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Straying from the Platonic Tradition: The case of Nicomachus and Euclid Date 2004-7-1 18:16 |
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Straying from the Platonic Tradition:
_____________________________________
The case of Nicomachus and Euclid
_________________________________
Thomas O'Neill
parmenides1@hotmail.com
(received: December 6, 1997)
Many in the present age look back on Plato as an "otherworldly"
philosopher, whose only serious concern is the "Forms" (*eidoi*) which
comprise the core of his philosophy. While there does exist an element of
truth to this frequently made observation, it undoubtedly fails to grasp the
completeness which characterizes Platonic philosophy (a completeness which
Plato almost certainly intended). As described traditionally (as well as
paraphrased above), Plato's discussion of, even concern for, the world of
appearances is ignored. In fact, it is even argued that Plato has no such
direction in his thoughts of the universe. The world of appearances is not
truly real, many would say of Platonic philosophy. The result is that reality
is "out there," in a world which we can know but never describe[1], while we
are grounded in a world of essential darkness, awaiting Prometheus and the
fire's freeing light. Knowledge, essentially, would have no relation to the
world of appearances, according to the above characterization of Plato's
philosophical stance.
This cannot be the case, as I shall seek to assert throughout the
course of this article. Plato does not fore sake one world (that of
appearances) in favor of another (that of the *eidoi*). Instead, he subtly
takes great pains to tie them together. Though he never does this explicitly,
one can find elements of this unification throughout Plato's works. One
example which comes to mind is Plato's discussion of artists as recreators, in
the *Republic*.[2] In this instance, Plato asserts that there is the artist,
who is "just such another - a creator of appearances."[3] For, the artist
seeks to recreate what he sees, and can never actually attain his goal.
Never, though, does Plato dismiss the appearance as unreal or unimportant; in
fact, he does quite the opposite, for what follows in the *Republic* is a
discussion of whether or not it is right (just, morally acceptable, etc.) to
recreate in such manners as painting because what the artist does may not
accurately portray his model; simply put, "philosophers would say that he [in
this case, the artist] is not speaking the truth."[4] Thus, one can perceive
in this facet of the *Republic* a salient concern for appearances, as well a
their implications for human beings (in addition to their implications for
human beings _as participants in the apparent world_). Plato is not solely an
otherworldly philosopher; the field of his concerns does reach into the realm
of appearances, traditionally claimed by many to be no more than a mere
footnote to the "true" nature of his philosophy.
What is seen in the paragraph above is that Plato ties the world of
appearances and the realm of the *eidoi* together; they are related to one
another; in the example above, the artist is that which brings the
relationship into being. He constitutes the point at which the appearance and
the *eidos* are related. The whole of reality is not merely "out there," for
it is directly related to us "down here," too. One can look in many of Plato's
works to see that this is the case, the most potent being the *Apology*, which
shows almost no concern at all for the otherworldly. Instead, Plato has
Socrates answer to charges made in *this* world, and Plato further has
Socrates provide explanations for the actions performed in *this* world, and
shows that the implications of the entire trial for *this* world are
profound.[5] This trend is further continued in the *Phaedo*, the dialogue
which details Socrates' death, for we see a man about to die for what he
taught (the content of which, in many aspects, was otherworldly, though not
entirely so) while at the same time giving insights as to leaving his worldly
life (which entails leaving the realm of appearances). The most striking
thought by Socrates in this dialogue must be that "a man should die in
peace,"[6] in that there is a way to live one's life (detailed in the
*Euthyphro*, *Crito*, and especially the *Apology*, as well as in the
*Symposium*) but that there is also a way to meet one's own death; one should
"[b]e quiet then, and have patience."[7]
The purpose of the above description of the way in which Plato
characterized the life and death of Socrates is to highlight the notion that
Plato's concerns were not entirely otherworldly. He did not deal solely with
Truth and Justice; he also dabbled in truth and justice. So far, one is
probably wondering what, in fact, this has to do with the Greek tradition of
mathematics, specifically as it regards the thus far unmentioned (except in
the title) Euclid and Nicomachus, two "Platonic" mathematicians (at least as
many would believe). Actually, the connection is striking, thought not bold
against the rest of ancient Greek culture. The issue becomes quite simple,
given the bringing back to Earth of Plato's philosophy which occurred above.
