If physics is the empirical study of the world as it is then mathematics is the study of the mind of god, it is gnosis; an attempt to know the absolute. Thus the basis of mathematical reasoning is not reason nor reasonable but metaphysics and mystical.
As documented below,the crisis in mathematics at the turn of last century was a crisis between German rationalist philosophy and Jewish mysticism; the Kabbalah. The German and rationalist mathematical philosophers, including thier British scion's like Russel and Whitehead, were attempting to map the mind of god; the absolute, while the Jewish mystic mathematician Georg Cantor had gone mad, it aggravated his existing depression , when he discovered the mind of God. He looked upon the face of the abyss and the face of the abyss looked back at him.
Wanting to avoid the problem of the abyss, which is the mystical journey known as the Conversation with the Holy Guardian Angel, the German school attempted to refute Cantor's theorem. They did this by adopting a different metaphysics and mystical philosophy, one that was still rooted in the occult knowledge and traditions of the pre-enlightenment. An analysis that did not require an appreciation of the absolute, god, by expansive reasoning, by looking at the enormity of infinity, rather they looked inward. And in looking inward they had to answer one question, can the infinite be finite.Can it be formulated as a set of finite principles.
It took another mystic Luitzen Egbertus Jan Brouwer, to challenge Cantor. Brouwer was a Dutch mathematician and mystic, influenced by the works of Meister Eckhart as well as by the alchemistJakob Boehme. His mathematical premises were thus founded on the alienated sense of man separated from God, and most math commentators below miss this crucial point. They divorce his philosophical world view from his principa mathematica. At least one article below focuses upon the importance of his mystical thinking to his mathematical philosophy, which he published in a pamphlet in 1903.
Like the Master Ekhardt he believes that the search for the mathematical absolute; god, is best done through asceticism, seclusion of the mind.
Luitzen Egbertus Jan Brouwer
by Johannes (Master) Eckhart
I have read many writings of both Pagan masters and the Prophets of the old and new Covenant, and have investigated seriously and with great zeal which would be the best and highest virtue by which Man could best become similar to God, and how he could resemble again the archetype such as he was in God when there was no difference between him and God until God made the creatures. If I go down to the bottom of all that is written as far as my reason with its testimony and its judgment can reach, I find nothing but mere seclusion of all that is created. In this sense our Lord says to Martha: "One thing is needed," this means: He who wants to be pure and untroubled has to have one thing, Seclusion.
His essay corresponds to the changing world view of Modernity that was occurring at that moment in history. Herman Hesse also reflected this change in thought that was occurring before and after WWI. and he too was influenced by Master
Brouwer's paean to an ascetic mathematical gnosis of the mind of god is also reminiscent of his contemporary aesthete; the Russian composer Scriabin. Scriabin believed music, which is mathematical, and art the highest form of gnosis.
It was the artist, and the artist alone – not the scientist or politician – who could offer to mankind a form of gnosis achieved through the experience of ecstasy and
the act of creation that brought it about. And it was to this mission of artistic
creation that Scriabin was unyieldingly faithful despite all else.
Brouwer's influence on Godel would lead the two down separate gnostic paths of interpretation of principa mathematica. And yet both these paths reflect the dualistic nature of actual gnosticism, between the deniers of the world as it is and those who embrace the world as it is. Between the ascetic and libertine, the Cathars and the Adamites. Master Eckhart himself was a member of the heretical sect the proto-communist Brethren of the Free Spirit.
In the end though they never truly refuted Cantor, they merely built on his theorems. Brouwer created a topology of the mind of god while Godel proved that no set theories can be proven, which led to the Heisenburg uncertainty principle.
David Hilbert opened ‘Axiomatic Thought’ with the observation that ‘the most important bearers of mathematical thought,’ for ‘the benefit of mathematicsitself have always [. . . ] cultivated the relations to the domains of physics and the [philosophical] theory of knowledge.’ We have in L.E.J. Brouwer and Kurt Godel two of those ‘most important bearers of mathematical thought’ who cultivated the relations to philosophy for the benefit of mathematics (though not only for that). And both went beyond philosophy, cultivating relations to mysticism for the benefit of mathematics (though not for that alone).
There is a basic conception of mysticism that is singularly relevant here.
