On the Infinite Divisibility and Composition

               ____________________________________________ 

     of One-Dimensional Continua: an Ancient and Medieval Prespective

     ________________________________________________________________



	                    Mark A. Wilner

                        Georgetown University

                     wilnerm@gusun.georgetown.edu



                    (received: December 16, 1996)



	Consider the following question that most people, on a pensive 

occasion, have asked themselves in one form or another:  In what sense do 

parts make up a whole?  When most people think about this issue, they do so 

with regard to everyday, three-dimensional objects.  For most of these people, 

their philosophical quest ends when they arrive at the physical level of atoms 

and molecules.  For the more scientifically inclined, it may be at the level 

of these highly elusive sub-atomic particles, like electrons, etc.   However, 

there is no reason to suppose that even these sub-atomic particles cannot be 

"split" as well, at least intuitively.  If they are spatial objects, then how 

could they not have parts?  If they do not take up space, how could they form 

normal, three-dimensional objects with magnitude?  Perhaps, they could be 

divided *ad infinitum*.  This, roughly speaking, is the problem of the 

continuum. [1]	

 

	I pick up the discussion concerning the nature of the continuum from 

the medievals, who were directly responding to Aristotle's position.  The 

following argument is illustrative of a possible position one may have taken 

at that time. [2]

 

	1.  	If something can be done, to suppose that it has been 			

		done involves no impossibility.  (assumption) 

	2.  	If a continuum can be infinitely divided, it involves no 

		impossibility to suppose it has been infinitely divided. (from 1) 

	3.  	But, if a continuum has been infinitely divided, then either 	

		(i) there actually exists an infinity of indivisible but nonetheless 

		still extended parts (i.e., atoms); or (ii) there actually exists 

		an infinity of unextended parts (i.e., indivisibles).  (assumption) 

	4.  	It is impossible (i) that there be an indivisible and yet extended 

		part, and it is impossible (ii) that a continuum be made up out 

		of an infinity of unextended parts.  (assumption) 

	5.	So, it does involve an impossibility to suppose a continuum to 

		have been infinitely divided. (from lines 3 and 4)     

	6.  	So, a continuum cannot be infinitely divided. (from lines 2 and 5)     

	7.  	A continuum cannot be infinitely divisible. (from 6)

 

	The infinite divisibility of continua is essentially an Aristotelian 

concept.[3]   Because this argument concludes by denying this concept, it is 

ultimately anti-Aristotelian.  In essence, Aristotle argued for the 

divisibilist position.  Divisibilism holds that there exists no such entities 

as indivisibles, whether they be indivisible parts with extent, e.g. atoms, or 

indivisible parts without extent, e.g. indivisibles[4].   Atomism is the 

position which states that continua are composed of a finite number of 

indivisible parts with magnitude -- atoms.  Indivisibilism, on the other hand, 

maintains that continua are composed of indivisible parts without magnitude -- 

points, instants, etc.  Divisibilism denies both positions.  It holds that 

continua are composed of ever divisible parts with magnitude -- lines, 

stretches, or periods of time, etc.  Although the above argument might seem 

like a negative argument for divisibilism by demonstrating, in premises (1) to 

(6), the implausibility of the other above mentioned perspectives, it aims to 

show that continua cannot be infinitely divisible (7).

 

	It will be my contention that this argument fails.  In spite of its 

alleged unsound outcome, the argument is useful.  An analysis of each of the 

premises serves as a stepping-stone to further explore this multi-faceted 

debate concerning the composition and divisibility of continua and other 

related issues. Premises (1) and (2), I conclude, are justified.  I attack 

premise (3) on the grounds that there remains a further possibility not 

considered, i.e. an infinitely divided line may consist of no parts.  Then, 

using the same objection to premise (3), in yet a slightly different manner, I 

also attack part (ii) of premise (4) on the grounds that an infinitely divided 

continuum may consist of an infinite number of indivisibles.  Moreover, I 

appeal to an atomistic objection to part (i) of (4), but conclude that in the 

end it remains insufficient.  While these objections would be adequate to 

override the entire argument, it is not the only place open for a possible 

attack.  The argument also can be seen to be wanting due to the invalid move 

from premise (6) to the conclusion in (7).  I will then conclude with some 

closing remarks concerning the endorsed Aristotelian position.

