An Examination of the Reality of Number: 
                    ________________________________________ 
                Is Number Part of the Universe or is it Imposed
                _______________________________________________
                       on the Universe by Human Beings?
                       _________________________________
 
                               Thomas O'Neill
 
                                Ripon College
 
                           oneillt@acad.ripon.edu
                                TJPAO@aol.com

                         (received: March 20, 1997)
 
 
	Number is familiar to nearly every human being.  But, when asked if 
number is real, one will respond (at best) with a perplexed look.  A common 
(verbal) response is:  "A number of what?"  Although people are familiar with 
number, they are not consciously familiar with the nature of number.  What 
number actually is constitutes a veritable mystery, and one which most people 
pass over in silence.  Going beyond the realm of the average person's 
thoughts, number's status does not become clearer.  Among philosophers, this 
issue is quite heatedly contested.  Some, such as Isaac Newton, John Stuart 
Mill,[1] and Pythagoras, have argued that number is fundamentally real.  
Number is that upon which the universe is based.  Others, including Immanuel 
Kant, H. Grassmann, and H. Henkel have argued that number is at best (in terms 
of its status as "real") a mental construct, and that it does not constitute a 
facet of what is ultimately real.[2]
 
	There is a significant metaphysical issue regarding the reality of 
number.  Assuming that number is more than mere nonsense, one must attempt to 
know the character of number in order to know whether or not one simply 
imposes[3] number on the universe, or if in fact number is a real part of the 
universe, independent of the minds of human beings.  Among the concerns which 
arise in such an undertaking are, assuming that number is real, whether or not 
number is embedded in matter, as Karl Popper asserts.[4]  In fact, one must 
determine whether in fact number is itself material, and if not, how it can 
exist at all.  Essentially, the metaphysical issues regarding number which 
arise when one inquires into the nature of number are the same which arise in 
any examination of universals.  Thus, this inquiry could be interpreted as a 
small examination of the metaphysical status of universals.  But, number is 
not the same as most other universals.
 
	In the world which is available to the senses, people sense examples 
of all universals; for example one sees a particular chair or smells a 
particular pie.  Number, unlike pies and chairs, is comprised of pattern 
(accounting for the relationship between number and logic which is discussed, 
for example, by Wittgenstein and Poincare[5]).  One does not sense it as one 
would a pie or a chair; instead, one arrives at an understanding of number 
through the exercising of reason.  This rational uncovering of number entails 
a step beyond sensation.  Number cannot be sensed.  But, this is not to say 
that the senses are useless in one's attempting to deal with number.
 
	Person A, for example, may claim to sense number by seeing "7" on the 
page of a book.  But, what Person A does is not sensation at all.  The act of 
sensation ends with the sensing of a symbol.  The number is represented by 
"7."  For, "7" is the same thing as "seven" and "x" in the equation "21-14=x." 
 Thus, number eludes the senses, but it can be reached from empirical 
sources.[6]  In fact, in the example given, empirical data constitutes the 
foundation from which one seeks to find the nature of number.  Since number 
seems so difficult to locate (metaphysically), further inquiry is necessary.  
It cannot be sensed, but it is known.  There must be some aspect of its nature 
which makes this possible.
 
	Another curiosity regarding number (including mathematical pattern) is 
that it always seems to "fit."  One is always followed by two, and the square 
root of 4 will always equal positive or negative 2.  In the apparent world 
(the world available to the senses, "apparent" is used as "that which 
appears"), non-numerical patterns do not reproduce themselves with this same 
accuracy.  A person pacing in a careful, calculated manner, cannot with complete 
accuracy retrace those steps which he had just made when he turns about so that 
he can walk back to his starting point.  It is easy to retrace the steps made by 
number:  1, 2, 3 becomes 3, 2, 1.
 