If Platonic philosophy is viewed in a slightly more worldly manner,then the
same is certainly possible for the perception of his mathematics, and as a
result, I will illustrate that it *is* the case that Platonic mathematics does
in fact have more worldly leanings than scholars are willing to admit. The
result of this is that such thinkers as Euclid and Nicomachus, often
considered disciples of Plato, will in fact have to be viewed as having
strayed from the Platonic tradition (as I have reinterpreted it). Of course,
this does seem a bit bizarre, given the traditional interpretation of Platonic
philosophy, as well as the typical characterization of Greek mathematics as
having its roots in Plato's thought, but a consideration of certain (often
overlooked) aspects of Plato's views will show that Euclid and Nicomachus
actually may not be followers; their works could instead indicate a reaction
to (or at least a deviation from) that which their forefather had penned.
Usually, Platonic mathematics, especially geometry, is seen as having
applications in the realm of appearances but that these mathematical ideas are
*eidetic* in nature (*eidetic*, here, meaning "of or among the *eidoi*").
thus, there is a certain elasticity of mathematics in Platonic philosophy.
What one would call mathematics (comprised mostly of *arithmos*, *logistikos*,
*geometros*) is, of the *eidoi*, the most connected (or directly related) to
the apparent world.[8] One can use geometry, for example, as a means to
attaining a knowledge of the *eidoi* (let us not forget the words suspended
above the door to Plato's Academy: "Enter not here unless you first have a
knowledge of geometry"), and in fact, geometry does represent the best tool
for attaining such knowledge. It represents, in a way, a bridge from the
apparent world to the *eidioi*. As a bridge must be grounded on either bank
of the river, so must ancient Greek geometry. One side must be firmly planted
among the *eidoi*, yet the other must allow for man's embarkation from the
world of appearances. Since this is the case[9], then there is an "apparent
world" aspect to Platonic mathematics. It can be applied to the apparent
world, and it works in the apparent world, but it is not entirely *of* the
apparent world. Geometry is like a cosmic "and;" it provides a connection
between the two parts of the universe, and it allows for interchange between
them.
This, I will admit, does seem to constitute a radical reinterpretation
of Plato's mathematical ideas (as well as a strange interpretation of Platonic
philosophy in general), but it certainly makes a great deal of sense. Many,
for example, attempt to avoid the apparent inconsistency exhibited in Platonic
philosophy by attributing these inconsistencies[10] to Plato's "changing his
mind as he grew older." what I have sketched so far[11] indicates that such a
decision is simply far too rash. Also, by allowing the notion that Plato is
simply inconsistent, Platonic philosophy is collapsed in order to permit a
misinterpreted mathematical tradition to retain its merit.
The tradition to which I refer (and to which I have referred
throughout this article) is the notion that later mathematicians[12], such as
Euclid and Nicomachus, are merely disciples of Plato, and that they simply
developed their mathematical systems upon a foundation of Platonic philosophy.
This, frankly, constitutes a gross oversimplification of the entire Greek
tradition of mathematics. According to the brief sketch which I have provided,
the respective roles of Euclid and Nicomachus must be reinterpreted. No
longer are they *myrmidons* (or, in the eyes of others, faithful disciples)
following in the wake of Plato's vast philosophical system; instead, one must
perceive them as deviating from the Platonic stance. I will not deny that
Euclid and Nicomachus did build upon some of Plato's ideas; they did so
blatantly. Nicomachus, for example, often cited the *Timaeus* in his
*Introduction to Arithmetic*.[13] But, each chose his own route, as opposed
to merely echoing their "master." Euclid and Nicomachus sought the
otherworldly, and this is what constitutes the the center of their deviation.
Plato went to great lengths to maintain the utility of his mathematics in the
apparent world[14], while in the works of Euclid and Nicomachus one finds no
such connection to the realm of appearances.