 

Premises (1) and (2)

____________________

 

	Premises (1) and (2) may seem plausible.  If doing X is a real 

possibility, then there seems to be no impossibility in *supposing* that X has 

in fact already been done.  That is to say, to posit that it has been done, 

for the sake of the argument, involves no logical impossibility.  For example, 

if it is possible that I could run a mile in under five minutes, then the 

supposition that I have already done so would not be logically impossible.  

And likewise, by respectively replacing the universal terms found in (1), e.g. 

`something' and `done', to the particular terms found in (2), e.g. `a 

continuum' and `infinitely divided', in the manner which they are replaced, as 

well seems plausible.[5]

 

	Let us, however, take a closer look at premise (1).  If we separate 

the antecedent from the consequent and analyze the respective verbs, we can 

begin to see how the entire claim may not be plausible in all cases.  The verb 

phrase in the antecedent -- `can be done' -- involves a modal auxiliary, i.e. 

`can', which does not refer to any tense.  If X can be done, then (possibly) X 

could have been done, can be being done, or could be done in the future.  

Whereas the verb phrase in the consequent -- `X has been done' -- clearly 

refers to the past tense.  So, the pressing question is whether (1) would 

remain true when the `something' with which it is concerned could only happen 

in either the present or the future (and not the past).

 

	One may claim, however, that there are sound objections which involve 

past facts.  A possible objection of this sort might proceed as follows:  Say, 

one was *in fact* sitting at time t1 (in the past).  Even if it is possible 

that he can walk (he is a normal human being with two legs, etc.), to suppose 

him to be walking at t1 is impossible.  Thus, premise (1) is not true in all 

cases.

 

	This objection however fails.  Should we be consistent, it outright 

denies the antecedent.  It is not even possible that he can walk at t1, since 

it is a *timeless fact* that he was then sitting.  It seems to me that all 

such objections to premise (1) involve this inconsistency.  As spelled out 

above, we consider words like `can', `would', `may', i.e. modal auxiliaries, 

to be timeless.  This is also true of *facts* -- whether they be facts about 

the past, present, or future.  Thus, if we posit a supposition concerning a 

timeless fact about the past (or for that matter the present and future as 

well), then something to its contrary occurring at the same time (and to/for 

the same subject, object, etc.) is not even a possibility.  I, therefore, 

believe all objections of this nature fail.

 

	We can however consider `somethings' which can only happen in either 

the present or the future. For example, if the universe can end, would the 

supposition that it has been ended involve an impossibility? Of course, this 

depends upon whether time ceases to exist when the universe does.  If we 

accept this assumption (say, time only relationally exists), then premise (1), 

in this case, would be nonsensical.  For the consequent involves a supposition 

which could only be possible coming from a future perspective.  But if time 

ended, then there would be no future perspective by which to look back into 

the past.  Thus, even such a supposition is an impossibility.  So, by this 

example alone, we can see that it is possible that premise (1) may not be 

universally true.

 

	The above objection nonetheless rests upon a rather arguable 

assumption.  It is my contention that there are no other possible candidates 

which could satisfy the requisite conditions for an adequate objection to (1). 

 There are no other `somethings' which could only be done in either the 

present or the future (and not the past).  Moreover, I believe all objections 

concerning timeless facts occurring in the past fail on the grounds of 

inconsistency.  Thus, premise (1), as I see it, remains plausible.

 

	Let us now consider the putatively sound move to premise (2).  Perhaps 

one might prematurely object here by claiming that an infinitely divided 

continuum could not possibly exist.  The objector might argue that the process 

of infinite division would never end, so supposing a continuum to have been 

infinitely divided is contradictory.  It should be noted that whether or not a 

continuum can or cannot be infinitely divided is a significant question that 

will surface later in this paper.[6] The argument at this point, however, 

remains conditional.  That is to say, if a continuum can be infinitely 

divided, then this (hypothetical) itself involves no impossibility to suppose 

that it has in fact already been infinitely divided.[7]

 

Premise 3 

_________

 

	Premise (3) concerns the ontology of an infinitely divided continuum.  