	Despite the differences between number and the apparent world which 
have been described above, number can be found "at work" in the apparent 
world.  Although number cannot be located empirically, a given person, using 
the apparent world as the starting-point, can employ reason to discover number 
"beneath" the apparent world.  Thus, it would seem as though number were 
enclosed in the apparent world.  But, this is where the debate enters.  Since 
the empirical world's patterns are not as crisp as those of number, some could 
argue that number is not a part of the reality which constitutes the base of 
the apparent world.  Instead, number (it could be argued) is simply imposed on 
reality by human beings in order to make some sense out of the universe, to 
provide some sort of order.
 
	Whether or not number is imposed on reality by human beings does not 
alter the fact that number must be real.  By real, I mean that number exists 
in what Popper has called "World 3."[7]  World 3 is that part of existence 
which houses theory,[8] mental constructs of all sorts,[9] and all varieties 
of art.  As stated earlier, number, just as any other World 3 entity, is not 
material.  But, this does not hinder number from interacting with that which 
exists in the sensory world.  In fact, World 3 entities must, by definition, 
be capable of interacting with World 1 entities, entities which comprise the 
empirical world.[10]  This means that number would have to be real in order to 
interact with the apparent world.  Therefore, number can (and must) be real if 
it imposed on the universe or if it is an intricate part of the universe 
without such an imposition.
 
	This presence of number, either imposed or embedded, in the apparent 
world leads to the problem of imposition and displacement.  If the universe 
were already established a certain way, it seems as though the imposition of a 
mental construct (in this case, number) on the universe would in some way 
disrupt the already present structure of that universe.  This would not, of 
course, be the case if the universe were already structured in such a way as 
to be receptive to this mathematical imposition.  As regards number, this 
would mean that in order for number to be imposed on the universe without 
disrupting the nature of that universe (or to keep with the language with 
which I started, in order not to displace any facet of reality which is not 
conducive to receiving the imposition), the universe would have to be 
structured mathematically.  This means that the universe would have to exhibit 
certain patterns.  As I established above, number is comprised of these same 
patterns.  Thus, for number to be imposed on universe, the universe would 
already have to contain number, independent of number's being imposed.
 
	An example will help to illustrate this point.  Number's being imposed 
on a universe which is already structured in the same way as number itself 
would be similar to dough's exhibiting the shape of a particular cookie cutter 
before the cookie cutter were pressed down upon the dough.  If, for the sake 
of example, a cookie cutter were shaped like a Kwanzaa mat, in order for this 
cookie cutter not to disturb the dough upon imposition, the dough, too, must 
be shaped like a Kwanzaa mat.  To return from the analogy, in order for number 
to be imposed on the universe without somehow changing the universe, the 
universe must be structured mathematically.
 
	The consequent of the structure of the universe as described above 
yields two possibilities.  Either the universe is mathematically structured, 
or it is not.  For this there can be no other alternatives; there must be 
either A or -A.Since the universe is structured mathematically, one of the 
following must be the case.  Either number is a part of the universe (a part 
of reality), or it is imposed on reality.  There does exist the possibility 
that the universe is not structured mathematically, and that the imposition of 
number on the universe by human beings forces the universe to adopt a 
mathematical structure.  This is possible even assuming the nature of number 
(as a World 3 entity) by Popper.  For, World 3 entities must be capable of 
interaction with those of World 1.[11]
 
	This ability of Popper's Worlds 3 and 1 to interact could be 
interpreted as humankind's "patterning" the universe, which will not yet be 
discredited as a possibility.[12]  It is so far an acceptable alternative 
because there are many things which people do to affect their environment.[13] 
 Walking through mud, for example, shapes the universe, albeit a small part of 
the universe, in a certain way.  Thus, pushing the universe into a 
mathematical mold would seemingly do the same.  This entails displacement.  
When one steps in the mud, the foot moves mud from where it was (the foot 
displaces the mud), thus leaving a footprint.  Imposing number on the universe 
would have the same effect.  Unless the universe were already structured  
mathematically (unless the footprint preceeded the person's foot), some parts 
of the universe would "spill over" upon imposition.  This portion of the 
universe which remains would be non-mathematical, creating a serious dilemma.  
The universe would be at once mathematical and non-mathematical.
 
