Mathematical Truth Research Articles

The Origins of Eternal Truth in Modern Mathematics: Hilbert to Bourbaki and Beyond

The ArgumentThe belief in the existence of eternal mathematical truth has been part of this science throughout history. Bourbaki, however, introduced an interesting, and rather innovative twist to it, beginning in the mid-1930s. This group of mathematicians advanced the view that mathematics is a science dealing with structures, and that it attains its results through a systematic application of the modern axiomatic method. Like many other mathematicians, past and contemporary, Bourbaki understood the historical development of mathematics as a series of necessary stages inexorably leading to its current state — meaning by this, the specific perspective that Bourbaki had adopted and were promoting. But unlike anyone else, Bourbaki actively put forward the view that their conception of mathematics was not only illuminating and useful for dealing with the current concerns of mathematics, but that this was in fact the ultimate stage in the evolution of mathematics, bound to remain unchanged by any future development of this science. In this way, they were extending in an unprecedented way the domain of validity of the belief in the eternal character of mathematical truths, from the body to the images of mathematical knowledge.Bourbaki were fond of presenting their insistence on the centrality of the modern axiomatic method as a way to ensure the eternal character of mathematical truth as an offshoot of Hilbert's mathematical heritage. A detailed examination of Hilbert's actual conception of the axiomatic method, however, brings to the fore interesting differences between it and Bourbaki's conception, thus underscoring the historically conditioned character of certain, fundamental mathematical beliefs.

MATHEMATICS. A SCIENCE OF PATTERNS

The present article aims at showing that it is possible to construct a realist philosophy of mathematics which commits one neither to dream the dreams of Platonism nor to reduce the word 'realism' to mere noise. It is argued that mathematics is a science of patterns, where patterns are not objects (or properties of objects), but aspects, or aspects of aspects, etc. of objects. (The notion of aspect originates from ideas sketched by Wittgenstein in the Philosophical Investigations.) The philosophical importance of this contribution is mainly in the successfulness of the attempt made to justify a view of mathematics which, holding on to a Tarskian/Aristotelian conception of mathematical truth, does not involve the postulation of entities which are beyond the bounds of experience.

Meaning, Models and Selection: A Review of Philosophical Naturalism

This is a book which tries to put it all together. Papineau begins with topics central to contemporary naturalistic philosophy, but then he moves on. By the end of the book we reach issues that naturalists often avoid: modality, morality and mathematical truth. This is an admirable venture, and I expect that many readers will, like myself, benefit a great deal from tracing out some of the more remote consequences of familiar naturalistic commitments. The journey is made enjoyable by the exceptional clarity of the presentation. There are bold moves and novel connections are made. For example, functionalism is adopted, but with its applicability much restricted: only theories which treat the products of selection processes can be expected to resist reduction. The package also includes a teleological theory of content, a reliabilist position in epistemology, and fictionalism about mathematics. In this review I will focus on what is perhaps the keystone theory of the book: the teleological theory of mental representation. As Papineau explicitly (and helpfully) stresses, the teleological theory can be understood as a development of the older idea that the content of some thoughts can be understood in terms of a link between thought and practical success: a belief is a belief that p if it tends to lead to actions that are successful if p obtains. A central attraction of this idea is that it breaks the link between what is believed and what is or can be justifiably believed. A state of the world that is very epistemically inaccessible can be the truth-condition of a belief, so long as it is relevant to the success of action in the appropriate way. Papineau, who is in the front lines in the ongoing struggles with verificationism and antirealism, recognizes the importance of this fact about success-linked views of content. The teleological theory takes the intuitive link between truth and success and places it in an historical/explanatory context. A type of inner state has the content p if it has the biological function to be present and produce its characteristic effects on the rest of the system when p obtains. The production

Some Comments on Rosenberg's Review

I am grateful to Philip Kitcher for inviting me to comment on Alexander Rosenberg's (1996) review of Philosophy of Biology (Sober 1993) and to Rosenberg for his kind words about my book at the very beginning and the very end of his review. However, I cannot help feeling that most of the material in Rosenberg's review describes a different book from the one I wrote. Of the four philosophical claims that he ascribes to me, only one of them is asserted or implied in Philosophy of Biology. Rosenberg is right that I have argued that many evolutionary laws turn out, when stated carefully, to be a priori mathematical truths. However, he is mistaken in thinking that I am a “historicist” who holds that “fundamental theory in biology is narrative.” Rosenberg also misconstrues my views on the meaning of probability statements in evolutionary theory.

Turing‐computability and artificial intelligence: Gödel’s incompleteness results

Establishes the claims that Gödel’s Incompleteness Theorems cannot prove the existence or the non‐existence of a machine proving all mathematical truths; there exists a machine proving all mathematical truths and the AI thesis is correct.

