Mathematics of Philosophy or Philosophy of Mathematics?

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Nonlinear Dynamics, Psychology, and Life Sciences [ndpl] PH009-291950 December 15, 2000 14:18 Style file version Oct 23, 2000

Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 5, No. 3, 2001

Mathematics of Philosophy or Philosophyof Mathematics?

Jeffrey Goldstein1

This article examines recent attempts to gain insight into philosophical para-doxes through using NDS models employing iterated difference equationsand resulting phase portraits and escape time diagrams. The temporal na-ture of such models is contrasted with an alternative approach based on thea-temporal and non-dynamical construct of a lattice. Finally, there is a discus-sion of how such strategies for understanding paradox transcend the realm ofempirical research and enter territory in the philosophy of mathematics.

KEY WORDS: difference equations; escape diagrams; lattice; logical paradox; model;nonlinear dynamical systems theory (NDS); phase portraits; philosophy of mathematics;self-reference.

NDS AND SELF-REFERENTIAL PARADOXES

Paradoxes involving self-reference have beguiled thinkers since ancienttimes. For example, the infamous paradox of Epimenides the Cretan, “AllCretans are liars,” results in the curious situation that it is “true if and onlyif it is false” (Barwise & Etchemendy, 1987). Such conundrums have notmerely remained idle philosophical curiosities but have played a crucial rolein much ground-breaking work in mathematics, logic, and computer scienceas in the work of Godel, 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 dynami-cal systems theory (NDS). Yet, although it is true that general philosophi-cal implications of chaos and complexity theories have not gone unnoticed(e.g., see Goldstein, 1996), only now is NDS beginning to be exploited to

1Adelphi University, Garden City, NY 11530.

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1090-0578/01/0700-0197$19.50/0 C© 2001 Human Sciences Press, Inc.

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enrich our understanding of self-referential phenomena in philosophy andlogic. Thus, Grim, Patrick, and St. Denis (1998) have recently employedNDS models to probe semantical self-referential paradoxes. I am familiarwith only one work predating this current thrust that has used dynamicsto directly model philosophical statements: Greeley’s (1995) NDS model ofGreek philosophical texts. However, Greeley modeled entire philosophicaltexts, not self-referential paradoxes themselves. Because, the use of NDS tomodel philosophical paradoxes is inaugurating an entirely new arena of ap-plication of NDS, in this article I will examine the specific modeling strategiesinvolved, and along the way, point out how these attempts at a mathematicsof philosophy may be better understood as falling within the domain of thephilosophy of mathematics rather than empirical research per se.

“DYNAMICAL SEMANTICS”

In The Philosophical Computer, Grim et al. (1998) develop a “dynamicalsemantics” which employs dynamical systems to model philosophical para-doxes. The first step of their modeling strategy consists in the conversion ofclassical bivalent logic into an infinite-valued one in which truth or falsity isconstrued not as an either/or decision but, instead, a matter of degree forwhich numerical values are assigned. The biconditional (p if and only if q),as classically understood, only holds in the case that there is no differencein truth value between p and q. The value of the Lukasiewicz biconditional,however, holds to the extent that there is no difference in truth value be-tween p and q; so it is expressed as 1 minus the absolute difference in valuebetween p and q. Thus, the value (V) of the assertion that p has the value vis considered untrue to the extent that p differs from v, v being the variableranging over all possible truth values in the interval [0,1] with 1 = true and0 = false, can be represented by:

|Vvp| = 1−Abs(v − |p|). (1)

Then, in order to capture greater semantic richness, fuzzy logic rules for al-locating 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 fortrue, whereas, “fairly,” a more hedged qualification, is modeled by squareroots. Next, deliberations about the truth and falsity of a paradoxical state-ment are transposed into iterations of difference equations. For example, thenotorious paradox of the Liar (“This statement is false”) can be reperesentedby a difference equation displaying iterations of estimates of the value (xn)of the sentence within the quotations:

xn+1 = 1−Abs(0− xn). (2)

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Each subsequent deliberation about the statement’s truth is the next itera-tion of the equation. Finally, phase portraits and escape diagrams emergingfrom the resulting dynamical equations are explored to gain insight into thedeeper structure of paradox.