It is assumed that if a continuum has been infinitely divided, then there must 

exist an infinite number of parts.  It is then understood that the only option 

for this infinite number of parts is that either (i) they are parts with 

magnitude like atoms, or (ii) they are parts without magnitude, like points or 

instants, i.e. indivisibles.  This distinction is designed to cover all of the 

possibilities.

 

	Perhaps, however, there is room to object to the assumption that if a 

continuum has been infinitely divided, then there necessarily would exist an 

infinite number of parts.  Let's assume that the process of infinitely 

dividing a finite line entails halving the original line, then halving the 

remaining two lines, then halving the remaining four lines, and so on.  Now, 

let's suppose that this process has sufficiently been completed, and as such, 

the line is correctly termed `infinitely divided'.  It is not clear that what 

remains would be an infinite number of parts (either with or without 

magnitude).[8]

 

	We can examine this process in a slightly different manner, which 

would suggest that if a continuum was infinitely divided, then there would 

exist *no* remaining parts.  Let's imagine a line being infinitely divided, in 

the manner as described above, but after each division, the remaining lines 

are *actually* separated (this potential/actual distinction will hopefully 

make my point easier to explicate).  Given the process-style as described 

above, one would first cut the line in half, thereby leaving two equal halves 

separated from each other.  Even though the line originally existed with the 

*potential* of being halved, these potential halves shared at least one part 

of the line.  This shared part is exactly what made the original line one line 

and not any more than one line in the first place.  Upon actually halving the 

line into two separate and equal lines, either (1) the remaining lines 

continue to share the part, or (2) the remaining lines cease to share the 

part.  (1) is impossible due to the fact that sharing the part was precisely 

the reason that the original line remained a coherent and unseparated whole.  

So, if two potential halves shared the part, then they would not be separated. 

 They are separated, so they do not anymore share the part.  Thus, we must 

conclude that (2) the remaining lines cease to share the part.  If (2), then 

either (2a) one of the remaining lines has the part and the other does not, or 

(2b) both of the remaining lines have parts of the part, but neither has the 

whole part, or (2c) both of the remaining lines do not either have the whole 

part or even parts of the part.  If (2a), then it would be impossible for the 

two remaining lines to have been halved equally, as one line would have one 

more part than the other.  Thus, (2a) is not an option.  If (2b), then we are 

correct to ask what part of the part was shared by the original line.  Upon 

recognition of this part of the part, we are faced with the initial problem 

once removed.  Either (1) the remaining lines continued to share the part of 

the part, or (2) the remaining lines ceased to share part of the part.  Thus, 

(2b) seems to be embarking on an infinite regress, and as such is 

unsatisfactory.  We are, therefore, left with (2c)--both of the remaining 

lines contain neither the whole part nor parts of the part.

 

	What would (2c) leave us with?  If both of the remaining lines do not 

contain any parts of the part, nor the whole part, we are forced to conclude 

that the original shared part between the two potential halves somehow ceases 

to exist when this potential is actualized, i.e. when potential parts are 

separated.[9]  Now, if we then consider the two remaining lines as actually 

two lines in themselves, and yet consider each of them containing two 

potential halves within each of them, then upon the subsequent separation of 

these potential halves, these two lines will both lose another part each.  So, 

it seems after every step in this process of halving lines, a part is lost.  

If this process continued to infinity and was completed, so as to correctly 

call the line `infinitely divided,' then an infinite number of parts would be 

lost. Thus, I believe it is possible to object to the assumption in premise 

(3) which holds that if a continuum is infinitely divided, then there will 

subsequently remain an infinite number of parts.[10]

 

	This above objection implicitly assumes that premise (3) holds that a 

continuum can be infinitely divided into *all* of its parts.  This is a 

necessary criterion for the objection to work.  The premise as it stands, 

however, holds that a continuum can be infinitely divided into parts, not all 

of its parts. Nevertheless, the latter *may* necessarily follow from the 

former.  This ultimately depends upon the meaning one attaches to the term 

`divisibility,' which will be further examined.

