inscription Let no one ignorant of mathematics
enter here, and then I said:
Plato saw in mathematics unshakable evidence
that there was an absolute standard for the
Universe. And where one such standard existed,
it was logical to assume that there were others.
Today humans regard mathematics principally
as an applied science, but in Plato’s time it was
considered by the Pythagoreans to be “pure”,
having nothing to do with the gross and
imperfect everyday world.
The Chimæra: Would you care to elaborate upon that?
The Sphinx: The best thing to do is to quote directly
from Thomas Stanley’s 1687 account of the
Pythagorean doctrines, which draws its material
from Porphyrus, Iamblicus, Strabo, etc. The
Stanley text materializes, and the Sphinx turns to
Part IX page #522. Consider the following: [sic]
The mind being purify’d [by Discipline] ought to
be applied to things that are beneficial; these he
procured by some contrived ways, bringing it by
degrees to the contemplation of eternal
incorporeal things, which are ever in the same
state; beginning orderly from the most minute,
lest by the suddenness of the change it should
be diverted, and withdrew itself through its
great and long pravity of nutriment.
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To this end, he first used the Mathematical
Sciences, and those Speculations which are
i n t e r m e d i a t e b e t w i x t C o r p o r e a l s a n d
Incorporeals, (for they have a Threefold
Dimension, like Bodies, but they are impassible
like Incorporeals) as Degrees of Preparation to
the Contemplation of the things that are;
diverting, by an artificial Reason, the Eyes of the
Mind from corporeal things (which never are
permanent in the same manner and estate)
never so little to a desire of aliment; by means
whereof, introducing the contemplation of
things that are, he rendered men truly happy.
This use he made of the Mathematical Sciences.
These Sciences were first termed μάθημα by
Pythagoras upon consideration that all Mathesis
(discipline) is Reminiscence, which comes not
extrinsecally to souls as the phantasies which
are formed by sensible objects in the Phantasie;
nor are they an advantageous adscititious
knowledg, like that which is placed in Opinion;
but it is excited from Phænomena’s, and
perfected intrinsecally by the cogitation
converted into it self.
The Chimæra: How very interesting. It would seem that
the recollective basis of knowledge, heretofore
assumed to be a Platonic concept, is in fact
Pythagorean.
The Sphinx: And the use of mathematics as a key to this
particular sort of knowledge, i.e. of the Forms.
The Chimæra: Who is this Stanley, and how reliable
can he be considered to be?
The Sphinx: Thomas Stanley graduated from
Cambridge at age 16 as a Master of Arts. He
practiced law; was fluent in French, Italian,
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Spanish, and the Classical languages; and issued
the first volume of his famous History of
Philosophy when he was only 30. The three
paragraphs cited above are all footnoted to original
Greek sources.
The Chimæra: So Plato used mathematics as a “place to
stand”, in an effort to make the Universe
intelligible by reason alone. And Platonists tend to
emphasize this, shielding Plato from the despised
title of “mystic”. See here: He indicates page #xv
in the Collected Dialogues .
[Huntington Cairns:] But the difference
between Plato and the mysticism that has
attached itself to his philosophy is essential.
Plato’s aim is to take the reader by steps, with as
severe a logic as the conversational method
permits, to an insight into the ultimate necessity
of Reason. And he never hesitates to submit his
own ideas to the harshest critical scrutiny; he
carried this procedure so far in the Parmenides
that some commentators have held that his own
doubts in this dialogue prevail over his
affirmations. But the beliefs of mystics are not
products of critical examination and logical
clarification; they are, on the contrary, a series
of apprehensions, flashes, based on feeling,
denying the rational order. The mystic’s reports
of his experiences are beyond discussion
inasmuch as they are subjective and emotional;
they must be accepted, by one who wishes to
believe them, as a matter of faith, not
knowledge. Plato’s view of the world is that of
an intelligible system that man can know by
disciplined intellect alone. He was, in fact, the
founder of logic, a logician and a poet, but he
was not a mystic, he never exalted feeling above
reason.
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The Sphinx: Well, well. What do you think Cairns would
say to the following quote from The Statesman?
He turns to page #1082.
