@SemprePhi drew my attention to the following book review on the 23rd November:
Phillip Sidney Horky, Plato and Pythagoreanism 2013.
Reviewed by Simon Trépanier bmcr.brynmawr.edu/2014/2014-05-1…
#brynmawr
#philosophy
I responded in four short posts, which I’ve now augmented
with further discussion.
@SemprePhi Hi. Thanks for the pointer to Horky's book and
the review. Where to start! You cannot rely on Aristotle for accurate
information about the Pythagoreans. Huffman has been occupying academic space
for thirty years, and won't cross the boundaries. 1/
Note the whole
argument is based on the idea that philosophy in Greece is an autocthonous
development. Not invented elsewhere. I've shown that Pythagoras derived many of
his ideas from Mesopotamia. And that these influences are reflected in Plato.
They just don't want to see. 2/
Or they fear to
step outside the accepted paradigm for fear of committing heresy, and having to
pay for their sin. There is much information about what Pythagoras brought back
from the east in Greek writing. But scholars don't know what it is, and why [it]
is important. 3/
In order to stay
within the acceptable paradigm, or 'episteme', they don't read the full range
of sources which are available. Consequently it is difficult to make sense of
the sources that they do read. If you read the full range of sources, it is an
eye-opener. 4/
Philosophers and
Historians are nervous about crossing the boundaries of their subjects, not
just because of the risk to their reputations. They are happiest when sense can
be made of what they are looking at. That sense isn’t always the sense that
things made in antiquity. Modern scholars make fictions, and sit upon the pile
they have made.
The philosopher
Adrian Moore wrote a history of the infinite in 1990, and presented a series on
BBC radio in 2016 on the same subject. Both discuss the problems and issues
around the human response to the idea of the infinite. However Moore’s idea of
the history of man’s relationship is strangely structured. Writing about the
broadcast series I pointed out that:
We get many clues about the Greek understanding
of the infinite and the unlimited from a number of Plato’s dialogues,
including The Timaeus, The Sophist, The
Republic, The Theaetetus, The Laws, and The
Parmenides. In skipping Plato, the first reference to Parmenides and his
notion of the universe as simply one and one alone, is as an introduction in
the first episode to his pupil Zeno of Elea, and his response to paradox. There
is no discussion of Plato’s demolition of Parmenides arguments, no discussion
of the Platonic forms, no discussion of the relationship of the forms to the
form of the Good, which is another way of talking about what is infinite, and
no discussion of what amounts to a different logical modality in the pages of
Plato (where he discusses things passing into one another by means of their
similitude), which is a way of understanding the relationship of finite things
to the infinite.
What Moore has constructed is a
Catholic perspective on the idea of the infinite, since it is viewed from the
perspective of Thomas Aquinas and Bishop Anselm. What made sense to those
scholars, makes sense to Moore. Plato was largely unavailable to any
scholars of that period (with the
exception of some sections of the Timaeus). But to write now about the infinite
as if the writings of Plato are unknown to us, or of no importance to our
understanding of the human response to the infinite, is difficult to fathom. I
summarised part of this first episode of the series as follows:
Essentially Aristotle’s rapprochement, which Moore
characterises as an attempt to make the concept of the infinite more palatable
to the Greeks, involved dividing the idea of the infinite into two. As already
mentioned, one of these was the potential infinite, and the second was the
actual infinite. As outlined in the first episode, Zeno’s paradoxes depended on
the idea of an infinite divisibility, which seemed to make the idea of any kind
of movement impossible, since that would require a universe of infinite
complexity. Zeno therefore regarded all forms of movement as illusion. Since in
order to travel a certain distance, you would have to travel half the distance
to your destination, and then half of the distance remaining, and then half of
that, and half of what still remained, and so on. Which would result in an
infinite number of steps. Which would be impossible.
Aristotle’s response was that though the various
stages of the journey could be understood in such a way, the stages were not
marked, and did not have to be considered in making a journey. The idea of
limit is however a crucial point. What Aristotle was saying is that there are
two ways of looking at the idea of what a limit is. Essentially there is
limitation which is defined by what a thing is, and there is limitation which
is not. In the first case the limit of a thing cannot be transcended without
the nature of that thing turning into something else.
