank you again for your time.

Best regards,

Matias

2023-08-19 13:04 GMT-03:00, Jon Alan Schmidt :
> Matias, List:
>
> Although I cannot offer "any information that traces the history of this
> problem" as requested, I can suggest Peirce's own explanation of it.
>
> CSP: 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. (CP 7.204, 1901)
>
>
> See also NEM 4:1-12 (1901), which begins with the second quotation below;
> CP 4.233 (1902), where Peirce proposes that "corollarial, or
> 'philosophical' reasoning is reasoning 

Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Evgenii Rudnyi]

Recently I have seen a paper below that could be of interest to this 
discussion as it shows that to work deductively even with the number 1 
is not that easy.


Best wishes, Evgenii

Mathias, Adrian RD. "A Term of Length 4 523 659 424 929." Synthese 133, 
no. 1 (2002): 75-86


"Bourbaki suggest that their definition of the number 1 runs to some 
tens of thousands of symbols. We show that that is a considerable 
under-estimate, the true number of symbols being 4 523 659 424 929, not 
counting 1 179 618 517 981 disambiguatory links."
_ _ _ _ _ _ _ _ _ _
► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON 
PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . 
► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu 
with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the 
body.  More at https://list.iupui.edu/sympa/help/user-signoff.html .
► PEIRCE-L is owned by THE PEIRCE GROUP;  moderated by Gary Richmond;  and 
co-managed by him and Ben Udell.

Aw: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Helmut Raulien]

 


Dear all,

 

one example, how great complexity arises from seemingly trivial, because obviously quite simple, equations, is the Mandelbrot-set with the appleman. Mathematics is based on axioms, and these seem trivial to us, because they are easy to understand, and they suit our experience perfectly and on first sight. If the paradox is, that a highly complex system deductively develops out of seemingly trivial premisses, then I deduce, that these premisses, the axioms, merely seem trivial to us, but in fact are not, but highly complex. How can that be? Must be, that our minds are constructed in a highly complex way, which we donot see, because our minds are constructed that way without having an in-fact-trivial reference. if this is so, and I donot see another solution for the paradox, then the physicochemical realm, the world we live in, is also based on high complexity, and there is no in-fact-triviality easily accessible. Maybe the real, in-fact-triviality, would we see it, would seem highly complex to us, and maybe we would not be able to tell it from real complexity. I would call this a challenge: The quest for the real triviality. I have the book from Spencer-Brown, have not understood it at all, as I donot see the difference from Boole with using merely NOT and OR, or the difference from Entitative Graphs by Peirce. But his calculus is based on distinction solely, and he claims to have derived from that the row of numbers, in a quite complicated way. I have not understood it, as I said, but take it as a hint, that something I see as trivial, the row of numbers or addition of ones, in fact is not trivial, but complicated or even complex.

 

Best

 

Helmut

 

Gesendet: Dienstag, 22. August 2023 um 12:25 Uhr
Von: "Ben Udell" 
An: "Peirce List" 
Betreff: Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics


Matias, Phyllis, all,

Peirce didn't talk a whole lot about novelty in deduction, and I doubt that he ever discussed non-triviality or depth in later mathematicians' sense of those ideas (which are allied, though not the same as, the idea of difficulty) though he did focus quite a bit, eventually, on verisimilitude which he also called likelihood in induction and plausibility which he analyzed as natural simplicity in abduction. Peirce was certainly aware of Mill's stated view of deduction, a view which Peirce, as both mathematician and logician, likely regarded as hyperbolic at best.  Peirce didn't spend a lot of time debating with people who seemed to belittle deduction.  He thought that induction and especially abductive inference needed general defense.  

I've now read through Gilman's article and I'm not so hopeful that he's solved the paradox.  I admit I don't expect a solution that clears the paradox away like morning mist, I'm all for seeing it as a natural part of a bigger pattern.  Gilman thinks that part of the answer is to see the problem as basically geometrical.  Now, some like visual shapes, some like algebraic expressions, and sometimes one way or the other seems definitely better.  It seems best (given enough time and energy) to do it both ways, and to be able to transform, to metamorphose, things between the ways.  It's good to see the valence of a relation as a dimension or degree, and some will remember that I like visual tables of ideas, still I think I somewhat prefer algebraic expressions generally but that may just be that I'm more used to them.  However, Gilman knew that logicians and related researchers didn't go far into visual diagrams and that such shortfall of interest had been a barrier faced by Peirce.  A decade or more ago, a peirce-l member (I forget who) told me off-list that half of logicians don't even want to look at tables, visual arrays of ideas.  Since Peirce's time, system-building has gone quite out of fashion in philosophy.