As more elaborate paradoxes are modeled, this “dynamical semantics”becomes progressively more interesting. Thus, the dynamical model of theLiar reveals merely a simple oscillation since any initial value v generates aperiodic alternation between the values v and 1− v (the one fixed point at-tractor is 1

2 since 12 = 1− 1

2 .). A similarly uninteresting dynamic is found evenwith the more complex-sounding paradox of the Half-sayer (“This sentenceis as true as half its estimated value”) which is modeled with the equation:

xn+1 = 1−Abs(1/2 · xn − xn). (3)

Like the Liar, the Half-sayer goes to a fixed point for the value 0.5. Aninteresting departure, however, occurs with the similar sounding Mini-malist (“This sentence is as true as whichever is smaller: its estimated valueor the opposite of its estimated value”). From an initial estimate of 0.6,the Minimalist diverges outward to a Liar-like oscillation between 0 and1, with 2

3 as a repeller. A comparison of the Minimalist and the Half-sayerdemonstrates that although very congruent in appearance, they turn-outto have opposite dynamics: The Half-sayer has an attractor right wherethe Minimalist has a repeller whereas in classical logic their semantic be-haviors are identical with an oscillation between 0 and 1 (0 for false, 1 fortrue).

A paradox with even more fascinating dynamics is the so-called ChaoticLiar (“This sentence is as true as it is estimated to be false”) which is mod-eled as:

xn+1 = 1−Abs(1− xn)− xn. (4)

This result is technically chaotic in the sense that it shows sensitive depen-dence on initial conditions, is topologically transitive, and the set of periodpoints is dense. Grim et al., also confirm a chaotic logistic map for the sen-tence: “It is very false that this sentence is as true as it is estimated to befalse.” With even more labyrinthian paradoxes, Grim et al., turn to escapediagrams which plot paths according to the amount of time it takes to escapesome 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 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

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gadget in the paradox of the Triplist :

Socrates: What Plato says is truePlato: What Socrates says is falseChrysippus: Neither Socrates nor Plato speak truly.

NDS AND LIMITATIVE THEOREMS ON CHAOS

One of the most intriguing and far-reaching implications of The Philo-sophical Computer’s “dynamical semantics” are its derivation of two limita-tive theorems (see Machover, 1996) on chaos. Here they are guided by theirinvestigations of the Strengthened Chaotic Liar :

Either the boxed sentence has a chaotic semantic behavior or it is as true asit 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 willbe true. But the semantic behavior of a completely true assertion will not bechaotic. And if the statement does not have a chaotic semantic behavior, itstruth value will depend on the second disjunct. But the latter will mimic thebehavior of the Chaotic Liar which is chaotic. Guided by their discoveriesof this paradox, and claiming fealty to a tradition in mathematical logic thatis inspired by paradoxes (e.g., Godel by the Richard Paradox, Tarski bythe Liar Paradox, and Chaitin by the Berry Paradox), Grim et al. (1998)prove a theorem in mathematical logic that shows there can be no effectivemethod for deciding whether an arbitrary expression of a system determinesa function chaotic on the interval [0,1]. A related second limitative theoremconcerns the noncalculability of chaos. I think these two limitative theoremsare quite telling in how a dynamical exploration of semantical paradoxescan lead to insights in more purely mathematical, i.e., nonempirical, areas.

THE DIAMOND LATTICE AND PARADOX

The Philosophical Computer justifies its NDS models by construingparadox in terms of a temporal sequence of deliberations with successive de-liberations represented by subsequent functional iterations. Indeed, it is thistemporal understanding of paradox which allows them to utilize NDS mod-els in the first place. Such a temporal perspective on paradox, however, is notuniversally shared. Thus, the influential work of Barwise and Etchemendy

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(1987) on the paradox of the Liar relied on a hybrid of a-temporal andnon-dynamical constructs including graph theory, “hyperset” theory, and thelinguistic philosophy of John Austin. And much more recently, Hellerstein(1997) modeled semantic paradoxes using the non-dynamical construct ofa lattice which is an array of points spaced with enough regularity that anypoint can be symmetrically transposed to any other point (e.g., the grid of in-tegers formed by the points of all integral Cartesian coordinates). Since all itsgrid points are simultaneously present, a lattice is decidedly a-temporal, itsmain properties having to do with translational symmetry—hence its expli-cation via group-theory (see Kramer, 1981). Contrasting such an a-temporaland non-dynamical perspective with The Philosophical Computer’s dynami-cal approach can lead to greater clarity about the advantages and disadvan-tages of the dynamical understanding of paradox.

