--- title: "Meaning & Key Takeaway of: It is well known that geometry…" description: "\"“It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in..." image: "https://sms4smile.com/quotes/i/d49/bernhard-riemann-79557_axioms_it-is-well-known-that-geometry-presuppos_3108543.jpg" --- # It is well known that geometry presupposes not only the… ![“It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship…” quote by Bernhard Riemann](https://sms4smile.com/quotes/i/d49/bernhard-riemann-79557_axioms_it-is-well-known-that-geometry-presuppos_3108543.jpg) [Download](https://sms4smile.com/quotes/i/d49/bernhard-riemann-79557_axioms_it-is-well-known-that-geometry-presuppos_3108543.jpg) [Open image](https://sms4smile.com/quotes/i/d49/bernhard-riemann-79557_axioms_it-is-well-known-that-geometry-presuppos_3108543.jpg) > ““It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre , to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.”” Bernhard Riemann ## About This Quote This interpretation was drafted with AI assistance. It is one reading of the quote, not the author's own explanation. The quote argues that geometry assumes concepts of space and construction without explaining their origins, leaving their necessity and possibility unclear despite centuries of study. **In simple terms:** Geometry assumes space concepts without explaining them. Key Takeaway Question the hidden assumptions in geometry. Themes philosophy mathematics epistemology Mood reflective inquisitive Type analytical philosophical When to use this quote - academic research - teaching geometry - philosophical debate - educational curriculum Key Concepts axiomatic foundations conceptual analysis Questions to Reflect On - What assumptions do we take for granted in other fields? - How can we make implicit foundations explicit? A Different Perspective The claim may overlook advances in modern geometry that address some of these issues. ★ ★ ★ ★ ★ No ratings yet [Axioms](https://sms4smile.com/quotes/about/axioms/) [Darkness](https://sms4smile.com/quotes/about/darkness/) [Euclid](https://sms4smile.com/quotes/about/euclid/) [Geometry](https://sms4smile.com/quotes/about/geometry/) [Mathematician](https://sms4smile.com/quotes/about/mathematician/) [Mathematics](https://sms4smile.com/quotes/about/mathematics/) [Philosophers](https://sms4smile.com/quotes/about/philosophers/) [Relationships](https://sms4smile.com/quotes/about/relationships/) [Science](https://sms4smile.com/quotes/about/science/) [Scientist](https://sms4smile.com/quotes/about/scientist/) [Space](https://sms4smile.com/quotes/about/space/) ## Related quotes > [ \[As a young teenager\] Galois read \[Legendre's\] geometry from cover to cover as easily as other boys read a pirate yarn. ](https://sms4smile.com/quotes/eric-temple-bell/as-a-young-teenager-galois-read-legendre-s-geometry-from-cover-to/) [Eric Temple Bell](https://sms4smile.com/quotes/author/eric-temple-bell/) > [“And so will I here state just plainly and briefly that I accept God. But I must point out one thing: if God does…](https://sms4smile.com/quotes/fyodor-dostoevsky/and-so-will-i-here-state-just-plainly-and-briefly-that-i-accept-god/) [Fyodor Dostoevsky](https://sms4smile.com/quotes/author/fyodor-dostoevsky/) > [“Yet there have been and still are mathematicians and philosophers who doubt whether the whole universe, or to speak more widely, the whole of…](https://sms4smile.com/quotes/fyodor-dostoyevsky/yet-there-have-been-and-still-are-mathematicians-and-philosophers-who/) [Fyodor Dostoyevsky](https://sms4smile.com/quotes/author/fyodor-dostoyevsky/) > [“The truth, he thought, has never been of any real value to any human being - it is a symbol for mathematicians and philosophers…](https://sms4smile.com/quotes/graham-greene/the-truth-he-thought-has-never-been-of-any-real-value-to-any-human/) [Graham Greene](https://sms4smile.com/quotes/author/graham-greene/) > [ For other great mathematicians or philosophers, he \[Gauss\] used the epithets magnus, or clarus, or clarissimus; for Newton alone he kept the prefix summus. ](https://sms4smile.com/quotes/anonymous/for-other-great-mathematicians-or-philosophers-he-gauss-used-the/) [Anonymous](https://sms4smile.com/quotes/author/anonymous/) > [The early study of Euclid made me a hater of geometry.](https://sms4smile.com/quotes/james-joseph-sylvester/the-early-study-of-euclid-made-me-a-hater-of-geometry/) [James Joseph Sylvester](https://sms4smile.com/quotes/author/james-joseph-sylvester/) > [I would say, if you like, that the party is like an out-moded mathematics...that is to say, the mathematics of Euclid. 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Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you…](https://sms4smile.com/quotes/farkas-bolyai/please-give-it-up-fear-it-no-less-than-the-sensual-passion-because-it/) [Farkas Bolyai](https://sms4smile.com/quotes/author/farkas-bolyai/) > [Regular geometry, the geometry of Euclid, is concerned with shapes which are smooth, except perhaps for corners and lines, special lines which are singularities,…](https://sms4smile.com/quotes/benoit-mandelbrot/regular-geometry-the-geometry-of-euclid-is-concerned-with-shapes/) [Benoit Mandelbrot](https://sms4smile.com/quotes/author/benoit-mandelbrot/) > [“When one day Lagrange took out of his pocket a paper which he read at the Académe, and which contained a demonstration of the…](https://sms4smile.com/quotes/jean-baptiste-biot/when-one-day-lagrange-took-out-of-his-pocket-a-paper-which-he-read-at/) Jean-Baptiste Biot ## More by Bernhard Riemann > [The greatest strategy is doomed if it's implemented badly.](https://sms4smile.com/quotes/bernhard-riemann/the-greatest-strategy-is-doomed-if-it-s-implemented-badly/) Bernhard Riemann > [If only I had the Theorems! Then I should find the proofs easily enough.](https://sms4smile.com/quotes/bernhard-riemann/if-only-i-had-the-theorems-then-i-should-find-the-proofs-easily-enough/) Bernhard Riemann Explore all 3 Bernhard Riemann quotes ## More Axioms quotes > [As a leader... 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