Genius and Species: Categoric and Axiomatic Understanding
	_________________________________________________________
 
                     Timothy Paul Smith
 
                      Department of Physics
                      University of New Hampshire
                      Durham, New Hampshire 03824
                      tps@fermi.unh.edu
 
 			(received: December 23, 1994)
 
	We grapple with nature and try to comprehend her through the 
collection of observations and then the subsequence synthesis of that 
raw data.  But at what stage does "understanding" take place?  It is 
sometimes argued that only in the collecting and categorizing of 
observation do we have a real handle on nature, only in the collection 
do we touch what is real and with substance.  Other will argue that only 
in the synthesis can we understand the observations.  The raw 
observation has too many details and the data is too vast.  This is our 
ideal model of inductive reasoning.  The power of this type of 
understanding is to be able to take an observation and place in the best 
and most descriptive context, to put it in the right "pigeon-hole".  I 
will refer to this as "categorical understanding".
 
observations			-->		synthesis
(real, but too		  [categorizing]	(abstract,
 vast to be to					  but easier
 really comprehended)	 			  to comprehend)
	
particulars			-->		general
			    [induction]
 
	There is a second tradition in reasoning; deduction, moving 
from a general concept to the particulars. I will refer to this as 
"axiomatic understanding". In the simplest sense it seems reasonable 
that if you could just learn a few axioms and then derive all the rest, 
you would have the simplest possible system to explain nature.  In its 
extreme form a zealot of the axiomatic approach view the axioms or laws 
of nature as being the most real aspect of nature, the most important 
description, and therefore the only realistic type of understanding.
 
general				-->		particulars
			    [deduction]
 
axioms				-->		observations
(most important)	    [axiomatic]		(confirmation)
 
	In this essay I will examine the role of "categoric" and 
"axiomatic" understanding.  First, however, I must define what I mean by 
"understanding".
 
	"Understanding" is usually defined simply as being able to 
reiterate or apply some information.  Perhaps an alternative definition 
is; an explanation which gives satisfaction.  One can imagine a 
conversation with a three or four year old child:
 
		"But why is it cold outside?"
		"It is winter and it is nighttime."
		"But why is it cold in the winter?"
		"Because the sun doesn't shine as much."
		"But why? What happens to the sun?"
 
If the line of question ever ended with "Oh I see now", then perhaps 
some type of understanding took place.  If, however, it only continues 
with "But why?" until one or both parties are exhausted it is 
questionable if there was any understanding. It is the experience of 
science that any answer will only raise new questions,  which perhaps 
marks the incomplete state of our understanding of nature.  However I 
maintain that the partial satisfaction which comes with a good 
explanation is also a mark of a partial understanding.
 
	Mythology answers questions in such a way that there is no 
need, no place for the "But why?".
 
		"Why does the sun move?"
		"It is the Chariot of Apollo."
 
In one sense the type of understanding you wish to obtain dictates what 
type of answers are possible and what type of questions are acceptable.
 
				_____
 
	"What is Man?"
 
	One answer might be, "he is the handy work of the gods", which 
has "plugged" man into his role in the cosmos,  he is above the animals, 
but below the angels.  Another answer might be; Kingdom: Animalia, 
Phylum: Chordata, (subphylum: Vertebrata), Class: Mammalia, Order:  
Primates, Family: Hominidae, Genus: Homo, Species: Homo sapiens.  This 
to completely answers the question, "What is Man?".  We can categorize 
to understand. [1]
 
	A place for everything and everything in its place. But that 
trivializes the role of a good categories. Categories are not just 
warehouses of information,  categories can tell us something unique 
about their contents.
 
	The medieval world view as described by Dante was a world of 
order. Everything of importance, ie. all ethics and morals, virtues and 
vices had their sphere.  Dante's "The Divine Comedy" is a Gazetteer to 
this orderly world. The order from the deepest pit of hell through the 
mountain peak of Purgatory, to the highest heights of the "Primum 
Mobile" (the prime mover, the outer-sphere) and finally to the 
"Empyreum", the seat of God, told us not only the order of vice and 
virtue,  but also the rate of ascension towards God.  If you see the 
world as having as its purpose the purification of souls, and the 
migration of souls towards the "Primum Mobile" then the system Dante 
describes is not so strange.  The categories make sense and are nearly 
inevitable.  The categories merely reflect the purpose for the existence 
(and why not structure?) of the cosmos.
 
	I suspect that part of the resistance to the Copernican 
Heliocentric model of the solar system was that it left in doubt the 
positions of the souls in heaven.  In the Dante - Ptolemaic - 
Aristotelian universe there is a particular place for souls which have 
indulged in every vice and virtue.  Tally up the acts of kindness and 
greed of a departed friend or relative and there was a prescription for 
calculating their position in heaven or hell.  If one debunks the 
geocentric universe,  one doesn't know where these souls are left.
 
