The axiology of Robert S. Hartman: A critical study

Formal axiology is based on the logical nature of meaning, namely intension, and on the structure of intension as a set of predicates. It applies set theory to this set of predicates. Set theory is a certain kind of mathematics that deals with subsets in general, and of finite and infinite sets in particular. Since mathematics is objective and a priori, formal axiology is an objective and a priori science; and a test based on it is an objective test based on an objective standard.

---

     

And so on . . . : reasoning with infinite diagrams

This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a ??pre?? form of this thesis that every proof can be presented in everyday statements-only form.     

Propositions, Sets, and Worlds

If we agree with Michael Jubien that propositions do not exist, while accepting the existence of abstract sets in a realist mathematical ontology, then the combined effect of these ontological commitments has surprising implications for the metaphysics of modal logic, the ontology of logically possible worlds, and the controversy over modal realism versus actualism. Logically possible worlds as maximally consistent proposition sets exist if sets generally exist, but are equivalently expressed as maximally consistent conjunctions of the same propositions in corresponding sets. A conjunction of propositions, even if infinite in extent, is nevertheless itself a proposition. If sets and hence proposition sets exist but propositions do not exist, then whether or not modal realism is true depends on which of two apparently equivalent methods of identifying, representing, or characterizing logically possible worlds we choose to adopt. I consider a number of reactions to the problem, concluding that the best solution may be to reject the conventional model set theoretical concept of logically possible worlds as maximally consistent proposition sets, and distinguishing between the actual world alone as maximally consistent and interpreting all nonactual merely logically possible worlds as submaximal. I am grateful to the Netherlands Institute for Advanced Study in the Humanities and Social Sciences (NIAS), Royal Netherlands Academy of Arts and Sciences (KNAW), for supporting this among related research projects in philosophical logic and philosophy of mathematics during my Resident Research Fellowship in 2005-2006.     

The Romantic Absolute

In this article I argue that the Early German Romantics understand the absolute, or being, to be an infinite whole encompassing all the things of the world and all their causal relations. The Romantics argue that we strive endlessly to know this whole but only acquire an expanding, increasingly systematic body of knowledge about finite things, a system of knowledge which can never be completed. We strive to know the whole, the Romantics claim, because we have an original feeling of it that motivates our striving. I then examine two different Romantic accounts of this feeling. The first, given by Novalis, is that feeling gives us a kind of access to the absolute which logically precedes any conceptualisation. I argue that this account is problematic and that a second account, offered by Friedrich Schlegel, is preferable. On this account, we feel the absolute in that we intuit it aesthetically in certain natural phenomena. This form of intuition is partly cognitive and partly non-cognitive, and therefore it motivates us to strive to convert our intuition into full knowledge.     

Hume on space, geometry, and diagrammatic reasoning

Hume??s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume??s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume??s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume??s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume??s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume??s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume??s most original contributions. First, Hume??s epistemological model invokes the ??confounding?? of indivisible minima to explain the appearance of spatial continuity. Second, Hume??s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the ??loose judgements?? of the vulgar.     

The Astounding Assumption of Infinite Life

The multimillennial philosophical discussion about life after death has received a recent boost in the prospect of immortality attained via technologies. In this newer version, humans generally are considered mortal but may develop means of making themselves immortal. If “immortal” means not mortal, thus existing for infinity, and if the proposed infinite‐existing entity is material, it must inhabit an infinite material universe. If the proposed entity is not material, there must be means by which it can shed its material substance and exist nonmaterially. The article examines arguments for how an infinite life would be possible given current physical understanding. The paper considers a Pascalian‐style wager weighing the likelihood of adjusting to existence wholly within a finite universe versus betting on there being some way to construe the universe(s) as a viable medium for infinite beings. Conclusion: the case for a finite being to exist infinitely has little viable support.     

Truth as a ‘Living Bond’: A Dialectical Response to Recent Rhetorical Theology

Abstract: In recent years, the Radical Orthodoxy movement (especially John Milbank) has developed an influential theological response to the putative nihilism inherent in modern philosophical tendencies to construe the relation between finite and infinite realities as utterly disjunctive and thus incapable of mediation. This response, which generally implies the championing of a ‘participatory’ ontology, has been very hostile to Protestant or ‘dialectical’ theology, whose insistence upon an ‘indirect’ rather than a ‘rhetorical’ form of truth is taken to implicate such theology in the nihilism of a ‘univocal’ ontology. In this article I offer another reading of the dialectical, via Søren Kierkegaard and René Girard, according to which its anti‐objectivism is due not to the inheritance of modern epistemological dilemmas but to a quite biblical existential rigor. I argue, contrary to Milbank, that this rigor is not finally gnostic, but instead that it alone can preserve the form of truth as a living, spiritual form.     