(’Mysticism’ labels that.) That corresponds to a basic conception of philosophy
(’Philosophy’), also singularly relevant here. Both Mystic and Philosopher begin
in a condition of seriously unpleasant, existential unease, and aim at a condition
of abiding ease. For Mystic and Philosopher the way to that ease is through
being enlightened about the real and true good of all things. Thus Mysticism
and Philosophy are triply optimistic: there is a real, true good of all things,
the Philosopher and Mystic can become enlightened about it, and being thus
enlightened would give them ease.
That Enlightenment sought comes from some sort of cognitive or intelligent
engagement with what we will here call ‘the Good’. Some use ‘the Absolute’
when it seems important to emphasize that ‘the Good’ is unconditioned—there
is nothing behind it, nothing above it. Others use ‘the One’; still others, ‘God’.
It is natural to regard the Good as somehow mind-like, or like something (permanently) in mind. It should in either case be in some way homogeneous with, or in sympathy with, our minds, for the Good must attract and support the intelligent engagement of it by our minds. In that way it can enlighten us.
We have seen that both Godel and Brouwer were looking for mystical experiences,
in which an openness of the mind to the Absolute is operative. What
is disclosed in such experiences has the air of being something imparted to the
person. The imparting is preceded by a preparation or transformation of the
person. The self must be brought into a condition to receive, support, and appreciate what is to be disclosed. This preparation we see mentioned by both Brouwer (the abandonment of mathematics) and Godel (closing off the senses, etc.)
However, they made very different claims as to how what is disclosed in
such experience is related to mathematics. What strikes us is how the bond
between mathematics and mysticism is equally tight in Godel and Brouwer, but
that the signs are different so to speak. According to both, mathematics relates
individual thought to ultimate reality, but Godel thinks of a positive relation
and Brouwer of a negative one.
For Godel, doing mathematics is a way of accessing the Absolute. For
Brouwer, doing mathematics precisely prohibits access to the Absolute.
Put differently, according to Godel, mathematical experience reveals (part
of) Reality; according to Brouwer, mathematical experience conceals Reality.
A mystical disclosure in the relevant sense has about it the phenomenological
character of being a form of knowing or enlightened understanding; it discloses
the Good, the significant, the important, fundamental values.
We would like to end by making the following two remarks. First, of course one could, and usually does, engage in mathematics for its own sake, without any interest in relating it, be it positively or negatively, to mysticism. From Godel’s and Brouwer’s point of view, that would probably be not unlike the possibility to perform a hymn for its own sake, without any interest in the religious meaning it may have.The second remark is related to the first. In spite of the incommensurability
of Brouwer’s and Godel’s positions, their respective motivations to take the
mystical turn may have much in common. Both were disgruntled with the
materialistic and formalistic philosophies prevalent at their times; both thought
that these philosophies could not do justice to the Good.
Draft, 26 July 2004
The foundational crisis is a well-known affair for almost all mathematicians. We
all know something about logicism, formalism, and intuitionism; about the hopes
to place mathematical theories beyond the shadow of any doubt; about the
impact of Gödel’s results upon our images of mathematical knowledge. But the
real outlines of the historical debate are not well known, and the subtler
philosophical issues at stake are often ignored. In the limited space available,
we shall essentially discuss the former, in the hopes that this will help bring the
main conceptual issues under sharper focus.
Usually, the crisis is understood as a relatively localized event, a heated
debate taking place in the 1920s between the partisans of “classical” (late 19th
century) mathematics, led by Hilbert, and their critics, led by Brouwer,
advocating strong revision of the received doctrines. There is however a second
sense and in my opinion a very important one, in which the “crisis” was a long
and rather global process, indistinguishable from the rise of modern
mathematics and the philosophical/methodological perplexities it created.
This is the standpoint from which the present account has been written.
Within this longer process one may isolate five noteworthy intervals:
1) around 1870, discussions about non-Euclidean geometries, function
theory, and the real numbers;
2) around 1885, fights in algebra, higher arithmetic, and set theory;
3) by 1904, debates on axiomatics and logic vs. intuition, the concept of the
continuum, and set theory;
4) around 1925, the crisis in the proper sense, transforming the main
previous views into detailed research projects;
5) in the 1930s, Gödel’s results and their aftermath.
