In
the Theaetetus, Plato examines the
manner and nature of human understanding, raising the implicit question whether
unhypothetical knowledge is possible. The third section of the Theaetetus demonstrates that unhypothetical
knowledge is impossible because an account of primary elements cannot be given.
The
third definition of knowledge posited in the Theaetetus is true judgement with an account (Theae. 201d)[1].
True judgement is distinguished from knowledge in that it is possible to have
true judgement without possessing an account. This implies that true judgement
alone cannot stand as knowledge. Knowledge is also distinguished from
perception or sensation, the initially rejected definition of ‘knowledge’.
Working
under this definition, one may categorize the components of reality according
to their ability to be judged and accounted for to determine what may be known
and what falls outside of knowledge. In the dream that Socrates recounts, it is
assumed that there are primary elements of reality of which everything else is
composed (Theae. 201e). A combination
of several primary elements gives rise to a ‘complex,’ which may either be the
same as the sum of its primary elements, or may constitute a ‘whole’ that is
distinguished from a mere sum of parts. If the ‘whole’ is different from the
‘sum,’ then the whole may not have parts. Since the whole has parts, it must be
the same as the sum (Theae.
204e-205a). As ‘sum’ and ‘whole’ have
the same meaning, a complex is identical to the sum of its primary elements.
Given
the definitions of primary elements and complexes, the next project is to
determine whether either or both of these categories of items may be objects of
knowledge. Primary elements may not be objects of knowledge as no account may
be given of them. If an account could be given of a primary element, there
would need to exist either information contained within the element that could
be extracted, or some outside information which could be applied to the
element. In the first case, allowing for information to be extracted from a
primary element would imply that the element is not indeed primary, as it would
be divisible into even more fundamental parts. These further fundamental parts
would then face the same dilemma of accountability that the original primary
element faced. If some outside information were applied to a primary element in
the creation of an account, then the element would be complex, not primary.
Even the attribution of being or non-being to a primary element is a further
attachment, which is inappropriate if the element is to be spoken of alone (Theae. 202a). Primary elements are
simply perceivable, and may be named (Theae.
202b).[2]
Complexes,
unlike primary elements, are knowable, given the assumption that an account is
a complex of names (Theae. 202b). In
the limited example of words and letters, the primary elements – the letters –
may be combined in various permutations to create words. Just as letters may be combined to create
words, names of primary elements may be woven together to create accounts of
complexes. An account of a complex is essentially a derivative of the language
used to speak about an object. Primary elements are perceivable, and named
using language. If a complex object is perceived, it may be understood by
describing it using the names of its most fundamental elements.[3]
There is an implicit limitation of perception in this analysis, since names are
a human invention and may only point to elements that are perceived. A
linguistic description, a string of names that lists the properties of the
object, constitutes the account.
There
is a problematic asymmetry that arises from this description of an account and
the argument that a complex is the same as the sum of its primary elements.
While complexes may be knowable since they are both accountable and objects of
true judgement, the primary elements that compose them are unknowable. The
account of a complex is a description of an object using named and perceived
but unknown components. Since the primary elements may not be known they remain
hypothetical – not objects of knowledge. Likewise the names used to talk about
primary elements and give accounts of complexes are not known. There is a
fundamental problem in asserting that a complex may be known despite the
unaccountability of its constituent parts. Knowledge of primary elements is
more fundamental than knowledge of complexes; it is “more decisive for the
mastery of any branch of study than knowledge of the complex” (Theae. 206b). If a complex is known, and
it consists of nothing more than a sum of its parts, it follows that both the
complex and its primary elements should be capable of being known. Since it has
already been demonstrated that primary elements may not be accounted for using
the previous definition of account, the definition of account should be
reevaluated.
Another
possibility is that a complex or ‘whole’ is not the same as the sum of its
parts. If a whole is different from the sum of its parts, it must contain no
parts; otherwise, it would be the same as a sum (Theae. 204e). If a complex is single and without parts, it is on
the same level as a primary element, and must be both unknowable and
unaccountable (Theae. 205e). This
resolves the asymmetry, but in a manner disclaiming the possibility of
knowledge. Another possibility is overlooked in this argument: the hypothesis
that complexes may consist of either primary elements or other complexes that
are composed of primary elements.[4]
This hypothesis regarding the formation of complexes leads to a certain ‘theory
of types’ in which complexes must be accounted for in terms of the primary or
complex elements they are immediately composed of as well as the primary
elements that are the most fundamental structures of composition. This
hypothesis results in an asymmetry despite the fact that it differentiates
between complexes composed of primary elements and complexes composed of other
complexes. The problem is the same as the one encountered in the supposition
that the whole is the same as the sum: complexes may be known and given
accounts, but primary elements may not.
