List,

This part of MS 466 explains why logic is a "positive science" while
mathematics is not. It's because the logician investigates "what positive
facts about the real universe of things and of thoughts it is from which the
necessity of the mathematician's reasonings and the validity of other kinds
of reasonings depend, and exactly what the nature of that dependence is." 

We already know that the factual truth of its premises is irrelevant to the
necessity of the mathematician's reasonings; the conclusions drawn by the
latter are "necessary" because they assert no positive facts about the real
universe of things, and are therefore immune to the unavoidable fallibility
of such assertions. But from the logician's point of view, there must be
some implicit rules or laws governing such reasoning, and they cannot be
arbitrary assumptions as the premises of the mathematician's argument can
be. They must be dependent on "positive facts about the real universe of
things and of thoughts" (including facts about the relations between things
and thoughts). 

Moreover, the validity of other kinds of reasonings must depend on those
same "positive facts." The mathematician per se is content to rely on the
implicit guidance of these "facts," but the logician per se wants to make
them explicit, to show what they are. A logician like Peirce is interested
in "examining the fundamental nature of mathematics" not because logic is
mathematical, but because mathematics must have a logical core, so to speak.
If we can show the positive facts which constitute the core of even the
simplest mathematics, we will have the key to the validity of reasoning
generally - that is, we will know something basic about cognition itself,
about how it is possible to learn from experience.

The simplest mathematics would have only two values, and says Peirce. "our
system of Existential Graphs is precisely an application of the mathematics
of two values." He also says that "there would be a Mathematics of a System
of Three Values which would not be without utility," but does not venture
here to begin development of this would-be Mathematics. So it would seem
that system of Existential Graphs - including the Gamma part, which Lecture
4 is supposed to introduce, but which is not mentioned in MS 466 - confines
itself to "the mathematics of two values" in its attempt to reveal the core
facts on which cognition depends. But maybe that should be posed as a
question; and maybe we can ask whether analysis of a System of Three Values
could tell us more about cognition than we can learn from a two-valued
system.

Gary f.

 

From: g...@gnusystems.ca [mailto:g...@gnusystems.ca] 
Sent: 18-Feb-18 15:45
To: 'PEIRCE L' <PEIRCE-L@list.iupui.edu>
Subject: [PEIRCE-L] Lowell Lecture 4.2

 

Continuing from Lowell Lecture 4.1,
https://www.fromthepage.com/jeffdown1/1903-lowell-lectures/ms-466-467-1903-l
owell-lecture-iv/display/13956:

 

The only reason I do not agree with Dedekind in making mathematics a branch
of logic is that logic is not a science of pure assumptions but is a study
of positive truth. The mathematician seeks only to trace out the
consequences of his assumptions in the readiest and speediest way. The
logician does not care much what the conclusions from this or that system of
assumptions may be. What he is interested in is in dissecting reasonings, in
finding out what their elementary steps are, and in showing what positive
facts about the real universe of things and of thoughts it is from which the
necessity of the mathematician's reasonings and the validity of other kinds
of reasonings depend, and exactly what the nature of that dependence is. I
have already pointed out that the characters that make a system of symbols
good for [mathematical] purposes are quite contrary to those which would
make it good for logical purposes. The truth is that the mathematician and
the logician meet in one department on a common highway. They meet; but one
is facing one way while the other is facing just the other way. Each of
them, it is true, finds it interesting to turn round occasionally and take a
glance in the opposite direction. 

The mathematician, however, has little or nothing to learn of the logician.
Mathematics differs from all the special sciences whether of physics or of
psychics in never encountering logical difficulties, which it does not lie
entirely within his own competence to resolve. The logician on the other
hand has everything to learn from the mathematician. Mathematics is, with
the exception of Phenomenology and Ethics, the only science from which he
can draw any real guidance. There is therefore every reason in the world why
we should not delay examining the fundamental nature of mathematics. 

The simplest possible kind of mathematics would be the mathematics of a
system of only two values. Do we in experience meet with any case in which
such mathematics could have any application? I reply that we do. There are
just two values that any assertion can have. It is either true when it has
all the value it can have, or it is false, and has no value at all. It
follows that our system of Existential Graphs is precisely an application of
the mathematics of two values. It will give you an idea of what Pure
Mathematics is to imagine the Existential Graphs to be described without any
allusion whatever to their interpretation, but to be defined as symbols
subject to the fundamental rules of transformation. Namely, 

1st, Graphs are figures drawn on a surface and composed of any finite number
including zero of each of these 3 kinds of elements: 1st oval cuts of which
no two intersect, 2nd heavy lines, and 3rd spots each with definite places
each of which is at an extremity of a heavy line. 

2nd, Any graph within an even number of cuts can be erased and any within an
odd number of existing cuts can be inserted. 

3rd, Any graph can be iterated or deiterated provided the iterated or
deiterated replica is not outside of any cut that the other is inside of. 

4th, Two cuts one inside the other with nothing between except heavy lines
passing from within the inner to outside the outer, can either be made or
destroyed anywhere. 

5th, A heavy line of which both ends connect with lines inside a cut which
lines do not connect spots connected inside the cut can anywhere be erased
or inserted. 

>From those assumptions everything universally true of existential graphs
could be deduced, and that would be the Pure Mathematical Treatment. I have
defined Mathematics in general as the science which draws necessary
conclusions and which formulates the assumptions from which such conclusions
can be drawn. I now define Pure Mathematics as that Mathematics which leaves
its assumptions entirely indeterminate in respects which have no bearing
upon the manner in which they can be combined to produce conclusions. 

The Pure Mathematics of the System of Two Values would leave us free to
regard the Graphs as representing anything for which their fundamental
transformations would hold good. 

In like manner there would be a Mathematics of a System of Three Values
which would not be without utility and which has been in some measure
developed. The theory of numbers furnishes partial developments of the
mathematics of every system having a finite multitude of values. 

http://gnusystems.ca/Lowell4.htm }{ Peirce's Lowell Lectures of 1903

 

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