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."
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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."
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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 sees less value in a deductive 

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 

[PEIRCE-L] LUW August 23, Chai Wah Wu. On rearrangement inequalities for triangular norms and co-norms in multi-valued logic

[jean-yves beziau]

The next session of the Logica Universalis Webinar will be Wednesday
August 23 at 4pm CET.

Speaker:  Chai Wah Wu
IBM Research / T. J. Watson Research Center, New York
Title:  On rearrangement inequalities for triangular norms and co-norms in
multi-valued logic
Abstract: The rearrangement inequality states that the sum of products of
permutations of 2 sequences of real numbers are maximized when the terms
are similarly ordered and minimized when the terms are ordered in opposite
order. We show that similar inequalities exist in algebras of multi-valued
logic when the multiplication and addition operations are replaced with
various T-norms and T-conorms respectively. For instance, we show that the
rearrangement inequality holds when the T-norms and T-conorms are derived
from Archimedean copulas.
https://link.springer.com/article/10.1007/s11787-023-00332-0

Associate Organization:
Theory of Computation and Information Group IBM Watson Research Center
presented by its manager Ken Clarkson

Chair:  Sayantan Roy
Assistant Editor Logica Universalis

Everybody is welcome to attend.
https://www.springer.com/journal/11787/updates/23910922

Jean-Yves Beziau
Editor-in-Chief Logica Universalis
Organizer Logica Universalis Webinar
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