Jews

"Will you really sweep away the righteous with the wicked? Suppose there are 50 righteous men within the city. Will you, then, sweep them away and not pardon the place for the sake of the 50 righteous who are inside it? It is unthinkable that you would act in this manner by putting the righteous man to death with the wicked one so that the outcome for the righteous man and the wicked is the same! It is unthinkable of you. Will the Judge of all the earth not do what is right?"

"Wilt Thou indeed sweep away the righteous with the wicked? Peradventure there are fifty righteous within the city; wilt Thou indeed sweep away and not forgive the place for the fifty righteous that are therein? That be far from Thee to do after this manner, to slay the righteous with the wicked, that so the righteous should be as the wicked; that be far from Thee; shall not the Judge of all the earth do justly? … Behold now, I have taken upon me to speak unto the LORD, who am but dust and ashes. Peradventure there shall lack five of the fifty righteous; wilt Thou destroy all the city for lack of five? … Oh, let not the LORD be angry, and I will speak yet but this once. Peradventure ten shall be found there?"

"I have lifted up mine hand unto the LORD, the highest God, the possessor of heaven and earth, That I will not take from a thread even to a shoe latchet, and that I will not take anything that is thine, lest thou shouldest say, I have made Abram rich…"

"Let there be no strife, I pray thee, between me and thee, and between my herdmen and thy herdmen; for we be brethren. Is not the whole land before thee? separate thyself, I pray thee, from me: if thou wilt take the left hand, then I will go to the right; or if thou depart to the right hand, then I will go to the left."

"People know what they love … Even those whose lives are floundering. If they're directionless, it's not because they lack knowledge of what they want. It's because they lack the courage to acknowledge that they want it."

"Capitalism is the greatest benefactor man has ever had. It is time for the thinking men and women of every nation to recognize that fact and to fully embrace the system of the mind and of individual rights. Men and women of all countries unite - in your support of capitalism. You have a world of joyous achievement to win."

"It cost Andrei Dmitrievich 10 months of complete isolation and two hunger strikes over two months, which had a terrible effect on his health. The effects are still felt to this day."

"Why was Cantor so vehemently opposed to infinitesimals? In his valuable essay, "The Metaphysics of the Calculus," Abraham Robinson suggests that Cantor already had enough problems trying to defend transfinite numbers. It seems likely that, consciously or otherwise, Cantor deemed it politically wise to go along with orthodox mathematicians on the question of infinitesimals. Cantor's stance might be compared to that of a pro-marijuana Congressional candidate who advocates harsh penalties for the sale or use of heroin."

"If we have only to classify a finite number of objects, it is easy to preserve these classifications without change. If the number of objects is indefinite, ...[i.e.,] if we are constantly liable to find new and unforeseen objects springing up, it may happen that the appearance of a new object will oblige us to modify the classification, and it is thus that we are exposed to antinomies. There is no actual infinity. The Cantorians forgot this, and so fell into contradiction. It is true that Cantorism has been useful, but that was when it was applied to a real problem, whose terms were clearly defined, and then it was possible to advance without danger. Like the Cantorians, the logicians have forgotten the fact, and they have met with the same difficulties. ...[B]elief in an actual infinity is essential in the Russellian logic, and this... distinguishes it from the Hilbertian logic. Hilbert takes the... view of extension... to avoid the Cantorian antimonies. Russell takes the... view of comprehension... to regard the infinite as actual. And we have not only infinite classes; when we pass from the genus to the species... the number of conditions is still infinite, for they generally express that the object... is in... relation with all the objects of an infinite class. But all this is ."

"In 1874 the German mathematician Georg Cantor made the startling discovery that there are more irrational numbers than rational ones, and more transcendental numbers than algebraic ones. In other words, rather than being oddities, most real numbers are irrational; and among irrational numbers, most are transcendental."

"No one shall expel us from the Paradise that Cantor has created."