Meanwhile, back in the 1900s, a young mathematician in the Netherlands
was beginning to find his way toward a philosophically coloured version of
constructivism. Egbertus Brouwer presented his strikingly peculiar (to some,
outrageous) metaphysical and ethical views in 1905, and started to elaborate a
corresponding foundation for mathematics in his thesis of 1907. His philosophy
of intuitionism derived from the old metaphysical view that individual
consciousness is the one and only source of knowledge.
Brouwer’s worldview was idealistic and tended to solipsism, he had an artistic temperament, his private life was eccentric; he despised the modern world, looking for the inner life of the self as the only way out (at least in principle, though not always in practice).
They all contributed to the ultimate alchemical paradox of modern physics that the observer influences what is observed, As Above-So Below, which we know today as quantum theory.
Modern mathematical philosophy is simply gnosis stripped of its religious iconography and poetry. In that it still remains kabbalistic ,as Cantor suggested, guided by mystics whose language is their own and who despite their philosophical differences remain of one mind. The purpose and outcome of their theories are an attempt to define and understand a pantheistic/monistic universe. Their failure to resolve the contradictions of their theories is their failure to embrace dialectics. Like their enlightenment counterparts, the Freemasons, they remain a secret society founded on mysticism.
THE METAPHYSICS OF MATHEMATICS
Historically, the starting point is Plato who proposed that mathematical reality consists of perfect forms independent of the physical world. This view of the subject matter of mathematics lies at one end of a spectrum of metaphysical views; towards the other end is the view is that the subject matter is a purely human artefact. Views towards the Platonic end are known as Platonist; towards the other end, anti- Platonist. That is a classification of metaphysical views. Epistemological views fall into two classes, roughly speaking mathematical truths are known (i) by reason, or (ii) by inference from the evidence of the senses supplemented by deduction. There are a few important epistemological views which fall into neither camp, notably those of Plato, Kant and Gödel.
- Do numbers and other mathematical entities exist independently of human cognition?
- If not then how do we explain the extraordinary applicability of mathematics to science and practical affairs? If so then what kind of things are they and how can we know about them?
- What is the relationship between mathematics and logic?
The first question is a metaphysical question with close affinities to questions about the existence of other entities such as universals, properties and values. According to many philosophers, if such entities exist then they do so outside of space and time, and they lack causal powers; they are often termed abstract (as opposed to concrete) entities. If we accept the existence of abstract mathematical objects then an adequate epistemology of mathematics must explain how we can know about them. Of course, proofs seem to be our main source of justification for mathematical propositions but proofs depend on axioms and so the question of how we can know the truth of the axioms remains.
It is usually thought that mathematical truths are necessary truths; how then is it possible for finite, physical beings inhabiting a contingent world to have knowledge of such truths? Two broad views are possible: either mathematical truths are known by reason; or they are known by inference from sensory experience. The former rationalist view is adopted by Descartes and Leibniz who also thought that mathematical concepts are innate. Locke and Hume agreed that mathematical truths were known by reason but they thought all mathematical concepts were derived by abstraction from experience. Mill was a complete empiricist about mathematics and held both that mathematical concepts are derived from experience and also that mathematical truths like 2+2=4 are really inductive generalisations from experience. (N.B. Kant’s views on mathematics are complex and important; see Kant.)
The discovery in the mid-nineteenth century of non-Euclidean geometry meant that philosophers were forced to reassess the status of Euclidean geometry which had previously been regarded as the shinning example of certain knowledge of the world. Many took the existence of consistent non-Euclidean geometries to be a direct refutation of both Mill’s and Kant’s philosophies of mathematics. By the end of the nineteenth century Cantor had discovered various paradoxes in the theory of classes and there was something of a crisis in the foundations of mathematics. The early twentieth century saw great advances in mathematics and also in mathematical logic and the foundations of mathematics.Most of the fundamental issues in the philosophy of mathematics are accessible to anyone who is familiar with geometry and arithmetic and who has had the experience of following a mathematical proof. However, some of the most important philosophical developments of the twentieth century were instigated by the profound developments that have taken place in mathematics and logic, and a proper appreciation of these issues is only available to someone who has an understanding of basic set theory and intermediate logic. To study philosophy of mathematics at an advanced level one ought really to have followed a course which includes proofs of Gödel’s incompleteness theorems.
Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a German mathematician. He is best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.
Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaréand later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God on one occasion equating the theory of transfinite numbers with pantheism. The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong".Cantor's recurring bouts of depression from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries, but these episodes can now be seen as probable manifestations of a bipolar disorder.
The harsh criticism has been matched by international accolades. In 1904, the Royal Society of London awarded Cantor its Sylvester Medal, the highest honor it can confer. Cantor believed his theory of transfinite numbers had been communicated to him by God.David Hilbert defended it from its critics by famously declaring: "No one shall expel us from the Paradise that Cantor has created."
CANTOR'S PHILOSOPHICAL WRITING
Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.
… Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real.
"… the essence of mathematics lies entirely in its freedom".
The best-selling author of Infinite Jest on the two-thousand-year-old quest to understand infinity. ONE OF THE OUTSTANDING VOICES of his generation. David Foster Wallace has won a large and devoted following for the intellectual ambition and bravura style of his fiction and essays. Now he brings his considerable talents to the history of one of math's most enduring puzzles: the seemingly paradoxical nature of infinity. Is infinity a valid mathematical property or a meaningless abstraction? The nineteenth-century mathematical genius Georg Cantor's answer to this question not only surprised him but also shook the very foundations upon which math had been built. Cantor's counterintuitive discovery of a progression of larger and larger infinities created controversy in his time and may have hastened his mental breakdown, but it also helped lead to the development of set theory, analytic philosophy, and even computer technology. Smart, challenging, and thoroughly rewarding, Wallace's tour de force brings immediate and highprofile recognition to the bizarre and fascinating world of higher mathematics.
by Amir Aczel Four Walls Eight Windows, New York, NY, 258 pp., 2000
Seeing a marked increase in the number of books on mathematics written for the general populace and published in the past few years has been nice, seeing so many of them take a historical view is even more exciting. The Mystery of the Aleph is a fine addition to this collection. Amir Aczel's topic is Georg Cantor and his discovery/invention of transfinite numbers. The book is a well-written nontechnical introduction to Cantor's life, set theory, transfinite numbers, the continuum hypothesis, and related mathematical and historical issues. While staying true to the mathematics Amir Aczel has written The Mystery of the Aleph with the attention to suspense and character development of a skilled story-teller.
The story begins with Cantor's death in a university mental clinic in 1918. Like a fine mystery writer Aczel draws us into the tale by concluding a short (9 pages) first chapter with the following:
"One fact is known about Georg Cantor's illness. His attacks of depression were associated with periods in which he was thinking about what is now known as 'Cantor's continuum hypothesis.' He was contemplating a single mathematical expression, an equation using the Hebrew letter aleph: 2... = ... . This equation is a statement about the nature of infinity. A century and a third after Cantor first wrote it down, the equation - along with its properties and implications - remains the most enduring mystery in mathematics. " (pp. 8-9)
From the mental clinic in Halle, The Mystery of the Aleph takes the reader back to the paradoxes of Zeno, to the Pythagoreans, and then to the Kabbalah, the Jewish system of secret mysticism, numerology, and meditations. Here Aczel introduces notion of the intense light of the infinity of God as a metaphor for the wonder of Cantor's infinities. The metaphor continues with good effect throughout the book. Although no actual clear connection between Cantor's work and the Kabbalah is established in The Mystery of the Aleph, the metaphorical connection is successful and contributes to the story.
Quickly the pace of the tale picks up and the reader is treated to wonderful discussions of Galileo's demonstration of the one-to-one correspondence between the natural numbers and the even natural numbers, Bolzano's pioneering work with infinite series, the mathematical hegemony of German universities in the late nineteenth century, and the powerful personalities of Weierstrass and Kronecker. Woven through it all we watch the development of Cantor as a mathematician, and the birth of modern set theory and transfinite numbers.