 

Premise (4) 

___________

 

	Let us now suppose that the first three premises are wholly justified. 

 Premise (4) has two claims: (i) it is impossible that there be an indivisible 

yet extended part, and (ii) it is impossible that a continuum be made up of an 

infinity of unextended parts.  The assumption behind the claim (i) is that all 

extended parts, as such, are divisible.  This claim, at the very least when 

considering extended entities such as lines, seems quite plausible.  The 

alternate view is atomism.  When considering physical entities atomism becomes 

more tenable, for perhaps it is impossible to divide a physical entity, yet it 

nonetheless remains extended. However, I believe it is entirely impossible for 

a continuum's parts to have this property of indivisibility.

 

	Atomism primarily emerged as an *a priori* solution to Zeno's 

paradoxes.  It asserts that the basic units of continua are indivisibles with 

extent.  The assumption is that Zeno's paradoxes are not wholly descriptive of 

reality.  In order to avoid their difficulties, the atomists argue that the 

existence of indivisibles with magnitudes must necessarily be posited.[11]   

For example, Sorabji writes, "Democritus . . . used the paradox of division 

everywhere as an argument for the existence of atomic magnitudes, that is, of 

magnitudes which, although having a positive size, are indivisible" (338).  A 

version of Democritus' argument could take the following form:

 

	(1)  	If a magnitude is divided everywhere, the resultant parts will 

		either be (i) indivisibles without magnitudes, or 

		(ii) indivisibles with magnitudes. 

	(2) 	If (i), then the original magnitude must have been composed 

		only of entities with no extent, which is impossible.

	(3) 	If (ii), then the original magnitude was not fully divided 

		everywhere, which contradicts (1). 

	(4)  	Thus, no magnitude can be divided everywhere. 

	(5)  	Thus, there must be indivisible entities with magnitude, i.e. atoms.

 

We should here make a few necessary clarificatory distinctions.  First, there 

is a distinction between types of divisibility.  On the one hand, there is the 

concept of physical divisibility or actual divisibility.  This refers to the 

ability or extent that one can actually "pull apart" segments from each other. 

For example, scientists may refer, in this sense, to the basic, constituent, 

and indivisible particles of matter. On the other hand, there is the concept 

of conceptual or potential divisibility.  This concept refers to the extent to 

which one may, in theory, mentally divide magnitudes.  In this mental sense, 

it seems plausible that even the basic building blocks of matter are possibly 

divisible into smaller parts.

 

	Another distinction is between `division everywhere' and `infinite 

divisibility'.  Division everywhere is the concept of division used by 

Democritus in his arguments for atomism.  For him, being divisible everywhere 

means simultaneous division, i.e. division into all constituent parts all at 

once. This form should be distinguished from the Aristotelian concept of 

`infinite divisibility'.  Aristotle's concept refers to the ability that a 

segment can continue to be divided into ever minute segments.

 

	Let's return now to Democritus' argument for atomism.  Democritus 

argues that division everywhere is not possible.  As such, one must posit 

indivisibles with extent, i.e. atoms (this is the move from 4 to 5).[12]  In 

spite of the fact that the Aristotelian position accords with the claim that 

division everywhere is impossible, he would argue against the subsequent 

necessary postulation division of atoms. That is, he would deny the conclusion 

follows from premise (4).  The reason is that when one applies the concept of 

infinite divisibility, there consequently remains an ever smaller magnitude.  

Democritus' argument "does not show any theoretical lower limit . . . .  And 

to this extent Aristotle is right to insist . . . that you can still divide 

*anywhere*" (Sorabji 341).  In other words, the postulation of atoms does not 

remain exempt from the problems caused by potential or conceptual 

divisibility.  Therefore, I believe that (i) in premise (4) of the original 

argument is justified.

