![The truth is that the mathematical sciences are growing in complete security and harmony. The ideas of Dedekind, Poincare, and Hilbert have been systematically developed with great success, without any conflict in the results. It is only from the philosophical point of view that objections have been raised. They bear on certain ways of reasoning peculiar to analysis and set theory. These modes of reasoning were first systematically applied in giving a rigorous form to the methods of the calculus. [According to them, ] the objects of a theory are viewed as elements of a totality such that one can reason as follows: For each property expressible using the notions of the theory, it is [an] objectively determinate [fact] whether there is or there is not an element of the totality which possesses this property. Similarly, it follows from this point of view that either all the elements of a set possess a given property, or there is at least one element which does not possess it. (Paul Bernays)](https://cdn.quotesdtb.com/img/quotes_images_webp/64/paul-bernays-analysis-bear-444564.webp)
The truth is that the mathematical sciences are growing in complete security and harmony. The ideas of Dedekind, Poincare, and Hilbert have been systematically developed with great success, without any conflict in the results. It is only from the philosophical point of view that objections have been raised. They bear on certain ways of reasoning peculiar to analysis and set theory. These modes of reasoning were first systematically applied in giving a rigorous form to the methods of the calculus. [According to them, ] the objects of a theory are viewed as elements of a totality such that one can reason as follows: For each property expressible using the notions of the theory, it is [an] objectively determinate [fact] whether there is or there is not an element of the totality which possesses this property. Similarly, it follows from this point of view that either all the elements of a set possess a given property, or there is at least one element which does not possess it.
Paul BernaysRelated topics
analysis bear calculus certain complete conflict determinate element fact form given giving great growing harmony least peculiar point property reason reasoning security set success theory totality truth using view elements hilbert sciences waysRelated quotes
Physical science comes nearest to that complete system of exact knowledge which all sciences have before them as an ideal. Some fall far short of it. The physicist who inveighs against the lack of coherence and the indefiniteness of theological theories, will probably speak not much less harshly of the theories of biology and psychology. They also fail to come up to his standard of methodology. On the other side of him stands an even superior being - the pure mathematician - who has no high opinion of the methods of deduction used in physics, and does not hide his disapproval of the laxity of what is accepted as proof in physical science. And yet somehow knowledge grows in all these branches. Wherever a way opens we are impelled to seek by the only methods that can be devised for that particular opening, not over-rating the security of our finding, but conscious that in this activity of mind we are obeying the light that is in our nature.
Arthur Eddington
The global reach of Smith's moral and political reasoning is quite a distinctive feature of his thought, but it is strongly supplemented by his belief that all human beings are born with similar potential and, most importantly for policymaking, that the inequalities in the world reflect socially generated, rather than natural, disparities.
There is a vision here that has a remarkably current ring. The continuing global relevance of Smith's ideas is quite astonishing, and it is a tribute to the power of his mind that this global vision is so forcefully presented by someone who, a quarter of a millennium ago, lived most of his life in considerable seclusion in a tiny coastal Scottish town. Smith's analyses and explorations are of critical importance for any society in the world in which issues of morals, politics and economics receive attention. The Theory of Moral Sentiments is a global manifesto of profound significance to the interdependent world in which we live.
Amartya Sen
The Definition in the Elements, according to Clavius, is this: Magnitudes are said to be in the same Reason [ratio], a first to a second, and a third to a fourth, when the Equimultiples of the first and third according to any Multiplication whatsoever are both together either short of, equal to, or exceed the Equimultiples of the second and fourth, if those be taken, which answer one another.... Such is Euclid's Definition of Proportions; that scare-Crow at which the over modest or slothful Dispositions of Men are generally affrighted: they are modest, who distrust their own Ability, as soon as a Difficulty appears, but they are slothful that will not give some Attention for the learning of Sciences; as if while we are involved in Obscurity we could clear ourselves without Labour. Both of 300 which Sorts of Persons are to be admonished, that the former be not discouraged, nor the latter refuse a little Care and Diligence when a Thing requires some Study.
Isaac Barrow
In the winter of 1841 Schelling gave his famous series of lectures at the University of Berlin before a distinguished audience including Kierkegaard, Burckhardt, Engels, Bakunin. Schelling set out to overthrow Hegel, whose vast rationalist system including the identification of abstract truth with reality and the bringing of all history into an "absolute whole,” held immense and dominant popularity in the Europe of the middle of the nineteenth century. – 1844 Kierkegaard published Philosophical Fragments, and two years later he wrote the declaration of independence for existentialism, Concluding Unscientific Postscript to Philosophical Fragments. Also in 1844 there appeared the second edition of Schopenhauer's The world as will and idea,.. central emphasis "will” along with "idea” -- two related works by Marx 1844-45 – "attacked abstract truth' as "ideology” "using Hegel as his whipping boy” "men and groups bring truth into being” "money economy turns people into things”.
Rollo May
Just as divorce according to the Saviour's word was not permitted from the beginning, but on account of the hardness of our heart was a concession of Moses to the human race, so too the eating of flesh was unknown until the deluge. But after the deluge, like the quails given in the desert to the murmuring people, the poison of flesh-meat was offered to our teeth. ... At the beginning of the human race we neither ate flesh, nor gave bills of divorce, nor suffered circumcision for a sign. Thus we reached the deluge. But after the deluge, together with the giving of the law which no one could fulfil, flesh was given for food, and divorce was allowed to hard-hearted men, and the knife of circumcision was applied, as though the hand of God had fashioned us with something superfluous. But once Christ has come in the end of time, and Omega passed into Alpha and turned the end into the beginning, we are no longer allowed divorce, nor are we circumcised, nor do we eat flesh.
Jerome
