Mysticism and Mathematics

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.

Life, Art, and Mysticism

Luitzen Egbertus Jan Brouwer

Like the Master Ekhardt he believes that the search for the mathematical absolute; god, is best done through asceticism, seclusion of the mind.

Seclusion

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
Eckhart.

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.

Mysticism and mathematics: Brouwer, Godel, and the common core thesis

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.

The Crisis in the Foundations of Mathematics

José Ferreirós
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).

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.

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.


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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.

Philosophy of Mathematics

Many philosophers have taken mathematics to be the paradigm of knowledge, and the reasoning employed in following mathematical proofs is often regarded as the epitome of rational thought. But mathematics is also a rich source of philosophical problems which have been at the centre of epistemology and metaphysics since the beginnings of Western philosophy; among the most important are the following:

1. Do numbers and other mathematical entities exist independently of human cognition?
2. 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?
3. 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 Cantor - Wikipedia, the free encyclopedia

Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – 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".

Everything and More: A Compact History of Infinity

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.

THE MYSTERY OF THE ALEPH: MATHEMATICS, THE KABBALAH, AND THE SEARCH FOR INFINITY

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

In gematria, aleph represents the number 1, and when used at the beginning of Hebrew years, it means 1000 (i.e. א'תשנ"ד in numbers would be the date 1754).

• Aleph number or cardinality, a measurement of mathematical sets
  • aleph null (), the cardinality of countable infinite sets
  • aleph one (), the cardinality of certain uncountably infinite sets (if the continuum hypothesis is true, then the set of all real numbers has this cardinality).
  • 
    
• 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.

Aleph is also the first letter of the Hebrew word emet, which means truth. In Jewish mythology it was the letter aleph that was carved into the head of the golem which ultimately gave it life.

Aleph also begins the three words that make up God's mystical name in Exodus, I Am That I Am, (in Hebrew, 'Ehye 'Asher 'Ehye), and aleph is an important part of mystical amulets and formulas.

Luitzen Egbertus Jan Brouwer

In 1905, at the age of 26, Brouwer expressed his philosophy of life in a short tract Life, Art and Mysticism described by Davis as "drenched in romantic pessimism" (Davis (2002), p. 94). Then Brouwer "embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions"; indeed his thesis advisor refused to accept his Chapter II " 'as it stands, ... all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics' " (Davis, p. 94 quoting van Stigt, p. 41). Nevertheless, in 1908:

"... 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

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.

Gödel's incompleteness theorems

From Wikipedia, the free encyclopedia

Learn more about citing Wikipedia •

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

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

Millikin University

Copyright Mathematics and Computer Education Spring 2006
Provided by ProQuest Information and Learning Company. All rights Reserved

On Gödel's Philosophy of Mathematics

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.

2.) The Vicious Circle Principle.

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.

Cantor Godel Brouwer Russell Frege Whitehead

A power point presentation of their contributions to Philisophica Mathematica.



See:

Godel, Cantor, Wiener and Schrodinger's Cat

Dialectics, Nature and Science

Kabbalistic Kommunism

For a Ruthless Criticism of Everything Existing

Goldilocks Enigma

9 Minute Nobel Prize

Is God A Cosmonaut

Cosmic Conundrum

My Favorite Muslim



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A Big But

When an explanation is not an explanation it usually ends with a "but".

As is in this case of the latest scientific explanation for why folks experience their astral body.

"Brain dysfunctions that interfere with interpreting sensory signals may be responsible for some clinical cases of out-of-body experiences," said Henrik Ehrsson, a neuroscientist formerly of University College London, and now at the Karolinska Institute in Sweden.

"Though, whether all out-of-body experiences arise from the same causes is still an open question," he added.

SEE:

Kabbalistic Kommunism

Snake Oil Saint

For a Ruthless Criticism of Everything Existing

New Age Libertarian Manifesto

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The War Against Secular Society

The right wing is expanding it's identity politics campaign claiming that Christianity is being oppressed and abused; with politically correct attacks on outspoken atheists of late.

As Barbara Kay in the National Post writes;

"atheists in democratic countries can't conjure up grim tales of the truncheon's midnight thud on the door,"

Once again proving that right wing political correctness is based on historical revisionism.

Indeed Ms. Kay atheism was considered a legal offense in Merry Olde England for the longest time.

Free Thinkers, as they were called, did have the police truncheon and worse put upon them. Indeed their publications were banned and their printing presses destroyed. Free Thought, indeed secularism, with its libertarian origins in Godwin, Bakunin, Carlisle, Tucker, Proudhon, Woodhull, etc. was began in the late 18th Century and was a 19th Century phenomena.

Richard Carlisle, a "freethinker," opened a lecturing, conversation, and discussion establishment, preached the "only true gospel," hung effigies of bishops outside his shop, and was eventually quieted by nine years' imprisonment, a punishment by no means undeserved.

Despite its origins in Greek Philosophies such as those of Heraclitus and Epicurus, atheism is a modern movement coincidental with the Enlightenment and the development of modern industrial/capitalist society.

