--- title: "Meaning & Key Takeaway of: ... there are those who believe that…" description: "\"... there are those who believe that mathematics can sustain itself and grow without any further contact with anything outside itself, and those who be..." image: "https://sms4smile.com/quotes/i/4cc/mark-kac-58181_adequate_there-are-those-who-believe-that-mathema_2106721.jpg" --- # ... there are those who believe that mathematics can… ![“... there are those who believe that mathematics can sustain itself and grow without any further contact with anything outside itself, and those who believe that nature is still and always will be one of the main (if not the main) sources of mathematical inspiration. The first group is identified…” quote by Mark Kac](https://sms4smile.com/quotes/i/4cc/mark-kac-58181_adequate_there-are-those-who-believe-that-mathema_2106721.jpg) [Download](https://sms4smile.com/quotes/i/4cc/mark-kac-58181_adequate_there-are-those-who-believe-that-mathema_2106721.jpg) [Open image](https://sms4smile.com/quotes/i/4cc/mark-kac-58181_adequate_there-are-those-who-believe-that-mathema_2106721.jpg) > “... there are those who believe that mathematics can sustain itself and grow without any further contact with anything outside itself, and those who believe that nature is still and always will be one of the main (if not the main) sources of mathematical inspiration. The first group is identified as "pure mathematicians" (though "purist" would be more adequate) while the second is, with equal inadequacy, referred to as "applied".” [Mark Kac](https://sms4smile.com/quotes/author/mark-kac/) ## About This Quote This interpretation was drafted with AI assistance. It is one reading of the quote, not the author's own explanation. Mathematics can be pursued in isolation or drawn from nature; both approaches have merit and limits. **In simple terms:** Pure math is self‑contained; applied math draws on the world. Key Takeaway Balance theory with real‑world observation. Themes theory creativity interdisciplinarity Mood curious analytical Type philosophical inspirational When to use this quote - research - education - problem solving - model building Key Concepts epistemology methodology inspiration Questions to Reflect On - How does nature inspire your work? - When should you seek external data? A Different Perspective Pure abstraction may miss practical relevance. ★ ★ ★ ★ ★ No ratings yet [Inadequacy](https://sms4smile.com/quotes/about/inadequacy/) [Inspiration](https://sms4smile.com/quotes/about/inspiration/) [Mathematician](https://sms4smile.com/quotes/about/mathematician/) [Mathematicians](https://sms4smile.com/quotes/about/mathematicians/) [Mathematics](https://sms4smile.com/quotes/about/mathematics/) ## Related quotes > [Pure mathematicians just love to try unsolved problems - they love a challenge.](https://sms4smile.com/quotes/andrew-wiles/pure-mathematicians-just-love-to-try-unsolved-problems-they-love-a/) [Andrew Wiles](https://sms4smile.com/quotes/author/andrew-wiles/) > [It's true we pure mathematicians are connected to a different world. But it is a very real world nevertheless.](https://sms4smile.com/quotes/isadore-singer/it-s-true-we-pure-mathematicians-are-connected-to-a-different-world/) Isadore Singer > [The inner circle of pure mathematicians will respond to the book with delight.](https://sms4smile.com/quotes/philip-j-davis/the-inner-circle-of-pure-mathematicians-will-respond-to-the-book-with/) [Philip J. Davis](https://sms4smile.com/quotes/author/philip-j-davis/) > [On the basis of my historical experience, I fully believe that mathematics of the 25th century will be as different from that of today…](https://sms4smile.com/quotes/george-sarton/on-the-basis-of-my-historical-experience-i-fully-believe-that/) [George Sarton](https://sms4smile.com/quotes/author/george-sarton/) > [If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.](https://sms4smile.com/quotes/john-von-neumann/if-people-do-not-believe-that-mathematics-is-simple-it-is-only/) [John von Neumann](https://sms4smile.com/quotes/author/john-von-neumann/) > [I don't believe in mathematics.](https://sms4smile.com/quotes/albert-einstein/i-don-t-believe-in-mathematics/) [Albert Einstein](https://sms4smile.com/quotes/author/albert-einstein/) > [I firmly believe that mathematics does not exist outside of humans. It is something we, as a species, invent.](https://sms4smile.com/quotes/keith-devlin/i-firmly-believe-that-mathematics-does-not-exist-outside-of-humans-it/) [Keith Devlin](https://sms4smile.com/quotes/author/keith-devlin/) > [I believe only in the mathematics.](https://sms4smile.com/quotes/claudio-ranieri/i-believe-only-in-the-mathematics/) [Claudio Ranieri](https://sms4smile.com/quotes/author/claudio-ranieri/) > [I am obliged to interpolate some remarks on a very difficult subject: proof and its importance in mathematics. All physicists, and a good many…](https://sms4smile.com/quotes/g-h-hardy/i-am-obliged-to-interpolate-some-remarks-on-a-very-difficult-subject/) [G. H. Hardy](https://sms4smile.com/quotes/author/g-h-hardy/) > [Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians.](https://sms4smile.com/quotes/edsger-dijkstra/programming-is-one-of-the-most-difficult-branches-of-applied/) [Edsger Dijkstra](https://sms4smile.com/quotes/author/edsger-dijkstra/) > [“I spoke of the 'real' mathematics of Fermat and other great mathematicians, the mathematics which has permanent aesthetic value, as for example the best…](https://sms4smile.com/quotes/g-h-hardy-2/i-spoke-of-the-real-mathematics-of-fermat-and-other-great/) [G.H. Hardy](https://sms4smile.com/quotes/author/g-h-hardy-2/) > [Relations between pure and applied mathematicians are based on trust and understanding. Namely, pure mathematicians do not trust applied mathematicians, and applied mathematicians do…](https://sms4smile.com/quotes/albert-einstein/relations-between-pure-and-applied-mathematicians-are-based-on-trust/) [Albert Einstein](https://sms4smile.com/quotes/author/albert-einstein/) > [It has been a fortunate fact in the modern history of physical science that the scientist constructing a new theoretical system has nearly always…](https://sms4smile.com/quotes/r-b-braithwaite/it-has-been-a-fortunate-fact-in-the-modern-history-of-physical/) R. B. Braithwaite > [When we pray, we must begin by conceiving God in full and vigorous images, but we must go on to acknowledge the inadequacy of…](https://sms4smile.com/quotes/austin-farrer/when-we-pray-we-must-begin-by-conceiving-god-in-full-and-vigorous/) [Austin Farrer](https://sms4smile.com/quotes/author/austin-farrer/) > [No matter the style, the farther one goes the more obstacles increase, and the more distant appears the object it is desired to attain.…](https://sms4smile.com/quotes/jean-georges-noverre/no-matter-the-style-the-farther-one-goes-the-more-obstacles-increase/) [Jean-Georges Noverre](https://sms4smile.com/quotes/author/jean-georges-noverre/) ## More by Mark Kac > [To exist (in mathematics), said Henri Poincaré, is to be free from contradiction. But mere existence does not guarantee survival. To survive in mathematics…](https://sms4smile.com/quotes/mark-kac/to-exist-in-mathematics-said-henri-poincar-is-to-be-free-from/) [Mark Kac](https://sms4smile.com/quotes/author/mark-kac/) > [There are surely worse things than being wrong, and being dull and pedantic are surely among them.](https://sms4smile.com/quotes/mark-kac/there-are-surely-worse-things-than-being-wrong-and-being-dull-and/) [Mark Kac](https://sms4smile.com/quotes/author/mark-kac/) > [In recent years we have become much more preoccupied with streamlining and organizing our subject than with maintaining its overall vitality. 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