Are the traditional philosophies of mathematics really incompatible?

Having been under the impression that most mathematicians do not care about the foundations of their subject, I was amazed by the heat generated by this topic in recent issues of the Mathematical Intelligencer, particularly in the letters to the editor (see, e.g., Paris [25]). The purpose of this article is to marshal a number of facts that support a certain philosophical thesis, which I hope to persuade at least some readers to share. I would like to argue that, contrary to widely held opinion, the traditional philosophies, logicism, formalism, Platonism, and intuitionism, if stated with sufficient moderation, do not really contradict each other, although I still have some reservations about logicism. This idea was first proposed in our book [17] by Phil Scott and me and elaborated for a philosophical audience in collaboration with Jocelyne Couture [5]. The present discussion owes a considerable debt to both co-authors. For background material on the traditional mathematical philosophies, the reader is referred to the standard references Benacerraf and Putnam [1], Hintikka [10], and van Heijenoort [30]. There are a number of problems a philosophy of mathematics should address. Perhaps the most important of these are: How is mathematical knowledge obtained (epistemology), and why can it be applied to nature? However, we shall here confine attention to another problem: What is the nature of mathematical entities (ontology) and of mathematical truth? The best-known mathematical philosophies have given different answers to this ontological question (see [5]), which we shall summarize here in rather abbreviated form.

Temporal and atemporal truth in intuitionistic mathematics

Temporal and atemporal truth in intuitionistic mathematics

DESCARTES AND THE DESTRUCTION OF THE ETERNAL TRUTHS

AbstractDescartes's view that the eternal truths of mathematics and logic have been established by God and depend on his will does not merely commit him (as some commentators have suggested) to denying that such truths are necessarily necessary; it abolishes their necessity altogether. For similar reasons, some contemporary views also unwittingly abolish necessity.

Natural internal forcing schemata extending ZFC: Truth in the universe?

Mathematicians have one over on the physicists in that they already have a unified theory of mathematics, namely, set theory. Unfortunately, the plethora of independence results since the invention of forcing has taken away some of the luster of set theory in the eyes of many mathematicians. Will man's knowledge of mathematical truth be forever limited to those theorems derivable from the standard axioms of set theory, ZFC? This author does not think so, he feels that set theorists' intuition about the universe of sets is stronger than ZFC. Here in this paper, using part of this intuition, we introduce some axiom schemata which we feel are very natural candidates for being considered as part of the axioms of set theory. These schemata assert the existence of many generics over simple inner models. The main purpose of this article is to present arguments for why the assertion of the existence of such generics belongs to the axioms of set theory.Our central guiding principle in justifying the axioms is what Maddy called the rule of thumb maximize in her survey article on the axioms of set theory, [8] and [9]. More specifically, our intuition conforms with that expressed by Mathias in his articleWhat is Maclane Missing? challenging Mac Lane's view of set theory.

RANDOMNESS AND COMPLEXITY IN PURE MATHEMATICS

One normally thinks that everything that is true is true for a reason. I’ve found mathematical truths that are true for no reason at all. These mathematical truths are beyond the power of mathematical reasoning because they are accidental and random. Using software written in Mathematica that runs on an IBM RS/6000 workstation, I constructed a perverse 200-page algebraic equation with a parameter N and 17,000 unknowns: [Formula: see text] For each whole-number value of the parameter N, we ask whether this equation has a finite or an infinite number of whole number solutions. The answers escape the power of mathematical reason because they are completely random and accidental. This work is an extension of the famous results of Gödel and Turing using ideas from a new field called algorithmic information theory.

The canonical artefact and its cosmological interpretations

We argue that there is a factor of at least 10 264 to somehow be accommodated to explain how it is possible that our universe produced a life-form with our advanced level of intelligence. The factor arises from an analytical development of the theme that very intelligent life-forms would give physical expression to a highly distinguished mathematical truth in a canonical way. This structure, which serves to flag the occurrence of a life-form possessing an advanced intelligence, is termed the canonical artefact (TCA). Using TCA we show that if the laws governing the universe’s evolution to the present are indifferent to the emergence of a life-form with advanced intelligence, then the chance of emergence is less than one part in 10 264 , and, if instead, the laws make the emergence a virtual certainty, then the emergence is greater than a 264 orders of magnitude effect. In the first case, emergence of a life-form with advanced intelligence is exceedingly unlikely, and, in the second case, where such a life-form is inevitable, physics faces the problem of giving even a semblance of a quantitative explanation for a 264 orders of magnitude effect. In either case physics is left with a puzzle: explain why our universe contains a life-form with advanced intelligence.