There are numerous ways in which one can perceive Euclid and
Nicomachus as having strayed from their allegedly Platonic roots. The most
fascinating of these is the difference reflected by their particular writing
styles. What immediately strikes even the only moderately attentive reader is
that Plato's style is conversational, while that of both Nicomachus and Euclid
is doctrinal; the latter write in what would presently be termed a "textbook"
manner. They simply present their ideas outright, and they subsequently
defend these ideas with the utmost rigor. In his proof of the assertion that
an equilateral pentagon can always be inscribed in a circle, for example,
Euclid provides a proof, using atomic propositions, in *excess of ten steps*
to illustrate his point. Each statement in his proof (as in others) is
assertive; each is a proposition. There exist no background to explain the
propositions or to facilitate the movement from one proposition to the
next.[15] Thus, one recognizes that he is dealing with philosophers who
desire to leave nothing to the mind of the reader. Such philosophers as
Nicomachus and Euclid have a distinct purpose in writing, and their sole
objective is to attain that goal.
Plato, on the other hand, employs a style which could quite easily be
termed conversational. His goals are not technical, and his style does not
exhibit the salient rigor which one finds in Nicomachus and Euclid. There
exists, though, in the works of Plato a more subtle rigor which is the basis
for even attempting to illustrate the cohesiveness of his philosophical
system, let alone base an argument upon this cohesiveness. The rigor is more
subtle than that presented in the works of Nicomachus and Euclid in that
Plato's rigor is embedded in the conversations, the "dialogues;" the rigor
itself does not constitute the entirety of the format of the written work. As
opposed to the skeleton (which would describe the rigor of Nicomachus and
Euclid), rigor in Plato's works would assume the role of the cardio-pulmonary
system. Rigor keeps each dialogue alive (as does the cardio-pulmonary
system), but it does not constitute the shape of the body of the work. A
passage from the Timaeus will help to illustrate this point.[16]
A) Timaeus, as the best *astronomer* is chosen to lead a discussion about God.
Critias declared that "[s]eeing that Timaeus is our best astronomer and has
made his task to learn about the nature of the universe, it seemed god to us
that he should speak first."[17]
B) The next step, if one follows the logical progression of the passage
*rigorously*, is the passage in which Timaeus invokes the aid of God in this
particular endeavor: "Nay, as to that, Socrates, all men who possess even a
small share of good sense call upon God always, at the outset of every
undertaking, be it small or great."[18]
For one who believes in God, understands the universe, its
relationship to God, and who is in fact "our best astronomer"[19], it would
make sense to invoke the deity; it indicates that this understanding of the
universe is legitimate and further that Timaeus is willing to act in
accordance with God/Cosmos (I am using "Cosmos" and "universe" interchangeably
in this article), which is a conscious decision not to thwart the nature of
the Cosmos.
C) Following the invocation of the deity is the main issue; all that has
preceded constitutes something of a preamble. This issue which Timaeus,
Socrates, and the others present wish to explore is the nature of God. They
are satisfied as to the existence of God, so the group involved in this
discussion has moved to the characterization of God. It had already been
decided that God is great, extensive, etc. for the participants have linked to
God "the origin of the Cosmos and . . . the generation of mankind."[20] the
transition to the stated issue is expedited by the phrase uttered by Socrates,
"it will be your task [said to Timaeus], it seems , to speak next, when you
have duly invoked the Gods."[21] Thus, what we see is not a rough movement
from proposition to proposition; instead, Plato's writing exhibits a movement
facilitated by colloquial expression.
D) The purpose of the discussion subsequently surfaces, "to deliver a
discourse concerning the nature of the universe."[22]
E) Plato believes that the first step in dealing with the nature of God must
be ontological.[23] "What is that which is Existent always and has no
Becoming? And what is that which is Becoming always and never Existent?"[24]
In a conversational, relaxed manner, Plato makes an important distinction,
without asserting anything *explicitly*. That which is Existent is
necessarily static, a notion common among the pre-Socratics and even
(evidently) to Plato himself. The use of "E" instead of "e" indicates the
intended universality of the claim. Becoming, on the other hand is
necessarily, universally ("B" versus "b") dynamic; it does not, like
Existence, stand still in the face of eternity. What has happened is that
Plato has provided definitions, much as Euclid did "common notions"[25], but
he did so with simple conversation. The point is made, but not at the expense
of the reader's interest.
