It is a widely known but little considered fact that Albert Einstein and Kurt Gödel were best friends for the last decade and a half of Einstein's life. The two walked home together from Princeton's Institute for Advanced Study every day; they shared ideas about physics, philosophy, politics, and the lost world of German science in which they had grown up. By 1949, Gödel had produced a remarkable proof: In any universe described by the Theory of Relativity, time cannot exist (...) . Einstein endorsed this result-reluctantly, since it decisively overthrew the classical world-view to which he was committed. But he could find no way to refute it, and in the half-century since then, neither has anyone else. Even more remarkable than this stunning discovery, however, was what happened afterward: nothing. Cosmologists and philosophers alike have proceeded with their work as if Gödel's proof never existed -one of the greatest scandals of modern intellectual history. A World Without Time is a sweeping, ambitious book, and yet poignant and intimate. It tells the story of two magnificent minds put on the shelf by the scientific fashions of their day, and attempts to rescue from undeserved obscurity the brilliant work they did together. (shrink)
This is an expansion of the author's 1991 work which investigates the implications of Gödel's writings on Einstein's theory of relativity as they relate to the fundamental questions of the nature of time and the possibilities for time travel.
The dead are gone. They count for nothing. Yet, if we count the dead, their number is staggering. And they account for most of what is great about civilization. Compared to the greatness of the dead, the accomplishments of the living are paltry. Which is it then: are the dead still there tobe counted or not? And if they are still there, where exactly is "there"? We are confronted with the ancient paradox of nonexistence bequeathed us by Parmenides. The mystery (...) of death is the mystery of nonexistence.A successful attempt to provide a metaphysics of death, then, must resolve the paradox of nonexistence. That is the aim of this study. At the same time, the metaphysics of death, of ceasing to exist, must serve as an account of birth, of coming to exist; the primary thesis of this book is that thisdemands going beyond existence and nonexistence to include what underlies both, which one can call, following tradition, "being." The dead and the unborn are therefore objects that lack existence but not being. Nonexistent objects - not corpses, or skeletons, or memories, all of which are existentobjects - are what are "there" to be counted when we count the dead. (shrink)
This is a book about the philosophy of time, and in particular the philosophy of the great logician Kurt Godel (1906-1978). It evaluates Godel's attempt to show that Einstein has not so much explained time as explained it away. Unlike recent more technical studies, it focuses on the reality of time. The book explores Godel's conception of time, existence, and truth with special reference to Plato, Aristotle, Kant, and Frege. In the light of this investigation an attempt is made to (...) shed light on such issues as the precise sense in which Godel believed in the possibility of time travel, the relationship of the reality of time to the objectivity of temporal becoming, and the significance of time for human existence.This is a book about the philosophy of time, and in particular the philosophy of the great logician Kurt Godel (1906-1978). It evaluates Godel's attempt to show that Einstein has not so much explained time as explained it away. Unlike recent more technical studies, it focuses on the reality of time. The book explores Godel's conception of time, existence, and truth with special reference to Plato, Aristotle, Kant, and Frege. In the light of this investigation an attempt is made to shed light on such issues as the precise sense in which Godel believed in the possibility of time travel, the relationship of the reality of time to the objectivity of temporal becoming, and the significance of time for human existence. (shrink)
Traditionally, skeptics as well as their opponents have agreed that in order to know that p one must be able, by some preferred means, to rule out all the alternatives to p. Recently, however, some philosophers have attempted to avert skepticism not (merely) by weakening the preferred means but rather by articulating a subset of the alternatives to p — the so-called relevant alternatives — and insisting that knowledge that p requires only that we be able (by the preferred means) (...) to rule out members of the set. In this paper I argue that a precise formulation of this new approach reveals it inadequate as a solution to skepticism. (shrink)
'You ask me about the idiosyncrasies of philosophers? There is their lack of historical sense, their hatred of even the idea of becoming, their Egyptianism. They think they are doing a thing honour when they dehistoricize it, sub specie aeternitatis — when they make a mummy of it.'Nietzsche, Twilight of the Idols.
