Howard, lists,
Responses interleaved.
On 9/20/2014 11:12 AM, Howard Pattee wrote:
>> BU: Epistemologies are not claims about special concrete phenomena
in the sense that they and their deductively implied conclusions
would be directly testable for falsity by special concrete
experiments or experiences.
> HP: Does your statement differ significantly from my statement,
"Epistemologies are not empirically decidable, e.g., not falsifiable."?
BU: Your statement suggests that epistemologies ought but fail to be
testable in the idioscopic manner of physics, chemistry, etc. Mine does
not suggest that and was specific about the kind of testability which
they lack - direct testability of them or their deductively implied
conclusions by concrete special experiments. Epistemologies are testable
as statistical principles are - in study and criticism. Likewise with
math. If somebody miscalculates pi (to some number of digits) and the
result is problems in manufacturing, then further investigation may
point doubts toward the employed calculated value. Yes, the application
was kind of test of the employed value of pi. But better math is what
disproves the employed value, i.e., shows that it was indeed a
miscalculation, bad math. If a statistical principle seems to be failing
in concrete applications, it needs to be investigated and, if need be,
corrected, at the level of theoretical statistics.
>> BU: That's also true of principles of statistics and of
statistical inference, yet such principles are not generally regarded
as requiring a leap of faith. Mathematics is also not directly
testable by special concrete experiments, yet mathematics, whether as
theory or language, is not generally regarded as requiring a leap of
faith.
> HP: I don't follow your logic. If A and B are both not empirically
testable that does not imply they both require a leap of faith. Of
course statistics and mathematics require much less faith than
epistemology. Math has rules. My next statement follows directly from
the first: "True belief in any epistemology requires a leap of faith."
Do you disagree?
Your next statement did not follow from the first, because your first
statement did not include the claim that statistics and mathematics have
better or stronger rules than epistemology and thus require less of a
leap of faith.
I agree that theoretical epistemology, as a discipline, has not had the
same evident success as theory of statistics. Of the sciences of
reasoning and beings that reason, only the mathematical ones, which I
take (in my not fully Peircean way) to be mathematical logic and theory
of structures of order, seem to have escaped serious disciplinal
dysfunctionality. Maybe it's because, like Peirce said, we homini
sapientes aren't rational and reasonable enough, at least not yet. Or
maybe there's an intractable problem with the reflexivity involved (I
have nothing original to say about this). However, that doesn't mean
that everybody needs to give up trying to do better than a leap of faith.
>> BU: What mathematics requires is leaps of transformational
imagination in honoring agreements (hypothetical assumptions) as
binding. Two dots in the imagination are as good an example of two
things as any two physical objects - better, even, since more
amenable for mathematical study.
> HP: That is fairly clear. Why do you think I would disagree?
Because in your original statement you said that ideas that are not
'empirically' (i.e., directly idioscopically) testable require a leap of
faith. I don't think that mathematics requires much of a leap of faith.
I brought up the idea of leaps of imagination as a contrast to the idea
of leaps of faith.
>> BU: Some sets of mathematical assumptions are nontrivial and lead
inexorably, deductively, to nontrivial conclusions which compel the
reasoner. If you think that mathematics is _/merely/_ symbols, still
that's to admit that mathematical symbols form structures that, by
their transformabilities, model possibilities.
> HP: Of course math is not /merely/ symbols. I said, "Mathematics is
the best /language/ that we use to describe physical laws. There is an
inexorability in physical laws that does not exist in the great
variety of mathematical concepts and rules." Do you disagree?
BU: I disagree. You say 'inexorable' but you mean inexorably
physically-forceful. It's a transference of sense. I'm thinking of
inexorable mathematical relations as a kind of inexorable. You're taking
the word 'real' in a sense much like that of 'actual'. This is why I
started asking you whether you thought numbers are objectively
investigable as numbers. I was taking the _/word/_ 'real' out of it, and
focusing on the meaning with which Peirce used it.
>> BU: Contrary to your claim, physical laws are not physical forces
and do not depend like forces on time and rates.
> HP: I did not say laws are forces. However, current fundamental laws
are /expressed/ as rate equations and involve one or more of the four
known forces.
BU: You said, "Symbolic rules are not like physical material forces.
Specifically, laws are inexorably time and rate-dependent." That's to
talk of laws as if they were the forces of which they are the laws. The
laws seem for all the world like mathematical rules nontrivially
operative as laws of physical quantities such as force, mass, velocity,
etc. That's why the laws can be formulated as mathematical rules, in
conventional mathematical symbols and formulas.
