Dear Jeff,
In your post you say, “Doing mathematics in a more scientific spirit
requires, it seems, an understanding of the purposes that govern the
activities and the methods that should be employed.” Why? Who cares, as
long as the math is good. Doing the math, animated by a passion to learn,
is not the same as reflectively understanding some philosophical conception
of pure science. The old saying that, “the dumbest farmer has the biggest
potatoes,” probably applies to many scientists animated by a passion to
learn, methods to realize that learning, but not much of a sense of what
the big picture is for theoretical science.
In your example of ancient math, you say that the mathematicians were
“called in,” as though “the ancient Egyptian property tax collector, a
Chinese astronomer, or an Indian engineer” were not also the
mathematicians. That is an imposition of modern disciplinary
specializations on ancient practices.
When the Indian mathematician and astronomer Madhava of
Sangamagrama (c. 1340 – c. 1425), who came up with the power series, did
mathematical astronomical calculations, his job was just beginning. He then
had to express the math in poetic form, in two different versions, as I
remember from a lecture I once heard by historian of science David Pingree.
Pingree published that talk, and gives an example of the second version:
“His most momentous achievement was the creation of methods to compute
accurate values for trigonometric functions by generating infinite series.
In order to demonstrate the character of his solutions and expressions of
them, I will translate a few of his verses and quote some Sanskrit…”
“…Another extraordinary verse written by Madhava employs the katapayadi
system in which the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 are
represented by the consonants that are immediately followed by a vowel;
this allows the mathematician to create a verse with both a transparent
meaning due to the words and an unrelated numerical meaning due to the
consonants in those words. Madhava's verse is [question marks are accent
marks not printed]: vidv?ms tunnabalah kav?sanicayah sar v?rthas?lasthiro
nirviddh?nganarendrarun The verbal meaning is : ‘The ruler whose army has
been struck down gathers together the best of advisors and remains firm in
his conduct in all matters; then he shatters the (rival) king whose army
has not been destroyed.’”
How do you like this math? You might enjoy taking a look at
Pingree’s article, “The Logic of Non-Western Science: Mathematical
Discoveries in Medieval India,” in *Daedalus*. Vol. 132, No. 4, On Science
(Fall, 2003), pp. 45-53
http://www.jstor.org/stable/20027880?seq=1#page_scan_tab_contents
Gene Halton
On Thu, Mar 9, 2017 at 10:50 AM, Jeffrey Brian Downard <
[email protected]> wrote:
> John S, List,
>
> The question of what kinds of activities counts as mathematical inquiry
> and what kinds of systems count as mathematical systems is a question that
> is answered (on Peirce's account), first, by looking at the purposes that
> properly guide the inquiries, and second by looking at the methods that
> should be used to accomplish those purposes. The character of the diagrams
> that are constructed on the basis of the precepts is a tertiary concern.
>
> When it comes to mathematics "I mean by its general hypothesis the
> substance of its postulates and axioms, and even of its definitions, should
> they be contaminated with any substance, instead of being the pure verbiage
> they ought to be. We have to make choice, then, between a division of
> mathematics according to the matter of its hypotheses, or according to the
> forms of the schemata of which it avails itself." (CP 4.246)
>
> We could ask the further question of where the mathematical study of the
> systems drawn from games such as chess, go, checkers and tic-tac-toe fit
> within its major branches. In order to answer this question, Peirce
> indicates that the division should proceed on the following grounds:
>
> Let us, then, divide mathematics according to the nature of its general
> hypotheses, taking for the ground of primary division the multitude of
> units, or elements, that are supposed; and for the ground of subdivision
> that mode of relationship between the elements upon which the hypotheses
> focus the attention. CP 4.248
>
> As such, the question of whether or not one or another kind of game counts
> as mathematics is determined, on Peirce's account, mainly by the purposes
> that guide the formation and use of the general hypotheses that serve as
> the starting points for the more deductive parts of inquiries. What is
> more, the primary division between the mathematical study of these games is
> made first on the grounds of the kinds of multitudes that are involved and
> second on the grounds of the kinds of relations that hold between the
> elements. So, we should ask: where does the mathematical study of games
> such as chess, go, checkers and tic-tac-toe fit within the main branches
> of mathematics. Which are the study of finite and discrete systems, which
> are the study of denumerable or first-abnumerable infinite and discrete
> systems, and which are the study of higher order abnumerable infinite and
> continuous systems?