And it was the philosopher Spinoza, a Jew, who began the attack on Christianity, Judaism, Islam and all the Abrahamic religions with his philosophical defense of atheism.

It is well known that Marx was familiar with Spinoza; indeed, he hand-copied whole passages of Spinoza's Tractatus Theologico-Politicus into his notebooks. Less clear is the significance of this fact, and the extent of Spinoza's influence on Marx's thought.

And it would be Marx who would proclaim that atheists needed to take one more step to truly be revolutionaries.

In the "Economic and Philosophical Manuscripts" (1844), Marx said:

Once the essence of man and of nature, man as a natural being and nature as a human reality, has become evident in practical life, in sense experience, the quest for an ALIEN being, a being above man and nature (a quest which is an avowal of the unreality of man and nature) becomes impossible in practice. ATHEISM, as a denial of this unreality, is no longer meaningful, for atheism is a NEGATION OF GOD and seeks to assert by this negation the EXISTENCE OF MAN. Socialism no longer requires such a roundabout method; it begins from the THEORETICAL and PRACTICAL SENSE PERCEPTION of man and nature as essential beings. It is positive human SELF-CONSCIOUSNESS, no longer a self-consciousness attained through the negation of religion. (Marx 1964A: 166-67)

In the famous Introduction to the Critique of the Hegelian philosophy of public law, Marx gives an even more explicit and elaborate formulation of this outlook. "Religious misery", he writes, "is at once the expression of real misery and a protest against it. Religion is the groan of the oppressed, the sentiment of a heartless world, and at the same time the spirit of a condition deprived of spirituality. It is the opium of the people. The suppression of religion as the illusory happiness of the people is the premise of its real happiness. It is first and foremost the task of philosophy, operating in the service of history, to unmask self-alienation in its profane forms, after the sacred form of human self alienation has been discovered. Thus criticism of heaven is transformed into criticism of the earth, criticism of religion into criticism of law, criticism of theology into criticism of politics". And just before: "Religion is the consciousness and awareness of man who has not yet acquired or who has again lost himself. But man is not an abstract being, isolated from the world. Man is the world of man, the State, society. This State and this society produce religion, an upside-down consciousness of the world, just because they are an upside-down world. Religion is the general theory of this world, its encyclopedic epitome, its logic in popular form, its spiritualistic point d'honneur, its enthusiasm, its moral sanction, its solemn completion, its fundamental reason of consolation and justification. It is the fantastic realization of human essence, since human essence does not possess a true reality. The struggle against religion is therefore indirectly the struggle against that world of which religion is the spiritual aroma" (K. Marx, Per la critica della filosofia del diritto di Hegel, Introduzione, Rome 1966, pp. 57-58).

Ironically in her attack on atheists Kay attacks Christopher Hitchens, the right wings favorite former Trotskyist turned pro war contrarian. She of course claims that atheists only want to ban Christianity and Judaism.

Aggressively marketed grievance has worked for women and gays. The same strategy for brights will doubtless end in a government-funded Status of Atheists Council to undo the iniquities of 10,000 years of theocratic hegemony and repression. After that, we may yet see -- don't laugh until you're sure it can't happen -- demands for reparations payout by churches and synagogues to redress the ignominy and shame now-atheist, former (involuntarily-designated) Christians and Jews suffered as children when force-fed the Ten Commandments in Sunday and Hebrew School. (Somehow, I do not envisage a similar campaign by Muslim atheists directed against the madrassas, not sure why ?)

While Christopher Hitchens has made it clear for many years that he opposes all theocracies and theocrats, Christian, Jewish or Muslim, heck he doesn't even like the Dali Lama. He has been outspoken against Islamism in fact his support for the war on terror is based upon his atheist opposition to Muslim Fascism.

So like her counterpart at the Sun; Michael Coren, she smears atheists with the Anti-Christian PC label, while failing to accurately point out that atheists oppose all religions and all belief systems that put faith in a supreme being.

Like Coren, Kay and other social conservatives, especially those of the evangelical Protestant faith, believe in the coming Rapture, the end times, and the Israel plays a role in this as predicted in the Book of Revelations in the New Testament.

They of course overlook the persecution and pogroms of the Jews by the Catholic, Orthodox, and Protestant Empires in Europe. They are not defending Jews or Jewish culture, which has had a major cultural impact on the West in developing secularism, socialism, and yes atheism as well as anarchism and libertarianism. Rather they are defending Israel and Zionism.

The phony war on Christianity is just so much bunkum. According to Stats Canada the dominant religion in Canada remains Christianity and its sects and cults.

For Kay, Coren and the Byfields, the supposed war on Christianity is being engaged in by the secularist elites and heathen pagans, whoever they are. Oh yeah the old 'Powers That Be'. Except that the PTB in North America are Christians. Once again the right embraces historical revisionism; the screed of conspiracy theorists.



h/t to Another Point of View




SEE:

Lou Dobbs New Enemy: The Church

Pauline Origins of Social Conservatism

Marxism and Religion

Secular Democracy

GoldilocksEnigma

American Polytheism

1666 The Creation Of The World

Snakes Alive

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Snake Oil Saint

This is a bit hard to swallow.