>> BU: Instead physical laws _/are/ _ those dependences on time and
rates and are expressed mathematically, which is to say that some
mathematics is instantiated in the actual . . .
> HP: I only said, "Mathematics provides the most exact /symbolic
language/ in which the laws are described."
BU: Mathematics is often called a 'language' but it's unlike any
conventional language. It grows through deductive proofs. In a way,
mathematics is deduction and vice versa. Doing creative mathematics
involves doing those proofs, not just fashioning a language. Applying
mathematics involves using deduction, not just expressing things with
mathematical formulas. In physical and other concrete applications,
mathematical deduction may seem like mere symbol manipulation because
often (but not always) less mathematical insight is exerted than in
doing creative, deep mathematics.
>> BU: . . . although you think that mathematical limit ideas of
absolute continuity and absolute discreteness should be instantiated
like photons, rocks, trees, or Socrates in order for mathematics to
be real.
> HP: That is certainly /not/ how I think. All I said was, "Logic and
mathematics do not involve time and rates." Peirce thought that logic
should involve time, but that in its (then) present state adding time
would be confusing.
BU: You said "I have no evidence that inanimate nature does even simple
mathematics. Human math began with geometry and numbers. No one has
discovered a point or a triangle or a number, the infinite or the
infinitesimal, in Nature." I have taken that in the general context of
your arguing that nominalism is the safest bet, although you are willing
to think like a realist when it's useful.
_/Doing/_ logic and mathematics involves physical times and rates in
_/a/ same sense as _/doing/_ physics does, insofar as the researcher is
physical. Physical theory is also specifically _/about/_ physical times
and rates, such that _/doing/_ physics involves _/doing/_ empirical
tests of them, while mathematics does not do such empirical tests and is
about physical times and rates only indistinctly, as possibilities, to
the extent that the math is applicable to physical times and rates, as
the same math may be applied to other things. This math is not merely
capable of linguistically 'expressing' physical times and rates, it is
implying predictions of how they will work and how they would work under
circumstances that have not yet been observed. One seeks to find which
math is applicable by testing the predictions. I've never seen anything
unreasonable about the effectiveness of mathematics in idioscopy.
Wondrous, yes, unreasonable, no, and I regard the wondrousness of it as
the wondrousness of mathematics' endlessly metamorphic inexorabilities,
some of which have been researched for their applicability to that
section of possibilities embodied as our actual universe. You suggest
that math is a lot freer than physics, it doesn't have the same
inexorability. But it is simply not confined to a particular kind of
inexorability, the kind involved in the operation of physical force.
The times that affect doing mathematics and logic by the nature of
mathematics and logic (as opposed to by the nature of physics) are
finitudes versus infinities. Some mathematical processes are infinite,
others are finitistic. That's inexorable. Some infinite processes
converge, some do not. The halting problem is inexorable. These logical
and mathematical properties also affect the doing of physics. But the
physical sheriff-like force doesn't always show up to enforce the rule
in a way that we can discern. By "real" you seem to mean "actual" or
"actual like a physical force." That's not what Peirce or most other
realists mean by "real".
Really, if any mathematics is applicable, without implying falsehoods,
to the actual, then all of it is in one way or another, to the extent of
the transformabilities that relate all of math. The applications vary in
nontriviality as well as in degree of unnecessary complication,
'epicycles' and the like; if the application is too trivial
(irrespectively of triviality/nontriviality of the math itself), we say
that the math is not really being applied, it's mostly about what would
happen if the actual situation were modified in some imaginative or
fantastical way, and we're really back in math's land of imagination.
But we also never completely leave the imagination even in math's
nontrivial applications to the actual, we merely confine ourselves to
the more 'realistic' imagined variations of the actual. Imagination and
daydream in their transformative capacities are the 'mathematical
sense,' they are our cognitive access to the possible, what can be, what
could be, just as concrete perception, exercising the familiar senses as
a single grip, is our cognitive access to the concrete actual. In a way
perception exercises all cognition in a single grip, and so it is that
Peirce can uphold the doctrine that everything in the mind enters
through the senses (including by observation of diagrams). Imagination
and daydream are in a way the most volition-like cognition, we can
freely imagine by whim and whimsy. But to honor assumptions, deduce,
actually to learn something from it that is not merely the product of
our whims, is to actually learn about what inexorably could be. This
turns out, as Peirce said, to involve considering things only in their
samenesses and diversities, not in their positive qualities, and I would
add, not in their positive concrete haecceities either. I would call
that a very high degree of generality.
