In Nonlinear Dynamics, Psychology, and Life Sciences, July, 2001 Vol 5, No. 3: 197-204.
Mathematics of Philosophy or Philosophy of Mathematics?
Jeffrey Goldstein, Ph.D.
Adelphi University
Garden City, NY 11530
Keywords: difference equations; escape diagrams; lattice; logical paradox; model; nonlinear
dynamical systems theory (NDS); phase portraits; philosophy of mathematics; self-reference.
Abstract: This article examines recent attempts to gain insight into philosophical paradoxes
through using NDS models employing iterated difference equations and resulting phase portraits
and escape time diagrams. The temporal nature of such models is contrasted with an alternative
approach based on the a-temporal and non-dynamical construct of a lattice. Finally, there is a
discussion of how such strategies for understanding paradox transcend the realm of empirical
research and enter territory in the philosophy of mathematics.
Abbreviated title: Mathematics of Philosophy
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NDS and Self-referential Paradoxes
Paradoxes involving self-reference have beguiled thinkers since ancient times. For
example, the infamous paradox of Epimenides the Cretan, “All Cretans are liars,” results in the
curious situation that it is “true if and only if it is false” (Barwise & Etchemendy, 1987). Such
conundrums have not merely remained idle philosophical curiosities but have played a crucial
role in much ground-breaking work in mathematics, logic, and computer science as in the work
of Gödel, Tarski, and Turing (Machover, 1996). Furthermore, self-reference is inherent in the
recursive functions at the heart of chaos, fractals, cellular automata, and similar phenomena in
nonlinear dynamical systems theory (NDS). Yet, although it is true that general philosophical
implications of chaos and complexity theories have not gone unnoticed (e.g., see Goldstein,
1996), only now is NDS beginning to be exploited to enrich our understanding of self-referential
phenomena in philosophy and logic. Thus, Grim, Patrick, and St. Denis (1998) have recently
employed NDS models to probe semantical self-referential paradoxes. I am familiar with only
one work predating this current thrust that has used dynamics to directly model philosophical
statements: Greeley’s (1995) NDS model of Greek philosophical texts. However, Greeley
modeled entire philosophical texts, not self-referential paradoxes themselves. Because, the use of
NDS to model philosophical paradoxes is inaugurating an entirely new arena of application of
NDS, in this article I will examine the specific modeling strategies involved, and along the way,
point out how these attempts at a mathematics of philosophy may be better understood as falling
within the domain of the philosophy of mathematics rather than empirical research per se.
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“Dynamical Semantics”
In The Philosophical Computer, Grim, Mar, and St. Denis (1998) develop a “dynamical
semantics” which employs dynamical systems to model philosophical paradoxes. The first step
of their modeling strategy consists in the conversion of classical bivalent logic into an infinitevalued one in which truth or falsity is construed not as an either/or decision but, instead, a matter
of degree for which numerical values are assigned. For instance, the proposition that p has the
value v is considered untrue to the extent that p differs from v, v being the variable ranging over
the interval [0,1]:
∣Vvp∣ = 1 - Abs(v - ∣p∣).
(1)
Then, in order to capture greater semantic richness, fuzzy logic rules for allocating numerical
values are applied to the semantic qualifiers “very” and “fairly”, e.g., “very” as in “very true” is
valued as the square of the value for true, whereas, “fairly,” a more hedged qualification, is
modeled by square roots. Next, deliberations about the truth and falsity of a paradoxical
statement are transposed into iterations of difference equations. For example, the notorious
paradox of the Liar (“This statement is false”) is repesented by the difference equation:
xn+1 = 1-Abs (0 - xn).
(2)
Each subsequent deliberation about the statement’s truth is the next iteration of the equation.
Finally, phase portraits and escape diagrams emerging from the resulting dynamical equations
are explored to gain insight into the deeper structure of paradox.