F) Of course, in any philosophical inquiry, one must use rational processes;
"one of these [in this case, the issue being discussed] is apprehensible by
thought with the aid of reason."[26]
G) Next, reason is applied to the examination of the issue. The discussion
of the ontological aspect of the nature of God commences; "everything which
becomes must of necessity becomes owning to some Cause."[27]
Further explication from this point in my examination is unnecessary.
It has been thoroughly illustrated that Plato does use to a significant extent
a salient rigor in his works, but not as a result of authoritative declarations
or doctrinal presentations. He strikes a balance between the two. Rigor is
preserved, and it is not used as a crutch.
Plato's style can of course be contrasted with that of Euclid and
Nicomachus. Both Nicomachus and Euclid are quite doctrinal in their styles,
though Nicomachus is less so than Euclid. What Euclid and Nicomachus do is
present their definitions, postulates, etc. and proceed toward their goals
without "softening" their writing with a connection to life, people, the
apparent world. Mathematics itself is their focus; applying it or using it
except within the framework of mathematics does not occur, making any
conclusion at which they arrive merely tautological.[28] Since Euclid's rigor
is the most salient, we will look at him before examining the rigor of
Nicomachus. To do this I will recreate on of Euclid's proofs, while
interjecting related commentary, as I did with the passage from Plato's
*Timaeus*. Thus, one will see the differences in style. Proposition 36 from
the *Elements*, which deals with parallelograms, will constitute the focus of
this examination. True to what I have labelled the "doctrinal style," Euclid
presents his postulate, the postulate being what his goal is (i. e. his goal
is to prove the postulate which he presents). Then, Euclid takes the proof
step-by-step, using his initial assertions (definitions, common notions, etc.)
to validate each step in the proof. Finally, a conclusion is reached; the
conclusion of course is identical to the postulate which began the ordeal.
A) The postulate: "*Parallelograms which are on equal base and in the same
parallels are equal to one another*."[29] What we have here is a universal
assertion which refers to nothing outside of Euclid's mathematical framework.
It looks inward on itself, as opposed to the writings of Plato, which look
outward.[30]
B) given that the postulate has been presented, Euclid begins his proof with
blatant rigor: "Let ABCD, EFGH be parallelograms which are on equal bases BC,
FG, and in the same parallels AH, BG."[31]
Euclid's first step was to lay the groundwork to his postulate.
Essentially, his goal here is to make them tangible by labeling them. But,
they are really no more than abstractions with attached labels.
.....A__________ D.....E__________H ........
\ \ / /
\ \ / /
\ \ / /
\ \/ /
\ /\ /
\ / \ /
\ / \ /
\ / \ /
.......\/________\/........
B F C G
Fig. 1 - An illustration of the parallelograms in Proposition 36.
C) Without any sort of transition or mediation, Euclid subsequently declares
the following: "I say that the parallelogram ABCD is equal to [the
parallelogram] EFGH."[32] What he does is 'legal," to be sure, but has Euclid
really done anything at all beside further describe these abstractions with
essentially meaningless labels? There is some distinguishing occurring here,
but the fact remains that Euclid is still playing with ideas.
D) "For let BE, CH be joined."[33] This is still more of the same, cf. "C"
above.
E) "Then, since BC is equal to FG, while FG is equal to EH, BC is also equal
to EH."[34] We see that, much like Timaeus had, Euclid invokes reason, and he
does so effectively. But, what Euclid does differs significantly from what
Plato had done in the Timaeus in that Euclid, in a sense, has "stacked the
deck." Euclid has set himself up for success by reasoning within his
pre-established (already rational) system,and also by not straying from the
use of universals instead of particulars.[35]
F) Next,we finally see another relationship,though like that above in "E," it
is a relationship among universals: "But they [meaning BC and HC] join
them."[36] For commentary, look above; this is similar to what is found in
"E."
G) "And EB, HC join them."[37] Pardon my yawn, but is it not obvious where
this is headed? Thus, I will spare the reader (as well as myself) any further
pain on Euclid's 36th postulate, for one can saliently see my point.