Three
possible definitions of ‘account’ are proposed in an attempt to resolve the
asymmetry in accountability between primary elements and complexes.[5]
The list of definitions is presented as exhaustive (Theae. 206c). Under the first definition, an account is considered
to be making one’s thought apparent through speech (Theae. 206d). This definition is rejected, as it would allow an
account to be applicable to any correct judgement. A true judgement would no
longer be found without an account (Theae.
206d); this would dissolve the distinction between true judgement and true
judgement with an account.
The
second proposed definition of ‘account’ holds that an account consists in the
ability to reference the elements of an object of knowledge (Theae. 207a). This definition resolves
the asymmetry problem by providing a standard of accountability that may be met
by both primary elements and complexes. An account of a complex would consist
in listing the names of the elements that compose it; an account of a primary
element would consist in naming the element.[6]
This definition is rejected because it is not stringent enough upon what may be
considered knowledge; it allows for true judgement and an account to result in
an incomplete understanding of the object of knowledge. In an example, one may
possess the ability to recount the letters of a word, yet may use those letters
incorrectly in composing another word. In this case the person possesses true
judgement and an account of the first word, yet does not possess knowledge.
This definition of account fails to provide for the generality that is
necessary to claim that one has knowledge of the elements of an object. An
implicit requirement for knowledge is the ability to utilize a true judgement
and an account such that the object of knowledge may be properly used or
defined in a different context.
The
third proposed definition of ‘account’ holds that an account consists in the
ability to distinguish one object from all others (Theae. 208c). The requirement to distinguish one object from all
others is that one must be able to discern a feature that differentiates it
from everything else. If a feature common to several objects is grasped, an
account is provided for the subset of objects that possesses that feature. For
an account to be given of an individual object, there must be some feature that
is possessed by nothing else. It is plausible that an account may be given of
primary elements: since all they possess is a name, the name itself is the
distinguishing feature. If two primary elements are items of true judgement and
possess different names, they are distinguished from one another by the
difference in name. If they possess the same name, they are indistinguishable
by any feature and are the same.[7]
The distinguishability of a particular primary element from all other primary
elements by virtue of name provides a plausible account of that primary element.
Working
backward from complexes, the third definition of account becomes problematic in
that all the features of a complex may not be readily grasped. The initial
requirement of an account under the third definition is that all the primary
elements that compose a complex should be distinguished; otherwise, there may
exist some unperceived primary element possessed by one complex and not the
other. This could result in a mistaken account, in which two items are judged
to be the same but are actually distinguishable by one or more primary
features. Such an account would constitute false judgement and hence would not
result in knowledge. In this respect, it becomes necessary to first know all
the primary elements so that they may be distinguished within complexes. If
primary elements may be given an account and if it is possible to use these
accounts to know all of them, then it may be possible to give an account of a
complex.
When
complexes are considered, another difficulty arises with respect to the accountability
of primary elements. The requirement for an account is some feature that
distinguishes the object of knowledge from all other objects; primary elements
fulfill this requirement with respect to one another. When complexes are
considered as well, primary elements lose their distinguishability. Since
primary elements compose all complexes, a given primary element contained in a
complex possesses no feature that distinguishes it from the complex it is
contained in. Further, if two given complexes share a certain set of elements
but the second contains one additional element, the first complex possesses no
feature which allows it to be distinguished from the second.[8]
The account of a primary element must then include not only what distinguishes
it from all other primary elements, but what features it does not possess that
some other objects do. The account of any given object invokes not only
knowledge of a feature that distinguishes it from all other objects, but
knowledge of features it lacks that other objects possess.
This
results in the final definition of knowledge, “correct judgement accompanied by
knowledge of the differentness” (Theae.
210a), still working under the assumption that knowledge is correct judgement
with an account. As posited, this definition invokes an infinitely regressive
argument. If knowledge requires correct judgement and knowledge of
differentness, then knowledge of differentness itself requires correct
judgement and knowledge of that differentness, and so on, ad infinitum. A
skeptical argument of this pattern would never allow the definition of
knowledge to be fulfilled, resulting in a conjecture, itself unprovable, that
there exist no objects which may be known.