Anyway Gilman shows how elements in reasoning can begin in separation from each other, and their being brought together is novel.  Indeed that novelty, or newness, is not nothing. The mind cannot mentally or physically observe a thing in all perspectives or aspects at once, and I think that that's the heart of a solution to such paradoxes.

Maybe I missed something and maybe it's just a matter of Gilman's phrasing, but it seems to me that Gilman neglects the fact that an argument's premises are logically conjoined, or are to be treated as logically conjoined, as Peirce says somewhere, into one big premise, and in that sense they _have_ been brought together, just not very spotlightedly.  If two premises are disjoined, as by the connective "OR", then they are not even called two premises, instead just one premise. Still, it does make sense to look closely at deductive conclusions' novelty as Gilman does (which Peirce would applaud), and to draw it out, as Gilman does, showing _some_ sense in which it involves into dimension beyond ones found in any particular premise.  He 

Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Ben Udell]

*Matias, Phyllis, all,**
*

*Peirce didn't talk a whole lot about novelty in deduction, and I doubt 
that he ever discussed non-triviality or depth in later mathematicians' 
sense of those ideas (which are allied, though not the same as, the idea 
of difficulty) though he did focus quite a bit, eventually, on 
verisimilitude which he also called likelihood in induction and 
plausibility which he analyzed as natural simplicity in abduction. 
Peirce was certainly aware of Mill's stated view of deduction, a view 
which Peirce, as both mathematician and logician, likely regarded as 
hyperbolic at best.  Peirce didn't spend a lot of time debating with 
people who seemed to belittle deduction.  He thought that induction and 
especially abductive inference needed general defense. **

*

*I've now read through Gilman's article and I'm not so hopeful that he's 
solved the paradox.  I admit I don't expect a solution that clears the 
paradox away like morning mist, I'm all for seeing it as a natural part 
of a bigger pattern.  Gilman thinks that part of the answer is to see 
the problem as basically geometrical.  Now, some like visual shapes, 
some like algebraic expressions, and sometimes one way or the other 
seems definitely better.  It seems best (given enough time and energy) 
to do it both ways, and to be able to transform, to metamorphose, things 
between the ways.  It's good to see the valence of a relation as a 
dimension or degree, and some will remember that I like visual tables of 
ideas, still I think I somewhat prefer algebraic expressions generally 
but that may just be that I'm more used to them.  However, Gilman knew 
that logicians and related researchers didn't go far into visual 
diagrams and that such shortfall of interest had been a barrier faced by 
Peirce.  A decade or more ago, a peirce-l member (I forget who) told me 
off-list that half of logicians don't even want to look at tables, 
visual arrays of ideas.  Since Peirce's time, system-building has gone 
quite out of fashion in philosophy.**

*

*Anyway Gilman shows how elements in reasoning can begin in separation 
from each other, and their being brought together is novel.  Indeed that 
novelty, or newness, is not nothing. The mind cannot mentally or 
physically observe a thing in all perspectives or aspects at once, and I 
think that that's the heart of a solution to such paradoxes.**

*

*Maybe I missed something and maybe it's just a matter of Gilman's 
phrasing, but it seems to me that Gilman neglects the fact that an 
argument's premises are logically conjoined, or are to be treated as 
logically conjoined, as Peirce says somewhere, into one big premise, and 
in that sense they _**/have/**_ been brought together, just not very 
spotlightedly.  If two premises are disjoined, as by the connective 
"OR", then they are not even called two premises, instead just one 
premise. Still, it does make sense to look closely at deductive 
conclusions' novelty as Gilman does (which Peirce would applaud), and to 
draw it out, as Gilman does, showing _/some/_ sense in which it involves 
into dimension beyond ones found in any particular premise.  He sees 
less value in a deductive conclusion that seems too much like its 
premises — too much verisimilitude, so to speak. If verisimilitude and 
plausibility are a non-deductive conclusion's seeming like it might be 
_/true/_ despite being unnecessary, then novelty and nontriviality are a 
deductive conclusion's seeming like it might be _/false/_ despite 
seeming deductive, and by that surprisingness or complexity, having more 
value than otherwise.  Well, that's enough from me.  Feel free to let us 
know what you think of Gilman's article.