Hellerstein’s lattice consists of four values/two components in the shapeof a diamond:

TRUE = T/T/ \

I = T/F J = F/T (5)\ /

FALSE = F/F

Appended to the customary T and F are two new paradoxical values: I (Truebut False) designating “undetermined” or “insufficient data for a definite an-swer”; and J (False but True) defined as “over-determined” or “contradictorydata” (I and J are interchangeable). Hellerstein believes his audacious inclu-sion of two new paradoxical values is analogous to the way complex numbersare 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 com-pared to the infinite-valued logic of Grim et al.’s dynamical approach, yet itis 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 ofdeliberations. And, once this simple solution is accepted, Hellerstein can useit to resolve more sophisticated paradoxes like those of Russell, Grelling, andQuine. Hellerstein remarks that it is only logicians trained to treat paradoxwith respect bordering on terror who think there must be more going onthan his simple solution. Thus, the logical paradox of “This statement is bothtrue and false,” modeled as the lattice

II — TF — JJ (7)

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202 Goldstein

suggests its self-referential similarity to the Liar, whereas, to discover a likesimilarity, Grim and company had to go to the much greater lengths of theirdynamical equations and phase portraits. Likewise, Hellerstein interpretsthe Dualist paradox

Tweedledee: “Tweedledum is a liar”Tweedledum: “Tweedledee is a liar”:

as the lattice:

TF/ \

II JJ (8)\ /

FT

This again hints at a similar symmetrical structure underlying disparate para-doxes. Moreover, like the forays of The Philosophical Computer into areasin pure mathematics, Hellerstein uses his lattice interpretations of paradoxto reinterpret orthodox Cantorian set theory.

There is a price, however, to be paid for the simplicity of the lattice, infact, a dynamical price, since to explicate the power of the lattice approach,Hellerstein introduces two explanatory devices that are replete with tempo-ral and dynamical associations. The first is an electric circuit with a phasedoperation, i.e., a spring-loaded relay, a switch, and a battery set-up so thatif current flows, the relay is energized to break the circuit, but then whenthere’s no current, the spring-loaded switch reconnects the circuit. The resultis a temporally oscillating “buzz” that is quite similar to The PhilosophicalComputer’s first oscillating dynamical model of the Liar. It is important tonote, however, that the phased manner in which the electric circuit workspoints to its temporal and dynamical nature. Similarly, Hellerstein resorts toharmonic functions iterated in order to find fixed points in a manner com-parable to the functional iteration of difference equations in NDS. Again,we see the entry of a temporal/dynamical-like notion.

CONCLUSION—LOGICAL PARADOX: EMPIRICAL DOMAINOR MATHEMATICAL STRUCTURE?

The very fact that the a-temporal lattice framework had to be supple-mented by temporal and dynamical constructs indicates that some sort ofdynamical perspective may be inescapable in the development of a morecomplete understanding of paradox. After all, since semantics consistsin processes of human meaning comprehension, paradox must include

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deliberations about truth values and, therefore, innately unfolds over time.And, of course, it is with such temporally unfolding processes where dynam-ical systems methods are so appropriate. Furthermore, although it may be aquestion of temperament, it seems to me that Hellerstein’s electric circuitsand harmonic functions are impoverished compared to the rich visual repre-sentations afforded by phase portraits and escape diagrams. Yet, it is also truethat the latter lack the striking simplicity of the a-temporal lattice models.Consequently, a melding of dynamical with non-dynamical constructs mayturn-out to be the most promising new avenue for studying philosophicalparadoxes.