The
problem of regress may be resolved by operating within a particular framework
in which certain elements are assumed to be the only features of reality. One
could assume that only those elements that have been named factor into the
ontology, and could consider every possible complex that could be created by combining
such elements. If every possible complex and every element is listed, then one
could provide an account of each object by comparing it to the others and
stating what features it possesses that others do not, and what features it
does not possess that others do.[9]
This would allow one to gain a certain sort of ‘knowledge’ within a relative
context. There always may exist the possibility that there are unlisted
elements outside the framework of discourse; there may likewise be elements
that are more fundamental than those assumed as the primary elements. This
possibility leaves the framework hypothetical. It does, however, allow for
hypothetical knowledge, as it allows assumed primary elements and the complexes
they compose to be given an account as well as constitute objects of correct
judgement.
It
may be objected that Socrates makes an unfounded assumption in asserting that
the three proposed definitions of account form an exhaustive listing. If an
account must be one of these three possibilities, and it may be demonstrated
that none of the three possibilities allow for non-hypothetical knowledge, then
it follows that knowledge may not be non-hypothetical. If the listing is not
necessarily exhaustive, one could conjecture a definition of account that would
allow for non-hypothetical knowledge.
The greater
problem is the asymmetry in giving accounts of primary elements versus giving
accounts of complexes. The greater objection is that Socrates fails to consider
a twofold definition of account: one applicable to complexes, the other
applicable to primary elements. The definition of account proposed in the dream
may be retained, given that it is possible to provide some account for primary
elements. If primary elements may be accounted for, then it would be possible
to gain knowledge through an account that names primary elements. If the
assumption that primary elements and complexes must be subject to the same type
of account is removed, there remains the possibility that an account may be
given of primary elements using some other tactic. This would allow a recovery
from the asymmetry problem, rather than eliminating all asymmetry.[10]
A question
that remains about the account that would suffice to allow primary elements as
objects of knowledge. One candidate for an account of a primary element would
consist in naming it and defining its use. This overcomes the problem in the
second definition of ‘account’, in that it does not allow for ‘knowledge’
without an understanding of how the object of knowledge is used. It would also
overcome the problem of knowledge of differences leading to the rejection of
the third definition of account. A primary element may be pointed to and given
a name, as well as a set of rules for use.[11]
If one perceives a primary element, one may account for it by telling its name
and its use or function. While the name of the element and the words used to
describe its function may be arbitrary and assigned by humans, that the element is named and functions
in a certain way is objective.
The flaw in
this definition of an ‘account’ of a primary element is that it still fails to
overcome the basic problem of considering a primary element alone in giving an
account. Given rules for the use of a primary element, those rules of use must
be either contained internally or attached externally. If the rules are
contained internally, then the primary element contains more than one part,
implying it is not primary. In this case one would be forced to give an account
of the rules of use.[12]
If the rules of use are attached externally, then something is added to a
primary element in giving its account. Given the description of primary
elements in the dream, this is contradictory, as elements may only be named. To
attach anything else – even being or non-being – to a primary element would
disqualify that element from being considered alone (Theae. 202). If it is accepted that primary elements must be
considered alone when giving an account, attaching rules for use to them is
contradictory. If primary elements are not considered alone when giving an
account, the object of knowledge becomes not the element by itself but the
element with its attachment.
On a more
general level, the attempt at giving an account of primary elements fails
because it is impossible to give an account of a primary element without either
attaching such an account externally or seeking an account contained within the
element. This problem does not arise for complexes, since complexes may be
analyzed to reveal an account contained internally. It is impossible have both
the primary element by itself and to provide an account for it. The three
proposed and rejected definitions of ‘account’ failed because they subjected
primary elements and complexes to the same standard of account, despite the different
compositions of such structures. Proposing different definitions of ‘account’
for primary elements and complexes does not resolve the dilemma, as the
fundamental problem of giving an account of a primary element by itself
remains.
One
solution to overcome this problem is to reject the definition of ‘knowledge’ as
true judgement with an account, the route taken by Socrates (Theae. 210b). Another solution is to
accept primary elements despite their unaccountability and go about building
relative ‘knowledge’ based upon such assumptions.[13]
Complexes within such a framework could be given an account using the primary
elements; primary elements, although essentially
unaccountable, would serve as the
constituents of everything within the framework.[14]
The framework itself might imply certain hypothetical rules for assumption,
such as the principle of non-contradiction. Since no account could be given of
the primary elements, they would remain hypothetical; all knowledge built upon
them would likewise be hypothetical, although it may still constitute true
judgement. If the definition of ‘knowledge’ as true judgement with an account
is preserved, hypothetical knowledge is the only form knowledge may take,
because it is impossible to account for primary elements by themselves outside
of an assumed framework.