As more elaborate paradoxes are modeled, this “dynamical semantics” becomes
progressively more interesting. Thus, the dynamical model of the Liar reveals merely a simple
oscillation since any initial value v generates a periodic alternation between the values v and 1- v
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(the one fixed point attractor is ½ since ½ = 1- ½.). A similarly uninteresting dynamic is found
even with the more complex-sounding paradox of the Half-sayer (“This sentence is as true as
half its estimated value”) which is modeled with the equation:
xn+1= 1 - Abs(1/2xn - xn).
(3)
Like the Liar, the Half-sayer goes to a fixed point for the value .5. An interesting departure,
however, occurs with the similar sounding Minimalist (“This sentence is as true as whichever is
smaller: its estimated value or the opposite of its estimated value”). From an initial estimate of
0.6, the Minimalist diverges outward to a Liar-like oscillation between 0 and 1, with 2/3 as a
repeller. A comparison of the Minimalist and the Half-sayer demonstrates that although very
congruent in appearance, they turn-out to have opposite dynamics: The Half-sayer has an
attractor right where the Minimalist has a repeller whereas in classical logic their semantic
behaviors are identical with an oscillation between 0 and 1 (0 for false, 1 for true).
A paradox with even more fascinating dynamics is the so-called Chaotic Liar (“This
sentence is as true as it is estimated to be false”) which is modeled as:
xn+1= 1 - Abs(1 - xn) - xn.
(4)
This result is technically chaotic in the sense that it shows sensitive dependence on initial
conditions, is topologically transitive, and the set of period points is dense. Grim, Mar, and St.
Denis also confirm a chaotic logistic map for the sentence: “It is very false that this sentence is as
true as it is estimated to be false”. With even more labyrinthian paradoxes, Grim, Mar, and St.
Denis turn to escape diagrams which plot paths according to the amount of time it takes to
escape some set threshold. Revised values for x and y are calculated simultaneously. For
example, an escape diagram of the Chaotic Dualist (”X is as true as Y” and “Y is as true as X is
false”) displays an intricately nested fractal. Yet, while the attractors for the variants of their
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Chaotic Dualist are different, the general shape of their corresponding escape diagrams are
clearly related. And, in a similar context, they discover the strange fractal of the Sierpinski
gadget in the paradox of the Triplist:
Socrates: What Plato says is true
Plato: What Socrates says is false
Chrysippus: Neither Socrates nor Plato speak truly.
NDS and Limitative Theorems on Chaos
One of the most intriguing and far-reaching implications of The Philosophical
Computer’s “dynamical semantics” are its derivation of two limitative theorems (see Machover,
1996) on chaos. Here they are guided by their investigations of the Strengthened Chaotic Liar:
Either the boxed sentence has a chaotic semantic behavior or it is as true as it is
estimated to be false.
Does the Strengthened Chaotic Liar have a chaotic semantic behavior or not? If it does, it will be
true in its first disjunct, therefore, the entire sentence will be true. But the semantic behavior of a
completely true assertion will not be chaotic. And if the statement does not have a chaotic
semantic behavior, its truth value will depend on the second disjunct. But the latter will mimic
the behavior of the Chaotic Liar which is chaotic. Guided by their discoveries of this paradox,
and claiming fealty to a tradition in mathematical logic that is inspired by paradoxes (e.g., Godel
by the Richard Paradox, Tarski by the Liar Paradox, and Chaitin by the Berry Paradox), Grim,
Mar, St. Denis prove a theorem in mathematical logic that shows there can be no effective
method for deciding whether an arbitrary expression of a system determines a function chaotic
on the interval [0,1]. A related second limitative theorem concerns the non-calculability of chaos.
I think these two limitative theorems are quite telling in how a dynamical exploration of
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semantical paradoxes can lead to insights in more purely mathematical, i.e., non-empirical, areas.