Up to one million people are set to gather in Brazil to watch Pope Benedict XVI canonise the country's first home-born saint, Friar Galvao.Friar Galvao, an 18th Century monk, is still a hugely influential figure. He is best remembered for producing Latin prayers written on tiny balls of paper that, when swallowed, had the apparent effect of curing a range of ailments.

Until you realize that this is the practice of Kabbalistic Magick developed during the Renaissance, whereupon the Jewish physicians would use Kabbalistic talismans as part of their healing practice. The medical texts and practices used were Islamic and introduced by Jewish scholars into Europe.

Once again the Catholic Church proves it's religious syncretic nature, in Brazil it blends Voodoo with Christianity and appoints a Saint who used Christian Kabbalistic magick.

Snake oil is a traditional Chinese medicine used to treat joint pain. However, the most common usage of the words is as a derogatory term for compounds offered as medicines which imply they are fake, fraudulent, or ineffective. The expression is also applied metaphorically to any product with exaggerated marketing but questionable or unverifiable quality. In short, it refers to a product sold as one part of a hoax.

"Snake oil." The expression has come to be synonymous with a quack remedy. But questions about the origins of the term provide the basis for an interesting investigation.

Although considered quintessentially American, patent medicines actually originated in England. The recipient of the first royal patent for a medicinal compound is unknown, but the second was granted to Richard Stoughton's Elixir in 1712. By the mid-eighteenth century an incomplete list included 202 "proprietary" medicines-those protected by patent or registration. Relatively few of the ready-made medicines were actually patented-which required disclosure of their ingredients-but rather had their brand name registered. Nevertheless, the term patent medicine has become a generic term for all self-prescribed nostrums and cure-alls.

Shipments of patent medicines were halted by the Revolutionary War, and American entrepreneurs took the opportunity to meet the demand. Post-war nationalism and cheaper prices of the non-imported medicines helped American vendors maintain their lead over English suppliers (Munsey 1970).

Snake Oil and Holy Water Richard Dawkins,

In 1996 the Vatican, fresh from its magnanimous reconciliation with Galileo, a mere 350 years after his death, publicly announced that evolution had been promoted from tentative hypothesis to accepted theory of science. This is less dramatic than many American Protestants think it is, for the Roman Catholic Church has never been noted for biblical literalism--on the contrary, it has treated the Bible with suspicion, as something close to a subversive document, needing to be carefully filtered through priests rather than given raw to congregations. The pope's recent message on evolution has, nevertheless, been hailed as another example of late-20th-century convergence between science and religion. Responses to the pope's message exhibited liberal intellectuals at their worst, falling over themselves in their eagerness to concede to religion its own magisterium, of equal importance to that of science, but not opposed to it. Such agnostic conciliation is, once again, easy to mistake for a genuine meeting of minds.

In any case, the belief that religion and science occupy separate magisteria is dishonest. It founders on the undeniable fact that religions still make claims about the world that on analysis turn out to be scientific claims. Moreover, religious apologists try to have it both ways. When talking to intellectuals, they carefully keep off science's turf, safe inside the separate and invulnerable religious magisterium. But when talking to a nonintellectual mass audience, they make wanton use of miracle stories--which are blatant intrusions into scientific territory. The Virgin Birth, the Resurrection, the raising of Lazarus, even the Old Testament miracles, all are freely used for religious propaganda, and they are very effective with an audience of unsophisticates and children. Every one of these miracles amounts to a violation of the normal running of the natural world. Theologians should make a choice. You can claim your own magisterium, separate from science's but still deserving of respect. But in that case, you must renounce miracles. Or you can keep your Lourdes and your miracles and enjoy their huge recruiting potential among the uneducated. But then you must kiss goodbye to separate magisteria and your high-minded aspiration to converge with science. The desire to have it both ways is not surprising in a good propagandist. What is surprising is the readiness of liberal agnostics to go along with it, and their readiness to write off, as simplistic, insensitive extremists, those of us with the temerity to blow the whistle. The whistle-blowers are accused of imagining an outdated caricature of religion in which God has a long white beard and lives in a physical place called heaven. Nowadays, we are told, religion has moved on. Heaven is not a physical place, and God does not have a physical body where a beard might sit. Well, yes, admirable: separate magisteria, real convergence. But the doctrine of the Assumption was defined as an Article of Faith by Pope Pius XII as recently as November 1, 1950, and is binding on all Catholics. It clearly states that the body of Mary was taken into heaven and reunited with her soul. What can that mean, if not that heaven is a physical place containing bodies? To repeat, this is not a quaint and obsolete tradition with just a purely symbolic significance. It has officially, and recently, been declared to be literally true. Convergence? Only when it suits. To an honest judge, the alleged marriage between religion and science is a shallow, empty, spin-doctored sham.

Homeopathy and other popular therapies demonstrate ancient and universal principles of magical thinking, which some recent research suggests are fundamental to human cognition, even rooted in neurobiology.