Friedrich albert langes bewundernswerte logische studien

Friedrich Albert Lange (1828‐1875) author of a famous History of Materialism and Critique of Its Present Significance(1866, English transi.I–III 1877–79, repr.1925 with introduction by Bertrand Russell), was also interested in the epistemological foundations of formal logic.Part I of his intended two‐volume Logische Studienwas published posthumously in 1877 by Hermann Cohen“head”of the Marburg school of neo‐Kantianism.Lange, departing from Kant, claims that spatial intuition is the source of the apodeictic character not only of the truths of mathematics, but also of the truths of logic.He aims at showing this by basing validity and invalidity of syllogistic inferences on an interpretation of the standard forms (of proposition in assertoric syllogistic) with the help of the five kinds of possible relations (in fact what is known today as the Gergonne‐Euler relations) between extensions of concepts given to us as areas in a plane, i.e.in space.Generality is achieved by considering all possible variations wi...

Truth in Images: From the Technical Drawings of Ibn al-Razzaz al-Jazari, Campanus of Novara, and Giovanni de'Dondi to the Perspective Projection of Leon Battista Alberti

"Truth in Images: From the Technical Drawings of Ibn al-Razzaz al-Jazari, Campanus of Novara, and Giovanni de'Dondi to the Perspective Projection of Leon Battista Alberti." When Alberti justified artist's perspective projection in mathematical and theoretical terms, he relied on a graphic tradition best exemplified not by Euclidean geometric optics, as most commentators would have it, but by technical illustrations used in the creation of exceedingly precise instruments of measure. Long before artists employed Alberti's method of projection to recreate the world illusionistically, late medieval instrument makers were creating drawings that functioned as graphic substitutes for the real thing. Beyond a concern for verisimilitude, however, much broader shared assumptions existed between Alberti and earlier scientists. These assumptions depended on mathematical truth to chart the perfections of time and space. The drawings of planetary clocks, dials, and astrolabes by the fourteenth-century philosopher/astronomer Giovanni de'Dondi, as well as those by the thirteenth-century Italian mathematician Campanus of Novara and the Arab engineer Ibn al-Razzaz al-Jazari, signaled a new response to an ever more complex technology. Taken together, they reflect a growing, cumulative commitment among erudite craftsmen to the twin standards of mathematical rigor and illusionistic credibility, the very hallmarks of Alberti's more famous method of projection.

Conditionals of Freedom and Middle Knowledge

The doctrine of middle knowledge was developed by the Spanish Jesuits Luis de Molina and Francisco Suarez in the late sixteenth century, and immediately became the subject of fierce controversy, being bitterly opposed by the Dominicans, especially Domingo Bafiez and Diego Alvarez, and requiring the intervention of the Vatican (in 1607) to cause the bitterness to subside. As recent work has shown, the intellectual interest of the doctrine extends far beyond that of an inhouse debate in the Catholic Church of the Counter-Reformation; for it touches on issues of central importance not only in analytic theology, but also in philosophical logic. According to Molina, there are three moments in God's foreknowledge of the history of the world which He creates.' These moments are not to be thought of as stages in a temporal series, but as ordered by the relation of logical or conceptual dependence of later on earlier.2 In the first moment, God knows by natural knowledge all metaphysically necessary propositions. These include not merely truths of logic and mathematics, and natural laws, but also facts about which contingent states of affairs are possible, since that a state of affairs is possible is itself a necessary truth. God's natural knowledge would suffice to equip Him with complete foreknowledge of the history of His creation, if the world He created were entirely deterministic, and if He undertook to refrain from free intervention in that world. The creation

