Lecture at Yale University, "Chemical Achievement and Hope for the Future." (October 1947) Published in Science in Progress. Sixth Series. Ed. George A. Baitsell. 100-21, (1949). 
1940s-1960s 
Context: Science cannot be stopped. Man will gather knowledge no matter what the consequences – and we cannot predict what they will be. Science will go on — whether we are pessimistic, or are optimistic, as I am. I know that great, interesting, and valuable discoveries can be made and will be made… But I know also that still more interesting discoveries will be made that I have not the imagination to describe — and I am awaiting them, full of curiosity and enthusiasm.
                                    
“While it is true that science cannot decide questions of value, that is because they cannot be intellectually decided at all, and lie outside the realm of truth and falsehood. Whatever knowledge is attainable, must be attained by scientific methods; and what science cannot discover, mankind cannot know.”
            Religion and Science (1935), Ch. IX: Science of Ethics. 
1930s 
Variant: "What science cannot tell us, mankind cannot know." (Attributed to Russell in Ted Peters' Cosmos As Creation: Theology and Science in Consonance [1989], p. 14, with a note that it was "told [to] a BBC audience [earlier this century]").
        
Help us to complete the source, original and additional information
Bertrand Russell 562
logician, one of the first analytic philosophers and politi… 1872–1970Related quotes
“One cannot attain divine knowledge till one gets rid of pride.”
                                        
                                        Source: The Gospel of Sri Ramakrishna (1942), p. 874 
Context: One cannot attain divine knowledge till one gets rid of pride. Water does not stay on the top of a mound; but into low land it flows in torrents from all sides.
                                    
                                        
                                        Two scientific activities are equally valid if they achieve results that are true. Now, how do you decide which activity is more valuable? The question of value is the basic question that the scientific administrator asks so that decisions can be made about funding priorities. 
 Interview http://www.ornl.gov/info/ornlreview/rev28-1/text/wbgbar.htm by Bill Cabage and Carolyn Krause for the ORNL Review (April 1995).
                                    
Interview http://www.ornl.gov/info/ornlreview/rev28-1/text/wbgbar.htm by Bill Cabage and Carolyn Krause for the ORNL Review (April 1995).
“Philosophy cannot be taught; it is the application of the sciences to truth.”
Source: The Count of Monte Cristo
                                        
                                        Book XXIII, ¶ 7,as rendered in the epigraph to Ch. 3 of The Lathe of Heaven (1971) by Ursula K. Le Guin, based upon the 1891 translation by James Legge, Le Guin was subsequently informed that this was a very poor translation, as there were no lathes in China in the time of Zhuangzi. The full passage as translated by Legge reads:
He whose mind is thus grandly fixed emits a Heavenly light. In him who emits this heavenly light men see the (True) man. When a man has cultivated himself (up to this point), thenceforth he remains constant in himself. When he is thus constant in himself, (what is merely) the human element will leave him, but Heaven will help him. Those whom their human element has left we call the people of Heaven. Those whom Heaven helps we call the Sons of Heaven. Those who would by learning attain to this seek for what they cannot learn. Those who would by effort attain to this, attempt what effort can never effect. Those who aim by reasoning to reach it reason where reasoning has no place. To know to stop where they cannot arrive by means of knowledge is the highest attainment. Those who cannot do this will be destroyed on the lathe of Heaven.
                                    
A Preliminary Discourse on the Study of Natural Philosophy (1831)
“That which is and that which cannot be are both outside the realm of becoming.”
                                        
                                        Source: Gravity and Grace (1947), p. 154 (1972 edition) 
Context: We must wish either for that which actually exists or for that which cannot in any way exist — or, still better, for both. That which is and that which cannot be are both outside the realm of becoming.