	A really good system of cataloging nature is not just about 
organizing nature,  but hopefully the categories tell us something else. 
 For example consider the classification of species,  such as the 
example of humans cited above; Kingdom, Phylum, Class, Order, Family, 
Genus, Species.  The "Binomial System" of  Karl Linne' [1707-1778].  
Species is not just a sub-sub..-sub-division, it has an independent 
meaning,  a population of potentially interbreed individuals.  The 
difference between other levels of this classification is not as clear 
to me,  however what they do mark is branching points in the tree of 
evolution.  Species which are closer in the binomial system evolved from 
a common stock more recently in the past.  For example a human and a 
monkey share the same classification from Kingdom to Order,  so our 
evolutionary paths have more recently diverged then say a human and a 
fish, who share classification from Kingdom to Phylum (and subphylum), 
but no more.
 
	When Mendeleev built his periodic table of the elements in 
1871, he first grouped chemical elements with shared characteristics 
onto famililies or groups. Then he ordered them by atomic mass.  When we 
look at the Periodic Table today we still see this organization:
 
   1A 2A 3B 4B 5B 6B 7B 8  8  8  1B 2B 3A 4A 5A 6A 7A 0
 
   H                                                  He
   Li Be                               B  C  N  O  F  Ne
   Na Mg                               Al Si P  S  Cl Ar
   K  Ca Sc Ti V  Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
 
		(... etc.)
 
The number/letter at the top of each column gives us a group's 
characteristic. For example the last column, 0 , are the "Noble- or 
Inert-gasses", they are all normally gasses and essentially 
non-interacting.  The thing which was embedded in this table is atomic 
structure. All elements in Group 1A contain one valence electron, all 
elements in Group 2A contain two valence electrons, where as the 
"noble-gasses" in Group 0 contain no valence electrons.  The number of 
valence electrons determines the type of chemical binding.  On the 
atomic level Hydrogen (H) and Helium (He) contain one or two electrons 
in the first "orbit".  Li through Ne contain two electrons in the first 
orbit, and 1 to 8 electrons respectively in the second orbit.
 
	The point is that Mendeleev's table were build just as a 
catalog of the element,  arranged mass and grouped by characteristics, 
but at their heart was atomic structure, quantum mechanics,  even 
electron spin and fermi-statics.
 
				_____
 
			
	In the axiomatic tradition thinkers and writers often tell us 
that they are trying to build a system as sound as Euclid's "Elements", 
his book of geometry.  Starting with such simple things as:
 
			BOOK ONE
 
			DEFINITIONS
 
	1. A point is that which has no part.
	2. A line is breadthless length.	[2]
 
and end up with the world of plane geometry. Euclid can move from 
axioms to propositions and theorems, to useful concepts and results.
 
	The axiomatic system has a great deal of appeal.  From a few 
well picked axioms a whole system is derived.  If one could come up with 
the "Laws of the Universe",  presumably a handful of laws,  all the rest 
would follow.  One would understand the whole range of phenomena.  The 
appeal of the axiomatic system even transcends the physical world. If 
one could isolate the essentials of mathematics (i.e.. Godel's theorem 
[3])? (In truth I expect that applying it to the physical world is  
relatively recent extension.) However there are intrinsic problems with 
axiomatic systems.  Where do you find your axioms?  What does one do if 
the system that one derives is in conflict with the observed world 
around us?
 
	The obvious place to look for your axioms is in the world 
around you. Also, if your axioms are completely build on observations,  
then the system one derives must be consistent with the world observed. 
However Francis Bancon warns us:
 
	But the best demonstration by far is experience,
	if it go not beyond the actual experiment. For if
	it be transferred to other cases which are deemed
	similar, unless such transfer be made by a just and
	orderly process, it is a fallacious thing. [4]
 
	The best known application of the axiomatic system to the study 
of nature must be Sir Isaac Newton's "Principia".  Newton prefaces his 
opus with a series of definitions; such as mass, motion, momentum, force 
and so forth.  He then adds a few Scholium (scholarly side notes),  and 
then, on page 13, chapter II -
 
		Axioms, or Laws of Motion      [5]
 
Newtons presents his three laws of motion.  That is it as far as new 
material is concerned.  The next nearly four hundred pages are full of 
"Scholiums", "Corollaries", "Lemmas", "Propositions" and "Theorems", but 
they are just results and consequence of these laws.  The rest of the 
phenomena of the everyday (non-relativistic, non-quantum, i.e. 
classical) world follows.  Almost.  Even Newton warns us of the 
problems:
 
			SCHOLIUM
 
		Hitherto I have laid down such principles as
	have been received by mathematicians, and are confirmed
	by abundance of experiments. By the first two Laws and
	the first two Corrollaries, Galileo discovered that
	the descent of bodies varied as the square of the time
	(in duplicata ratione temporis) and that the motion of
	projectiles was in the curve of a parabola; experience
	agreeing with both, unless so far as these motions are
	a little retarded by the resistance of the air.   [6]
 
Perhaps the resistance of the air can be accounted for in Newtonian 
mechanics. Still the mechanics has its limitations.
 