Can the new indispensability argument be saved from Euclidean rescues?

The traditional formulation of the indispensability argument for the existence of mathematical entities (IA) has been criticised due to its reliance on confirmational holism. Recently a formulation of IA that works without appeal to confirmational holism has been defended. This recent formulation is meant to be superior to the traditional formulation in virtue of it not being subject to the kind of criticism that pertains to confirmational holism. I shall argue that a proponent of the version of IA that works without appeal to confirmational holism will struggle to answer a challenge readily answered by proponents of a version of IA that does appeal to confirmational holism. This challenge is to explain why mathematics applied in falsified scientific theories is not considered to be falsified along with the rest of the theory. In cases where mathematics seemingly ought to be falsified it is saved from falsification, by a so called ??Euclidean rescue??. I consider a range of possible answers to this challenge and conclude that each answer fails.     

Mental models and probabilistic thinking

This paper outlines the theory of reasoning based on mental models, and then shows how this theory might be extended to deal with probabilistic thinking. The same explanatory framework accommodates deduction and induction: there are both deductive and inductive inferences that yield probabilistic conclusions. The framework yields a theoretical conception of strength of inference, that is, a theory of what the strength of an inference is objectively: it equals the proportion of possible states of affairs consistent with the premises in which the conclusion is true, that is, the probability that the conclusion is true given that the premises are true. Since there are infinitely many possible states of affairs consistent with any set of premises, the paper then characterizes how individuals estimate the strength of an argument. They construct mental models, which each correspond to an infinite set of possibilities (or, in some cases, a finite set of infinite sets of possibilities). The construction of models is guided by knowledge and beliefs, including lay conceptions of such matters as the “law of large numbers”. The paper illustrates how this theory can account for phenomena of probabilistic reasoning.     

Aging as a Social Form: The Phenomenology of the Passage

If philosophers have discussed life as preparation for death, this seems to make aging coterminous with dying and a melancholy passage that we are condemned to survive. It is important to examine the discourse on aging and end of life and the ways various models either limit possibilities for human agency or suggest means of being innovative in relation to such parameters. I challenge developmental views of aging not by arguing for eternal life, but by using Plato’s conception of form in conjunction with Simmel’s work and Arendt’s meditation on intergenerational solidarity, to evoke a picture of the subject as having capacities that offer avenues for improvisational action. This paper proposes a method for analyzing any social form as a problem-solving situation where the real “problem” is the fundamental ambiguity that inheres in the mix between the finite characteristics of the action and its infinite perplexity. I work through the most conventional chronological view of aging to show how it dramatizes a fundamental ethical collision in life that intensifies anxiety under many conditions, always raising the question of what is to be done with respect to contingency, revealing such “work” as a paradigm of the human condition.     

From Modal Discourse to Possible Worlds

The possible-worlds semantics for modality says that a sentence is possibly true if it is true in some possible world. Given classical prepositional logic, one can easily prove that every consistent set of propositions can be embedded in a ‘maximal consistent set’, which in a sense represents a possible world. However the construction depends on the fact that standard modal logics are finitary, and it seems false that an infinite collection of sets of sentences each finite subset of which is intuitively ‘possible’ in natural language has the property that the whole set is possible. The argument of the paper is that the principles needed to shew that natural language possibility sentences involve quantification over worlds are analogous to those used in infinitary modal logic.     

Infinite Reasoning

Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform infinite inferences. I argue that we have this ability. My argument looks to our best current theories of inference and considers examples of apparent infinite reasoning. My position is controversial, but if I'm right, our theories of truth, mathematics, and beyond could be transformed. And even if I'm wrong, a more careful consideration of infinite reasoning can only deepen our understanding of thinking and reasoning.     

Quantum measurements and supertasks

This article addresses the question whether supertasks are possible within the context of non-relativistic quantum mechanics. The supertask under consideration consists of performing an infinite number of quantum mechanical measurements in a finite amount of time. Recent arguments in the physics literature claim to show that continuous measurements, understood as N discrete measurements in the limit where N goes to infinity, are impossible. I show that there are certain kinds of measurements in quantum mechanics for which these arguments break down. This suggests that there is a new context in which quantum mechanics, in principle, permits the performance of a supertask.     

Are Mathematical Theories Reducible to Non-analytic Foundations?

In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process.     