Nicomachus follows suit with Euclid, in a certain sense, though he
does not abuse his readers' eyes with lengthy (and quite honestly, boring)
proofs. The doctrinal style is nonetheless present in Nicomachus'
*Introduction to Arithmetic*. Unlike Euclid, Nicomachus does not use a series
of atomic statements in his reasoning; the writing is a bit more natural, i.
e. Nicomachus uses normal prose. What characterizes Nicomachus, style,
though, is the assertiveness of his statements (very similar to those of
Euclid) with no grounding for these assertions in the apparent world.
Nicomachus, like Euclid, works within a pre-established framework; his
mathematical works look inward, in contrast to Platonic mathematics.
Nicomachus seems to feature the notion of "definition" in his works. As I
demonstrated above, Euclid's interest was distinction, separating and
classifying things.[38] Nicomachus seems to have preferred labeling things
instead.[39] This is indicated by his discussion of "pentagonal number[s],"
found in his *Introduction to Arithmetic*.[40] A sketch of Nicomachus' views
regarding pentagonal numbers will help show that Nicomachus adheres to the
"doctrinal style."
A) Nicomachus starts with a long, involved definition of pentagonal numbers;
note that this is a definition, the starting-point of any philosophical or
mathematical system. What will be shown in regards to Nicomachus, though, is
that all he seems to have is starting-points. Nonetheless, here is his opening
assertion: "The pentagonal number is one which likewise upon its resolution
into units and depiction as a plane figure assumes the form of an equilateral
pentagon."[41]
So far,this is not unacceptable, but it sets a certain tone for the
rest of the passage. What we have so far is a universal statement.
B) This definition is followed by an example, but the example is misleading.
For, like Euclid's mathematics, that of Nicomachus looks inward. The example
uses numbers (labeled universals), but not numbers *of* anything: "1, 5, 12,
22, 35, 51, 70 and analogous numbers are examples [of pentagonal
numbers]."[42]
C) Nicomachus continues to explain this notion of pentagonal numbers by
declaring that "in general the side contains as many units as are the numbers
that have been added together to produce the pentagon, chosen out of the
natural arithmetical series set forth in a row."[43]
This constitutes more of the same, showing that Nicomachus believes in
continuously asserting things, definitions. The style is not at all
conversational; it is declarative The use of a prose style provides for more
of a transition than one finds in Euclid's Elements, but it is still not
nearly as smooth as that, for example which we saw in the passage from Plato's
*Timaeus*.
This fairly lengthy discussion of the writing style which
characterizes the respective works of Plato, Euclid, and Nicomachus serves a
distinct purpose in this article. The style which each author uses helps to
highlight his views regarding the worldliness/otherworldliness of mathematics.
Plato's conversational style shows a concern for things of this (the apparent)
world. Consequently, Plato's mathematics can be (and are) applied to the
apparent world, and this seems to have been one of his purposes in devising
the mathematical system. Euclid and Nicomachus write in generalizations
which, though they are labeled, are not labeled in a way which adds to them
any meaning. The labels are random and help only for distinction within the
framework of their respective systems. The mathematical systems of Nicomachus
and Euclid can be applied to the apparent world, but this does not seem to be
the intent of the respective thinkers; this result seems to be more an
unintended consequence.
It is therefore evident that Euclid and Nicomachus cannot be
considered complete disciples of Plato. It is true that there are some ways
in which Euclid and Nicomachus are similar to Plato. It must be noticed,
however, that the extent of this similarity is not as great as that which is
portrayed in the usual interpretation of the Greek mathematical tradition.
The differences do outweigh the similarities. Most striking among the
differences is the focus of the system. Euclid and Nicomachus focus on the
otherworldliness of mathematics, number-as-idea for example. Plato, on the
other hand, does not isolate mathematics from the apparent world. Instead, he
goes to great lengths to ensure their interaction.
What is needed therefore is a profound reexamination of the ancient
Greek mathematical tradition, for the currently accepted notions[44] of
ancient Greek mathematics certainly do not tell the entirety of the situation.