	Kant's description of number as being imposed on reality accounts for 
the imposition without resulting in the problematic distinction between the 
mathematical and the non-mathematical.  He gives an example of a plate, in the 
Critique of Pure Reason.[14]  The plate is nothing more than a common plate.  
It becomes circular when a particular person imposes this mathematical concept 
(circularity) on the plate.[15]  This is not to say that the plate changes 
when a mathematical concept is imposed on it.  It does not change at all; the 
imposition is simply a means to understanding what the plate is.  This seems 
like a relatively harmless explanation, and therefore at first difficult to 
overcome.  For, according to Kant, the universe exists in a certain manner, we 
simply apply number to it.  Mathematics, therefore, would be no more than a 
hermeneutical tool, guiding people in their attempt to understand the 
universe.  Numbers would be mere labels applied to keep track of different 
things which one enounters through experience.
 
	This is simply impossible.  Number is more than an instrument to be 
applied to one's perceptions in order to better understand the phenomenal 
world.  In fact, within Kantian philosophy, this is inconsistent.  The 
concepts which one applies to the phenomenal world, according to Kant, are 
real; "[t]hat which coheres with the material conditions of experience 
(sensation), is real."[16]  This means that these hermeneutical labels are 
real.  There is no problem yet.  But, Kant further states that those concepts 
which correspond to phenomena are "necessary."[17]  The connection between the 
two is necessary, which means the one (for human beings at least) cannot exist 
without the other.  Phenomenal entities cannot exist without their conceptual 
counterparts.  This means that concepts must be imposed, resulting in a 
necessary connection.  What Kant portrays as two distinct entities lose the 
distinction here, for there exists *a necessary connection*.
 
	The noumenal and phenomenal worlds can no longer be completely 
separate, at least in regards to the status of number.  There exists a 
necessary connection between circularity and a certain plate (probably with 
most plates), even though the plate is a phenomenal concept.  For, the plate 
derives its structure (to each person who perceives the plate) from the 
imposition of circularity.  This circularity (to be imposed) comes to each 
human being intuitively; there is no process by which a person learns this 
(and other noumenal) concept.  People need such mathematical concepts to 
understand the phenomenal world in general, including the plate.  Thus, from 
the perspective of each human being (although this may not be something of 
which each human being is cognizant) numerical concepts are essential to their 
making sense of reality.  People only know the phenomenal world in terms of 
the concepts which they use to understand it.  This does not mean that people 
know "number."  One can glean the pattterns which comprise number through 
maninpulating the examples of number in the phenomenal world, but by 
definition, one cannot at all know number in the noumenal sense.
 
	One knows the plate discussed above in terms of the circularity which 
is necessarily imposed on the plate.[18]  For the circularity to be imposed on 
the plate, the plate must already be circular.  Kant even agrees with this; 
"the conception must contain  that which is represented in the object."[19]  
Kant further states that "the representation of the object must be homogeneous 
with the conception."[20]  If the representation of the object and the 
conception are to be homogeneous, they must be intertwined; they must exist 
together within the same entity.  This would mean that they could no longer be 
separated.  This inseparability destroys the distinction between phenomenal 
and noumenal.  As a result, Kant's distinction of number from empirical 
reality fails.
 
	Even though Kant's attempt to declare that number is imposed on 
reality did not succeed, there are other options.  Before the discussion of 
Kant's ideas, the problem of imposition not being able to concurrently 
encompass all of reality was discussed.  This would  result in some of 
reality's "spilling over."  Thus, the universe would at once be mathematical 
and non-mathematical, a dichotomy which invites difficulties.  If there 
existed a clear division between the two (mathematically imposed reality and 
that which does not have mathematical structures imposed on it), and no 
fluctuation between the two were possible, then the problem would be solved.  
But, if mathematical structure is imposed by people, then the distinction is 
not necessarily this simple.  The method of the imposition of mathematical 
structures becomes an issue.  If structure is imposed as one perceives the 
universe, then the structure of the universe would be in constant flux.  
Nobody's field of perception remains the same for every moment of life.  The 
result of this is a sort of "mathematico-structural relativism."  Essentially, 
the universe would only be structured mathematically for each individual 
person, and what is not perceived would be relegated to a Berkeleyan 
irrationality.[21]
 
