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Article 5312 of comp.ai.philosophy:
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>From: mp508504@ucunix.san.uc.edu (Michael Poremba)
Subject: Mathematics and Cognition
Message-ID: <1992Apr28.174351.14511@ucunix.san.uc.edu>
Organization: University of Cincinnati
Date: Tue, 28 Apr 92 17:43:51 GMT


The following has been brewing in me for quite some time. I typed it up after
having read a couple of threads on the nature of numbers and mathematics.


                       MATHEMATICS AND COGNITION:
                  A Meditation on the Nature of Numbers
 

DISSATISFACTION

	I doubt that I'm alone in thinking that the typical conversation 
concerning the existence of mathematical entities is almost never 
satisfying. As an undergraduate I have been disappointed by the 
inadequacies of technically versed members of academia in approaching 
the metaphysical or ontological basis underlying their field of study 
and professing. I've found this to hold too well within all fields--from 
music, to psychology, to engineering, to philosophy, to mathematics I 
repeatedly have encountered *specialists* unable to provide insightful 
direction in understanding human intellectual and "cognitive" endeavors. 
Oddly enough, or perhaps not, the best teacher I have found on 
approaches to these matters is a sculptor by trade.
	Whenever I encounter talk of the existence of mathematical 
entities I'm reminded of what I have learned concerning the nature of 
intelligence, language, rationality, and perhaps *cognition*. (I'm 
hesitant to use this word for fear of being mistaken as hopping on some 
bandwagon.) 
	I'm sure that anyone who would care to take the time to 
investigate the nature of the "existence" of entities such as the number 
3 would find that existence is actually quite different from how we 
ordinarily understand it. It  might be helpful to consider how some 
object such as a chair is ordinarily perceived.


CONSIDERING PERCEPTION

	A newborn, we can reasonably believe, does not perceive a chair as 
we normally do. So, there must be something developmental and acquired 
here since we all were infants. We are certainly not born with any sound 
notion of 'chair' or anything else. Considering the nature of our neural 
systems, it would be safe to bet that our familiarity with chair arises 
out of repeated stimulation by our environment. This stimulation takes 
many different forms--physical, visual, and  linguistic, to name some 
important ones. (It's also interesting to consider the perception of an 
already socially initiated human. Sensory deprivation leads to 
instability in consciousness, so there must be not only initial training 
but sustained interaction in order to maintain or ground the world-view 
that we develop.)


CONSIDERING PHYSICAL THERIES OF THE UNIVERSE

	If we adopt a quantum mechanical perspective of the same chair, we 
see that there is nothing definite which we can identify as 'the chair'. 
It seems to be "made up of" a locally predicted grouping of particles, 
which is quite unspecific. (The results of Bell's theorem suggest we 
haven't even this much.) The chair is actually ever-evolving on the sub-
atomic level; there is no static entity which we can easily pin as 
"existing." But a second look at the chair will show us that this change 
is ubiquitous. As it is used there is wear on the fabric covering it, or 
the varnish on the wood. Dirt accumulates on it and is removed from it. 
Even the paint can be removed with-out us having to refer to it as a 
different chair, although it is different in some way.


CONSIDERING ABSTRACTION

	What remains is it's use to us in our lives. It still relates to 
us and the things that we do and don't so with it. Something remains 
with this--the Platonic form. The chair remains an instantiation of the 
form 'chair' despite its having been altered through usage, decay, and 
alteration. This is the abstraction from the object itself. It's been 
quite some time since people has considered seriously the existence of 
Platonic forms. (But there have been many hardships in removing our feet 
from this thick but inviting muck. For example, observe the theories of 
this century which postulated characteristic and defining features of 
objects.) Abstraction seems to be something quite natural for us, yet we 
know that there are some inherent problems with it.
	So how is it that we claim the existence of the chair, yet what 
the chair "actually is" seems to elude us? This question has been 
approached for quite some time in philosophy and science with many 
enlightening answers having emerged. Unfortunately, such answers are 
rarely encountered but in the literature or by looking at the literature 
as a whole and combining bits and pieces from disparate sources and 
fields. This area of study is actually a specialty in itself, yet the 
answers found in it affect the general philosophizing that we all do, 
want to do, need to do, should be doing, or perhaps find our selves 
doing with varying degrees of competency and self-satisfaction.
	We have seen just from the above that with everyday objects such 
as chairs, existence arises through complicated neural training and 
physical interaction during which we learn to discriminate between 
different sorts of definite objects, with a finite character despite the 
infinite nature of the physical universe, and involves persistence 
arising from a non-static world. The dynamics are far more complicated 
than we usually recognize or care to acknowledge.