*

*Best, Ben**
*

*On 8/21/2023 9:52 PM, Matias wrote:**
*
*Ben, Phyllis, Thank you both for your answers. I appreciate your 
insights. Ben, I will check out the Gilman article you mentioned. I 
didn't know about it, but it sounds like it could be helpful. I 
believe that Peirce's answer to the paradox lies in his notion of 
theorematic deduction. However, I'm also having trouble understanding 
what he means by that. I'm hoping that the Gilman article will shed 
some light on this. Furthermore, I think it would be helpful to put 
his answer in perspective, taking into account the history of the 
problem and the subsequent development of logic. Best regards, Matias 
El sáb, 19 de ago de 2023, 09:24, Ben Udell  escribió: *
** I just found B.I. Gilman's article at Google Books. The whole 
article was accessible to me here in the USA. 
https://books.google.com/books?id=dPhl9SLIU54C=PA38=PA38 
 
I'll try to see (not immediately!) what to think of it. Best, Ben On 
8/19/2023 7:22 AM, Ben Udell wrote: Matias, Phyllis, One does often 
start with guessing, retroduction, etc., in trying to solve a 
mathematical problem, be the problem trivial or deep. However this 
guesswork or the like is usually not formalized in 

Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Matias]

Ben, Phyllis,

Thank you both for your answers. I appreciate your insights.

Ben, I will check out the Gilman article you mentioned. I didn't know about
it, but it sounds like it could be helpful. I believe that Peirce's answer
to the paradox lies in his notion of theorematic deduction. However, I'm
also having trouble understanding what he means by that. I'm hoping that
the Gilman article will shed some light on this.

Furthermore, I think it would be helpful to put his answer in perspective,
taking into account the history of the problem and the subsequent
development of logic.

Best regards,

Matias

El sáb, 19 de ago de 2023, 09:24, Ben Udell  escribió:

>
>
>
>
>
>
>
> * I just found B.I. Gilman's article at Google Books.  The whole article
> was accessible to me here in the USA.
> https://books.google.com/books?id=dPhl9SLIU54C=PA38=PA38
>  I'll try
> to see (not immediately!) what to think of it. Best, Ben On 8/19/2023 7:22
> AM, Ben Udell wrote: Matias, Phyllis, One does often start with guessing,
> retroduction, etc., in trying to solve a mathematical problem, be the
> problem trivial or deep.  However this guesswork or the like is usually not
> formalized in publications.  Occasionally a mathematician publishes a
> mathematical conjecture, and some have been pretty important. One of
> Peirce's students Benjamin Ives Gilman whom Peirce got published in Studies
> in Logic (1883)
> https://archive.org/details/studiesinlogic00gilmgoog/page/n15/mode/2up?ref=ol=theater
> 
> did not make a career in logic but did author a published (1923) article
> "The Paradox of the Syllogism Solved by Spatial Construction" Mind, New
> Series, Vol. 32, No. 125 (Jan., 1923), pp. 38-49 (12 pages) Published By:
> Oxford University Press https://www.jstor.org/stable/2249497
>  and I've meant to get hold of it and
> read it because the general question interests me. Peirce thought highly of
> Gilman; and Gilman in that article may reflect, explicitly or implicitly,
> Peirce's views on novelty in deduction.  Gilman claimed to have solved the
> problem!  It certainly is a problem.  Who would bother with explicit,
> deliberately weighed deduction if it did not produce conclusions with
> aspects at least mildly surprising or with at least a jot of depth,
> nontriviality?  It's an instance of a broader paradox.  Induction actually
> (as opposed to seemingly like some deduction) adds claims; in Peirce's
> later view it should conclude with verisimilitude a.k.a likelihood
> http://www.commens.org/dictionary/term/verisimilitude
>  - which, as far as
> I can tell, is to say that it ought to seem UNsurprising despite going
> beyond the premises, or as Peirce put it, resemble the facts already in the
> premises.  Similar remarks can be made about abductive inference.  I tend
> to think that all reasoning depends for its value in part on
> characteristics that resist being exactly quantified or exactly defined and
> which are in some sort of tension, some sort of counterbalance, with the
> inference mode's distinctive or definitive entailment-related structure.
> I've noticed that the question of "seeing" or "not seeing" deductive
> implications is sometimes discussed as the question of logical omniscience
> and the lack thereof, for example by Sergei Artemov and Roman Kuznets in
> "Logical omniscience as infeasibility", Annals of Pure and Applied Logic,
> Volume 165, Issue 1, January 2014, Pages 6-25
> https://www.sciencedirect.com/science/article/pii/S0168007213001024
>  .
> Best, Ben  On 8/18/2023 9:08 PM, Phyllis Chiasson wrote: *
>
> * Wouldn't this be true for all of nature versus the all of discovery?
> Discovery is human and therefore retroductive (as are "newspapers and great
> fortunes"). Nature is. On Fri, Aug 18, 2023, 4:14 PM Matias
>   wrote: *
>
> * Dear list members, I am trying to contextualize Peirce's reference to
> the long-standing conflict between the notion of mathematical reasoning and
> the novelty of mathematical discoveries. I would appreciate any information
> that traces the history of this problem. Here are two citations in which
> Peirce mentions such a conflict: "It has long been a puzzle how it could be
> that, on the one hand, mathematics is purely deductive in its nature, and
> draws its conclusions apodictically, while on the other hand, it presents
> as rich and apparently unending a series of surprising discoveries as any
> observational science. Various have been the attempts to solve the paradox
> by breaking down one or other of these assertions, but without success."
> (Peirce, 1885, On the Algebra of Logic, p. 182) "It was because those
> logicians who were mathematicians saw that the notion 

Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Jerry LR Chandler]

Matias, Jon:

First, I am very curious, Matias, on where your critical question emerges from? 
 What are the sources of your curiosity?  The fuller the ascriptions of your 
cognitive status, the better I will be able to respond to this simple but 
daring question.

Jon, in your numerous posts that seek to intertwine your world wisdom with 
CSP’s philosophy, I have often questioned the origins of your interpretations 
of CSP’s descriptions of the relationships between graph theory and modern 
mathematics.  Thank you for these quotes from CSP as they illuminate two 
questions - or at least a tantalizing fragment - of why CSP’s usage of the 
corollary is rather distance from the classical language of geometry as well as 
my interpretation of your views.

More later.

Cheers

Jerry 







> On Aug 19, 2023, at 11:04 AM, Jon Alan Schmidt  
> wrote:
> 
> Matias, List:
> 
> Although I cannot offer "any information that traces the history of this 
> problem" as requested, I can suggest Peirce's own explanation of it.
> 
> CSP: 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. (CP 7.204, 
> 1901)
> 
> See also NEM 4:1-12 (1901), which begins with the second quotation below; CP 
> 4.233 (1902), where Peirce proposes that "corollarial, or 'philosophical' 
> reasoning is reasoning with words; while theorematic, or mathematical 
> reasoning proper, is reasoning with specially constructed schemata"; and 
> especially CP 4.612-616 (1908), where he discusses at length "the step of so 
> introducing into a demonstration a new idea not explicitly or directly 
> contained in the premisses of the reasoning or in the condition of the 
> proposition which gets proved by the aid of this introduction," which he 
> calls "a theoric step." As he writes in another contemporaneous manuscript ...
> 
> CSP: Everybody knows that mathematics, which covers all necessary reasoning, 
> is as far as possible from being purely mechanical work; that it calls for 
> powers of generalization in comparison with which all others are puny, that 
> it requires an imagination which would be poetical were it not so vividly 
> detailed, and above all that it demands invention of the profoundest. There 
> is, therefore, no room to doubt that there is some theoric reasoning, 
> something unmechanical, in the business of mathematics. I hope that, before I 
> cease to be useful in this world, I may be able to define better than I now 
> can what the distinctive essence of theoric thought is. I can at present say 
> this much with some confidence. It is the directing of the attention to a 
> sort of object not explicitly referred to in the enunciation of the problem 
> in hand. (NEM 3:622, 1908)
> 
> Regards,
> 
> Jon Alan Schmidt - Olathe, Kansas, USA
> Structural Engineer, Synechist Philosopher, Lutheran Christian
> www.LinkedIn.com/in/JonAlanSchmidt 
>  / twitter.com/JonAlanSchmidt 
> 
> On Fri, Aug 18, 2023 at 6:14 PM Matias  > wrote:
>> Dear list members, 
>> 
>> I am trying to contextualize Peirce's reference to the long-standing 
>> conflict between the notion of mathematical reasoning and the novelty of 
>> mathematical discoveries. I would appreciate any information that traces the 
>> history of this problem. 
>> 
>> Here are two citations in which Peirce mentions such a conflict: 
>> 
>> "It has long been a puzzle how it could be that, on the one hand, 
>> mathematics is purely deductive in its nature, and draws its conclusions 
>> apodictically, while on the other hand, it presents as rich and apparently 
>> unending a series of surprising discoveries as any observational science. 
>> Various have been the attempts to solve the paradox by breaking down one or 
>> other of these assertions, but without success." (Peirce, 1885, On the 
>> Algebra of Logic, p. 182) 
>> 
>> "It was because those logicians who were mathematicians saw that the notion 
>> that mathematical reasoning was as rudimentary as that was quite at war with 
>> its producing such a world of novel theorems from a few relatively 

Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Jon Alan Schmidt]

Matias, List:

Although I cannot offer "any information that traces the history of this
problem" as requested, I can suggest Peirce's own explanation of it.

CSP: 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. (CP 7.204, 1901)


See also NEM 4:1-12 (1901), which begins with the second quotation below;
CP 4.233 (1902), where Peirce proposes that "corollarial, or
'philosophical' reasoning is reasoning with words; while theorematic, or
mathematical reasoning proper, is reasoning with specially constructed
schemata"; and especially CP 4.612-616 (1908), where he discusses at length
"the step of so introducing into a demonstration a new idea not explicitly
or directly contained in the premisses of the reasoning or in the condition
of the proposition which gets proved by the aid of this introduction,"
which he calls "a theoric step." As he writes in another contemporaneous
manuscript ...

CSP: Everybody knows that mathematics, which covers all necessary
reasoning, is as far as possible from being purely mechanical work; that it
calls for powers of generalization in comparison with which all others are
puny, that it requires an imagination which would be poetical were it not
so vividly detailed, and above all that it demands invention of the
profoundest. There is, therefore, no room to doubt that there is *some *theoric
reasoning, something unmechanical, in the business of mathematics. I hope
that, before I cease to be useful in this world, I may be able to define
better than I now can what the distinctive essence of theoric thought is. I
can at present say this much with some confidence. It is the directing of
the attention to a sort of object not explicitly referred to in the
enunciation of the problem in hand. (NEM 3:622, 1908)


Regards,

Jon Alan Schmidt - Olathe, Kansas, USA
Structural Engineer, Synechist Philosopher, Lutheran Christian
www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt

On Fri, Aug 18, 2023 at 6:14 PM Matias  wrote:

> Dear list members,
>
> I am trying to contextualize Peirce's reference to the long-standing
> conflict between the notion of mathematical reasoning and the novelty of
> mathematical discoveries. I would appreciate any information that traces
> the history of this problem.
>
> Here are two citations in which Peirce mentions such a conflict:
>
> "It has long been a puzzle how it could be that, on the one hand,
> mathematics is purely deductive in its nature, and draws its conclusions
> apodictically, while on the other hand, it presents as rich and apparently
> unending a series of surprising discoveries as any observational science.
> Various have been the attempts to solve the paradox by breaking down one or
> other of these assertions, but without success." (Peirce, 1885, On the
> Algebra of Logic, p. 182)
>
> "It was because those logicians who were mathematicians saw that the
> notion that mathematical reasoning was as rudimentary as that was quite at
> war with its producing such a world of novel theorems from a few relatively
> simple premisses, as for example it does in the theory of numbers, that
> they were led,--first Boole and DeMorgan, afterwards others of us, -to new
> studies of deductive logic, with the aid of algebras and graphs." (NEM 4:1)
>
> I know that I am asking a basic question, but thank you for your time.
>
> Best regards,
>
> Matías A. Saracho
>
_ _ _ _ _ _ _ _ _ _
► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON 
PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . 
► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu 
with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the 
body.  More at https://list.iupui.edu/sympa/help/user-signoff.html .
► PEIRCE-L is owned by THE PEIRCE GROUP;  moderated by Gary Richmond;  and 
co-managed by him and Ben Udell.

Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Ben Udell]

*

I just found B.I. Gilman's article at Google Books.  The whole article 
was accessible to me here in the USA.


https://books.google.com/books?id=dPhl9SLIU54C=PA38=PA38 



I'll try to see (not immediately!) what to think of it.

Best, Ben

On 8/19/2023 7:22 AM, Ben Udell wrote:

Matias, Phyllis,

One does often start with guessing, retroduction, etc., in trying to 
solve a mathematical problem, be the problem trivial or deep.  However 
this guesswork or the like is usually not formalized in publications.  
Occasionally a mathematician publishes a mathematical conjecture, and 
some have been pretty important.


One of Peirce's students Benjamin Ives Gilman whom Peirce got published 
in /Studies in Logic/ (1883)
https://archive.org/details/studiesinlogic00gilmgoog/page/n15/mode/2up?ref=ol=theater 


did not make a career in logic but did author a published (1923) article
"The Paradox of the Syllogism Solved by Spatial Construction"
/Mind/, New Series, Vol. 32, No. 125 (Jan., 1923), pp. 38-49 (12 pages)
Published By: Oxford University Press
https://www.jstor.org/stable/2249497

and I've meant to get hold of it and read it because the general 
question interests me. Peirce thought highly of Gilman; and Gilman in 
that article may reflect, explicitly or implicitly, Peirce's views on 
novelty in deduction.  Gilman claimed to have solved the problem!  It 
certainly is a problem. Who would bother with explicit, deliberately 
weighed deduction if it did not produce conclusions with aspects at 
least mildly surprising or with at least a jot of depth, nontriviality?  
It's an instance of a broader paradox.  Induction actually (as opposed 
to seemingly like some deduction) adds claims; in Peirce's later view it 
should conclude with verisimilitude a.k.a likelihood 
http://www.commens.org/dictionary/term/verisimilitude - which, as far as 
I can tell, is to say that it ought to seem UNsurprising despite going 
beyond the premises, or as Peirce put it, resemble the facts already in 
the premises.  Similar remarks can be made about abductive inference.  I 
tend to think that all reasoning depends for its value in part on 
characteristics that resist being exactly quantified or exactly defined 
and which are in some sort of tension, some sort of counterbalance, with 
the inference mode's distinctive or definitive entailment-related 
structure.


I've noticed that the question of "seeing" or "not seeing" deductive 
implications is sometimes discussed as the question of //logical 
omniscience// and the lack thereof, for example by Sergei Artemov and 
Roman Kuznets in "Logical omniscience as infeasibility", Annals of Pure 
and Applied Logic, Volume 165, Issue 1, January 2014, Pages 6-25

https://www.sciencedirect.com/science/article/pii/S0168007213001024 .

Best, Ben

 On 8/18/2023 9:08 PM, Phyllis Chiasson wrote:

*

*
Wouldn't this be true for all of nature versus the all of discovery?
Discovery is human and therefore retroductive (as are "newspapers and great
fortunes"). Nature is.

On Fri, Aug 18, 2023, 4:14 PM Matias  wrote:
*

*
*Dear list members, I am trying to contextualize Peirce's reference 
to the long-standing conflict between the notion of mathematical 
reasoning and the novelty of mathematical discoveries. I would 
appreciate any information that traces the history of this problem. 
Here are two citations in which Peirce mentions such a conflict: "It 
has long been a puzzle how it could be that, on the one hand, 
mathematics is purely deductive in its nature, and draws its 
conclusions apodictically, while on the other hand, it presents as 
rich and apparently unending a series of surprising discoveries as 
any observational science. Various have been the attempts to solve 
the paradox by breaking down one or other of these assertions, but 
without success." (Peirce, 1885, On the Algebra of Logic, p. 182) "It 
was because those logicians who were mathematicians saw that the 
notion that mathematical reasoning was as rudimentary as that was 
quite at war with its producing such a world of novel theorems from a 
few relatively simple premisses, as for example it does in the theory 
of numbers, that they were led,--first Boole and DeMorgan, afterwards 
others of us, -to new studies of deductive logic, with the aid of 
algebras and graphs." (NEM 4:1) I know that I am asking a basic 
question, but thank you for your time. Best regards, Matías A. Saracho*

*

**_ _ _ _ _ _ _ _ _ _
► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON 
PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . 
► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu 
with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the 
body.  More at https://list.iupui.edu/sympa/help/user-signoff.html .
► PEIRCE-L is 

Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Ben Udell]

Matias, Phyllis,

One does often start with guessing, retroduction, etc., in trying to 
solve a mathematical problem, be the problem trivial or deep. However 
this guesswork or the like is usually not formalized in publications.  
Occasionally a mathematician publishes a mathematical conjecture, and 
some have been pretty important.


