On 9/20/2018 5:33 PM, Jerry LR Chandler wrote:
I ask that you clearly state the meaning of your notion of “pure
mathematics”. I have taken it to mean the usual undergraduate
level of mathematical philosophy...
The scope of pure mathematics, as Peirce defined it, is infinitely
larger than whatever was or ever will be discovered, taught, or
applied by anyone anywhere. That includes all intelligent aliens
in any galaxy anywhere in the universe.
in [John's] posts here when viewed from the traditional views of
the philosophy of Vienna Circle.
Absolutely not! the logical atomism of Russell and the logical
positivism of the Vienna Circlers were a low point in the logic
of the early 20th century. They made the "grave errors" (schwere
Irrtümer) that Wittgenstein spent the second half of his life
trying to correct.
A chemical atomic number... can never be a variable in the sense
of the Cartesian axis system, which is > the sense of Quine's
categorical error.
The word 'variable', as used in mathematics, is a metalevel term
about the notation. It just means that letters like x, y, z may
be used to refer to different things on different occasions. If
you use x to refer to something, that does not imply that the
thing you designate by x would vary.
To emphasize the point, you can replace the word 'variable' with
a synonym, such as 'index', 'label', or 'line of identity'.
Definition: For Peirce and modern mathematicians, pure mathematics
may be defined as the totality of all provably true statements of
the form "If [list of axioms], then [conclusion]."
For more, Peirce's CP has 49 instances of "pure mathematics".
In CP 1.636, for example, he says that the goal of pure mathematics
is to discover pure possibilities: "that real potential world" of
which actual existence is "nothing but an arbitrary locus":
if you enjoy the good fortune of talking with a number of
mathematicians of a high order, you will find that the typical
pure mathematician is a sort of Platonist... The eternal is for
him a world, a cosmos, in which the universe of actual existence
is nothing but an arbitrary locus. The end that pure mathematics
is pursuing is to discover that real potential world. (CP 1.646)
He also said that there is no danger of pure mathematics "evaporating
into an airy nothingness... spun from the stuff dreams are made of"
because that is "precisely what mathematics ought to be":
mathematics is distinguished from all other sciences except only
ethics, in standing in no need of ethics. Every other science...
is in its early stages in danger of evaporating into airy
nothingness, degenerating, as the Germans say, into an anachrioid
film, spun from the stuff that dreams are made of. There is no
such danger for pure mathematics; for that is precisely what
mathematics ought to be. (CP 4.242).
For 67 definitions of pure mathematics (which include CP 4.242),
https://todayinsci.com/QuotationsCategories/P_Cat/PureMathematics-Quotations.htm
Peirce would agree with most them, quibble with some, and reject
a few. I think he would like the following by Albert Einstein:
Pure mathematics is, in its way, the poetry of logical ideas.
One seeks the most general ideas of operation which will bring
together in simple, logical and unified form the largest possible
circle of formal relationships. In this effort toward logical
beauty, spiritual formulas are discovered necessary for the
deeper penetration into the laws of nature.
JLRC
This usage [in physics and chemistry] is much wider than predicate
logic but excludes the various forms of para-consistent logics.
Pure mathematics excludes nothing. Peirce rejected the idea that
mathematics depends on logic. Instead, every version of logic of
any kind is a special case of some theory of mathematics.
In fact, every humanly conceived version of pure mathematics,
which includes every version of logic as a subset, can be
specified with predicate calculus or existential graphs as the
metalanguage.
The nature of the metalanguage does not, in any way, constrain
or restrict the nature of the object language. For example,
the two-valued FOL can be used to specify 3-valued logic,
many-valued logic, fuzzy logic, modal logic(s), nonmonotonic
logic, intuitionistic logic, paraconsistent logic, etc.
John
-----------------------------
PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L
to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . To
UNSUBSCRIBE, send a message not to PEIRCE-L but to l...@list.iupui.edu with the
line "UNSubscribe PEIRCE-L" in the BODY of the message. More at
http://www.cspeirce.com/peirce-l/peirce-l.htm .