 

	In the original argument, the second claim of premise (4) is based 

upon the assumption that no number of unextended parts (indivisibles), even if 

it is an infinite number, will ever give rise to a magnitude.  I believe this 

claim also is quite plausible.  The alternative to this view is 

indivisibilism. Indivisibilists hold that continua are composed of parts with 

no magnitude -- indivisibles.  Some of them, e.g. Robert Grosseteste and Henry 

of Harclay, hold that an infinite number of indivisibles make up continua, and 

others, e.g. Walter Chatton, hold that continua are composed of a finite 

number of indivisibles.

 

	In *Physics* VI (231a 21-231b 8), Aristotle argues against 

indivisibilism by proposing that any number of indivisibles could not be 

continuous or in contact with each other.   For Aristotle, two things are 

continuous only when their extremities are one.   Since indivisibles have no 

parts, they can have no extremities (by definition).  Thus, no two 

indivisibles could have extremities which are one.  Moreover, Aristotle 

suggests the only manner that indivisibles could be in contact with one 

another would be in such a manner so as to produce no extent.  Since 

indivisibles have no parts, the only way they could touch each other would be 

whole to whole (as opposed to part to part).  If, however, they touched each 

other whole to whole, they could not produce any magnitude.

 

	The medieval indivisibilists, thus, had to deal with these powerful 

arguments (especially the `touching argument'), in order to propound their own 

views.  Harclay, for example, attempted to refute the idea that if 

indivisibles were in contact with one another, they would not produce any 

quantum.  In *Determinatio Il*, he argues that although indivisibles cannot 

produce an increase in size if touching whole to whole "in the same local 

position", if they touched whole to whole "in distinct positions immediately 

next to one another" a quantum would result.  However, it seems to me in order 

to produce an increase in size, not only do things need to be immediately next 

to each other, but both things must already have magnitude. This objection 

strengthens Aristotle's position for it suggests that indivisibles cannot 

touch each other in any other respect than whole to whole, superimposed upon 

one another.  Because indivisibles entirely lack extent, to be touching whole 

to whole in distinct positions immediate to one another is just touching whole 

to whole in the same local position (superimposed upon one another).  In spite 

of Harclay's endeavors, many other philosophers have attempted to form 

plausible indivisibilist arguments, probably to explain angelic motion or the 

existence of unequal infinities. [16]

 

	Perhaps, my prior objection to premise (3) might be better constructed 

as a kind of indivisibilist argument itself, an argument which would attempt 

to show, like Grosseteste and Harclay, that an infinite number of indivisibles 

make up continua.  It was assumed throughout that argument that parts of lines 

must have extent.  But if assumed, it becomes unclear how if one of these 

parts is lost in the process of, i.e. the conclusion from (2c), how the 

summation of remaining lines, even when considering just the first step in the 

process, could be equal to that of the original line.  Thus, I should have 

also considered another alternative to (2c), i.e. that the parts in question 

may be parts without magnitude -- indivisibles.

 

	This picture of the remaining parts of infinite division on this 

indivisibilist account would be significantly different from the conclusion 

drawn from (2c) in my above objection to premise (3).  The main difference for 

the indivisibilist is that at each step in the process of infinitely halving a 

line, the part that originally was shared has no extent or magnitude (this is 

the further alternative to 2c).  Thus, when the two potential halves are 

actually separated, they do not lose any of their total magnitude, yet the 

shared part ceases to be shared nonetheless.  So, what remains after the first 

division is two lines and an unshared part without magnitude.  As this process 

is repeated, the magnitudes of the remaining lines continue to get smaller and 

smaller, and the number of remaining unshared parts without magnitude gets 

larger and larger. Thus, if taken to its infinite extent and completed, it 

would seem to follow that ultimately what remains from this process would be 

an infinite number of unshared parts without magnitude.

 

	Regardless of which form of indivisibilism one might hold, there 

ultimately remains the significant problem of how entities with no magnitude 

can possibly compose an entity with magnitude.  This issue understood in a 

mathematical context best explicates the problem.  Zero multiplied by any 

finite number, or infinity, will necessarily produce zero.  Hence, something 

with zero magnitude, even if multiplied an infinite number of times, will not 

yield something with non-zero magnitude.  Therefore, in spite of the above 

argument, there are further reasons to suppose indivisibilism in any form to 

be implausible.  We should, I believe, conclude that both claims (i) and (ii) 

in (4) are justified assumptions to make.

