were given their alogonal complexities by the Pythagoreans’
discovery of the incommensurable numbers, and as will be discussed in this study,
became radically questioned and rethought by modern mathematics, especially in
view of Gödel’s incompleteness theorems, and quantum theory, which radically
redefined the concept of measurement, to the point of making a quantum “measure-
ment” something to which no previous concept of measurement could any longer
apply... This [ALOGON] aspiration is efined in this study as Platonism because it defined Plato’s thought.......
the logos of philosophy as a logos without the alogon within it, as
against the mathematical, Pythagorean, logos of his time, a logos that emerged, even
if only as a possibility, with the discovery of the incommensurables. The subject is
discussed in several dialogues, such as Meno, Theaetetus, Statesman, and the
Republic. Plato conceived of the ultimate architecture of the ideal realm on the
mathematical model, ............
as coextensive with radical Pythagorean mathematics in the twentieth
century....radical Pythagorean theories, such as quantum mechanics or quantum
field theory, has never divorced itself from mathematics. ...not, thus, be confused with that of “mathematical Pythagoreanism,” defined by the role of numbers in the harmony of the cosmos, which harmony implied a logos without an alogon with it.
This aspiration has been equally
dominant in mathematics, physics, and philosophy throughout their history. It is a
separate question whether any of the Pythagoreans ever embraced the alogonal logos
without aiming to overcome the alogon in it, which, as I said, appears unlikely.
the possibility of a logos with the
alogon within it that reveals the presence of Pythagoras’s ghost or, in the present
view, Pythagoras’s spirit, and hence inspiration, in modern mathematics, or in
twentieth-century physics and philosophy, as against “Plato’s ghost.”
In radical Pythagorean mathematics, the structure of concepts is defined, in
addition, by the presence of the unthinkable in mathematical thinking, the alogon
in the logos of mathematics. This is a philosophical, including epistemological,
component of their structure, which, as noted, is not limited to Pythagorean con-
cepts, and could be found in algebraic concepts or those (in essence still algebraic,
however) of mathematical logic........
may be seen as a (qualified) return of the Pythagorean original view of
numbers, led to the mathematics of irrational numbers, to real and complex numbers,
to quaternions, to nonstandard numbers and Conway’s “surreal numbers,” and
beyond.......... radical Pythagorean mathematical thinking, which, against Plato-
nism, affirms the unthinkable as a friend rather than an enemy of thought, the
alogonal as a friend of a logos. In other words, this thinking does not preclude,
but, on the contrary, helps, the development of new concepts and theories
this alogonal nature is essentially defined by abstract algebra, thusp. 188
making quantum theory essentially algebraic in spirit, although it contains new,
more algebraic, forms of geometry, which makes it Pythagorean as well. This
contrast is reflected in the other mathematical features of these theories, in particular
the role of (noncommuting) operators in Hilbert spaces over ℂ as variables only used
to predict the probabilities of the outcomes of quantum experiments, without
representing the physical properties and behavior of quantum objects, as against
the (commuting) functions of real variables, representing physical quantities and
behavior of physical objects considered in classical physics and relativity
Heisenberg (who always had some
interest in Plato, whose Timaeus he was reading on his way to his discovery of
quantum mechanics), eventually, in part in view of quantum field theory and the role
of group theory there, especially in considering elementary particles, adopted a more
Platonist view of quantum theory....the algebraic
noncommutativity of QM, discovered by Heisenberg, supports this argument
Alain Connes:
the mathematician develops a special sense, I think—irreducible to sight, hearing, or touch—that allows him to perceive a reality every bit as constraining as physical reality, but one that’s far more stable than physical reality for not being located in space and in time. (Changeux and Connes 1998, pp. 27–28)
- I am an Orangutan and I disapprove of this story!