>> BU: But "real" in a Peircean context just means capable of being
objectively investigated such that various intelligences would
converge sooner or later, but still inevitably, on the same
conclusions, rather than on some set of mutually incompatible
conclusions. You think that some sort of dynamicism is a safer and
more skeptical bet than realism about generals and modalities. But
the idea that varied intelligences will not tend toward agreement
about mathematical conclusions is no safe bet.
So the question is, again, do you think that numbers can be
objectively investigated as numbers? - such that (individually,
biologically, etc.) various intelligences, proceeding from the same
assumptions, would reach the same conclusions. If you do think so,
then you are a nominalist or anti-realist in name only.
> HP: As I have said, numbers can be interpreted by all of the
well-known epistemologies. I agree that various intelligences, various
trained animals, and various computers, proceeding from the same
axioms, strings rewriting rules, and programs will reach the same
conclusion. This has nothing to do with how I view numbers or
epistemologies.
BU: You seem like you may be limiting it to the 'automatic' or
corollarial kind of inference, but supposing that you're not, then what
you say is very nearly A: You agree with Peirce about the reality of
generals, and very nearly B: that Peirce's realist view of generals and
modalities has nothing to do with how you view numbers or epistemologies
- not in the sense that you disagree with Peirce's realism, but in the
sense that you and he are not discussing the same question in the first
place. In that sense, his and your conclusions about epistemology etc.
are not necessarily incompatible at all.
>> BU: The idea that mathematics' real end is to help physics, with
which your wording suggests agreement, was put forth by some
positivists, one of whom went so far as to say that mathematicians
who thought themselves to have some other or broader purpose should
discount their subjective feelings about it as merely illusory and
due to their choice of profession.
> HP: I think you are stretching to create straw men. I made no
suggestion that mathematics' real end is to help physics.
BU: I'm sorry I got you wrong. But it's hard to understand what you
think mathematics' theoretical aim is, if you regard it only as a
language and not as theory about something more than the way nerves act,
or something like that. But maybe none of that reflects your view either.
>> BU: I could go on, but the question is, do you think that numbers
can be objectively investigated as numbers? If so, then you are a
nominalist or anti-realist in name only, and a realist in the
Peircean sense. If not, then you do not believe that there is a
reliable mathematical expression of physical phenomena.
> HP: You do not seem to understand my point about the necessity of
complementary views, and epistemological name-calling is not going to
help. My teacher, George Polya, said always look at number in as many
way as possible. Platonism is helpful in thinking about pure math, in
the style of G. H. Hardy <http://en.wikipedia.org/wiki/G._H._Hardy>;
formalism and constructivism are helpful in computer design, intuition
is helpful in thinking about the continuum and infinity, empiricism is
necessary in cognitive studies, etc. One can always view number as
both objective and subjective.
As I said in my first (9/18/14) post answering Frederik, "What is
preposterous is to claim that any one view of the Foundations of
Mathematics <http://en.wikipedia.org/wiki/Foundations_of_mathematics>
is the only non-preposterous view." Historically, the so-called
/foundational crisis/ in math was only between epistemologies, like
Platonism, Intuitionism, Logicism, Formalism, etc. Today most
mathematicians and physicists feel secure proving theorems and trying
theories without being shackled by such undecidable and unproductive
arguments.
BU: It's very often useful to look in things in many ways, even in
incorrect ways, if one can find an interesting and fruitful pattern.
Peirce looked at various orderings of the propositions in the Barbara
syllogism, only one of which orderings was deductive, and one of which
is still called a fallacy in deduction today, and in them he saw the
deduction-induction-abduction trichotomy, and this seems to have
conformed or contributed to his view that all mental action has the form
of valid inference, valid inference merely sometimes mis-identified as
to mode. I've often thought in the classic formalist way of math as
mere marks on paper. Yet, in their following mathematical rules, they
behave as diagrams or parts of diagrams, and their comparative
'bareness' seems to reflect their power to establish fantastically
general lessons. Presumably you are not arguing that one should not take
sides in such questions, but are arguing only that one should be able to
see many perspectives even if one does not adopt them as one's own. I
don't know why you think that I don't do that. You argue for a
nominalist viewpoint - the "safest bet," etc. I'm arguing back, that's all.
Best, Ben
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