>
> Let us apply Peirce's distinction between the practical arts and sciences
> to this classification of mathematical systems. In a sense, the
> counting, measuring and calculating that was done for many centuries by
> ancient Chinese, Egyptian, and Indian mathematicians was certainly
> mathematical in character, but some of these activities (like much of our
> own) is really mathematics as a practical art. Doing mathematics in a
> more scientific spirit requires, it seems, an understanding of the purposes
> that govern the activities and the methods that should be employed. The
> main purpose of scientific inquiry in mathematics is the same as the
> guiding purpose of all scientific inquiry which is to conduct experiments
> that are designed to help us learn the truth concerning general questions
> about which we are currently experiencing real doubt. (CP 8.115)
>
> In "The Logic of Mathematics, an attempt to develop my categories from
> within," Peirce asks three questions:
>
> 1. 1. what are the different systems of hypotheses from which
> mathematical deduction can set out,
>
> 2. 2. what are their general characters,
>
> 3. 3. why are not other hypotheses possible, and the like?
>
> The answers to these questions--including especially the third--will help
> us sort out what kinds of purposes and methods are really mathematical as
> sciences and which are not. When it comes to the character of systems
> of hypotheses that lie at the basis of the *scientific* study of games
> such as chess, go, checkers and tic-tac-toe, what is necessary in order for
> the mathematical deduction of theorems to proceed?
>
> In his discussions of how mathematical inquiries often get started, Peirce
> points out that someone such as an ancient Egyptian property tax collector,
> a Chinese astronomer, or an Indian engineer asks a practical question that
> couldn't be answered by the normal means, so the mathematicians were called
> in for assistance. At this point, the mathematician went to work
> constructing diagrams that could serve as little "models" of the problem.
> In order for the models to be of any service, they had to be simpler than
> the real problem. Once the "models" and the rules for manipulating the
> diagrams were established, the mathematicians tries to discover what
> follows--necessarily--from those diagrams and rules. At this point, the
> mathematician is guided by a very different purpose than the tax collector,
> astronomer or engineer because the mathematician cares, first and foremost,
> about the purpose of finding the truth about what really does or doesn't
> follow consistently from the idealized hypotheses. It is a
> different question altogether--having a different purpose--when we ask what
> can be learned about the actual world from the results of the mathematical
> inquiries concerning this idealized systems.
>
> So, I am drawn to the conclusion that most people who play games such as
> chess and go are not engaged in mathematical inquiry--at least not in a
> scientific spirit. They just want to win the game. Perhaps I am being
> uncharitable to most people who play these games, or perhaps I'm being too
> much of a stickler about what is requisite to be guided by the proper aims
> and methods, but this is the way I interpret the texts cited above.
>
> --Jeff
>
>
> Jeffrey Downard
> Associate Professor
> Department of Philosophy
> Northern Arizona University
> (o) 928 523-8354 <(928)%20523-8354>
> ________________________________________
> From: John F Sowa <[email protected]>
> Sent: Thursday, March 9, 2017 7:09 AM
> To: Jerry LR Chandler; Peirce-L; Clark Goble
> Cc: Benjamin Udell; Frederik Stjernfelt; Jeffrey Brian Downard; Jeffrey
> Goldstein; Jon Alan Schmidt; Ahti-Veikko Pietarinen
> Subject: Re: [PEIRCE-L] Truth as Regulative or Real; Continuity and
> Boscovich points.
>
> Jerry, Clark, list,
>
> In my response to Jeff B.D., I was defending the claim that board
> games are versions of mathematics. But I definitely do *not* restrict
> math to board games or to set-theoretic models.
>
> Jerry
> > Many mathematicians reject set theory as a foundation for mathematics
>
> Yes. Peirce did and so do I. The four board games I cited illustrate
> diagrammatic reasoning. But those diagrams use only discrete set
> theory. Peirce also considered continuous diagrams, and so do I.
> I would also allow diagrams for any mathematical structures anyone
> might propose or discover -- including quantum-mechanical diagrams.
>
> >>> JFS
> >>> Thanks for the reference. On page 134, Béziau makes the
> >>> following point, and Peirce would agree:
> >> JYB
> >> Universal logic is not a logic but a general theory of different
> >> logics.
> > Jerry
> > Analyze this quote. Is [JYB] saying anything more beyond
> > a contradiction of terms?
>
> Peirce's semiotic is a general theory of all kinds of sign systems.
> Those systems include, as special cases, all natural languages and
> all versions of formal logic. I agree with Montague that the
> underlying semantics of NLs and formal logics are essentially the
> same, but I would add that formal logics are weaker than NLs.