The essence of this argument is that there are forms
of limit which can be ignored. One of which is the actual infinite: instead we
should deal with the potential infinite. The actual infinite, by its nature, is
always there. But we cannot deal with it. The potential infinite we can work
with, since it is not always there, and spread infinitely through reality. So
we can count numbers without ever arriving at infinity, or ever being in danger
of arriving there. Moore mentioned that this conception of infinity more or less
became an orthodoxy after Aristotle, though not everyone accepted that his
argument against actual infinity was solid. Which is something of an
understatement. Aristotle’s distinction between the potential infinite and the
actual infinite is between what is, in practical terms, something we can treat
as finite, and what is actually infinite.
Moore has defined
himself as an Aristotelian finitist, meaning that, since (he argues), man
cannot deal with the actual infinite, only the potential infinite can make any
sense to us. And so, much ancient discussion is swept away, as of very little
interest or importance. This is why we cannot easily understand much of the
intellectual world of antiquity. Instead we choose to write unflattering
fictions about it.
I said that we
cannot rely on Aristotle for accurate information on the Pythagoreans. This is
not because I regard him as a poor scholar. Both Plato and Aristotle taught in
the Academy in Athens. They were both dealing with a body of traditional
doctrine (there are many passages where a comparison shows this – their discussion
of the importance of the liver, for example). But they had quite different ways
of discussing doctrine. Plato gives the reader real information about the
subject, but hedges it about with other arguments, and sometimes talks in terms
of images and myth (the account of the prisoners in the cave, in the Republic,
for example). So his work makes sense to those who already know the doctrine,
and intrigues those who don’t. Aristotle on the other hand, seems to have had
the job of sifting through students to find those who might have the
intelligence to be able to grasp the
essence of the doctrine (when properly instructed). He did this sometimes by
constructing complex sophistical arguments which actually contradicted
doctrine, and sometimes even rational sense.
Two examples: The
first is Aristotle’s Nicomachean Ethics, originally a series of
lectures, ends up concluding the gods cannot act in the world, but only
contemplate. Imagine the response to that argument in the ancient world! Why
did Aristotle argue like this? He was looking for students who could provide critical
rational responses to the argument, and who could see that it did not make any sense in a reality
which was (at the time) populated by divinities expected to play a constructive
role in the world. The second example is Aristotle’s comments on logical
modality (mostly in the Metaphysics), which I’ve discussed elsewhere (in
‘Logical Modality in Classical Athens’). This also contradicts traditional
doctrine which underpinned the human relationship with the divine. And not just
in Greece. Plato discusses the logical modality which enables contact and
engagement with the divine, and other authors do too. We ignore all of this information
concerning doctrine, because we prefer the unfathomable shadows on the wall.
However, despite
the occasional tricksiness of Aristotle, he tells us something important,
which, if we are familiar with relevant texts by other authors, we can figure out.
I quote again from my critical review of Adrian Moore’s broadcast History of
the Infinite, concerning the arguments of Zeno:.
The idea of limit is however a crucial point. What
Aristotle was saying is that there are two ways of looking at the idea of what
a limit is. Essentially there is limitation which is defined by what a
thing is, and there is limitation which is not. In the first case the limit of
a thing cannot be transcended without the nature of that thing turning into
something else.
Aristotle’s discussion references the doctrinal view which
is also discussed by Plato. Which is that there is an important connection
between the idea of limit and the infinite. The infinite is just another way of
specifying what is unlimited, and beyond the physical world. Paradoxically, it
is the actual limit of what is, and what can be. This does not represent a
retreat from commerce with the actual infinite, but actually tells us how that
commerce was supposed to work.
However the physical and finite world is also full of
limits. These sometimes function as boundaries, and serve to close off access. Some
limits you can choose to pass beyond, and there are others which you cannot
pass. And in some cases, because of the nature of the limit, it is the
nature of the limit itself which allows commerce with the ultimate limit of everything,
which is where the Gods were once understood to have their existence.
This is the most important thing to understand about
antiquity, both east and west. For Moore, the actual infinite is simply
something which defies our understanding. In antiquity, the actual infinite was
something of vital importance, and which we could have commerce with through
its earthlly analogues (totalities, completions, limits. etc). Aristotle, in
talking about Zeno’s paradox, is referencing the key doctrinal point, which is
that reality has a double nature. And that we have (if we are properly
informed), a choice about how we respond to that double nature.
In modern times, we no longer have this choice, since the
doctrine concerning actual infinity has been mostly lost, and in fact entirely
lost to those who function in the modern successors of Plato’s Academy. We are
stuck in a world that imagines it must deal with everything in terms of
calculable finitudes. Effectively we are, to quote the Mesopotamian king
Esarhaddon, “blind and deaf ” for the whole of our lives.
It was not always so.
