Jon AS, Gary F, List,
JAS: the question is simply whether
Peirce ever explicitly states that
"mathematics is the universe
of all possibilities." If not, then nobody
can definitively
claim that this is what he intended.
The assumption that
mathematics specifies all of what Peirce called
"real
possibilities" is the working hypothesis of all scientists from
the ancient Greeks to the present -- independently of what words they
use to express their opinions. Peirce's many years of work and
writings
about science and engineering show that he was fully
convinced of the
power of mathematics. His many criticisms of
"metaphysicians" who don't
use mathematics show his belief
in the importance of mathematics for
every branch of philosophy.
JAS: As I have pointed out before, Peirce's "three
Universes" do not
consist of possibilities, actualities, and
necessities; they consist of
Ideas, Brute Actuality, and Signs (CP
6.455, EP 2:435, 1908) or
Possibles, Existents, and Necessitants (EP
2:478-479, 1908). Moreover,
Peirce does not claim that all
possibilities are real, only that some
possibilities are real (CP
5.453, EP 2:354, 1905; CP 4.581, 1906); and
pure mathematics is not
concerned at all with whether any given
possibility is real, only
with what would follow necessarily from it.
The first sentence
is inconsistent with the quotations cited in the
middle of that
paragraph. The last sentence is true, and it shows why
pure
mathematics is the basis for phaneroscopy. It's the first stage in
the identification and classification of the elements of experience.
After phaneroscopy, further evaluation depends on the normative
sciences.
To see the similarities, compare those quotations
to my version:
Ideas, Brute Actuality, Signs (CP 6.455, EP
2:435, 1908)
Possibles, Existents, Necessitants (EP 2:478-479,
1908)
Possibilities, Actualities, Necessities (AKA Laws)
(JFS)
The only difference between my version and Peirce's is in
the choice of
suffixes. As a compromise, I would be happy with a
selection from both of
Peirce's versions: Possibles, Actuality,
Necessitants.
As for the issue about which possibles are real
possibles, that depends
entirely on the applications. As scientists
make new discoveries,
different versions of mathematics may represent
different possibles that
may be considered real. There is no way of
knowing which of the
infinitely many possible mathematical theories
may someday prove to be
useful.
JFS: {This discussion]
shows the power of mathematics to specify
anything that exists in the
universe and beyond. That includes any
imaginable or even
unimaginable level of infinities.
jas: This is quite a
sweeping claim. How could mathematics specify
something that is
unimaginable?
Very easily. Cantor's hierarchies of infinities
include infinitely many
unimaginable structures and patterns of
structures. The mathematics of
quantum mechanics has a huge number
of surprises. Physicists have
learned how to do the math. But after
a century of research, they are
still stumbling across surprising
results that defy intuition. There
are fractals and multidimensionl
patterns that are important for various
applications, but only the
simplest examples can be imagined.
JFS: Those examples [of
birds, bats, moths, and Olympian gymnasts] show
the huge amount of
processing that takes place in seconds or less. And
all of it can be
specified in some version of mathematics, even though
it may take
many years for neuroscientists to discover the proper kind
of
mathematics to use.
JAS: This is a statement of faith, not
fact.
No. It's a brief summary of the ongoing work in
neuroscience, which has
made enormous progress in the past 30 years.
The artificial neural
networks implemented in computer systems have
been successful in complex
perception, such as recognizing faces in a
crowd or driving an
automobile. But the neuroscientists are working
diligently on the far
more complex brains from nematodes to fruit
flies to birds to humans.
They have learned a lot, and mathematics is
key to every stage of the
science. But there is much more work to be
done.
In a previous note, I commented on the following points
by Gary, but I'd
like to add another remark, based on the above.
JFS: Phenomenology/phaneroscopy analyzes experiences in the
phaneron in
order to classify and determine the elements of
experience. But as
Peirce said, the same kinds of experiences may
come from external
sensation, from imagination, or from memories.
GF: The phrase kinds of experiences is ambiguous. My specific
answer
to the question I posed would be that the distinction between
the actual
world and a world of imagination arises from awareness of
the difference
between Secondness and Firstness, together with the
recognition that
these distinct elements of experience are equally
elementary.
It's not ambiguous. It's sufficiently general to
include the others, but I agree that it's important to distinguish
them.
GF: The question is not about distinctions in general,
but about a
specific distinction which normative science cannot rely
on mathematics
to supply. Your general statement glosses over the
fact that according
to Peirces classification of sciences, normative
logic depends for its
principles on phenomenology/phaneroscopy (as
well as on mathematics,
from which it cannot inherit this distinction
in principle).
To illustrate the issues, consider an extremely
vivid dream. The
initial stages of phaneroscopy for analyzing the
experience would be the
same as the analysis of sensations from a
similar situation while awake.
But the evidence that it is a
dream might not come from any particular
experience, but from an
unusual transition between experiences. An
obvious example would be
hiking in the woods and suddenly waking up in
bed. Other examples
may be physically impossible actions or features
that are known to be
false. People may also have false memories or deja
vu
experiences that require detailed methodeutic or even clinical
therapy to untangle.
The initial interpretations would occur
in phaneroscopy. But more
complex analyses would require normative
logic and semeiotic.
John
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