The Diamond Lattice and Paradox
The Philosophical Computer justifies its NDS models by construing paradox in terms of
a temporal sequence of deliberations with successive deliberations represented by subsequent
functional iterations. Indeed, it is this temporal understanding of paradox which allows them to
utilize NDS models in the first place. Such a temporal perspective on paradox, however, is not
universally shared. Thus, the influential work of Barwise and Etchemendy (1987) on the paradox
of the Liar relied on a hybrid of a-temporal and non-dynamical constructs including graph
theory, “hyperset” theory, and the linguistic philosophy of John Austin. And much more
recently, Hellerstein (1997) modeled semantic paradoxes using the non-dynamical construct of a
lattice which is an array of points spaced with enough regularity that any point can be
symmetrically transposed to any other point (e.g., the grid of integers formed by the points of all
integral Cartesian coordinates). Since all its grid points are simultaneously present, a lattice is
decidedly a-temporal, its main properties having to do with translational symmetry– hence its
explication via group-theory (see Kramer, 1981). Contrasting such an a-temporal and nondynamical perspective with The Philosophical Computer’s dynamical approach can lead to
greater clarity about the advantages and disadvantages of the dynamical understanding of
paradox.
Hellerstein’s lattice consists of four values/two components in the shape of a diamond:
TRUE = T/T
/
\
I = T/F
J = F/T
\
/
FALSE = F/F
(5)
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Appended to the customary T and F are two new paradoxical values: I (True but False)
designating “undetermined” or “insufficient data for a definite answer”; and J (False but True)
defined as “over-determined” or “contradictory data” (I and J are interchangeable). Hellerstein
believes his audacious inclusion of two new paradoxical values is analogous to the way complex
numbers are a 2-dimensional extension of the real number line that solve x2 = -1.
At first sight, the Diamond lattice’s four values may seem paltry compared to the infinitevalued logic of Grim and company’s dynamical approach, yet it is the very simplicity of the
Diamond-lattice that is its strength. For example, Hellerstein’s simple model of The Liar
I ------------------- J
(6)
has two solutions, I and J, neither of which require a dynamical unfolding of deliberations. And,
once this simple solution is accepted, Hellerstein can use it to resolve more sophisticated
paradoxes like those of Russell, Grelling, and Quine. Hellerstein remarks that it is only logicians
trained to treat paradox with respect bordering on terror who think there must be more going on
than his simple solution. Thus, the logical paradox of “This statement is both true and false,”
modeled as the lattice
II ------- TF ------ JJ
(7)
suggests its self-referential similarity to the Liar, whereas, to discover a like similarity, Grim and
company had to go to the much greater lengths of their dynamical equations and phase portraits.
Likewise, Hellerstein interprets the Dualist paradox
Tweedledee: “Tweedledum is a liar”
Tweedledum: “Tweedledee is a liar” :
as the lattice:
TF
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/
\
JJ
II
\ /
FT
(8)
This again hints at a similar symmetrical structure underlying disparate paradoxes. Moreover,
like the forays of The Philosophical Computer into areas in pure mathematics, Hellerstein uses
his lattice interpretations of paradox to reinterpret orthodox Cantorian set theory.
There is a price, however, to be paid for the simplicity of the lattice, in fact, a dynamical
price, since to explicate the power of the lattice approach, Hellerstein introduces two explanatory
devices that are replete with temporal and dynamical associations. The first is an electric circuit
with a phased operation, i.e., a spring-loaded relay, a switch, and a battery set-up so that if
current flows, the relay is energized to break the circuit, but then when there’s no current, the
spring-loaded switch reconnects the circuit. The result is a temporally oscillating “buzz” that is
quite similar to The Philosophical Computer’s first oscillating dynamical model of the Liar. It’s
important to hote, however, that the phased manner in which the electric circuit works points to
its temporal and dynamical nature. Similarly, Hellerstein resorts to harmonic functions iterated in
order to find fixed points in a manner comparable to the functional iteration of difference
equations in NDS. Again, we see the entry of a temporal/dynamical-like notion.