Another issue concerns the seemingly empirical nature of these mod-eling strategies. Typically, in empirical research, the adequacy of models areevaluated through their fit with research data, e.g., Guastello’s (1995) struc-tural equations are shown to be better models when they fit the data betterthan alternatives. But, when it comes to the study of paradoxes, what exactlyis 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 forthcomingfrom initial measurements but arises only afterwards as an outcome of themodeling steps described above. We are not seeing some kind of measure-ment of cognitive processes involved with such deliberations, but, instead,an interpretation using a mathematical structure. This means that it wouldinappropriate to evaluate the resulting models by their ability to make bettersense out of the data since it is these very models that create the data in thefirst place! Hence, what does the finding of chaos and fractals in models ofparadox really amount to if the data itself arises from models using iterateddifference equations which we now know from years of research will yieldchaotic dynamics at appropriate parameter values? That is, where have Grimet al. (1998) Denis actually found chaos and fractals? In the logic of paradoxor in their specifically chosen models of paradox? A similar problem is goingon with Hellerstein’s lattices.

This is not to suggest that these attempts at modeling paradox mustnecessarily fit the mold of conventional empirical research. But it does leadto the crucial question of how we are to assess their findings. Such musingsseem to imply that the mathematical modeling of paradox is not merely amathematics of philosophy but touches on basic issues in the philosophyof mathematics. Such a conclusion is buttressed by the fact that both thedynamical and the lattice perspectives are used to delve further into othermathematical arenas such as mathematical logic and set theory. Indeed, thevery issue of whether to interpret paradox more in a temporal or an a-temporal manner is itself an issue that transcends a purely empirical point ofview. Again, we are forced to enter the realm of philosophical intuitions thatgo beyond empirical research as such. It is for these reasons the assessment

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of these two books requires something different than that for empiricalresearch: they seem to be in a different genre altogether, one, I suggest, morerelated to mathematics per se, including the philosophy of mathematics, thanempirical research.

In conclusion, in my opinion, we can better appreciate these examplesof modeling philosophical paradox by considering them as following a time-honored strategy in mathematical research: using what is known about onestructure to explore a different structure which is isomorphic to the first,e.g., in algebraic topology where theorems in abstract algebras are usedto explore topological phenomena isomorphic to those abstract algebras.Mathematics is not science and exploring mathematical structures is not thesame as empirical research. Accordingly, it seems more appropriate to viewmodeling strategies discussed in this paper as the development of new typeof mathematics, such as new nonstandard logics (e.g., see Haack, 1996), thantraditional empirical research.

ACKNOWLEDGMENTS

I wish to thank Terry Marks-Tarlow for her helpful comments on anearlier draft of the section on The Philosophical Computer.

REFERENCES

Barwise, J., & Etchemendy, J. (1987). The liar : An essay on truth and circularity. NY: OxfordUniversity Press, 1987.

Goldstein, J. (1996). Causality and emergence in chaos and complexity theories. In W. Sulis andA. Combs (Eds.), Nonlinear dynamics in human behavior (Studies of Nonlinear Phenom-ena in Life Sciences—Volume 5), pp. 161–190, Singapore: World Scientific Publishing.

Greeley, L. (1995). Complexity in the attention system of the cognitive generative learningprocess. In A. Albert (Ed.), Chaos and society, pp. 371–386. Amsterdam: IOS and Pressesde l’Universite du Quebec.

Grim, P., Mar, G., & St. Denis, P. (1998). The philosophical computer: Exploratory essays inphilosophical computer modeling. Cambridge, MA: MIT Press.

Guastello, S. (1995). Chaos, catastophes, and human affairs : Applications of nonlinear dynamicsto work, organizations, and social evolution. Mahwah, NJ: Lawrence Erlbaum Associates.

Haack, S. (1996). Deviant logic fuzzy logic : Beyond the formalism. Chicago: University ofChicago Press.

Hellerstein, N. (1997). Diamond: A paradox logic. By Singapore: World Scientific, 1997.Kramer, E. (1981). The nature and growth of modern mathematics. Princeton: Princeton Uni-

versity Press.Machover, M. (1996). Set theory, logic, and their limitations. Cambridge, England: Cambridge

University Press.

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