The Mystery of the Aleph then takes us to the questions of the foundations of mathematics that have haunted generations of mathematicians from Peano, Russell, Frege, Zermelo, Hilbert and Brouwer, to Godel, Turing, and Cohen. The story culminates in Cohen's proof of the independence of the continuum hypothesis from the axioms of Zermelo-Fraenkel set theory, Godel's incompleteness theorem, and Turing's argument for the undecidibility of the halting problem. Throughout this grand tour of the key issues of mathematics and infinity, The Mystery of the Aleph never lets us lose sight of the humanity (and the inevitable failures and successes that go with it) of these giants of mathematics. The book ends with a quote on a plaque in Halle commemorating Georg Cantor. It reads "The essence of mathematics lies in its freedom." (p. 228)
The Mystery of the Aleph is not a source of details on the mathematics of Cantor, Godel, and Cohen, but it is a wonderful source for a quick historical overview of the issues of infinity in modern mathematics, biographical information on Cantor and Godel, and a good introduction to the politics of mathematics in the nineteenth century. This book would be a valuable addition to a school library or a text to share with a student who has begun to wonder about infinity.
Reviewed by James V. Rauff Millikin University
Copyright Mathematics and Computer Education Spring 2001
Provided by ProQuest Information and Learning Company. All rights Reserved
- Aleph number or cardinality, a measurement of mathematical sets
- In set theory, The Hebrew aleph glyph is used as the symbol to denote the aleph numbers, which represent the cardinality of infinite sets.
Aleph is the subject of a midrash which praises its humility in not demanding to start the Bible. (In Hebrew the Bible is begun with the second letter of the alphabet, Bet.) In this folktale, Aleph is rewarded by being allowed to start the Ten Commandments. (In Hebrew, the first word is 'Anokhi, which starts with an aleph.)
In the Sefer Yetzirah, The letter Aleph is King over Breath, Formed Air in the universe, Temperate in the Year, and the Chest in the soul.
- "... Brouwer, in a paper entitled "The untrustworthiness of the principles of logic", challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384--322 B.C.) have an absolute validity, independent of the subject matter to which they are applied" (Kleene (1952), p. 46).
"After completing his dissertation [year?], Brouwer made a conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert -- the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict -- admired the young man and helped him receive a regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism " (ibid).
Kurt Gödel (April 28, 1906 Brünn, Austria-Hungary (now Brno, Czech Republic) – January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher.
One of the most significant logicians of all time, Gödel's work has had immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were attempting to use logic and set theory to understand the foundations of mathematics.
Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
He also showed that the continuum hypothesis cannot be disproved from the accepted axioms of set theory, if those axioms are consistent. He made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.
From Wikipedia, the free encyclopedia
In mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest.
The theorems are also of considerable importance to the philosophy of mathematics. They are widely regarded as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert's second problem. Authors such as J. R. Lucas have argued that the theorems have implications in wider areas of philosophy and even cognitive science, but these claims are less generally accepted.
INCOMPLETENESS: THE PROOF AND PARADOX OF KURT GÖDEL by Rebecca Goldstein Atlas Books, 2005, 296 pp. ISBN: 0-393-05169-2
On page 253 of Incompleteness: The Proof and Paradox of Kurt Gödel, there is a photograph of Albert Einstein and Kurt Gödel walking together on the grounds of the Institute for Advanced Study (IAS). I doubt that any student of mathematics could fail to be moved by this photograph. What wonderful ideas are being exchanged? What new areas of mathematics, physics, logic, or philosophy were born in the conversations between these two giants of twentieth-century thought?
Incompleteness: The Proof and Paradox of Kurt Gödel provides many tantalizing glimpses at the life and work of Kurt Gödel. Rebecca Goldstein follows Gödel from his early days with the Vienna Circle to his last days at the IAS. Although there are several recent books about Gödel and/or his incompleteness theorems, Goldstein's stands out on three fronts.
First, Incompleteness is absolutely beautifully written. The style is conversational and the reader is carried along by the author's obvious joy in her subject matter. I read Incompleteness in three consecutive evenings. It was truly difficult to put down. Undergraduate students in mathematics, physics, or philosophy will find Incompleteness exciting. It will reaffirm their choice of study.