Paracelsus rejected Gnostic traditions, but kept much of the Hermetic, neoplatonic, and Pythagorean philosophies from Ficino and Pico della Mirandola; however, Hermetical science had so much Aristotelian theory that his rejection of Gnosticism was practically meaningless. In particular, Paracelsus rejected the magic theories of Agrippa and Flamel; Paracelsus did not think of himself as a magician and scorned those who did, though he was a practicing astrologer, as were most, if not all of the university-trained physicians working at this time in Europe. Astrology was a very important part of Paracelsus' medicine. In his Archidoxes of Magic Paracelsus devoted several sections to astrological talismans for curing disease, providing talismans for various maladies as well as talismans for each sign of the Zodiac. He also invented an alphabet called the Alphabet of the Magi, for engraving angelic names upon talismans.

How old is Kabbalah?

The earliest documents which are generally acknowledged as being
Kabbalistic come from the 1st. Century C.E., but there is a suspicion
that the Biblical phenomenon of prophecy may have been grounded in a
much older oral tradition which was a precursor to the earliest
recognisable forms of Kabbalah. Some believe the tradition goes back
as far as Melchizedek. There are moderately plausible arguments that
Pythagoras received his learning from Hebrew sources. There is a
substantial literature of Jewish mysticism dating from the period
100AD - 1000AD which is not strictly Kabbalistic in the modern sense,
but which was available as source material to medieval Kabbalists.

On the basis of a detailed examination of texts, and a study of the
development of a specialist vocabulary and a distinct body of ideas,
Scholem has concluded that the origins of Kabbalah can be traced to
12th. century Provence. The origin of the word "Kabbalah" as a label
for a tradition which is definitely recognisable as Kabbalah is
attributed to Isaac the Blind (c. 1160-1236 C.E.), who is also
credited with being the originator of the idea of sephirothic
emanation.

Prior to this (and after) a wide variety of terms were used for those
who studied the tradition: "masters of mystery", "men of belief",
"masters of knowledge", "those who know", "those who know grace",
"children of faith", "children of the king's palace", "those who know
wisdom", "those who reap the field", "those who have entered and
left".
+++

History of Kabbalah

The word Kabbalah, simply means "tradition". Its root is the Hebrew word for "receive". It implies a received tradition. There have been traditions handed down, orally and in writing, throughout the three thousand and more years of Jewish history. From its very inception Judaism had different paradigms of leadership that sometimes overlapped and sometimes conflicted. Moses gave way to Joshua, who was succeeded by judges, and then kings. The priesthood was initially the repository of the religious tradition, but it sometimes failed in its role and either judges or prophets stepped in to fill the gap. There have been alternative, mystical traditions, too, from the period of the prophets--those spiritual outsiders who railed against the betrayals of the established religious structures. Mysticism has always been an essential part of the Jewish spiritual tradition. Some even suggest the mystical goes back to Abraham. A fascinating Midrash suggests that the Wisdom of the East originated from the teachings he passed on to the sons of his concubines.

Academic convention assumes that the technical term "Kabbalah" applies exclusively to a body of esoteric literature that emerged in Medieval Spain, and Provence in France, and went on flourishing from there. It is true that two thousand years ago the rabbis of the Talmud did not use this word but rather spoke about "nistar", the secret world of Torah that paralleled the "niglah", the revealed. But I believe the roots of what is called Kabbalah go back to the very beginning of the Jewish tradition.

As the Christian world based itself on Greek philosophy, and its power and influence spread, in general, within Judaism too, alternative approaches were sidelined. Sefer Yetzirah, the first book that defines mainstream Kabbalah, appears somewhere between the third and the fourth century. It is referred to in the Midrash . However, many academics argue that the text we have today is another one of later provenance.

In Sefer Yetzirah we find the first clear statement of an alternative way of looking at the world, life, and God, based on the Sephirot and the Hebrew alphabet. (Incidentally, the symbolic power of letters and numerology, was something Pythagoras had already written about.)

New writings--Sefer Raziel ("The Book of Raziel, the Angel"), Sefer Bahir ("The Book of Enlightenment"), and then the Zohar ("Bright Light")--emerged into the public domain. The Zohar was discovered, some say written, by Moses De Leon (about 1290 in Spain) but attributed to Shimon Bar Yochai. It is a multi-volumed collection of monologues and commentaries on the Torah that creates a totally different atmosphere from the rational commentators. It became the most widespread and accepted book of the Kabbalah.

As life under Christian monarchs in Spain became unstable and God seemed to retreat from the Jews, a non-rational world became both an escape and a comfort. Mystics such as Abraham Abulafia (born in Spain at the end of the thirteenth century) preached messianism and a new world order. They courted danger. (Abulafia was imprisoned by the Pope, and Shabbetai Zvi, much later, in Constantinople, was imprisoned by the Sultan.)

The expulsion of Jews from Spain caused great chaos and upheaval. But the establishment of a "city of refuge" in Safed in Galilee created a dynamic centre for a new wave of Kabbalistic innovation. Moses Cordovero, Isaac Luria, and Chayim Vital, all expanded the ideas found in Sefer Yetzira and Sefer Bahir, and combined them with ecstatic mystical practices and experiences. They popularized Kabbalah as a way of reaching God and living a fuller, more spiritual life.