Sets are multitudes which are also unities. It is surprising that the fact that multitudes are also unities leads to no contradictions: this is the main fact of mathematics.Kurt Gödel (Hao Wang,A Logical Journey: From Gödel to Philosophy)In what sense can something be at the same time one and many? The problem is familiar since Plato (for example,Republic524e). In recent times, Whitehead and Russell, inPrincipia Mathematica,have been struck by the difficulty of the problem: ‘If there is such an object as (...) a class, it must be in some senseoneobject, yet it is only of classes that many can be predicated. Hence, if we admit classes as objects, we must suppose that the same object can be both one and many, which seems impossible.' It is, however, in Frege's great work,The Foundations of Arithmetic(henceforth,Grundlagen),that many see the final resolution of the old question: how can something be at the same time one and many? (shrink)
Frege's definition of the natural number n in terms of the set of n-membered sets has been treated rudely by history. It has suffered not one but two crippling blows. The discovery of Russell's Paradox revealed a fatal flaw in the ‘naive’ conception of set. In spite of its intuitive appeal, Frege's Basic Law V turned out to be impermissible, leaving us only with the etiolated concept of set that survives in the axiomatic treatments initiated by Zermelo. The independence results, (...) however, of Godel and Cohen, concerning Cantor's Continuum Hypothesis, have left us adrift in choosing between Cantorian and non-Cantorian set theories, which has induced in some logicians a skepticism in regard to the very idea of set-theoretic platonism. (shrink)
DAVID KAPLAN HAS ARGUED THAT RUSSELL'S THEORY OF\nDESCRIPTIONS IN EFFECT EXPLAINS AWAY THE VERY IDEA OF\nDEMOTING. IN THIS PAPER I TRY TO SHINE SOME LIGHT ON THE\nPUZZLE KAPLAN HAS UNCOVERED, AND IN THE PROCESS SHOW THAT\n1) RUSSELL DID NOT HIMSELF FULLY REALIZE EXACTLY WHAT HIS\nTHEORY ACCOMPLISHED, AND 2) IN EFFECT, RUSSELL'S THEORY IS,\nSURPRISINGLY, A KIND OF DEFINITIONAL VARIANT OF FREGE'S.
Is a mathematical theorem proved because provable, or provable because proved? If Brouwer’s intuitionism is accepted, we’re committed, it seems, to the latter, which is highly problematic. Or so I will argue. This and other consequences of Brouwer’s attempt to found mathematics on the intuition of a move of time have heretofore been insufficiently appreciated. Whereas the mathematical anomalies of intuitionism have received enormous attention, too little time, I’ll try to show, has been devoted to some of the temporal anomalies (...) that Brouwer has invited us to introduce into mathematics. (shrink)
Gödel's first incompleteness theorem shows that no axiomatic theory can prove all mathematical truths, while Gödel's second incompleteness theorem shows that a specific mathematical result is unprovable. A famous mathematician of the time, David Hilbert, had asked for a proof that an important axiomatic theory was consistent, and Godel showed that such a proof could not be carried out within the axiomatic theory itself, and presumably could therefore not be established in a convincing way outside of the theory either.
A comparison of a recent paper by Giora Hon in this journal with a book I wrote several years ago, on Gödel's philosophy of time, reveals that the substance, and indeed many of the words themselves, appearing in Hon's essay are in fact original to my book—the ideas of which he sadly failed to understand.
In his Kurt Gödel Award 2021 essay, “The Philosophical Meaning of the Gödel Universe,” Prof. Kahle takes a fresh look at the philosophical ramifications of Gödel’s cosmology and helps clarify what Gödel’s intentions were and what the significance is of his arguments. I have several reservations, however, concerning Kahle’s discussion which will be discussed in this essay.