	This notion of mathematical structure relying directly on the scope of 
one's senses is absolutely absurd.  Patterns necessarily entail consistency, 
and this shifting of structure from the numerical to the non-numerical as a 
result of one's changing perspective adheres to no pattern.  In fact, this 
perspective necessitates an absence of pattern.  The ramifications of this 
fact are extensive.  Since patterns entail consistency, and this empirically 
based notion of imposing mathematical structure cannot result in the 
consistency which patterns entail, then either the universe is not (and cannot 
be) mathematically structured or the mathematical structure of the universe 
must have a source other than a mind's imposing structure on it.
 
	The "mathematico-relativism" described above causes concerns more 
extensive than  the disrupting of patterns.  Taken to its absolute conclusion, 
this relativism can only result in chaos.  If Person A and Person B were to 
stand in the same field and focus on different aspects of the universe, 
regardless of how minute the differences appear, Person A would be imposing 
mathematical on one aspect of reality and Person B would not be aware of the 
structure that Person A imposed, as Person A would not be aware of the 
imposition of Person B.  Person A, for 
the purposes of example, has chosen a 
tree as the focus of his attention (and therefore of his imposition).  Person 
B has chosen a rock.  Both are aware that the other's focus does exist.  
Person A does not deny (and, in fact, is quite aware of) the existence of the 
rock, as Person B is of the tree.[22]  But, if mathematical structure is 
imposed by particular minds, then Person B does not impose mathematical 
structure on the tree.[23]  The result of these numerous and varied 
perceptions and impositions is that certain things are at once structured 
mathematically and non-mathematically.  This of course is irrational, and 
therefore antithetical to the nature of mathematical structure.  The result of 
this would be chaos, in that what is assumed by all to be real (and imposed as 
such) really has no absolute meaning, in a universe which is absolute.
 
	This conclusion does not guaruntee that the universe is structured 
mathematically, for there does exist another alternative.  As opposed to 
numerous minds imposing mathematical structure on the universe, which leads to 
the "mathematico-relativism" described above, the possibility of an "absolute 
mind," God, etc. which constantly imposes mathematical structure on the 
universe has not been discounted.  This does take the power/ responsibility of 
imposition away from human beings.  Essentially, it is no different from 
simply asserting that the universe is mathematically structured.  For, making 
God the "absolute imposer" ties him to the universe in such a way that God 
becomes what is ultimately real.  Since part of what it means to be God (in 
this context) entails the constant imposition of mathematical structure on 
reality, then this would make number real as a part of reality.  God would 
have to contain mathematical pattern.  There is also another possibility, 
namely that this absolute being can constitute the actual source of the 
universe.
 
	If the former is the case (an absolute being which is a part of the 
universe constantly imposes mathematical structure on the universe), then this 
absolute being must be structured mathematically, for mathematical structure 
cannot be imposed by a mind which is not capable of structruing things 
mathematically. In order to be capable of structuring things mathematically, 
this absolute mind must, itself, be structured mathematically.  But, if 
mathematical structure is something which is imposed, then mathematical 
structure must be imposed on this absolute mind by some source outside of this 
mind.  The only possible source of this imposition is the unvierse itself; the 
absolute imposing mind would have to constitute one facet of an already 
mathematically structured universe.  Thus, would the imposition of 
mathematical structure mean anything?  It could not, for it would be similar 
to placing a given cookie cutter over a piece of dough which is already shaped 
in the same way as that particular cookie cutter.  In fact, it is debatable as 
to whether or not such an "absolute mind" would really be capable of imposing 
any sort of mathematical structure at all.[24]  For, it seems as though this 
mind would have to have mathematical structure imposed on it to be able to 
impose that same sort of structure on the universe.  This is absurd, for the 
structure which it seeks to impose would already be there.
 