The fact that they did, indeed, encourage a wider, non-academic audience to join them, and the fact that they elevated experience over scholarship, drew down opposition from the mainstream rabbinate. To make matters more confusing, many of the other marginal, magical, superstitious, esoteric and fringe movements of Jewish life pinned their colours to Kabbalah. The excesses of some of these movements led to a campaign to uproot and expunge mystical writings from Jewish life, particularly in Europe after the rationalism of the seventeenth century began to spread.

Shabbetai Zvi was a highly charismatic mystic who was born in Turkey in the seventeenth century. He succeeded in convincing most of the Jewish world that he was the Messiah. But when he got to Istanbul he converted to Islam and the whole movement collapsed. The Shabbetai Zvi debacle discredited Kabbalah. Indeed, Moshe Hagiz, from Jerusalem, went on a voyage around the Jewish world campaigning against the Sabbatean heresy, and as a result Kabbalists were all but driven underground. The Enlightenment also led to the marginalization of Kabbalah.

It was Chassidism, the eighteenth century charismatic revolution in Eastern European Jewry that popularized, and to some extent legitimized, the Kabbalistic approach to life and brought it back towards the mainstream. The early Chassidic masters drew inspiration, both in prayer and ideology, from Lurianic Kabbalah. Initially the free, experimental mood of Safed mysticism suffused the Chassidic masters of the second and third generation. But then, like many revolutionary movements, it lost its anti-establishment and innovative character and became part of the structured religious life of Orthodoxy. It lost its creative identity.

Harvey Hames notes that Elijah del Medigo (1440–ca. 1490), the teacher of Pico della Mirandola and a much sought-after translator of Averroes's works into Latin, composed his theological treatise only after he returned to his native Crete where he could more freely critique the rise of Christian kabbalah and the blending of Neoplatonism, Christianity, and magic he encountered in Florence.

Avicenna (al-Husain, b. Abdallah Ibn Sina, d. 1037), Avicennae canonis libri
In Latin
Translated from Arabic by Gerard of Cremona
Fourteenth century

The papal library also acquired copies of standard medical works used in the Middle Ages and Renaissance. Portions of the twelfth-century Latin translation of Avicenna's medical encyclopedia were used as textbooks in universities, and the work as a whole served as a medical reference tool. In this copy, numerous miniatures vividly depict patient problems with which the medical practitioner was likely to be confronted. Here a patient has hemorrhoids.

Islamic Medical Manuscripts, Magical/Astrological Medicine 5

The first item (fols. 1b-38a) contains the anatomical sections from the Qānūn of Avicenna (MS A 27, item 1); the second item (fols. 38b-39b) is Kashf ba‘d al-lughah min al-Qānūn wa-ghayrihi, an anonymous commentary on terms in the Qanun (MS A 27, item 2); and the third item on fol. 41a is a short anonymous essay on oxymel (MS A 27, item 3). The fourth item (fols. 41b-75a) is an anonymous treatise on prognostics (MS A 27, item 4), and the final item (fols. 75b-76b) contains magical procedures and invocations useful for illness here catalogued. Folios 40a is blank, and fol. 40b is blank except for an owner's note. Fol. 77 is a very different, more recent, paper, and is blank except for some owner's annotations.

The Exodus of a Medical School -- Nevins 123 (12): 963 -- Annals ...

During the fifteenth century, Padua became a haven for hundreds of Jewish medical students from all over Europe, but the first did not graduate until 1409 [4]. As the Renaissance progressed, the social climate became more hospitable and, particularly in Italy, Jewish physicians found it easier to integrate into the general community. These physicians were still excluded from most other occupations and from public office, and their success in medicine was an example of taking advantage of opportunity despite societal intolerance.

Kabbalistic Physiology: Isaac the Blind, Nahmanides, and Moses de Leon on Menstruation

---

Sharon Koren a1 a1 Hebrew Union College, New York, New York

Science and faith were inextricably intertwined in the Latin Middle Ages. Clerics would attend to both spiritual and physical needs because the need to care for the body coincided with the need to care for the soul. Until the rise of universities in the twelfth century, monasteries were the centers of scientific knowledge. And, even after the professionalization of medicine in the thirteenth century, Christian physicians continued to look to the Bible, in addition to their license, as the source of their authority. Indeed, many Christian physicians who received medical degrees went on to pursue higher degrees in theology. It is therefore not surprising that several Christian theologians used medical theories in the service of theology.