>
> I interpreted JYB as saying that universal logic is a theory about
> logics in the same sense that CSP's semiotic is a theory about logics.
> But JYB's notion of universal logic is weaker than CSP's semiotic.
>
> >> JYB
> >> This general theory is no more a logic itself than is
> >> meteorology a cloud.
> > Jerry
> > What the hell is this supposed to mean? Merely an ill-chosen metaphor?
>
> My interpretation of JYB: Universal logic is to any particular logic
> as meteorology is to clouds.
>
> Jerry
> > Chemical isomers are not mathematical homomorphisms because of the
> > intrinsic nature of chemical identities. Thus, this reasoning is
> > not relevant to the composition of Boscovichian points.
>
> I would not impose any restrictions on the kinds of diagrams or the
> mappings that define similarity. If you can define a Boscovichian
> diagram for chemistry, I believe that Peirce's notion of diagrammatic
> reasoning could accommodate that diagram.
>
> Implication: Instead of defining a special kind of logic for every
> kind of subject matter, I would just change the kinds of diagrams
> -- quantum mechanical diagrams, Boscovichian diagrams, or whatever
> mathematical structures anyone might discover or imagine.
>
> JLRC
> > Semiotics is not, in my view, a foundation for logic which is
> > grounded on antecedent and consequences.
>
> That is a Fregean view of logic, not a Peircean view. For his
> Begriffsschrift, Frege chose implication, negation, and the
> universal quantifier as his primitives.
>
> For his algebraic logic, Peirce started with Boolean algebra and
> added quantifiers. But he later switched to existential graphs.
> The early version distinguished Alpha (Boolean) from Beta (which
> added the line of identity). But he later started with relational
> graphs (existence and conjunction) and added ovals for negation.
>
> For beginning students, Boolean algebra is too abstract. It just
> represents an NL sentence with a single letter like 'p'. Peirce's
> relational graphs are a better starting point because they can be
> translated to and from actual NL sentences. As a pedagogically
> sounder approach, I follow Peirce's later tutorials (circa 1909).
> See the first 25 slides of http://www.jfsowa.com/talks/egintro.pdf
>
> Note slides 3 and 4 which come from Peirce's own intro in MS 145.
> In slide 8, I discuss one of CSP's examples that has a direct
> mapping to and from RDF -- the basic notation for the Semantic Web.
>
> Many people believe RDF is a good starting point for logic. I hate
> the RDF notation, but I use the comparison to show semantic webbers
> how a real logic can be defined on top of something like RDF.
>
> Also note CSP's rules of inference (slide 25). They are grounded
> in the need to preserve truth (as determined by endoporeutic). And
> they apply equally well to Kamp's Discourse Representation Structures,
> which Kamp designed for NL semantics.
>
> Note slide 31, which presents two *derived rules of inference*
> that are implied by the rules in slide 25. These derived rules
> emphasize generalization and specialization. I believe that it is
> more appropriate to say that logic is a theory of generalization
> and specialization. That includes implication as a special case
> (p implies q iff p is more specialized than q).
>
> There is much more to say, some of which I say in the slides
> http://www.jfsowa.com/talks/ppe.pdf . See slides 39 to 60.
> In particular, note slide 59 about Turing oracles.
>
> Clark
> > The problem with the game theoretical view of mathematics is
> > the question of realism.
>
> I'm not sure what you mean by "game theoretical view".
> There are three options, with some similarities among them:
>
> 1. The idea that games like chess are mathematical systems.
>
> 2. The point that Peirce's endoporeutic may be characterized
> as an example of Hintikka's game theoretical semantics.
>
> 3. Wittgenstein's debate with Turing. (I prefer LW's side.)
>
> Clark
> > there’s a difference between how we use the language of
> > mathematics and what the objects of mathematics are. That
> > is what are the relationship between the game and reality.
>
> The issues of nominalism vs. realism are orthogonal to all three
> of these kinds of games.
>
> Clark
> > I think we have to think through carefully what sort of game we
> > are playing if we’re going to use that as our metaphor.
>
> Yes. But I believe that both nominalists and realists could adapt
> any of the three "game theoretical views" to metaphors that are
> compatible with their ways of thinking and talking.
>
> Summary: What I'm trying to emphasize is the fundamental
> importance of diagrammatic reasoning for logic, mathematics,
> language, science, and everyday life. The model-theoretic
> semantics used to define truth in formal logics is a special
> case of diagrammatic reasoning.
>
> John
>
>
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