Howard, lists,
I regard math's effectiveness in idioscopy, especially in physics, as
wondrous but _/not unreasonable/_, probably because I don't regard
either mathematical theory or the mathematical subject matter itself as
the mere creation, free or otherwise, of the brain of _/homo sapiens/_,
likewise as I don't regard physical theory, much less the physical
subject matter itself, as the mere creation, free or otherwise, of the
brain of _/homo sapiens/_. I don't want to repeat this a whole lot like
I'm thumping a table. I'm just saying that if one regards mathematics
mainly as a neural activity, then mathematics would seem absurdly
effective in physics and other special sciences, as if an average child
by doodling had invented a rocket ship.
I should say at some point that the idea of the _/unreasonable/_ seems a
lot like the idea of the absurd to me. Maybe it doesn't have as strong a
sense of the absurd to you. Many physical phenomena have been anomolous,
surprising, not reasoned out by people, but I don't think of them as
literally unreasonable. Well, one can't reason with a brute force, a
brute force is unreasonable in that sense.
In your quote of Peirce, he is not discussing the effectiveness of
mathematics in the special sciences. He is saying that mathematical
necessity is just as occult, mysterious, hidden from direct
consciousness, as is the natural necessity in chemical reactions; and
that both kinds of necessity compel certain outcomes in which one learns
something over and above the principles in conformity to which one set
up the chemical experiment or mathematical observation.
[Quote Peirce 1887, W 6:37-38]
I wish to begin by giving you some general idea of the nature of
reasoning. All reasoning involves observation. A chemist sets up an
apparatus of flasks and tubes, he puts certain substances in the
former, he applies heat, and then he watches closely to see what the
result will be. The procedure of the mathematician is closely
analogous to this. He draws a diagram, for example, conforming to
certain general conditions, and then he observes certain relations
among the parts of this diagram, over and above those which were
used to determine the construction of it. The result that the
chemist observes is brought about by Nature, the result that the
mathematician observes is brought about by the associations of the
mind. But this does not constitute so radical a distinction as it
seems at first sight to do; for were the laws of nature not
intelligible, that is, were they not such as naturally occur to the
human mind, the chemist's observation could never teach him
anything, while on the other hand the power that connects the
conditions of the mathematician's diagram with the relations he
observes in it is just as occult and mysterious to us as the power
of Nature that brings about the result of the chemical experiment.
You do not quite see the truth of this? No, why should you: I have
stated it in abstract terms, which give you nothing to observe this
fact in, and the mind can see no truth except by observation. To
enable you to see it, I will give two instances of simple
mathematical reasoning; you will please first see that they are
proofs, and then remark the fact that even after you know the proofs
you have no direct consciousness of the necessity that binds the
conclusion to the conditions supposed.
[End quote]
This parallelism of necessities is a condition, part (but hardly the
whole) of an explanation of why math would be effective in special
sciences.
In my previous post I indicated most of the rest of how I look at it, so
most of this will be a restatement. Mathematical considerations compel
mathematicians to agreed conclusions such that one can say that
mathematicals can be objectively investigated, and that's realness
enough. And, as Peirce said, "no imagination is _/mere/_" More of the
explanation of math's effectiveness in idioscopy can be seen in math's
extreme generality: the mathematical form studied is left quite
indefinite in all but certain respects. Peirce said,
[CP 5.550, "Basis of Pragmaticism," 1906, quote]
A _/mathematical form/_ of a state of things is such a
representation of that state of things as represents only the
samenesses and diversities involved in that state of things, without
definitely qualifying the subjects of the samenesses and
diversities. [....] The complete mathematical form of any state of
things, real or fictitious, represents every ingredient of that
state of things except the qualities of feeling connected with it.
It represents whatever importance or significance those qualities
may have; but the qualities themselves it does not represent.
[End quote]
I would add, without representing actual concrete haecceitous
individuals either; if we're considering _/two/_ things in general, they
don't need to be Aristotle and Socrates, so the mathematical form keeps
just the aspect important for the form under consideration, their twoness.