STRANGER: When there arises in the soul of
men a right opinion concerning what is good,
just, and profitable, and what is the opposite of
these - an opinion based on absolute truth and
settled as an unshakable conviction - I declare
that such a conviction is a manifestation of the
divine occurring in a race which is in truth of
supernatural lineage.
YOUNG SOCRATES: It could not be more
suitably described.
The Chimæra: (dryly) He would probably say that,
since sphinxes and chimæras do not really exist,
nothing we say is to be taken seriously.
The Sphinx: So, where Plato is concerned, a great deal
hinges upon the basis for mathematics itself. Is it
acquired through reason or through mystical
vision, so to speak?
The Chimæra: This is rather curious. Plato actually sets
his dialectic process in contrast to mathematics,
almost as though the object of the Dialogues is to
arrive at a Form greater than that of mathematics.
He turns to page #746.
I understand, he said, not fully, for it is no slight
task that you appear to have in mind, but I do
understand that you mean to distinguish the
aspect of reality and the intelligible, which is
contemplated by the power of dialectic, as
something truer and more exact than the object
of the so-called arts and sciences whose
assumptions are arbitrary starting points. And
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though it is true that those who contemplate
them are compelled to use their understanding
and not their senses, yet because they do not go
back to the beginning in the study of them but
start from assumptions you do not think they
possess true intelligence about them although
the things themselves are intelligibles when
apprehended in conjunction with a first
principle. And I think you call the mental habit
of geometers and their like mind or
understanding and not reason because you
r e g a r d u n d e r s t a n d i n g a s s o m e t h i n g
intermediate between opinion and reason.
The Sphinx: It’s all very well for Plato to say that, and
I’m sure that dialecticians are not displeased to
consider themselves more intellectual than
mathematicians. Yet we have found, in both The
Sophist and The Statesman, that Plato cannot
proceed with his arguments unless he assumes the
divinely-inspired ability to perceive not only
greater, but absolute perfection when he is
confronted with it. That is not reason; it is
revelation. Plato does mathematics an injustice:
While mathematicians openly admit that their
conclusions are originally based upon assumptions
(axioms), Plato pretends that his are not. And of
course they are. Just as Einstein required a
concrete assumption - a constant speed of light -
upon which to build his mathematical philosophy,
so Plato must have an assumption - the ability to
recognize absolute perfection - upon which to
build his dialectic philosophy.
The Chimæra: Plato seems to be caught in a trap
between the relativistic Sophists on one hand -
who denied the reliability of intuitive assumptions
- and the Pythagoreans on the other - who
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permitted original assumptions via revelation/
intuition. Plato rejects the notion that axioms are
necessary for reason, yet he cannot reason without
them. No wonder he was so touchy about the
Sophists.
The Sphinx: Note the very precise manner in which the
Pythagoreans discussed the original assumptions
of mathematics: Again he indicates page #522 of
the Stanley text.
The whole science of Mathematicks, the
Pythagoreans divided into four parts,
attributing one to Multitude, another to
Magnitude; and subdividing each of these into
two. For Multitude either subsists by it self, or is
consider’d with respect to another; Magnitude
either stands still, or is moved. Arithmetick
contemplates Multitude in it self: Musick with
respect to another: Geometry, unmoveable
magnitude; Sphaerick, moveable.
These Sciences consider not Multitude and
Magnitude simply, but in each of these that
which is determinate: for Sciences consider this
abstracted from infinite, that they may not (in
vain) attempt in each of these that which is
infinite. When therefore the wise persons say
thus, we conceive it is not to be understood of
that multitude which is in the sensible things
themselves, nor of that magnitude which we
perceive in bodies, for the contemplation of
these I think pertains to Physick, not to
Mathematick. But because the Maker of all
things took Union and Division, and Identity,
and Alterity, and Station and Motion to
compleat the soul, and framed it of these kinds,
as Timæus teacheth, we must conceive that the
Intellect, consisting according to the diversity
thereof, and the division of proportions and
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multitude, and knowing it self to be both one
and many, proposeth numbers to it self, and
produceth them and the Arithmetical knowledg
of them. According to the union of multitude
and communication with it self, and colligation,
it acquireth to it self Musick: for which reason
Arithmetick excels Musick in antiquity, the soul
it self being first divided by the Maker, then
collected by proportions. And again establishing
the operation within it self, according to its
station, it produceth Geometry out of it self, and
one figure, and the principles of all figures, but
according to its motion, Sphaerick: for she is
moved by circles, but consists always in the
same manner according to the causes of those
circles, the straight and the circular: and for this
reason likewise Geometry is precedent to
Sphaerick, as Station is to Motion.