 

Premises (5) and (6) 

____________________

 

	Premises (5) and (6) are mere logical consequences from the first four 

premises in the argument. Thus, if we feel that these prior premises are 

justified, by modus tollens, we must necessarily hold that it does, in fact, 

involve an logically impossibility to suppose a continuum to have been 

infinitely divided. Hence, a continuum cannot be infinitely divided.

 

Conclusion (7) 

______________

 

	If the argument intended to prove that a continuum cannot be 

infinitely divided, then it would be a valid argument.  However, this is not 

the case.  The argument originally set out to prove that a continuum cannot be 

infinitely *divisible*.  The move from premise (6) to the conclusion (7) 

reflects the idea that the latter claim is somehow deducible from the former.  

But, it is altogether unclear how the impossibility of an infinitely divided 

continuum entails that a continuum cannot be infinitely divisible.

 

	On Aristotle's account, the infinite divisibility of a continuum means 

a continuing and never- ending process of division.  That is to say, an 

infinite process of division *can be* occurring, not that it has already been 

or could be completed (infinitely divided).  However, if defined as such, then 

the entire argument fails (the conclusion does not follow from premise (6)).  

It would conflate the impossibility of the infinite divisibility of a 

continuum with the impossibility of a continuum being infinitely divided.  

While it would necessarily be the case that if a continuum were infinitely 

divided, it would also be infinitely divisible, the converse nonetheless 

remains dubious.

 

	The reason that Aristotle holds this view of infinite divisibility and 

not an alternate view, like `division everywhere', can be understood by noting 

his views on the ontology of the infinite found in *Physics* III.  It is here 

that Aristotle denies the existence of actual infinities and argues the 

infinite can only potentially exist.  He describes infinity in terms of 

approaching limits or finitude.[17] Thus, the infinite is "not the outside of 

which there is nothing, but that outside of which there is always something" 

(III, 6, 207a 2-3).  Seen in this light, the infinite can never *actually* be 

more than finite, yet potentially it continues to ever extend.  Thus, infinity 

can only potentially exist.[18]

 

	Infinite division, likewise, can only potentially be completed.  Only 

a finite number of divisions could actually occur within a given magnitude, 

yet there will always remain the potential of more actual divisions.  

Concerning infinite divisibility, Aristotle writes, "as that magnitude is seen 

to be divided to infinity, the sum of the parts taken appears to tend toward 

something definite"(III 206b 6-8).  In this manner, we can see that 

Aristotle's notion of infinite division is similar to his conception of the 

infinite defined as approaching a limit.

 

	In this paper, I have attempted to show that the initial argument 

above fails.  First of all, I believe there are reasons for denying either the 

assumption in premise (3) or part (ii) of premise (4).  If a continuum has 

been infinitely divided, then there is the further possibility that actually 

there would exist either no remaining parts (contra premise (3)) or an 

infinite amount of indivisibles (contra part (ii) of premise (4)). Even if one 

does not accept this objection to (3) or to part (ii) of (4), the argument 

clearly fails elsewhere. In spite of the fact that the argument from (1) to 

(6) is sound (given that one accepts the assumptions made in premises (1), 

(3), and (4)), the move from premise (6) to the conclusion (7) is unwarranted. 

 The impossibility of the infinite divisibility of a continuum is simply not 

deducible from the impossibility of a continuum being infinitely divided.  

Hence, by assessing the original argument and consequently the many 

perspectives from which people have traditionally looked at the problems 

concerning the infinite divisibility and composition of continua, an 

Aristotelian form of divisibilism is ultimately endorsed.

 

=====================

 

[1]	  The above example concerning normal objects is illustrative of 

the problem of the continuum in three dimensions.  The problem in three 

dimensions concerns how two-dimensional planes form an object of three 

dimensions.  Similarly, there is the problem at the level of two dimensions, 

namely how lines form planes.  In this paper, however, I will be dealing with 

the most basic form of the continuum problem, the level of one dimension.  A 

relevant question would be the following:  In what sense do points constitute 

lines?