	Thus, it is clear that the idea that number is a structure which is 
imposed on reality is not sufficient to explain how number exists.  The 
alternative, consequently, must be pursued.  The alternative is that as 
opposesd to being imposed on the universe, number exists as a part of the 
universe.  One of the philosophers who most ardently defends this view of 
number and mathematics is Plato.  For Plato, number is an essential part of 
the universe.  Number is not real merely as a mental construct to be imposed 
on the universe; it consititutes an integral part of the universe.  This 
represents an important distinction, for not all of the philosophers who 
believe that number is imposed on reality assert that number is not in some 
way real.  Kant, for example, willingly admits the reality of number.[25]  
But, this view does not include number in the universe.  Thus, for Plato, 
number is not simply real.  It is included.[26]
 
	The best starting point for an examination of what Plato holds to be 
the nature of number in the universe is the Timaeus.  There, Plato asserts 
that the world resembles "something else," because it was constructed on an 
external pattern.[27]  Immediately, this sounds as though Plato holds the 
position that number is imposed on reality, as opposed to the notion that it 
is a part of reality.  Closer examination will show how this reflex cannot 
become a legitimate criticism.  The most important term to define is "world."  
By "world,' Plato means the apparent (or empirical) world.  According to 
Platonic philosophy, the apparent world is the world of "shadows," or examples 
of the Forms, for the Forms are basically Universals.[28]  Thus, in the 
apparent world, one can find examples of number, for instance 1, 5, 805, 
22/13, etc.  These examples of number are not imposed on the apparent world; 
they are simply in the apparent world.
 
	Further, in Platonic philosophy, the empirical world is only one facet 
of reality.  In fact, of the two parts into which Plato divides reality (the 
world of the Forms and that of examples of the Forms[29]), that of examples is 
secondary to the Forms.  It relies on the Forms in that the Forms constitute 
the sources of the examples.  It still seems as though the Forms are imposing 
number on reality, but this is impossible because the Forms constitute what is 
ultimately real.[30]  Thus, number is a fundamental part of the universe, 
because it constitutes part of what provides the basis for the emprircal 
world.  Imposition of number in Platonic philosophy is impossible, for it 
entails something outside placing a structure on something else.  To clarify 
this, in order for a structure of any kind (in this case mathematical) to be 
placed on the Forms, something would have to exist outside of the Forms, 
outside of reality.  According to Plato, this is impossible.[31]
 
	As opposed to simply showing how Platonic philosophy is defensible 
against criticism, I intend to demonstrate that it in fact does accurately 
portray reality.  This Form, "number," does represent the source of examples 
of number (or particular numbers) in the apparent world.  Otherwise, no actual 
proof is provided.  By demonstrating that one is not wrong does not constitute 
a substitute for showing that one is correct.  Plato's philosophy of 
mathematics does "work;" it can be applied to the apparent world successfully.
 
	Number, as I stated earlier, is defined by the myriad patterns of 
which it is comprised.  This is consistent with the Platonic definition of 
Form (as described in the Timaeus and Parmenides).  For, Plato asserts that 
Form is based on pattern.[32]  Number, of course, must be a Form.  This is 
evident from the fact that there are examples of this Form in the apparent 
world, and all of these examples share the same essential characteristics, 
namely the adherence to the necessary patterns.  Further, the mathematical 
patterns which comprise the Forms are "abstract ideas, which are intellectual 
conceptions."[33]  One thinks about number-as-Form, for according to Platonic 
philosophy, one cannot work with the Forms in the apparent world.[34]  Since 
number-as-Form cannot be discussed in the apparent world, as a result of its 
nature as an idea which refers to a universal concept, it must be an 
abstraction.  Since "number" fits the criteria which Plato established for 
determining whether or not something is a Form, then it must in fact be a 
Form.  Thus, that it is possible that number can be a Form has been 
established.
 
