Ben, lists,
The connection you drew between the first and the fourth definitions of
theorematic reasoning is quite interesting; I had not thought of conceptual
analysis in quite that way. At least, though, the complexity of the diagram
or icon is likely more complicated in the case of theorematic reasoning
than in corollarial reasoning. I suppose I somehow think that a theorematic
reasoning is often a previous corollarial reasoning but with something
novel introduced, which would make the theorematic reasoning
straightforwardly more complicated than the corollarial reasoning.
Part of my concern about the relationship between theorematic reasoning and
abductive inference is that Frederik isn't just attempting to discuss
mathematics when treating of theorematic diagrammatic reasoning. Rather,
the significance is for all knowledge. Because the mathematical-diagrams
are ubiquitous, and because Frederik takes the mathematical diagrams to be
a priori, this means that all knowledge includes the a priori as a
constituent element. This is a very Kantian move, repeated by C.I. Lewis in
his *Mind and the World-Order*. I am quite wary of this move.
I think it very important the way you put the following: "The conclusions
are aprioristically true only given the hypotheses, but the hypotheses
themselves are not aprioristically true nor asserted to be true except
hypothetically, and this hypotheticality is what allows such assurance of
the conclusions, although even the hypothesis is upended if it leads to
such contradictions as render the work futile". And then part of your quote
from Peirce: "Mathematics merely traces out the consequences of hypotheses
without caring whether they correspond to anything real or not. It is
purely deductive, and all necessary inference is mathematics, pure or
applied. Its hypotheses are suggested by any of the other sciences, but its
assumption of them is not a scientific act." There are two things to be
said about this. The first is that the hypotheses are originally suggested
by experience. The second is that, even once assumed, a hypothesis could
lead to a contradiction, which is a kind of experience, or so it seems to
me; there is a sense of brute fact, or Secondness, about a contradiction.
For hypotheses that continue to work, and turn out to apply to everyday
experience and sciences other than mathematics, it is the application which
proves the ultimate efficacy of the hypotheses and the necessary
conclusions drawn from them, and thus makes them a posteriori, a matter of
being really accepted as knowledge only when proved in application.
Here's an example of Frederik's take on a priori reasoning from p.287-8,
discussing Jared Diamond's *Guns, Germs, and Steel*: "Domestication
presupposes the presence of easily domesticated species and the stable
human settlement over many generations in the environment favoring the
survival of these species. But local domestications only get the ability to
deeply influence the development of human civilization if they are able to
spread from there to other areas and cultures...Most favourably it spreads
in the overall East-West direction, along isotherms, keeping climate
conditions approximately constant--as opposed to traveling in the
North-South direction where climate changes drastically with latitude. By
this piece of a priori diagram reasoning--based on the combination of
biogeographical ontology and the ontology of human culture
development--Eurasia stands out as a privileged site for the original
domestication of agricultural species...Empirical findings subsequently
corroborate this piece of theorematic reasoning".
The biogeographical ontology and the ontology of human culture development
cannot themselves be a priori, but rather the deliverance of scientific
inquiry. A whole host of information is brought forward, which is
inadequately reflected in the reference to two ontologies rather than to
two fields of scientific inquiry, which understanding as dealing not simply
with ontologies but with sciences would make not only ontology but also the
previously gathered information acknowledged as relevant. The diagrammatic
reasoning cannot be considered as an a priori affair. It is, so far as I
understand Peirce's placement of deduction in the order of inquiry, a
deductive development of ideas received through hypotheses, as would occur
in the typical abduction-deduction-induction approach to scientific method.
I don't doubt that pure mathematics is possible. I only doubt that it is
somehow to be conceived as reasoning which happens prior to all experience.
That's just not true. What is true is that its conclusions do not
immediately have to do with reality; for that, experiment in experience is
required.
I guess this is all related to my wondering about how diagrammatic
experimentation relates to experimentation generally, and the place of the
mathematical diagrammatic reasoning in the context of scientific method. It
seems to me that Frederik's take on the a priori is only made possible by
ignoring these considerations. To put it another way, I'm starting to feel
that his epistemology is a little too concerned with diagrams, and not
concerned enough with a holistic account of the process of knowledge
acquisition.
Well, but maybe I'm making too much out of it. It's just that, in reading
through the book, and finding a few different references to the a priori, I
suppose my empiricist sentiment got rattled.
-- Franklin
On Sun, Apr 19, 2015 at 6:09 PM, Benjamin Udell <bud...@nyc.rr.com> wrote:
> Franklin, lists,
>
> Again, thanks for your opening post.
>
> Some further comments. (I just noticed your reply to my previous message
> as I added finishing touches to my message below, which addresses other
> points than my first message did, so I figure that I'm not about to get the
> discussion crossed up).
>
> You wrote,
>
> As Frederik restates it, mathematics has to do with hypothetical abstract
> objects.