Conjuring

What arguments-drawing upon forgotten eighteenth-century religious, educational, medical, and artistic polemics-might be mustered today to justify perceptual expertise? What proof can be offered to show that specialization in visualization is not just deceitful craft? In other words, does it take distinctive art or knowledgeable skill to understand the precise techniques, histories, and rhetorical strategies of seeing and interacting with images?' A positive answer to such pressing contemporary questions requires the revision of commonly held view. This scholarly divisionism splits the early modern period into prophetic or interpretive Romanticism thought to be fundamentally at odds with rational or philosophical Enlightenment. Yet it can be demonstrated that the deepest concerns of these supposedly opposite eras frequently coincided.2 Eighteenth-century British empiricists and French Cartesians as well as nineteenth-century creators of aniconic aesthetics were simultaneously fascinated and repelled by trompe l'oeil trickery and new sensuous technology emitting fake simulations and delusive special effects (fig. 1).3 Broadly speaking, this distrust of misleading or fanatical visions led, on one hand, to an illuminating criticism determined to expose ignorant and optically fueled superstition (fig. 2).4 On the other hand, it gave rise to JudeoProtestant biblical, and eventually secular, hermeneutics. An informed scrutiny of texts was to help eradicate the spellbinding idolatry believed endemic to papist Baroque oral-visual culture. 5 In particular, visual quackery-castigated as empty conjuring and voyeuristic display6-was the target of Neoclassical systematics and of Romantic algebraicism directed alike against mere mechanical deception, as William Hazlitt put it.' These theoretical, and essentially. linguistic, methods aimed to demolish the juggling or ingenious artisans responsible for manufacturing the trifling tricks of Rococo painting. Analogies to the miraclemongering empiric, the legerdemain charlatan, and the duping quack eddied around the manually produced and reproduced, or conjured, image. Performance challenges and defies discourse.8 Thus the Romantic essayist Hazlitt, in The Indian Jugglers, noted that intellectuals can fool the public. Brahmins can build up their dogmas and philosophies end and not be detected. A juggler, rope-dancer, or knife-thrower, however, cannot persuade the audience at the Olympic Theater that he performs astonishing feats without actually giving proofs of what he says. To be able to catch four balls in less than second and to repeat this activity requires precision of movement akin to a mathematical truth.9

New foundations of ontology

This posthumous work by Gustav Bergmann was essentially complete before his death in 1987. In it, he proposes an ontological system that would account for all basic areas of human thought and experience within an extended framework of logical atomism. Bergmann's approach to traditional problems of ontology seeks to balance competing demands of phenomenology, which emphasizes reality presented to us by experience, and of metaphysics, which delineates most general kinds of existents given in experience and most general kinds of relationships they bear to one another. Beginning with atomic facts composed of phenomenally presented qualities, Bergmann goes on to develop an ontology that can account for ordinary objects of everyday experience, mental state through which we become aware of and acquire knowledge of these objects, and even truths of logic and mathematics that allow us to extend our thought and discourse about ordinary objects beyond what may be phenomenally apparent. Many ontologists will be particularly interested in attention Bergmann pays to concept of logical form. In his earlier works, Bergmann claimed that the form of world is in world; that a thing or a complex has a certain logical or syntactic form, he argued, is itself one more fact of our experienced reality, rather than a contribution of mind or of linguistic conventions. Critics of this claim have suggested that paradoxes and contradictions result form it. In New Foundations of Ontology Bergmann responds, arguing that his concept of logical form does not necessarily create problems noted in earlier critiques.

Constructibility and Mathematical Existence.

PART I: THE CONSTRUCTIBILITY THEORY: The Problem of Existence in Mathematics The Constructibility Quantifiers Constructibility and Open-Sentences The Deductive System Cardinality and Number Theory Measurable Quantities and Analysis. PART II: PHILOSOPHICAL DEVELOPMENTS: Mathematical Structuralism Science Without Numbers Why Burgess is a Moderate Realist Maddy's Solution to the Problem of Reference Kitcher's Ideal Agents Deflationism and Mathematical Truth. Appendix Field's Nominalistic Logical Theory.

Legislating the Transcendental: Von Hirsch's Proportionality

In 1897 the Indiana House of Representatives was persuaded by a mathematician (of sorts) to entertain a bill “introducing a new mathematical truth”. The new truth was a method, or more precisely methods, for computing the value of π. Since π is a transcendental number the methods could at best have achieved an approximation. In fact they yielded several inaccurate values, one as wild as 4. The legislators were doubtful enough to refer the bill to a committee (the Committee on Canals) and, when this reported favourably, to the Committee on Temperance. Eventually they had the sense to defer it sine die.

Causality, Reliabilism, and Mathematical Knowledge

Causality, Reliabilism, and Mathematical Knowledge

Characteristics of Classroom Mathematics Traditions: An Interactional Analysis

In this paper, we attempt to clarify what it means to teach mathematics for understanding and to learn mathematics with understanding. To this end, we present an interactional analysis of transcribed video recordings of two lessons that occurred in different elementary school classrooms. The lessons, which are representative of a much larger data corpus, were selected because both focus on place value numeration and involve the use of similar manipulative materials. The analysis draws on Much and Shewder’s (1978) identification of five qualitatively distinct types of classroom norms and pays particular attention to the mathematical explanations and justifications that occurred during the lessons. In one classroom, the teacher and students appeared consistently to constitute mathematics as the activity of following procedural instructions in the course of their moment by moment interactions. The analysis of the other classroom indicated that the teacher and students constituted mathematical truths as they coconstructed a mathematical reality populated by experientially real, manipulable yet abstract mathematical objects. These and other differences between mathematical activity in the two classrooms characterize two distinct classroom mathematics traditions, one in which mathematics was learned with what is typically called understanding and the other in which it was not.