One of Peirce's students Benjamin Ives Gilman whom Peirce got published 
in Studies in Logic (1883)
https://archive.org/details/studiesinlogic00gilmgoog/page/n15/mode/2up?ref=ol=theater 


did not make a career in logic but did author a published (1923) article
"The Paradox of the Syllogism Solved by Spatial Construction"
Mind, New Series, Vol. 32, No. 125 (Jan., 1923), pp. 38-49 (12 pages)
Published By: Oxford University Press
https://www.jstor.org/stable/2249497

and I've meant to get hold of it and read it because the general 
question interests me. Peirce thought highly of Gilman; and Gilman in 
that article may reflect, explicitly or implicitly, Peirce's views on 
novelty in deduction.  Gilman claimed to have solved the problem!  It 
certainly is a problem.  Who would bother with explicit, deliberately 
weighed deduction if it did not produce conclusions with aspects at 
least mildly surprising or with at least a jot of depth, nontriviality?  
It's an instance of a broader paradox.  Induction actually (as opposed 
to seemingly like some deduction) adds claims; in Peirce's later view it 
should conclude with verisimilitude a.k.a likelihood 
http://www.commens.org/dictionary/term/verisimilitude - which, as far as 
I can tell, is to say that it ought to seem UNsurprising despite going 
beyond the premises, or as Peirce put it, resemble the facts already in 
the premises.  Similar remarks can be made about abductive inference.  I 
tend to think that all reasoning depends for its value in part on 
characteristics that resist being exactly quantified or exactly defined 
and which are in some sort of tension, some sort of counterbalance, with 
the inference mode's distinctive or definitive entailment-related structure.


I've noticed that the question of "seeing" or "not seeing" deductive 
implications is sometimes discussed as the question of /logical 
omniscience/ and the lack thereof, for example by Sergei Artemov and 
Roman Kuznets in "Logical omniscience as infeasibility", Annals of Pure 
and Applied Logic, Volume 165, Issue 1, January 2014, Pages 6-25

https://www.sciencedirect.com/science/article/pii/S0168007213001024 .

Best, Ben

 On 8/18/2023 9:08 PM, Phyllis Chiasson wrote:

Wouldn't this be true for all of nature versus the all of discovery?
Discovery is human and therefore retroductive (as are "newspapers and great
fortunes"). Nature is.

On Fri, Aug 18, 2023, 4:14 PM Matias  wrote:


Dear list members,

I am trying to contextualize Peirce's reference to the long-standing
conflict between the notion of mathematical reasoning and the novelty of
mathematical discoveries. I would appreciate any information that traces
the history of this problem.

Here are two citations in which Peirce mentions such a conflict:

"It has long been a puzzle how it could be that, on the one hand,
mathematics is purely deductive in its nature, and draws its conclusions
apodictically, while on the other hand, it presents as rich and apparently
unending a series of surprising discoveries as any observational science.
Various have been the attempts to solve the paradox by breaking down one or
other of these assertions, but without success." (Peirce, 1885, On the
Algebra of Logic, p. 182)

"It was because those logicians who were mathematicians saw that the
notion that mathematical reasoning was as rudimentary as that was quite at
war with its producing such a world of novel theorems from a few relatively
simple premisses, as for example it does in the theory of numbers, that
they were led,--first Boole and DeMorgan, afterwards others of us, -to new
studies of deductive logic, with the aid of algebras and graphs." (NEM 4:1)

I know that I am asking a basic question, but thank you for your time.

Best regards,

Matías A. Saracho
_ _ _ _ _ _ _ _ _ _
► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON
PEIRCE-L to this message. PEIRCE-L posts should go to
peirce-L@list.iupui.edu  .
► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to
l...@list.iupui.edu  with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the
message and nothing in the body.  More at
https://list.iupui.edu/sympa/help/user-signoff.html  .
► PEIRCE-L is owned by THE PEIRCE GROUP;  moderated by Gary Richmond;  and
co-managed by him and Ben Udell.