Jews and Healing in the Middle Ages

Indeed, practical texts often show the interaction between members of Jewish and Christian communities in actual practice. For example, Christian and Jewish women appear to have shared similar knowledge and have used the same techniques regarding childbirth. It has been shown by historians that despite the differences with regard the use of plants (used according local availability), the techniques found in Western Hebrew texts were not different to those included in Latin texts (and Arabic). The similitude in remedies and techniques might be explained if we consider that, while the theory and notions in physiology are in general textually transmitted, techniques and recipes are more likely part of actual experience and belong largely to the province of orallity. In fact, there is evidence – for example - that Jewish midwives attended Christian women in labour, and vice versa, despite the prohibitions of the Church. This kind of interaction was a sure source of exchange of healing knowledge, and it is in the origin of the common substratum that we often discover in magic formulae and other healing methods and procedures included in sources of different provenance. Jews integrated these common practices, but it seems that they maintained their religious and cultural identity through the resource to Hebrew and to their own cultural background, as show the continuous allusions to practical Kabbalah in magic healing.

Our Sephardic Medical Roots

When I first began to study my Jewish medical roots, I presumed naively that I could start in Eastern Europe where my grandparents had come from and then work backwards. To my surprise, I soon learned that Jewish doctors were scarce in pre-Revolutionary Russia and that such medical care as existed most likely was delivered by a melange of healers, empirics, magicians, bath-house attendants and the like. If a 19th century Jewish mother proudly spoke of "my son the doctor", more likely she was bragging about a partially trained paramedic (feldsher), than a physician in the modern sense.

Except for purveyors of folk medicine, prior to the middle of the last century Jewish medicine substantially was a Sephardic enterprise, its practitioners either personally or spiritually descended from the erudite rabbi-philosopher-physicians who practiced in Medieval Spain and Portugal. Luminaries such as Judah Halevi, Maimonides and Nachmanides were among the first outflow of Jewish physicians from Spain during the 12th through 14th centuries and after the Expulsion of 1492, the trickle became a deluge. The emigres first went to Portugal and from there fanned out to Amsterdam, Hamburg, Italy, Poland, Greece, the Ottoman Empire, Goa and the Americas. Although their lives were not uniformly comfortable, many Conversos resumed practicing their former religion in these less hostile lands. They were an intellectual elite, and adept at integrating into the new societies in which they found themselves. Many were professionally successful, but with very few exceptions they were not in the forefront of emerging new medical ideas being exponents of the prevailing Galenic old-school.

In this brief essay, I offer four points that I suspect are not widely known. The first is that the University of Padua in northern Italy was a particularly receptive locale where Sephardim joined with exiles from France and Germany to participate in the intellectual ferment of the Renaissance. In this melting pot, hundreds of students not only learned medicine, but partook of the new humanistic ideas of the day. As Professor David Ruderman has described, when they returned to their countries of origin, they served as a vanguard for the Jewish Enlightenment that would emerge in the 18th century. Some of these Paduan graduates were exponents of Maimonides' rationalistic approach to medicine, others were enamored with astrology or the magic of Kabbalah, while still others attempted to reconcile traditional Jewish teaching with secular ideas which were heady and seductive. Indeed, one famous Jewish physician, Toviah Cohen, warned that before tasting the new science, a Jew first should fill his belly with Torah.

A second point worth noting is how frequently Sephardic expatriate physicians were sought after by the politically powerful. Kings and Popes, nobles and commoners, all favored Jewish doctors. Even in the 16th and 17th centuries when Jewish fortunes were in eclipse, the Queens of France, Russia, England and Sweden were attended respectively by Drs. Elijah Montalto, Antonio Ribera Sanches, Rodrigo Lopes and Benedict de Castro. How can we explain the remarkable acceptance by Christian society of this generally despised remnant? It's unlikely that their appeal can be attributed merely to medical acumen or to superior ethical principles. More likely it was that the Jews were perceived by Christian Europe as having skills greater than those of their gentile competitors.

Astrology, Astral Magic, & The Quest for Good Health
© 2002 Lauran Fowks
The longing for health and vitality is timeless. Whether to cure an existing illness or to ensure one’s continued good health, a great variety of methods have been employed over the ages. This paper will explore those techniques used by health practitioners in Medieval and Renaissance Europe that drew on astrology and astral magic - the manipulation of astrological influences - for their healing power. Particular attention will be given to the underlying principles and use of talismans for health and healing

Medieval Ceremonial Magic - The Gnostic Society ...

The term "magic" is etymological derived from an ancient Indo-Aryan root, which we find both in Greek and Latin, consisting of the three letters M, A, and G, or mag, meaning greatness or the bringing about of greatness. We discussed that many attempts have been made to define magic in the past. Entire volumes of anthropological writings have grappled with parsing the distinctions between magic, religion and science. But in the Medieval Period of Europe, these distinctions were not very clearly drawn. For the purposes of our discussion (that is, understanding the tenuous transmission of the Gnostic Tradition over an immense span of history) we defined magic as a particular ecstatic spirituality which seeks the expansion of consciousness through various sacred means. In particular, we are concerned with Theurgy (from Greek: θεουργί α, meaning "divine-working"), which describes the practice of rituals, sometimes seen as magical in nature, performed with the intention of invoking the action of God (or other personified supernatural power), especially with the goal of uniting with the divine, achieving henosis, and perfecting oneself. This was contrasted with other magical practices of the period, many of which may be seen as vestiges of pagan folk magic, and concerned with such issues as fertility, healing and protection from sorcery. Medieval Christianity coopted or assimilated many of these practices and promoted its own forms of magical thinking through the transformation of pagan deities and heroes into saints, adoption of pagan festivals and holidays as holy feast days, claiming sacred sites for cathedrals and pilgrimages, the Cult of Relics, and assigning of certain mystical powers to the Holy Sacraments, especially the Eucharist. However, in the minds of the people of this period, these would not be viewed as magical practices in the same way we may view them today. Dr. Karen Louise Jolly writes in her introduction to Magic in the Middle Ages: A Preliminary Discussion:

"Within Medieval Christendom, magic was the opposite of religion, and therefore defined by those who were in a position to define Christianity: church leaders and religious authors. In that sense "medieval magic" is whatever practices church leaders condemned as not of God. These authorities usually associate magic with the devil, paganism, heresy, and witchcraft or sorcery...."

Sorcery or malific forms of magic, were often called witchcraft or necromancy by people of the Middle Ages, though both terms had somewhat different meanings in Late Antiquity and in modern anthropology. With pressures from reformist and anticlerical groups intensifying during the Late Medieval Period, the Church increasingly equated magic and witchcraft as the most extreme forms of heresy, and was a frequent charge against those whom it sought to eliminate. This view would of course intensify later during the Reformation and Counter Reformation Periods, among both Catholic and Protestant religious and civil authorities. However, despite this negative association with magic, other forms of esoteric practice such as alchemy and astrology were tolerated during this period and received royal and papal patronage.

The Renaissance and Christian Kabbalah

Kabbalah was a growing force in Judaism throughout the late medieval period and by the beginning of the Renaissance had gained general acceptance as the true Jewish theology, a standing it maintained (particularly in the Christian view) into the eighteenth century.¹⁸ Only in the last several decades of the twentieth century, however, have historians begun to recognize the importance of Kabbalah in both the history of religion and in the specific framework of Renaissance thought. Frances Yates, one of this century's preeminent historians of the period, emphasized "the tremendous ramifications of this subject, how little it has been explored, and how fundamental it is for any deep understanding of the Renaissance." She continued,

Cabala reaches up into religious spheres and cannot be avoided in approaches to the history of religion. The enthusiasm for Cabala and for its revelations of new spiritual depths in the Scriptures was one of the factors leading towards Reformation. . . . The Cabalist influence on Renaissance Neoplatonism . . . tended to affect the movement in a more intensively religious direction, and more particularly in the direction of the idea of religious reform.¹⁹

Yates has delineated how understanding Kabbalah and its penetration into Christian culture are essential not only for comprehending Renaissance thought but also for studies of the Elizabethan age, Reformation religious ideals, the seventeenth-century Rosicrucian Enlightenment, and much that followed, including the emergence of occult Masonic societies in mid-seventeenth century England.

From its early medieval development in Spain, Jewish Kabbalah existed in close proximity to the Christian world and inevitably aroused notice among gentile observers.²⁰ During the fourteenth and fifteenth centuries, Kabbalists increasingly established a presence in several areas of Europe outside Spain, the most consequential of these perhaps being Italy, where Kabbalah soon touched the vanguard of Renaissance life. Then in 1492 came one of the great tragedies in Jewish history: the violent expulsion of Jews from newly unified Christian Spain. Forcibly expelled from their homeland, they fled to Italy, France, Germany, to the England of Henry the VII, and to Turkey, Palestine, and North Africa. With them went Kabbalah.

European culture in the fifteenth century had been animated by explorations, sciences, and bold visions reborn. Man stepped out from the shadow of the Creator and found himself master of worlds, capable of knowing God's handiwork. He discovered himself: the jewel of creation, the measure of all things. Perhaps no place was ablaze in this creative fire more than the Florentine courts of Cosimo and Lorenzo de' Medici. Cosimo had assiduously collected the rediscovered legacies of Greek and Alexandrian antiquity (an effort facilitated by the exodus west after the Turkish conquest of the Byzantine Empire in 1453). But most important, in 1460 he acquired and had brought to Florence the Corpus Hermeticum, a collection of fourteen ancient religious treatises on God and man. Authoritatively mentioned in the early Christian patristic writings of St. Augustine and Lactantius, these "lost" texts were thought to have been authored in antiquity by one Hermes Trismegistos ("Thrice Great Hermes"), an ancient Egyptian prophet older than Moses, a knower of God's ancient but forgotten truths, and a seer who foretold the coming of Christ.²¹ Though eventually dated to the Gnostic milieu of the second century C.E., sixteenth-century scholars believed that Hermes Trismegistos and the Hermetica were an occult source that nurtured true religion and philosophy from Moses to the Greek philosophers of late antiquity.²²

The influence of the Corpus Hermeticum was remarkable, its diffusion among intellectuals immense; it epitomized the Renaissance world view, a reborn prisca theologia, "the pristine font of ancient and Divine illumination." In a variety of ways, Renaissance thought was radically transformed by the Hermetic doctrine that man was infused with God's light and divinity: "You are light and life, like God the Father of whom Man was born. If therefore you learn to know yourself . . . you will return to life."²³ Man was a divine, creative, immortal essence in union with a body, and man reborn "will be god, the son of God, all in all, composed of all Powers."²⁴