Conclusion -- Logical Paradox: Empirical Domain or Mathematical Structure?
The very fact that the a-temporal lattice framework had to be supplemented by temporal
and dynamical constructs indicates that some sort of dynamical perspective may be inescapable
in the development of a more complete understanding of paradox. After all, since semantics
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consists in processes of human meaning comprehension, paradox must include deliberations
about truth values and, therefore, innately unfolds over time. And, of course, it is with such
temporally unfolding processes where dynamical systems methods are so appropriate.
Furthermore, although it may be a question of temperament, it seems to me that Hellerstein’s
electric circuits and harmonic functions are impoverished compared to the rich visual
representations afforded by phase portraits and escape diagrams. Yet, it is also true that the latter
lack the striking simplicity of the a-temporal lattice models. Consequently, a melding of
dynamical with non-dynamical constructs may turn-out to be the most promising new avenue for
studying philosophical paradoxes.
Another issue concerns the seemingly empirical nature of these modeling strategies.
Typically, in empirical research, the adequacy of models are evaluated through their fit with
research data, e.g., Guastello’s (1995) structural equations are shown to be better models when
they fit the data better than alternatives. But, when it comes to the study of paradoxes, what
exactly is to included within the set of data to which the equations are then applied? In the case
of The Philosophical Computer, the data set is not forthcoming from initial measurements but
arises only afterwards as an outcome of the modeling steps described above. We are not seeing
some kind of measurement of cognitive processes involved with such deliberations, but, instead,
an interpretation using a mathematical structure. This means that it would inappropriate to
evaluate the resulting models by their ability to make better sense out of the data since it is these
very models that create the data in the first place! Hence, what does the finding of chaos and
fractals in models of paradox really amount to if the data itself arises from models using iterated
difference equations which we now know from years of research will yield chaotic dynamics at
appropriate parameter values? That is, where have Grim, Mar, and St. Denis actually found
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chaos and fractals? In the logic of paradox or in their specifically chosen models of paradox? A
similar problem is going on with Hellerstein’s lattices.
This is not to suggest that these attempts at modeling paradox must necessarily fit the
mold of conventional empirical research. But it does lead to the crucial question of how we are to
assess their findings. Such musings seem to imply that the mathematical modeling of paradox is
not merely a mathematics of philosophy but touches on basic issues in the philosophy of
mathematics. Such a conclusion is buttressed by the fact that both the dynamical and the lattice
perspectives are used to delve further into other mathematical arenas such as mathematical logic
and set theory. Indeed, the very issue of whether to interpret paradox more in a temporal or an atemporal manner is itself an issue that transcends a purely empirical point of view. Again, we are
forced to enter the realm of philosophical intuitions that go beyond empirical research as such.
It is for these reasons the assessment of these two books requires something different than that
for empirical research: they seem to be in a different genre altogether, one, I suggest, more
related to mathematics per se, including the philosophy of mathematics, than empirical research.
In conclusion, in my opinion, we can better appreciate these examples of
modeling philosophical paradox by considering them as following a time-honored strategy in
mathematical research: using what is known about one structure to explore a different structure
which is isomorphic to the first, e.g., in algebraic topology where theorems in abstract algebras
are used to explore topological phenomena isomorphic to those abstract algebras. Mathematics is
not science and exploring mathematical structures is not the same as empirical research.
Accordingly, it seems more appropriate to view modeling strategies discussed in this paper as the
development of new type of mathematics, such as new nonstandard logics (e.g., see Haack,
1996), than traditional empirical research..
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ACKNOWLEDGMENTS
I wish to thank Terry Marks-Tarlow for her helpful comments on an earlier draft of the section
on The Philosophical Computer.
REFERENCES:
Barwise, J. & Etchemendy, J. (1987). The liar: An essay on truth and circularity. NY: Oxford
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