ASSIMILATION

	So, the nature of the existence of mathematical entities such as 
the number 3 is likely to be related to these other forms of existence. 
(Actually, I'm fudging here. This follows from some additional 
understandings about the nature of consciousness, cognition, perception, 
and all that.) Assuming this, the number three must somehow arise out of 
experience, include some natural kind of persistence, arise out of 
abstraction, and remain meaningful only within a systematic relationship 
to other entities. (That last one I pulled out of a hat, I realize, bur 
it's getting late. I believe this can be adequately shown and has been 
elsewhere. Accept a fudge for now.)
	Moreover, mathematical thinking for human beings must be quite 
natural (and, I believe it's reasonable to add, is fundamentally linked 
to linguistic practice). As we know, writing initially arose from the 
tabulation of goods in commercial transactions. Also, there was found 
(sorry, I haven't a reference at hand) a 35,000 year old wolf bone with 
about 45 parallel etched lines in it, with a pair of extended lines at 
25 (five five's). Is it dangerous to speculate that perhaps the ability 
for mathematics existed even then, despite the fact that this predates 
the foundation of any major civilization? Let's leave that one alone, 
presently. It's clear that simple arithmetic is older than writing 
itself, so we see the importance of numbers in understanding the nature 
of language, perception, and 'cognition'.


NUMBERS

	It seems reasonable, after all this talk, to suggest that the 
number three is an abstraction arising from our natural ability 
(tendency) to discriminate between objects (which itself arises from 
practicality as opposed to there being *any* justification or method for 
our discriminating between any two physical objects--see *some* 
interpretations of the results of Bell's theorem, although this issue 
has not been settled.) Abstraction and discrimination are inherent in 
our perceptual abilities. Most animals have the ability to discern, 
discriminate, and attach relational significance to distinguished 
entities. With linguistic activity we develop the conceptual skills to 
abstract further than we would without language. Counting "discrete" 
"objects" can be seen as a result of discernability skills which are 
inherent in our perception and abstraction skills which arise from 
language usage (easier than they would any other way). (There are some 
wonderfully interesting connections between early number systems, 
calendrical systems, and primitive vocabulary for things such as man, 
arm and finger. Many early counting systems were base 5. A man was 
called ten (ten fingers). The Mayan counting system contained 20 ones-
digits (maybe they used their toes!) and 18 tens-digits (relating to the 
number of months in their calendar. Eyes, arms and legs had similar 
names based on the word for 2. I heard it on NPR--it *must* be true.)
	Perhaps this is easy to believe for small integers, but what about 
the others? Systematic relationships arise out of being able to discern 
qualities of individual linguistic items. It the digit 3 has meaning, 
then the unsophisticated successor function will give you all of the 
others. More realistically concerning the development of mathematics, I 
would speculate that addition spurred the systemization of the positive 
integers. It's easy to verify through direct perception some qualities 
of addition. From these qualities, generalizations (abstractions) can be 
formed and tested through more involved experiences. Systemization of 
the integers seems only natural with this, in order to make the process 
of addition easier to manage.


MATHEMATICS

	From addition, subtraction, and multiplication seem to almost fall 
out. I propose that all of mathematics is based on seeing other 
relationships within this wholly abstracted, simplistic system, which is 
grounded in ordinary experience. The negative integers, reals, the 
number zero, division, square roots, logarithms were all follow from 
this basis. Similar types of abstraction from what we perceive as the 
real world led to geometry, but we won't go into this. Modern 
mathematics, as we know it, comes largely from developments in the past 
few centuries.
	So, I'm proposing here that we look at mathematics as a devised 
system which arose (and continues to develop) out of abstraction from 
our general experiences in "the world". The number three should not be 
taken to exist independently, somehow, from our minds. I want to go so 
far as to say that all existence is developed within our cognitive 
systems, but that it is severely conditioned by our perceptual 
experiences involving discrimination and abstraction (with the property 
of persistence resulting from the nature of our memory and the important 
part it plays in cognition).


RATIONALITY

	If this is clear, it should be easy to see that rationality is not 
a clear or definite practice or activity. We roughly approximate 
somewhat rational activity in our thought, but this is largely a result 
of the loose systemization of natural language but is helped along 
considerably by the rigor of linguistic systemization, including 
mathematics. (Think about the word deFINITion when you're trying to get 
to sleep tonight. Constructed systems.)

	It's weird how some people see this as so plain (ignoring the 
inaccuracies and slop) that it hardly seems worth mentioning. Others 
might nod along, accepting the general gist of it but failing to 
actually realize the full implications of this and be able to appreciate 
it. Some others will fight it and hold strong to century-old ideas which 
arose from fundamental mistakes, or newer ideas derived from this 
enormous tradition.


HEY YOU!

	I'd like some feedback on this stuff. It's not often that I get to 
voice it, but it's often that I hear things which don't fit into this 
conception of language, thought or understanding. There are tons of 
interesting related topics, but those too seem too rarely spoken of. 
Those who can speak of them are mostly way out of my ball park, and 
those that can't too often ignite frustration in me, because of the 
resulting communication gap.
	Much of what I said is rough, touched upon and vague, but some of 
you will, hopefully, be able to directly relate with it. Please respond. 
Write back personally, if you'd prefer. I welcome all criticism, but 
request that MZ in the Harvard mathematics department refrain from 
voicing any insults. Feedback will be used in thinking about a paper 
that I'm writing on this topic related to the sociology of science and 
the nature of scientific investigation.

Thanks for your time.


Michael Poremba
UnGr Studying Philosophy and Computer Science
mp508504@ucunix.san.uc.edu
mror323@pluto.csm.uc.edu