Kabbalah made a dramatic entry on the Renaissance stage at almost precisely the same time the rediscovered Hermetic writings were gaining wide dissemination in the elite circles of Europe. The initial impetus for study of Kabbalah as a Christian science and for its integration with Hermeticism came from Florentine prodigy Pico della Mirandola (1463-94). Pico's philosophical education was initiated under the Hermetic and Platonic influence of the Medici Academy and court, of which he became an intellectual luminary. About age twenty he began his studies of Kabbalah, a pursuit furthered by Jewish Kabbalists who assisted him in translating a considerable portion of Kabbalistic literature into Latin and then aided his understanding of their occult interpretations.²⁵ In 1486 Pico penned the "Oration on the Dignity of Man"--one of the seminal documents of the Renaissance--as an introduction to the famous 900 theses which he intended to debate publicly in Rome that year. More than a hundred of these 900 theses came from Kabbalah or Pico's own Kabbalistic research.²⁶ "The marrying together of Hermetism and Cabalism, of which Pico was the instigator and founder," notes Yates, "was to have momentous results, and the subsequent Hermetic-Cabalist tradition, ultimately stemming from him, was of most far-reaching importance."²⁷

Hermeticism found a perfect companion in Kabbalah. Sympathies that can be drawn between the two occult sciences, both supposed ancient and divine, are remarkable, and it is easy to see how they would have impressed themselves upon sixteenth-century philosophers: Kabbalah originated with God's word to Adam and the ancient Jewish prophets after him; Hermeticism was the sacred knowledge of the ancient Egyptian Gnosis, the legacy of a thrice-great prophet, transmitted to the greatest pagan philosophers, and foretelling the coming of the divine Word (Logos). Both placed considerable interest in a mystical reinterpretation of the Creation; the Hermetic text Pimander, often called "the Egyptian Genesis," complimented the new vision gained from a Kabbalistic revisioning of the Hebrew Genesis.²⁸ Each taught the great "Art" of Divine knowledge based on the tenet that man is able to discover the Divine, which he reflects within himself through direct perceptive experience. And both offered paths to God's hidden throne, the divine intellect, where humankind might find revealed the secrets of heaven and earth. Element after element of Renaissance thought and culture is linked to the force of a new religious philosophy born of these two Gnostic traditions intermingling in the cauldron of Western culture's rebirth. Indeed, Yates suggests that the true origins of the Renaissance genius may be dated from two events: the arrival of the Corpus Hermeticum in Florence and the infusion of Kabbalism into Christian Europe by the Spanish expulsion of the Jews.²⁹

Christian Kabbalah advanced an innovative reinterpretation of the Jewish tradition. For Pico and many influential Christian Kabbalists after him this ancient Gnostic tradition not only was compatible with Christianity but offered proofs of its truth. Many early Christian Kabbalists were, like Pico, not only scholars but Christian priests investigating remnants of a holy and ancient priesthood, rife with power and wisdom endowed by God. Their cooptation of the tradition was of course disavowed by most Jewish Kabbalists--though some aided and encouraged the development and a few converted to Christianity. But to the Christian scholars and divines who embraced it, Kabbalah was

a Hebrew-Christian source of ancient wisdom which corroborated not only Christianity, but the Gentile ancient wisdoms which [they] admired, particularly the writings of "Hermes Trismegistus". Thus Christian Cabala is really a key-stone in the edifice of Renaissance thought on its "occult" side through which it has most important connections with the history of religion in the period.³⁰

This was not just a speculative philosophy, but a new (though cautious and often occult) religious movement which radically reinterpreted normative Christianity. In some fashion it touched every important creative figure of the Renaissance. To an age seeking reformation and renewal, there had come forgotten books by prophets of old--pagan and Hebrew--who foresaw the coming of the Divine creative Logos, who knew the secret mysteries given to Adam, who taught that man might not only know God, but in so knowing, discover a startling truth about himself. These ideas reverberated in the creative religious imagination of the Western world for several centuries, perhaps even touching--though illusively and attenuated by time--the American religious frontier of the 1820s.

Jewish Magic Bibliography

The following bibliography is meant as an aid to the student of Jewish magic, ranging from biblical to modern times. It was compiled with the assistance of a Mary Gates Undergraduate Research Grant at the University of Washington and under the guidance of Prof. Scott Noegel. It is organized both chronologically and topically, with many entries repeated for ease of use. It is my hope that this bibliography will become a great asset in the further development of the study of Jewish magic. While this list is far from exhaustive, I have attempted to present the most up to date and relevant material for research in Jewish magic. Accordingly, I hope to continue to update this bibliography in order to make it as current as possible.

Alex Jassen and Scott Noegel

University of Washington

SEE:

Kabbalistic Kommunism

Passover Song

My Favorite Muslim

For a Ruthless Criticism of Everything Existing



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