	Establishing that number can be a Form in the context of Platonic 
philosophy is not enough to establish it is the alternative to the notion that 
number is imposed on reality.  It must be demonstrated that this Form, 
"number," and the particulars which result from it actually correspond to the 
reality which Plato sought to explain.  I established at the beginning of the 
paper that number is real, and that one can locate (through a process 
involving experience and reason) the patterns which comprise "number."[35]  
Such patterns include the sequence which comprises the counting numbers, 
algebraic equations, and all varieties of calculus.  These patterns are not 
unfamiliar, therefore, to most human beings.  Almost all human beings above 
the age of four can count the natural numbers, and many people have been 
exposed to algebra.[36]  People know that these patterns exist.  But, what 
people know must be demonstrated as being real (and in this case, a Form), and 
real as what it is, the Form "number."  Essentially, what must be shown is 
that the patterns which comprise number are consistent.
 
	Number, as Plato describes it does correspond to the way in which 
number actually exists.  This can be demonstrated by testing the consistency 
of these numerical patterns.  One does not need to test every possible pattern 
in order to confirm that mathematical structure is internally consistent.  The 
demonstration of a limited number of simple patterns will establish this fact 
with equal proficiency.  The first to be tested is one which deals with square 
numbers.  There is a pattern which results from squaring natural numbers:  
Take "1" for example.  12=1, this adheres to the necessary rules of 
mathematics, and the ability to square numbers was one which the ancient 
Greeks did possess (in order to keep within the context of Plato).  The next 
natural number is 2, and 22=4.  The result of the operation 4-1 is 3.  Next, 
take 3, the natural number which follows 2.  32 results in 9, and the 
difference between 9 and 4 (the square of the preceding natural number, the 
preceding natural number being 2) is 5.  One more difference is needed.  The 
operation 42 yields 16, and 16-9=7.  The differences which have been gathered 
are 3 (the result of 22-12), 5 (the result of 32-22), and 7 (the result of 
42-32).  The difference between any two adjacent differences is 2.  Adjacent 
differences are those whose sources (an example of a source in this case would 
be 22-12) share an element.  Adjacent differences are, for example, 7 and 5.  
The element which they share is 32.  This can be applied at any level of the 
natural numbers.  252 is 625; 262 is 676, and 272 is 729.  729-676 yields 53, 
and 676-625 results in 51.[37]  The difference between these adjacent 
differences (53-51) is 2.  Thus, it is established that one of the patterns 
which comprises the Form "number" is consistent.  Further, this pattern relies 
on other patterns, namely those which make mathematical operations possible.  
Those essential to this example were multiplication and subtraction.  Thus, 
one example demonstrated the validity of numerous patterns, all of wich are 
essential to defining the Form "number."

	The Form which Plato calls number, therefore, does correspond to 
reality.  Plato did not simply create a system which is effective as no more 
than a mental exercise.  Number is consistent with the nature of the universe, 
as Plato stated that it would be.  Further, the patterns which comprise number 
can be known to human beings, and they can be useful as more than a mere 
mental game.  Plato's philosophical system (at least that portion which is 
pertinent here) provides an alternative to the notion that number constitutes 
a structure which human beings impose on  the universe.  It does not simply 
demonstrate that the notion of imposition is  incorrect; it also illustrates 
its own validity and utility.
 
	It has been shown that number is a part of reality; it is not a 
structure which people impose on reality.  But, the seemingly flippant yet 
incredibly pertinent question remains:  "who cares?"  Most people would state 
that regardless of whether or not number is imposed or already there, they 
experience the universe in a mathematically structured way.  Yet, this issue 
is actually of great importance.  Without number's being a part of the 
universe, geometry is doomed to uselessness.  Geometry is intimately tied to 
number.  Thus, if number does not correspond to reality, then neither does 
geometry, and geometry is useful and necessary in the world and lives of human 
beings.  It could still be argued that geometry, if it is imposed, would not 
lose its utility.  But, in fact, it would.
 