 

[2]	  The argument was presented to me as part of an essay question for 

the class "Classical Metaphysics of Time".  The ultimate source remains 

unknown.



[3]	  As opposed to `division everywhere' which derives from Zeno and 

its utilization by the presocratic atomists, e.g. Democritus.  This 

distinction will be explained in the discussion of premise (4).

 

[4]	  Actually, Aristotle might be understood better as a moderate 

divisibilist, as he held that the boundaries of continua are indivisible.  

Nonetheless, concerning the concept of infinite divisibility and the 

composition of continua, labeling him as a divisibilist does his thought no 

disservice.



[5]	  In spite of the fact that modal auxiliaries can often introduce 

fallacies when they are mixed with quantifiers, the move from (1) to (2) seems 

to me but a variant of the universal elimination rule, which states that from 

any universal proposition with the form (x)Fx, one can derive any particular, 

e.g. Fm.



[6]	  Especially when considering the move from premise (6) to the 

conclusion in (7).



[7]	  It is precisely on these grounds that I chose not raise the 

objection to premise (1) concerning on-going change, i.e. change which never 

ends.  Take, for example, the futile project of determining the exact value of 

pi.  The fact that it seems an impossible endeavor is irrelevant to the claim 

as it exists in its conditional form.  If it is true that one can determine 

the exact value of pi, then premise (1) would encompass this situation.  If it 

is false, then the scope of premise (1) would lie outside of such 

considerations.  However, whether it can actually be done or not, is 

irrelevant to our present purposes.

 

[8]	  First of all, I must admit that this is an assumption that 

infinite division should occur in this fashion.  It certainly accords with the 

principle nonetheless.  Secondly, one might suggest that it involves a logical 

impossibility to suppose that this process could be completed.  This idea is 

perhaps implicitly lurking behind my subsequent objection to step (3).  A 

possible reply might be to the extent that from the eternal perspective, God 

would be able to have completed the process.  Nevertheless, since the argument 

has already supposed the process to have ended, it forces us to conceive what 

an infinitely divided continuum would look like.

 

[9]	  It might here be objected that perhaps the part does not cease to 

exist, rather it exists unshared and without magnitude. This further 

alternative will be examined later (in the examination of step 4).

 

[10]	  The above objection may seem to a variant of an argument by Adam 

of Wodeham (c. 1295-1358).  In his Tractatus de indivisibilibus, he argued 

against the existence of indivisibles on grounds that one cannot actually 

divide a continuum into them.  For after the first division, the resultant 

would be two different continua.  Thus, what one would be continuing to divide 

would be something other than the original continuum.  Although my objection 

to premise (3) might seem to have characteristics of this argument, it should 

remain distinct due to differences in purpose.  Adam of Wodeham was interested 

in showing the implausibility of actually dividing a continuum, whereas I am 

interested in the ontology of a continuum given that it has already been 

infinitely divided.

 

[11]	  Sorabji, Richard, Time, Creation, and the Continuum:  theories 

in antiquity and the early middle ages (New York: Cornell UP, 1983) 338.

 

[12]	  This of course depends upon whether one accepts the prior 

premises.  It seems to me that the only arguable prior premise is (2).  

Whether (2) is an impossibility or not is an issue that will be further 

examined later in this paper when concerning arguments for indivisibilism.

 

[13]	  He does, however, argue that indivisibles could be in succession 

to one another, but not in such a manner to form a continuum.

 

[14]	  If continuity is defined as such, i.e. extremities are one, it 

seems unclear how there originally existed two distinct things.  However, even 

if we discard his argument against the continuity of indivisibles, Aristotle's 

`touching argument' remains pressing.

 

[15]	  At least any magnitude greater than one indivisible -- which is 

nonetheless defined as extensionless.

 

[16]	  John E. Murdoch, "Infinity and continuity," The Cambridge 

History of Later Medieval Philosophy, ed. Norman Kretzmann, Anthony Kenny, and 

Jan Pinborg (Cambridge: Cambridge UP, 1982), 576-577

 

[17]	  This is the position that Sorabji takes on the matter.  He 

writes, "Infinity is an extendible finitude?(210).