[1] All references to the Theaetetus are from Plato 1997. Theaetetus, tr. M.J. Levett, revised by Myles Burnyeat. In Complete Works, John M. Cooper, ed. Pp. 157-234. Cambridge: Hackett.
[2] Since nothing may be attached to a primary element, a unique name is all that it has. The name must be unique: if the name is all that signifies the element, and two elements share the same name, it follows that those two elements are the same. Further, it is interesting to note that the perceivablility of primary elements is not attacked in the third section of the Theaetetus. For more on this topic, see Myles Burnyeat 1990, The Theaetetus of Plato pp. 181-187 Indianapolis: Hackett.
[3] The complex ‘book,’ for example, may be known in this manner. The account of ‘book’ may be given by naming its elementary properties, the ‘primary elements’ of perception that compose it: green, hard, extended, etc. The composite of all such properties results in the complex called ‘book,’ and the names of the properties constitute its account. The names of the properties may not be broken down further and are hence unaccountable.
[4] Using primary elements named a, b, c, and d, it would be possible to create a number of complexes, including {a, b, c, d}, {a, b}, and {c, d}. If complexes may be further combined to create other complexes, {a, b} and {c, d} could be combined to result in {{a, b}, {c, d}}. This complex could, in turn, be an element in yet another complex. While {a, b, c, d} and {{a, b}, {c, d}} are composed of the same primary elements, their accounts will vary due to the different parts they are directly composed of.
[5] For an in-depth analysis of Socrates’ account of accounts, see Christopher Shields 1999, “The Logos of ‘Logos”: Theaetetus 206c-210b,” Apeiron 32:4,107-124.
[6] Since it is primary, it has no further elements; only a name.
[7] Primary elements are also distinguished from nothing (the null set of features) by virtue of the singular feature they do possess.
[8] Assume primary elements a, b, and c. The complex {a, b} is distinguished from a by b, given that a is not the same as b. a is not distinguished from b since it possesses no feature which differentiates it from {a, b}. The same follows when considering the complex {a, b, c}: while {a, b, c} is distinguished from {a, b} by virtue of c, {a, b} possesses no feature which distinguishes it from {a, b, c}. The only differentiation in these cases is negative, in that the primary elements and complexes composed of less elements do not possess certain features.
[9] Consider a simple ontology in which only elements a and b are assumed. The only objects to consider, then, are a, b, and the complex {a, b}. The account of a would state that a is different from b and does not include b; a similar argument would account for b. {a, b} could be accounted for by distinguishing it from both a and b.
This solution will work if complexes may be composed only of primary elements. If complexes may be composed of other complexes as well as primary elements, it would become impossible to name all the complexes that could be constructed. Given elements a and b, the complex {a, b} could be constructed. If the element a is added to this complex, the result is {a, {a, b}}. Further, {a, {a, {a, b}}} could constitute another complex, and so on. This demonstrates that the argument that the ‘whole’ is nothing more than the sum of its parts must not fail if there is to be a possibility of resolving the infinite regress of knowledge using a relative, hypothetical framework.
[10] This would also allow the preservation of the conclusion that the whole is the same as the sum of its parts.
[11] At section 45 of the Philosophical Investigations, Wittgenstein notes that a name is explained by the gesture of pointing. See Ludwig Wittgenstein 1997, Philosophical Investigations tr. G.E.M. Anscombe. Englewood Cliffs, New Jersey: Prentice Hall.
[12] This argument could easily fall into infinite regress: if the ‘primary elements’ contain rules of use they are not primary, so an account must be given for the rules of use. If usage rules are not primary, they must have an account as well, and so forth.
[13] This is precisely what the mathematicians described in Book VI of the Republic do: they hypothesize certain figures and relations without giving an account of them (R. 510c5-6). Further discussion is given by Dirk C. Baltzly 1996, “‘To an Unhypothetical First Principle’ in Plato’s Republic” History of Philosophy Quarterly, 13(2): 149-165.
[14] Although assumed primary elements may not be given an absolute account, they may possibly be given a relative account, under the third proposed definition of account. Whether or not they may be given such a relative account depends upon whether a ‘theory of types’ (as elucidated in footnote 3) is invoked. If complexes may be composed of only primary elements, then it would be possible to enumerate the features within the framework that an object possesses that no other object has, as well as list those features it lacks that other objects do have. If complexes may be composed of other complexes as well as primary elements, then the cardinality of the listing of complex and simple objects would be infinite, making even a relative account impossible.