> [End quote]
>
> We should say, mathematics is about hypothetical abstract
> not-conventionally-linguistic objects (compare with Quine's saying that
> math deals with 'abstract nonlinguistic objects'). If _c*onceptual
> definitions*_ are understood as definitions either in, or lending
> themselves to, more-or-less conventional language, this makes more sense of
> saying that corollarial inference is deduction merely from conceptual
> definitions. Peirce says ("Truth (and Falsity and Error): *Logical*",
> Baldwin Dictionary, 1902, http://www.gnusystems.ca/BaldwinPeirce.htm#Truth
> CP 5.567) that mathematical statements are never pure enough to live up to
> the name 'pure mathematics' because they keep using words like 'points' and
> 'lines' which need to be understood more generally than the words
> themselves say. Elsewhere (CP 7.467) he says, "A concept is the living
> influence upon us of a diagram or icon, with whose several parts are
> connected in thought an equal number of feelings or ideas." Anyway, the
> above seems pertinent to where you summarized Frederik:
>
> 1. Theorematic reasoning is not reducible to inferences from conceptual
> definitions, i.e. conceptual analysis, in the way that corollarial
> reasoning is (though both require observation).
> [....]
> 4. Theorematic reasoning requires complex, or specially constructed,
> schemata, as opposed to simple schemata in corollarial reasoning; a matter
> of difference in degree of complexity.
> [....] [End quote]
>
> You wrote,
>
> The issue here is: how do theorematic reasoning and abductive inference
> relate to each other? How can we distinguish which cases are theorematic
> deductions and which cases are abductive inferences? Frederik mentions the
> relationship on p.276, but that discussion does not make the relationship
> very clear: "But in the course of conducting the experiment, an abductive
> phase appears when investigating which experimental procedure, among many,
> to follow; *which* new elements or foreign ideas to introduce". Is
> abduction somehow adventitious to theorematic reasoning, or is it in fact
> its inclusion in diagrammatic reasoning that marks the difference between
> corollarial and theorematic diagrammatic reasoning? If it is a necessary
> component, this throws doubt on Frederik's claims elsewhere that such
> reasoning can be a priori; abduction is always an answer to some experience
> calling for explanation.
> [End quote]
>
> Peirce indeed speaks in some places of the use of abductive inference in
> mathematics. Now, generally, the choosing of a deductive procedure is not
> itself a deductive act except for families of problems with established
> procedures; but that is just what 'creative' mathematics (Anellis's
> alternate term for 'pure' mathematical work) lacks; but this is just to say
> that we don't know how to program a computer to do creative mathematics.
> The conclusions are aprioristically true only given the hypotheses, but the
> hypotheses themselves are not aprioristically true nor asserted to be true
> except hypothetically, and this hypotheticality is what allows such
> assurance of the conclusions, although even the hypothesis is upended if it
> leads to such contradictions as render the work futile (when a
> contradiction can be safely 'cordoned off', then I guess it's like a
> birthmark of the system). According to Peirce in 1904 in his drafts of an
> intellectual autobiography
> http://www.degruyter.com/view/books/9783050047331/9783050047331.35/9783050047331.35.xml
> (Ketner 2009, among other places), the choosing of hypotheses for (pure)
> mathematical exploration "is not a scientific act", and Peirce is speaking
> of mathematics as itself a science:
>
> This classification (which has been worked out in minute detail) is to be
> regarded as simply Comte’s classification, corrected. That is to say, the
> endeavor has been so to arrange the scheme that each science ought to make
> appeal, for its general principles, exclusively to the sciences placed
> above it, while for instances and special facts, it will find the sciences
> below it more serviceable. Mathematics merely traces out the consequences
> of hypotheses without caring whether they correspond to anything real or
> not. It is purely deductive, and all necessary inference is mathematics,
> pure or applied. Its hypotheses are suggested by any of the other sciences,
> but its assumption of them is not a scientific act.
> [End quote]
>
> Yet can we confine mathematics to just the deductive part? Peirce himself
> did not always do so. Consider his remarks quoted within a quote from a
> Professor Fiske on page 7 in A Semicentennial History of the American
> Mathematifcal Society 1888-1938:
>
> "At a meeting of the Society in November 1894 in an eloquent oration on
> the nature of mathematics, C.S. Peirce proclaimed that the intellectual
> powers essential to the mathematician are 'Concentration, imagination, and
> generalization.' Then, after a dramatic pause, he cried, 'Did I hear some
> one say demonstration?' 'Why, my friends,' he added, 'demonstration is but
> the pavement on which the chariot of the mathematician rolls.'"
>
>
> http://books.google.com/books?id=sOGifU-L_coC&pg=PA7&lpg=PA7&dq=%22Peirce%22+%22pavement%22+imagination
>
> (Note that Peirce usually meant by (deductive) demonstration a particular
> kind of deduction.) Anyway, Peirce discussed the importance of
> concentration, imagination, generalization elsewhere too, as I recall. It's
> hard to reconcile those different views. If we take mathematics as all and
> only the deductions, then mathematics becomes 'but the pavement on which
> the chariot of the mathematician rolls'.
>
> I'm not sure what conclusion to draw here. I might add that Peirce
> believed that mathematics is not the science OF deductive conclusions, but
> instead just the science which draws deductive conclusions. Conclusions
> about the deductive relations among statements that constitute a theory -
> when we consider _*theory*_ narrowly as a system of logically related
> statements - might be considered a more 'theoretical' than 'hypothetical'
> science, and if this more purely 'theoretical' science were itself
> deductive, then it would seem to be a kind of applied but quite general
> mathematics - mathematical logic. Here one gets to the question of how to
> classify mathematical logic in terms of Peirce's distinction between
> 'mathematics of logic' and philosophical deductive logic, and, finding
> myself in over my head (as I so often am), I've become rather unsure about
> it.
>
> Best, Ben
>
>
>
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