The tradition is very intricate, and can quite easily be misconstrued. Once
the academic world accepts certain views, it is quite difficult for them to be
overcome. That is what makes a large-scale reinterpretation of the tradition
essential, especially in regards to the understanding of Plato. Plato,
really, is the first truly prolific philosopher of western civilization. As
such, an understanding of western philosophy is necessarily contingent upon a
solid understanding of Platonic philosophy, for it is the cornerstone of
western thought.
______________________________________________________
Notes
1) Who is to say that our "apparent-world-language" could ever allow us to
discuss the *eidoi*?
2) Plato. "Republic." Dialogues of Plato. transl. Benjamin Jowett
(Washington Square Press 1950), p. 373. (This bibliographic information is
the same for the Apology and the Phaedo.)
3) Ibid.
4) Ibid.
5) "*This*" world is in this instance the apparent world. It should be
noted that Socrates was accused of having committed two crimes, both of which
were saliently grounded in the world of appearances. The first was corrupting
the youth of Athens, and the second was (ironically) atheism. Plato.
"Apology," p. 15.
6) Phaedo, p. 159.
7) Ibid.
8) For a defense of this notion, please see my article in the June edition of
the Metaphysical Review.
9) Cf. note 8.
10) The inconsistencies to which I refer here are centered on the notion that
Plato started out as a puppet of Socrates, whose concerns were grounded in the
world of appearances, and that Plato later became the otherworldly thinker
which we see in such works as the *Republic*, the *Sophist*, and the
*Parmenides*.
11) I call this a sketch, for that is all that was possible in a work this
constricted in scope.
12) By "later," I mean later in the context of the Greek tradition.
13) Cf. the anecdote in Plato's Meno which details the conversation in which
the slave-boy worked through geometric puzzles.
14) Nicomachus. "Introduction to Arithmetic." transl. Martin L. D'Ooge.
Euclid, Archimiedes, Apollonius of Perga, Nicomachus. ed. William Benton
(Encyclopaedia Britannica, Inc. 1952), p. 811. (This bibliographic
information is the same for the entries regarding Euclid.)
15) Euclid. "Elements." transl. Sir Thomas Heath, p. 78.
16) The following passage is taken from Tufts University's "Perseus Project"
website (http://hydra.perseus.tufts.edu), which makes primary sources in the
classics available (in Greek, Latin, and English) through the worldwide web.
Plato. Timaeus, para. 26c-29e.
17) Timaeus, para. 27a.
18) Timaeus, para. 27c.
19) Timaeus, para. 27a. The fact that the "astronomer" is the one who
understands the universe and God concurrently (as God is intertwined with the
Cosmos) demonstrates the Greek view of science and religion.
20) Ibid.
21) Timaeus, para. 27b.
22) Timaeus, para. 27c.
23) This, of course, is not out of character for Plato. One can see quite
plainly in the *Sophist*, *Parmenides*, and *Theaetetus*, especially, a
salient concern for ontological issues. It is therefore safe to assert that
Plato believes ontology to be the gateway to metaphysics.
24) Timaeus, para. 27d-28a.
25) Euclid, p.2.
26) Timaeus, para. 28a.
27) Ibid.
28) I state "within the framework of mathematics" because they do *use* their
ideas, but they do not use them in examples regarding the world of
appearances.
29) Euclid, p. 22.
30) I should explain what I mean by "inward" and "outward." When I state
that Euclid's postulate "looks inward on itself," I mean that it refers back
to the established mathematical system from which it was generated. Plato's
assertions "lookoutward" in that, instead of referring back to themselves,
they are projected from the system to the world of appearances.
31) Euclid, p. 22.
32) Ibid.
33) Ibid.
34) Ibid.
35) As one can see plainly, Euclid is foreign to the inductive method of
logic.
36) Euclid, p. 22.
37) Ibid.
38) The word "things," here, refers to ideas.
39) Cf. note 38.
40) Nicomachus, p. 834.
41) Ibid.
42) Ibid.
43) Ibid.
44) I say "notions" (plural) because there is more than one interpretation of
the ancient Greek tradition, but those which are given any meaningful
attention whatsoever usually share certain basic elements.
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