On Finitism and the Beginning of the Universe: A Reply to Stephen Puryear

Stephen Puryear argues that William Lane Craig's view, that time as duration is logically prior to the potentially infinite divisions that we make of it, involves the idea that time is prior to any parts we conceive within it (Priority of the Whole with respect to Time: PWT). He objects that PWT entails the Priority of the Whole with respect to Events (PWE), and that it subverts the argument, used by proponents of the Kalam Cosmological Argument (KCA) such as Craig, against an eternal past based on the impossibility of traversing an actual infinite sequence of events. I argue that proponents of KCA can affirm that time is not discrete, nor is it continuous with actual infinite number of parts or points, but rather that it is a continuum with various parts yet without an actual infinite number of parts or points. I defend this view, and I reply to Puryear's other objections.     

The equivalence of two ways of computing distances from dissimilarities for arbitrary sets of stimuli

Given a set endowed with pairwise dissimilarities, the Dissimilarity Cumulation procedure computes the (quasi)distance between any two elements of the set as the infimum of the sums of dissimilarities across all finite chains of elements connecting the two elements. For finite sets, this procedure is known to be equivalent to recursive corrections for violations of the triangle inequality in any sequence of ordered triads of points which contains every triad a sufficient number of times. This paper extends this equivalence to infinite set.     

Strict finitism

Conclusion Dummett's objections to the coherence of the strict finitist philosophy of mathematics are thus, at the present time at least, ill-taken. We have so far no definitive treatment of Sorites paradoxes; so no conclusive ground for dismissing Dummett's response — the response of simply writing off a large class of familiar, confidently handled expressions as semantically incoherent. I believe that cannot be the right response, if only because it threatens to open an unacceptable gulf between the insight into his own understanding available to a philosophically reflective speaker and the conclusions available to one confined to observing the former's linguistic practice; for an observer of our linguistic practice could never justifiably arrive at the conclusion that red, child, etc., are governed by inconsistent rules. But the Sorites is not the subject of this paper. The points I hope to have made plausible are: that a generalized intuitionist position cannot be so much as formulated and that even a most local intuitionism, argued for the special case of arithmetic, is hard pressed effectively to stabilize and defend itself; that strict finitism remains the natural outcome of the anti-realism which Dummett has propounded by way of support for the intuitionist philosophy of mathematics; that it is powerfully buttressed by the ideas of the latter Wittgenstein on rule-following; and that there is no extant compelling reason to suppose that its involvement with predicates of surveyability calls its coherence into question. The correct philosophical assessment of strict finitism, and its proper mathematical exegesis, remain absolutely open, almost virgin issues. This is not a situation which philosophers of mathematics should tolerate very much longer.The term was introduced by Kreisel in [6] to denote what he took to be an aspect of Wittgenstein's later philosophy of mathematics; and taken over by Kielkopf (Strict Finitism, Mouton 1970) — misunderstanding, as it seems to me, both Kreisel and Wittgenstein — as a label for Wittgenstein's later philosophy of maths. in its entirety. It is not a happy label for the ideas I am concerned with, since it is only from non-strict finitist points of view that the strict finitist can be straightforwardly seen as stressing the finitude of human capacities, countenancing only finite sets, etc. (See subsections 5 and 6 below). But we need a labeel; and Dummett in [3] has already followed Kreisel's lead. Anyway, a rose by any other name,...     

Fundamentality,Effectiveness, and Objectivity of Gauge Symmetries

Much recent philosophy of physics has investigated the process of symmetry breaking. Here, I critically assess the alleged symmetry restoration at the fundamental scale. I draw attention to the contingency that gauge symmetries exhibit, that is, the fact that they have been chosen from an infinite space of possibilities. I appeal to this feature of group theory to argue that any metaphysical account of fundamental laws that expects symmetry restoration up to the fundamental level is not fully satisfactory. This is a symmetry argument in line with Curie’s first principle. Further, I argue that this same feature of group theory helps to explain the ‘unreasonable’ effectiveness of (this subfield of) mathematics in (this subfield of) physics, and that it reduces the philosophical significance that has been attributed to the objectivity of gauge symmetries.     

Finite Reasons Without Foundations

This article develops a theory of reasons that has strong similarities to Peter Klein's infinitism. The view it develops, Framework Reasons, upholds Klein's principles of avoiding arbitrariness (PAA) and avoiding circularity (PAC) without requiring an infinite regress of reasons. A view of reasons that holds that the “reason for” relation is constrained by PAA and that PAC can avoid an infinite regress if the “reason for” relation is contextual. Moreover, such a view of reasons can maintain that skepticism is false by the maintaining that there is more to epistemic justification than can be expressed in any reasoning session. One crucial argument for Framework Reasons is that justification depends on a background of plausibility considerations. The final section of the article applies this view of reasons to Michael Bergmann's argument that any nonskeptical epistemology must embrace epistemic circularity.