	The notion of imposing mathematical structure on reality entails the 
use of perception.  One must perceive something in order for it to be 
structured.  This would turn geometry into a subjective tool, for it would 
only be used by individuals within the field of their perceptions.  This could 
easily result in relativism, for a structuring which relies on individuals to 
impose the structure is open to the criticism that a place in which there are 
not any human beings is by nature irrational.  This means that a place which 
was temporarily inhabited by people loses all its mathematical structure once 
the people leave.  This has devastating ramifications in the field of 
knowledge, for if something's mathematical structure relies on human 
imposition, and if human beings understand things (in part) mathematically, 
then some human knowledge is actually meaningless, for it is knowledge of 
something real only while there is a human being there to maintain its 
mathematical structure.  Thus, a mathematical structure which exists 
independent of human beings saves a large portion of human knowledge and 
maintains order throughout the universe.
 
               ====================================
 
[1]  Frege, Gottlob.  The Foundations of Arithmetic.  transl. J. L. Austin 
(Northwestern University Press 1980), p.9. Also, the fragments about 
Pythagoras are taken from Readings in Ancient Western Philosophy by George F. 
McLean and Patrick J. Aspell (Apple, Century, Crofts 1970), 23-8.
 
[2]  Kant, Immanuel.  Critique of Pure Reason, p. 120.
 
[3]  Impose, here, means structure or organize.  If one imposes mathematical 
structure on the universe, one structures the universe mathematically, for 
one's self alone.  One does not impose mathematical patterns on the universe 
for other people.
 
[4]  Popper, Karl and John C. Eccles.  The Self and Its Brain:  An Argument 
for Interactionism (Routledge 1990), p. 40.
 
[5]  Poincare, Henri.  The Science of Value.  transl. Bruce Halsted (The 
Science Press 1907).
Wittgenstein, Ludwig.  Remarks on the Foundations of 
Mathematics.  eds.  G. H. von Wright, R. Rhees, and G. E. M. Anscombe.  
transl.  G. E. M. Anscombe (Basil Blackwell 1964).
 
[6]  Mill supported this view.  Quoted in Frege, p. 8.
 
[7]  Popper, p, 39.
 
[8]  Theory does not necessarily have to be "thought."  This means that 
theories which exist, but not as the objects of a mind (or multiple minds) can 
nonetheless be real.
 
[9]  Among these mental constructs, according to Popper, is language.
 
[10]  Popper, p. 39.
 
[11]  Popper, p. 16.
 
[12]  This patterning is similar to what Kant discusses in Critique of Pure 
Reason.  Specifically, he states that number is a "conception of the 
understanding" and is applied to phenomena. Kant, p. 120.
 
[13]  This, of course, involves the assumption that people can affect their 
environment.  It is a distinct possibility that, instead, people constitute 
intrical parts of their respective environments, and their actions do not 
"affect" their surroundings at all.  Instead, they merely constitute a cog in 
a greater machine, which includes both people and their surroundings.
 
[14]  Kant, p. 120.
 
[15]  Circularity may be called a geometrical construct, but geometrical form 
has been considered numerically based since before Aristotle. See Metaphysics. 
  trans. Richard Hope (Ann Arbor Press 1960), pp. 107, 274-5.
 
[16]  Kant, p. 166.
 
[17]  Ibid.
 
[18]  According to Kant, it is imposed necessarily.
 
[19]  Kant, p. 117.
 
[20]  Ibid.
 
[21]  This is not an assertion that Berkeley was in any way irrational.  What 
is intended is merely a play on Berkeley's statement (through Philonous):  
"[S]ensible things are those which are immediately perceived by sense."  A 
slight alteration of this assertion would yield:  Numerical (numerical 
entailing sensible) things are those which are immediately perceived by sense. 
 Essentially, what one does not perceive is, for that person, not structured 
mathematically. The quote is taken from: Berkeley, George.  "Three Dialogues." 
 From Plato to Nietzsche eds. Walter Kaufmann and Forrest E. Baird (Prentice 
Hall 1994), p. 625.
 