"[T]here exist no other sets than finite and denumerably infinite sets and continua... [I]n mathematics we can create only finite sequences, further by means of... 'and so on' the order type ω, but only consisting of equal elements... but no other sets. Cantor and his disciples... think they have knowledge of all sorts of further sets; their fundamental principle... comes to about the same as the axiomaticians. ...[T]his principle is unjustified and... we assert that the several paradoxes of the 'Mengenlehre'... have no right to exist... [I]t would have been the duty of Cantorians, immediately to reject a notion which gives rise to contradictions, because it is... not built... mathematically."

"If there is some determinate succession of defined whole real numbers, among which there exists no greatest, on the basis of this second principle of generation a new number is obtained which is regarded as the limit of those numbers, i.e. is defined as the next greater number than all of them."

"The essence of mathematics lies entirely in its freedom."

"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."

"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."

"What I declare and believe to have demonstrated in this work as well as in earlier papers is that following the finite there is a transfinite (transfinitum)--which might also be called supra-finite (suprafinitum), that is, there is an unlimited ascending ladder of modes, which in its nature is not finite but infinite, but which can be determined as can the finite by determinate, well-defined and distinguishable numbers."

"Infinity, in its first form (the improper-infinite) presents itself as a variable finite [veranderliches Endliches]; in the other form (which I call the proper infinite [Eigentlich-unendliche]) it appears as a thoroughly determinate [bestimmtes] infinite."

"As for the mathematical infinite, to the extent that it has found a justified application in science and contributed to its usefulness, it seems to me that it has hitherto appeared principally in the role of a variable quantity, which either grows beyond all bounds or diminishes to any desired minuteness, but always remains finite. I call this the improper infinite [das Uneigentlich-unendliche]."

"I entertain no doubts as to the truths of the transfinites, which I recognized with God’s help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science."

"The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly dissimilar to, and I might even say in principle the same as, my method described above of introducing transfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite."

"What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending ladder of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers."

"My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things."

"I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers."

"I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. I was de-Bourbakized, stopped believing in sets, and was expelled from the Cantorian paradise."

"After being relegated to an obscure mid-tier university, blocked from leading journals and openly mocked by his peers, including his former mentor, the late 19th century German mathematician found refuge for his groundbreaking work on infinities in, of all places, the Roman Catholic Church … Catholic theologians welcomed Cantor's ideas, which provided a workable way of understanding mathematical infinities, as evidence that humans could grasp the infinite and could also, therefore, have a greater understanding of God, himself infinite. What a welcome relief this must have been to the chronically depressed Cantor! As John D. Barrow writes in The Infinite Book: A Short Guide to the Boundless, Timeless and Endless, Cantor "started to tell his friends that he had not been the inventor of the ideas about infinity that he had published. He was merely a mouthpiece, inspired by God to communicate parts of the mind of God to everyone else.""

"In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi."

"This view [of the infinite], which I consider to be the sole correct one, is held by only a few. While possibly I am the very first in history to take this position so explicitly, with all of its logical consequences, I know for sure that I shall not be the last!"

"That from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they are to be considered in any form at all, must (in their contrast to the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research, but not of our arbitrariness or prejudices."

"The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type."

"The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds."

"A set is a Many that allows itself to be thought of as a One."

"I have never proceeded from any Genus supremum of the actual infinite. Quite the contrary, I have rigorously proved that there is absolutely no Genus supremum of the actual infinite. What surpasses all that is finite and transfinite is no Genus; it is the single, completely individual unity in which everything is included, which includes the Absolute, incomprehensible to the human understanding. This is the Actus Purissimus, which by many is called God. I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not — I do not say divisible — but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures."

"Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number."

"The totality of all alephs cannot be conceived as a determinate, well-defined, and also a finished set. This is the punctum saliens, and I venture to say that this completely certain theorem, provable rigorously from the definition of the totality of all alephs, is the most important and noblest theorem of set theory. One must only understand the expression "finished" correctly. I say of a set that it can be thought of as finished (and call such a set, if it contains infinitely many elements, "transfinite" or "suprafinite") if it is possible without contradiction (as can be done with finite sets) to think of all its elements as existing together, and to think of the set itself as a compounded thing for itself; or (in other words) if it is possible to imagine the set as actually existing with the totality of its elements."

"Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde."

"The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great."

"In order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus this domain is a definite, actually infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite."

"There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated."

"The old and oft-repeated proposition "Totum est majus sua parte" [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts "totum" and "pars". Unfortunately, however, this "axiom" is used innumerably often without any basis and in neglect of the necessary distinction between "reality" and "quantity", on the one hand, and "number" and "set", on the other, precisely in the sense in which it is generally false."

"Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand!"

"To govern is to choose."

"The Mutakallemim... apply the term non-existence only to absolute non-existence, and not to absence of properties. A property and the absence of that property are considered by them as two opposites, they treat, e.g., blindness and sight, death and life, in the same way as heat and cold. Therefore they say, without any qualification, non-existence does not require any agent, an agent is required when something is produced."

"As a corollary we have to add that the Guide cannot be called a theological work, for Maimonides does not know of theology as a discipline distinct from metaphysics. Nor is it a book of religion, for he expressly excludes religious, together with ethical topics from the subject matter of his work. Until we shall have rediscovered a body of terms which are flexible enough to fit Maimonides' thought, the safest course will be to limit the description of the Guide to the statement that it is a book devoted to the explanation of the secret teaching of the Bible."

"Maimonides states that the intention of his work is to explain the meaning of Biblical words of various kinds, as well as of Biblical parables. Such an explanation is necessary, because the external meaning of both lends itself to grave misunderstanding. Since the internal meaning, being hidden, is a secret, the explanation of each such word or parable is the revelation of a secret."

"From Moses to Moses there was none like Moses."

"Maimonides, the great Jewish theologian and historian, who at one time was almost deified by his countrymen and afterward treated as a heretic, remarks, that the more absurd and void of sense the Talmud seems the more sublime is the secret meaning. This learned man has successfully demonstrated that the Chaldean Magic, the science of Moses and other learned thaumaturgists was wholly based on an extensive knowledge of the various and now forgotten branches of natural science. Thoroughly acquainted with all the resources of the vegetable, animal, and mineral kingdoms, experts in occult chemistry and physics, psychologists as well as physiologists, why wonder that the graduates or adepts instructed in the mysterious sanctuaries of the temples, could perform wonders, which even in our days of enlightenment would appear supernatural? It is an insult to human nature to brand magic and the occult science with the name of imposture. To believe that for so many thousands of years, one-half of mankind practiced deception and fraud on the other half, is equivalent to saying that the human race was composed only of knaves and incurable idiots. Where is the country in which magic was not practised? At what age was it wholly forgotten? p. 19"

"Teach your tongue to say "I do not know" and you will progress."

"As regards circumcision, I think that one of its objects is to limit sexual intercourse, and to weaken the organ of generation as far as possible, and thus cause man to be moderate. Some people believe that circumcision is to remove a defect in man's formation; but every one can easily reply: How can products of nature be deficient so as to require external completion, especially as the use of the fore-skin to that organ is evident. This commandment has not been enjoined as a complement to a deficient physical creation, but as a means for perfecting man's moral shortcomings. The bodily injury caused to that organ is exactly that which is desired; it does not interrupt any vital function, nor does it destroy the power of generation. Circumcision simply counteracts excessive lust; for there is no doubt that circumcision weakens the power of sexual excitement, and sometimes lessens the natural enjoyment: the organ necessarily becomes weak when it loses blood and is deprived of its covering from the beginning. Our Sages (Beresh. Rabba, c. 80) say distinctly: It is hard for a woman, with whom an uncircumcised had sexual intercourse, to separate from him. This is, as I believe, the best reason for the commandment concerning circumcision."

"Those who wash their body and cleanse their garments whilst they remain dirty by bad actions and principles, are described by Solomon as "a generation that are pure in their own eyes, and yet are not washed from their filthiness; a generation, oh how lofty are their eyes!" &c. (Prov. xxx. 12-13). Consider well the principles which we mentioned... as the final causes of the Law; for there are many precepts, for which you will be unable to give a reason unless you possess a knowledge of these principles..."