Second, Incompleteness is an excellent introduction to the personalities and philosophies of the iconic members of the Vienna Circle (Moritz Schlick, Rudolph Carnap, Otto Neurath, Hans Hahn, Herbert Feigl, Karl Menger, Kurt Godel) and celebrated visitors and participants (John von Neumann, Willard van Orman Quine, Carl Hempel, Alfred Tarski, and the very influential Ludwig Wittgenstein). The reader can't help but imagine the intense level of intellectual activity going on in a single location. Goldstein skillfully shows us how Gödel was influenced by and influenced the Circle, and contrasts his Platonism with Wittgenstein's philosophy of mathematics, formalism, and logical positivism. These passages are superb introductions to the state of the philosophy of science in the first half of the twentieth century.
Finally, Goldstein presents one of the best non-technical outlines of Gödel's proof of the incompleteness of arithmetic. Any mathematics teacher would do well to begin their students' understanding of Gödel's results with a reading from Incompleteness.
Incompleteness: The Proof and Paradox of Kurt Gödel is a delightful introduction to the life, work, and times of Kurt Gödel. Written in a captivating conversational style, true to its mathematical, philosophical, and historical content, and just plain fun to read, this book deserves a spot on the recommended reading list for undergraduates.
Reviewed by James V. Rauff
Copyright Mathematics and Computer Education Spring 2006
Provided by ProQuest Information and Learning Company. All rights Reserved
by Harold Ravitch, Ph.D.
Chairman, Department of Philosophy
Los Angeles Valley College
(i) In thinking that the paradoxes were devastating mathematics, various restrictions on the usual methods of mathematical reasoning were imposed.
(ii) No paradox has been discovered which Involves entities which are strictly speaking mathematical: the "set of all sets," the "greatest ordinal number," "sets which are elements of themselves," etc. are logical and epistemological entities which do not belong to classical mathematics proper.
(iii) The concepts of classical mathematics are meaningful, precise, and are capable of being understood because they meet standards of clarity and exactitude which are adequate for their purpose.
(iv) Hence, there is no justification for applying unnecessary restrictions to classical mathematics.
The search for a once-and-for-all solution to the paradoxes led Russell, Poincaré, and others to the observation that each of the paradoxes trades on a vicious circle in defining an entity which ultimately creates the paradox. Questions of circularity are as old as philosophy., but it was never realized how deeply they could permeate logic and mathematics. Indeed Gödel himself remarked that "any epistemological paradox" could have been employed to yield an undecidable statement of arithmetic. Of course many nontechnical works on logic warn us about circular definitions.
In axiomatic set theory, one of the legislative functions of the axioms is to prohibit the existence of sets which would cause trouble, and the various axiom systems can be classified according to the manner in which the paradoxes are blocked. If one however wishes to derive totally his mathematics from his logic, it is found that the process of Dedekind Cuts, the fundamental method of establishing the real number system, is badly in violation of the vicious circle principles.Hermann Weyl attempted a development of analysis in Das Kontinuum which adhered to the vicious circle principle, but he was unable to obtain the whole of classical analysis. Recent research has shown that more can be squeezed out of these restrictions than had been expected:
all mathematically interesting statements about the natural numbers, as well as many analytic statements, which have been obtained by impredicative methods can already be obtained by predicative ones.
We do not wish to quibble over the meaning of "mathematically interesting." However, "it is shown that the arithmetical statement expressing the consistency of predicative analysis is provable by impredicative means." Thus it can be proved conclusively that restricting mathematics to predicative methods does in fact eliminate a substantial portion of classical mathematics.
Gödel has offered a rather complex analysis of the vicious circle principle and its devastating effects on classical mathematics culminating in the conclusion that because it "destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a good deal of modern mathematics itself" he would "consider this rather as a proof that the vicious circle principle is false than that classical mathematics is false."
The vicious circle principle as usually stated is dissected by Gödel into four forms:
(1) No totality can contain members definable only in terms of this totality.
(2) No totality can contain members involving this totality.
(3) No totality can contain members presupposing this totality.
(4) Nothing defined in terms of a propositional function can be a possible argument of this function.
The core of Gödel's rejection of the vicious circle principle reduces to his rejection of the view that mathematical entities are "constructed by ourselves." We shall see that this argument hinges an the interpretation of 'construction', and on Gödel's faith in the consistency of the axioms of set theory underlying classical analysis.
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