[22]  A strict empiricist view is not being promulgated here.  An object's 
existence (for a given person) does not depend on its being perceived.
 
[23]  If Person B recalls the memory of the tree, then he imposes mathematical 
structrue on that particular memory, not the object of his memory, in this 
instance the tree.  A memory of an object and the object itself cannot be 
assumed to be the same thing.  In fact, a third distinction must be made: 1)  
the object itself 2)  the perception of the object 3)  the memory of the 
object. The perception is actually the "location" of the imposition of 
mathematical structure.  But, the structure is related to the object itself.  
This is why, assuming that human beings impose mathematical structure on 
reality, "mathematico-relativism" is the only possible result.  Mathematical 
structure is limited to one's perception of a particular object.
 
[24]  In fact, it would be debatable as to whether or not such an "absolute 
mind" were really absolute.
 
[25]  Kant, p. 120 cf. p. 166.
 
[26]  Included, that is, by the universe.  Kant, for example, places number 
outside of what he calls reality (empirical reality), as a concept which is 
imposed on reality.  Plato, though, treats number (and mathematical 
relationships) as a Form (eidos).  Thus, it is a part of the "world of the 
Forms" and therefore a part of the universe.
 
[27]  Plato.  "Timaeus."  Timaeus and Critias.  (Penguin 1983), p. 41.  cf. 
Parmenides:  "[T]hese ideas exist in nature as patterns." Plato.  
"Parmenides."  Plato:  Cratylus, Parmenides, Greater Hippias, Lesser Hippias.  
transl. H. N. Fowler (Loeb 1992).
 
[28]  Still, there could be the tendency to say that number is being imposed 
on teh empirical "world."  This, though, would be wrong.  from the perpective 
fo human beings (to provide an analogy), the sun does not impose shadows on 
the empirical world.  Shadows happen as a result of the sun's role in the 
universe.  There is no conscious (or even unconscious) imposition.  This is no 
different from the "shadows" which comprise the empirical world in Platonic 
philosophy.  The Forms do not impose the "shadows."  The "shadows" are simply 
there.
 
[29]  Plato.  "Republic."  Great Dialogues of Plato.  transl. W. H. D. Rouse.  
(Mentor 1984), p. 30.
 
[30]  Republic, pp. 312-2.
 
[31]  Parmenides, p. 205:  "[A]ll is one."If "all is one," then nothing can 
exist outside of "all" to impose structures on it.
 
[32]  Timaeus, p. 41.  cf. Parmenides, p. 219.
 
[33]  Parmenides, p. 209.
 
[34]  This is demonstrated by the fact that in dialogues such as the Sophist 
and the Parmenides, the discussion of Fomrs which takes place between the 
participants in the dialogue becomes difficult to follow, for  people in the 
apparent world who are engaged in a discussion of  Form use language.  The 
only language to which human beings have access, according to Plato, is a 
language which is a function of the apparent world.  There do exist "true 
names" to things, but even if they can be known as abstractions, they cannot 
be discussed.  For, that would entail the dealing of Forms in terms of the 
apparent world, and that is impossible.  This subject is discussed at greater 
length by Plato in the Cratylus.
 
[35]  Plato would disagree with the means by which numerical patterns are 
found.  Empirically gathered information was by Plato considered to be 
useless.  Instead, he posited that knowledge of the Forms is intrinsic, and 
can be attained only through self-exploration and reason.  But, regardless of 
the means, the result remains the same.  Number is comprised of patterns and 
is therefore what Plato considered to be a Form.
 
[36]  Calculus, though it is not as widely known as counting and algebra, is 
nonetheless becoming an increasingly studied field of mathematics.
 
[37]  "25" was chosen at random.  "26" and "27," of course had to be chosen 
because of "25."  But, the random manner in which "25" was chosen preserves 
the randomness of the example.