But forasmuch as the Soul produced these
Sciences, not looking on the excitation of Ideas,
which is of infinite power, but upon the
boundure of that which is limited in their
several kinds, therefore they say that they take
infinite from multitude and magnitude, and are
conversant only about finite: for the mind hath
placed in her self all principles both of
multitude and magnitude, because being wholly
of like parts within her self, and being one and
indivisible, and again divisible, and producing
the world of Ideas, it doth participate essential
finiteness and infiniteness from the things
which it doth understand: But it understands
according to that which is finite in them, and
not according to the infiniteness of its life. This
is the opinion of the Pythagoreans, and their
division of the four Sciences. Hitherto Proclus.
The Chimæra: In the final analysis, whether Cairns
would enjoy the idea or not, Plato must be classed
with the Pythagoreans as a “mystic”, in that he
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assumed that humans possess a supernatural
power beyond reason to recognize perfection/
The Sphinx: Yes. The Sophists were the only ones who
could claim to be “non-mystics”, because they
would not admit to revealed accuracy of any sort.
Plato tried to strike a balance between the Sophists
and the Pythagoreans, but there is just no halfway
position that holds water. The Pythagoreans would
have been amused by Plato’s laborious
argumentative process, holding it to be a waste of
time, in that the final answer to a given problem
could be known only by revelation/recollection. As
for the Sophists, they would have faulted Plato’s
arguments by denying the primary assumptions/
revelations in them.
The Chimæra: All of which leaves us where?
The Sphinx: Well, I think we have pretty well finished
with The Statesman. But our discussion
concerning the Pythagorean aspects of “Plato’s”
philosophy raises yet another question: To what
extent was Plato an original thinker?
The Chimæra: On that thorny little problem I will let
you take the lead.
The Sphinx: I think we would be wise to start with some
observations about time - not just the way most
humans regard it, but the way Plato himself
perceived it. I recall a pertinent comment of G.J.
Whitrow’s in his book The Nature of Time:
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The first question to consider is the origin of the
idea that time is a kind of linear progression
measured by the clock and the calendar. In
modern civilization thi
s conception of time so
dominates our lives that it seems to be an
inescapable necessity of thought. But this is far
from true ... Most civilizations, prior to our own
of the last two or three hundred years, have
tended to regard time as essentially cyclic in
nature. In the light of history, our conception of
time is as exceptional as our rejection of magic.
The Chimæra: Well said. Modern academicians are
conditioned to an essentially Newtonian attitude
towards time. They regard it as a simple
progression of events. The past may be referred to,
and visions of the future may be projected, but
neither past nor future has any intrinsic effect
upon the present - nor do they exist objectively at
all.
The Sphinx: In a cyclical system of time, by contrast,
past, present, and future would all be part of a
single continuum. This wouldn’t necessarily mean
that “history repeats itself” either. Rather the
components of fourth-dimensional existence
would continue to exist, although they might be
undergoing periodic rearrangement and
recomposition. One might draw an analogy to the
interchangeability of matter and energy; a
seemingly-endless variety of transmutation takes
place, but the “sum of the whole” remains the
same.
The Chimæra: Take the Platonic notion of the
transmigration of souls. It wouldn’t make much
sense if entirely new souls could come into being
“out of nothing”, would it? Yet the transmigration
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theory has been ridiculed on the grounds that (a)
world population is expanding, and (b) past
incarnations have not been recalled to standards of
scientific proof. If “the stuff of which souls are
made” can transmutate from other components of
a unified time-continuum, then the first objection
disappears. And limited recomposition [short of
transmutation] would account for the second.
The Sphinx: And this would put the concept of the
recollective basis of knowledge in a new light as
well. Instead of knowledge being cumulative or
progressive [again a purely-linear concept] with
the linear “passage” of time, it can be considered
“circular” - rearrangements and recombinations of
certain all-inclusive principles which are part of
the cyclical continuum, i.e. “timeless”.
The Chimæra: A provocative way of approaching the
Theory of the Forms.