	In the twentieth century, Einstein's mechanics (law of motion) 
is build on the two principles (axioms) of relativity; the invariance of 
the laws of physics, and the constant speed of light measured in any 
reference frame. Again there is a caveat; special relativity is valid in 
any inertial reference frame.  That is a reference frame with no linear 
or angular acceleration.
 
	The axiomatic system is powerful. We can start with such simple 
principles as conservation laws / continuity equations or  symmetries or 
something called "Action", which roughly is a type of  conservation,  
and derive far reach results.
 
		It is a most beautiful and awe-inspiring fact
	that all the fundamental laws of Classical Physics can
	be understood in terms of one mathematical construct
	called Action. ... In addition, as Dirac and Feynman
	have shown, the Action acquires its full importance
	in Quantum Mechanics. [7]
 
But they are not infallible.  They have their regions of application. 
Newton's laws apply to slow moving (non-relativistic), large 
(non-quantum) systems. Even within the range of application these 
systems have their limits.  The three examples which we have examined 
all have their caveats;  Euclid: flat space, Newton: frictionless 
systems (?), Einstein's Special Relativity: inertial reference frames. A 
question worth asking is: "are there any real world systems which 
satisfy these caveats exactly?".  In fact, in the real world these 
conditions are never perfectly satisfied. In reality Newtonian mechanics 
make predictions about results in a universe where Newtons word is the 
law.  It just so happens that these laws do a very good job, but not 
perfect job, of describing what we see.
 
				_____
 
	When considering the roles of categoric and axiomatic 
understanding I have saved my most recent example for last.  The story 
of the quark model of subatomic particles combines both these 
traditions.  Before the advent of the quark model, high energy physicist 
had build an extensive list of subatomic particle, neutron, proton, 
pion, kaons, eta, rho, omega, etc.  When enough of the important 
attributes of these particles were measured (and the important 
attributes identified) the particles were arranged in patterns and 
tables which has been compared to the periodic table of Mendeleev.
 
 
 
Mesons:
			    K0 --- K+
^ strangeness		  /   \   /  \
|			pi-    pi0&    pi+	   eta'
|----> isospin	  	  \   /eta\  /
	   	  	    K- --- Kbar0
 
Baryons:
 
 
	       neutron --  proton	* - * - * - *  Delta
	     /    \       /    \	 \ / \ / \ /
	Sigma-	   Sigma0&      Sigma+	  * - * - *  Sigma*
	     \    / Lambda\    /	   \ / \ /
	       Xi-  -----  Xi0		    * - *  Xi*
					     \ /
					      *  Omega-
 
 
When the relevant attributes were identified as isospin and 
strangeness, Gell-Mann saw something called the su(3) symmetry, and 
postulated quarks (1964: "Eight-fold way"). I of course oversimplify, 
and I must point out the continued extension of the theory to include 
charm, bottom and top(?) quarks, as well as strong and weak interaction, 
etc.. My point is that modern theories of sub-atomic particles are 
rooted in the collection and organization of raw data, but it is a final 
axiomatic theory which we are striving for.  Some writers think we are 
near the roads end (Chris Quigg is a leading particle theorist from 
FermiLab/SSC);
 
	A qualitatively satisfactory, if not entirely satisfying,
	unified theory may already exist. The most sanguine
	observers accept this sort of "grand unification" as
	a fait accompli and argue that a complete Theory of
	the World awaits only the incorporation of gravitation. [8]
 
other are more reserved (Gottfried and Weisskopf from  Cornell and 
MIT);
		On the other hand, the Grand Unified Theories
	also have a number of very unsightly blemishes. They
	offer not insight whatsoever into the mass spectrum
	of the quarks and leptons. Indeed, these masses must
	be inserted into the theory "by hand". [9]
 
	Perhaps before any theory is deemed "final",  we must ask the 
question of how would a final theory satisfy us?  What type of 
understanding are we seeking?  As scientist we no longer ask about the 
purpose of the universe,  we seem to be looking for "building blocks".  
What every the final theory is,  I expect it will be presented in terms 
of axioms, to be comprehensible,  and in terms of categories of data to 
be believable.
 
References
__________
 
[1] Delmont C. Smith, personal communication.
 
[2] Euclid,  Elements, p. 1.
 
[3] Michael O'Connor, Metaphysical Review - August 1994.
 
[4] Fransics Bacon, Novum Organum, LXX.
 
[5] Newton, Principia, p. 13.
 
[6] Newton, Principia, p. 21.
 
[7] Pierre Ramond, Field Theory: A Modern Primer, p. 1.
 
[8] Chris Quigg, Gauge Theories of Strong, Weak, and
	Electromagnetic Interactions, p. 1.
 
[9] Gottfried and Weisskopf, Concepts
	of Particle Physics, Vol I, p.164.