So mathematics is much more general than the special sciences. Pure
mathematics is about as general as one can get. It's about what can be,
what could be, what could have to be, potential necessities, potential
inexorabilities, if you will. Mathematics is rife with
transformabilities, often enough _/wondrous/_ (though _/not
unreasonable/_) in depth and capacity to bridge the seemingly most
disparate things. This gives it power to range over possibilities widely
enough to capture in its net those possibilities nontrivially
instantiated in the actual world, and such possibilities have been much
researched for that reason; Newton developed calculus to deal with
mechanics. We all learn arithmetic for obvious reasons. Our imagination
and mathematics involve many generalizations and abstractions _/from/_
our actual world. Physical laws instantiate some mathematical relations
or rules particularly well. Peirce called "nomology" the study of
universal laws and elements, physical or psychological, in the special
sciences. "Nomological" is a lot like "nomothetic." It means pertaining
to universal idioscopic laws which hold for reasons that we don't fully
understand, laws that have _/not/_ been derived in their specific
characters, the specific values in conventional units, etc., of their
constants, etc., from entirely philosophical, statistical, or
mathematical considerations. Instead they need to be established by
experiments on nature. (Peirce held that the laws developed by
habit-taking). That's the difference that I see between physical
laws/inexorabilities and mathematical rules/inexorabilities. That there
is something idiosyncratic, "just so," about physical laws in comparison
to mathematical rules does not stop physical laws from instantiating
mathematical rules. And, as I said, many mathematical rules, such as the
lack of a dynamic chiral transformation of a rigid body, are
instantiated in physics without being rules of time and rates such that
brute forces governed by those rules arise like sheriffs to enforce
those times and rates inexorably. In pure mathematical forms that
bruteness is not represented, and the pure form represents the forces
only indistinctly along with endless other things together as a
generality of things; it takes indices to tie the form to specifically
representing the physical which is itself a physical representation of
the form. I think that in your sense of 'inexorable' you are combining
the bruteness of the force with the law's character as a necessity. I
see no reason to regard as a quirk or happenstance the world's alliance
of bruteness of force and physical law together, and I can see that
physical law involves bruteness in a way that mathematical rules do not,
if that's what you're getting at. While it's wondrous that necessities
operate in the natural world, and that mathematics gets so deep, I don't
think that any of it is unreasonable, or that it is unreasonable that
physical necessities instantiate mathematical ones.
Beyond that, I'd say that indeed there is something even more
mathematical about physics, especially mechanics (which Einstein called
that part of physics which has been reduced to a mathematical
formalism), than about other special sciences. However, my approach to
making some sort of sense of it is, first of all - not solely, but still
first -, to try to see it as an instance of a pattern, a tree in a
forest. To me what would seem unreasonable would be that it were not
part of a larger pattern. Since I seem to be the only person I know of
who sees it that way, maybe it's just an odd turn of my mind. If
theoretical physics is the most mathematical of idioscopy, what is the
most idioscopic of idioscopy? I see two axes along which to try this.
First, there is the axis along which Peirce divides idioscopic sciences
into nomological, classificatory, and descriptive/explanatory. I would
do it a bit differently, but not with a difference that matters to my
example. Geology and human history are examples that Peirce gave of
descriptive/explanatory studies, and I'd agree, and add that they are
particularly idioscopic areas of idioscopy. Human history in particular
is full of idiosyncrasies and involves many webs of explanatory
hypotheses and lots of detective work. Then there is another axis, that
along which Peirce divides idisocopy into studies of physical phenomena
and studies of mental phenomena. Again, I'd do it somewhat differently,
but I'm not convinced that it matters in this example. I'd say that the
studies of people, society, intelligent beings, etc., have generally
more 'idioscopic' quality than the physical sciences have. Then I'd try
the same sort of thing with departments of pure math. If a consistent
pattern results, it's still kind of mysterious, but at least then the
high effectiveness of math in idioscopy, in physics most of all, is not
a standalone mystery.
Best, Ben
On 9/23/2014 8:29 PM, Howard Pattee wrote:
At 12:45 PM 9/22/2014, Benjamin Udell wrote:
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.
Ben, I am trying to understand how you distinguish laws and rules if
"laws seem for all the world like mathematical rules." You say, That's
why laws can be described by rules. Why does that work?
Like many physicists, I do not see very clearly why the free creations
of pure mathematicians, like complex numbers, matrices, infinite
dimensional vector spaces, and Lie groups, have turned out to so
accurately and elegantly model physical systems -- systems that are
themselves beyond our common sense and even our logic. I agree with
Wigner, math appears "unreasonably effective," and with Peirce, " . .
. it is probable that there is some secret here which remains to be
discovered."
Do you not see a categorical difference between laws and rules?
Howard
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