_ _ _ _ _ _ _ _ _ _
► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON 
PEIRCE-L to this message. PEIRCE-L posts should go topeirc...@list.iupui.edu  .
► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but tol...@list.iupui.edu  
with UNSUBSCRIBE PEIRCE-L in 

Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics

[Phyllis Chiasson]

Wouldn't this be true for all of nature versus the all of discovery?
Discovery is human and therefore retroductive (as are "newspapers and great
fortunes"). Nature is.

On Fri, Aug 18, 2023, 4:14 PM Matias  wrote:

> Dear list members,
>
> I am trying to contextualize Peirce's reference to the long-standing
> conflict between the notion of mathematical reasoning and the novelty of
> mathematical discoveries. I would appreciate any information that traces
> the history of this problem.
>
> Here are two citations in which Peirce mentions such a conflict:
>
> "It has long been a puzzle how it could be that, on the one hand,
> mathematics is purely deductive in its nature, and draws its conclusions
> apodictically, while on the other hand, it presents as rich and apparently
> unending a series of surprising discoveries as any observational science.
> Various have been the attempts to solve the paradox by breaking down one or
> other of these assertions, but without success." (Peirce, 1885, On the
> Algebra of Logic, p. 182)
>
> "It was because those logicians who were mathematicians saw that the
> notion that mathematical reasoning was as rudimentary as that was quite at
> war with its producing such a world of novel theorems from a few relatively
> simple premisses, as for example it does in the theory of numbers, that
> they were led,--first Boole and DeMorgan, afterwards others of us, -to new
> studies of deductive logic, with the aid of algebras and graphs." (NEM 4:1)
>
> I know that I am asking a basic question, but thank you for your time.
>
> Best regards,
>
> Matías A. Saracho
> _ _ _ _ _ _ _ _ _ _
> ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON
> PEIRCE-L to this message. PEIRCE-L posts should go to
> peirce-L@list.iupui.edu .
> ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to
> l...@list.iupui.edu with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the
> message and nothing in the body.  More at
> https://list.iupui.edu/sympa/help/user-signoff.html .
> ► PEIRCE-L is owned by THE PEIRCE GROUP;  moderated by Gary Richmond;  and
> co-managed by him and Ben Udell.
>
_ _ _ _ _ _ _ _ _ _
► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON 
PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . 
► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu 
with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the 
body.  More at https://list.iupui.edu/sympa/help/user-signoff.html .
► PEIRCE-L is owned by THE PEIRCE GROUP;  moderated by Gary Richmond;  and 
co-managed by him and Ben Udell.

[PEIRCE-L] Conflict between deduction and discovery in mathematics

[Matias]

Dear list members,

I am trying to contextualize Peirce's reference to the long-standing
conflict between the notion of mathematical reasoning and the novelty of
mathematical discoveries. I would appreciate any information that traces
the history of this problem.

Here are two citations in which Peirce mentions such a conflict:

"It has long been a puzzle how it could be that, on the one hand,
mathematics is purely deductive in its nature, and draws its conclusions
apodictically, while on the other hand, it presents as rich and apparently
unending a series of surprising discoveries as any observational science.
Various have been the attempts to solve the paradox by breaking down one or
other of these assertions, but without success." (Peirce, 1885, On the
Algebra of Logic, p. 182)

"It was because those logicians who were mathematicians saw that the notion
that mathematical reasoning was as rudimentary as that was quite at war
with its producing such a world of novel theorems from a few relatively
simple premisses, as for example it does in the theory of numbers, that
they were led,--first Boole and DeMorgan, afterwards others of us, -to new
studies of deductive logic, with the aid of algebras and graphs." (NEM 4:1)

I know that I am asking a basic question, but thank you for your time.

Best regards,

Matías A. Saracho
_ _ _ _ _ _ _ _ _ _
► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON 
PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . 
► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu 
with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the 
body.  More at https://list.iupui.edu/sympa/help/user-signoff.html .
► PEIRCE-L is owned by THE PEIRCE GROUP;  moderated by Gary Richmond;  and 
co-managed by him and Ben Udell.