--- title: "Meaning & Key Takeaway of: I think mathematics is a vast territory…" description: "\"I think mathematics is a vast territory. The outskirts of mathematics are the outskirts of mathematical civilization. There are certain subjects that p..." image: "https://sms4smile.com/quotes/i/f30/william-thurston-39710_bounds_i-think-mathematics-is-a-vast-territory_2613806.jpg" --- # I think mathematics is a vast territory. The outskirts of… ![“I think mathematics is a vast territory. The outskirts of mathematics are the outskirts of mathematical civilization. There are certain subjects that people learn about and gather together. Then there is a sort of inevitable development in those fields. You get to the point where a certain theorem…” quote by William Thurston](https://sms4smile.com/quotes/i/f30/william-thurston-39710_bounds_i-think-mathematics-is-a-vast-territory_2613806.jpg) [Download](https://sms4smile.com/quotes/i/f30/william-thurston-39710_bounds_i-think-mathematics-is-a-vast-territory_2613806.jpg) [Open image](https://sms4smile.com/quotes/i/f30/william-thurston-39710_bounds_i-think-mathematics-is-a-vast-territory_2613806.jpg) > “I think mathematics is a vast territory. The outskirts of mathematics are the outskirts of mathematical civilization. There are certain subjects that people learn about and gather together. Then there is a sort of inevitable development in those fields. You get to the point where a certain theorem is bound to be proved, independent of any particular individual, because it is just in the path of development.” [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) ## About This Quote This interpretation was drafted with AI assistance. It is one reading of the quote, not the author's own explanation. Mathematics expands naturally; certain results become inevitable as the field matures. **In simple terms:** Math grows and some theorems are bound to appear. Key Takeaway Recognize the inevitable progress in mathematical research. Themes growth inevitability creativity Mood analytical curious Type educational philosophical When to use this quote - research planning - education curriculum - interdisciplinary projects Key Concepts mathematics philosophy of science history of ideas Questions to Reflect On - What areas seem most ripe for inevitable discovery? - How can we balance inevitability with creative exploration? A Different Perspective Predictability may limit openness to unexpected breakthroughs. ★ ★ ★ ★ ★ No ratings yet [Bounds](https://sms4smile.com/quotes/about/bounds/) [Certain](https://sms4smile.com/quotes/about/certain/) [Civilization](https://sms4smile.com/quotes/about/civilization/) [Development](https://sms4smile.com/quotes/about/development/) [Fields](https://sms4smile.com/quotes/about/fields/) [Independent](https://sms4smile.com/quotes/about/independent/) [Individual](https://sms4smile.com/quotes/about/individual/) [Inevitable](https://sms4smile.com/quotes/about/inevitable/) [Mathematical](https://sms4smile.com/quotes/about/mathematical/) [Mathematical Civilization](https://sms4smile.com/quotes/about/mathematical-civilization-2050837/) [Mathematics](https://sms4smile.com/quotes/about/mathematics/) [Mathematics Outskirts](https://sms4smile.com/quotes/about/mathematics-outskirts-2050834/) [Mathematics Vast](https://sms4smile.com/quotes/about/mathematics-vast-2050838/) [Outskirts Mathematical](https://sms4smile.com/quotes/about/outskirts-mathematical-2050836/) [Outskirts Mathematics](https://sms4smile.com/quotes/about/outskirts-mathematics-2050835/) [Particular](https://sms4smile.com/quotes/about/particular/) [Path](https://sms4smile.com/quotes/about/path/) [People](https://sms4smile.com/quotes/about/people/) [Subjects](https://sms4smile.com/quotes/about/subjects/) [Territory](https://sms4smile.com/quotes/about/territory/) [Theorems](https://sms4smile.com/quotes/about/theorems/) [Thinking](https://sms4smile.com/quotes/about/thinking/) [Together](https://sms4smile.com/quotes/about/together/) ## Related quotes > [Mathematics is a vast adventure; its history reflects some of the noblest thoughts of countless generations.](https://sms4smile.com/quotes/dirk-jan-struik/mathematics-is-a-vast-adventure-its-history-reflects-some-of-the/) [Dirk Jan Struik](https://sms4smile.com/quotes/author/dirk-jan-struik/) > [Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated…](https://sms4smile.com/quotes/william-thurston/mathematicians-can-and-do-fill-in-gaps-correct-errors-and-supply-more/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) > [The product of mathematics is clarity and understanding. Not theorems, by themselves. ... In short, mathematics only exists in a living community of mathematicians…](https://sms4smile.com/quotes/william-thurston/the-product-of-mathematics-is-clarity-and-understanding-not-theorems/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) > [All I remember about the examination is that there was a question on Sturm's theorem about equations, which I could not do then and…](https://sms4smile.com/quotes/louis-j-mordell/all-i-remember-about-the-examination-is-that-there-was-a-question-on/) [Louis J. Mordell](https://sms4smile.com/quotes/author/louis-j-mordell/) > [There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of…](https://sms4smile.com/quotes/niels-henrik-abel/there-are-very-few-theorems-in-advanced-analysis-which-have-been/) [Niels Henrik Abel](https://sms4smile.com/quotes/author/niels-henrik-abel/) > [For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that…](https://sms4smile.com/quotes/seymour-papert/for-what-is-important-when-we-give-children-a-theorem-to-use-is-not/) [Seymour Papert](https://sms4smile.com/quotes/author/seymour-papert/) > [Mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation…](https://sms4smile.com/quotes/imre-lakatos/mathematics-does-not-grow-through-a-monotonous-increase-of-the-number/) [Imre Lakatos](https://sms4smile.com/quotes/author/imre-lakatos/) > [It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something,…](https://sms4smile.com/quotes/g-h-hardy/it-is-a-melancholy-experience-for-a-professional-mathematician-to/) [G. H. Hardy](https://sms4smile.com/quotes/author/g-h-hardy/) > [ In 1975, ... \[speaking with Shiing Shen Chern\], I told him I had finally learned ... the beauty of fiber-bundle theory and the profound… ](https://sms4smile.com/quotes/chen-ning-yang/in-1975-speaking-with-shiing-shen-chern-i-told-him-i-had-finally/) [Chen-Ning Yang](https://sms4smile.com/quotes/author/chen-ning-yang/) > [The scientist has to take 95 per cent of his subject on trust. He has to because he can't possibly do all the experiments,…](https://sms4smile.com/quotes/christopher-zeeman/the-scientist-has-to-take-95-per-cent-of-his-subject-on-trust-he-has/) [Christopher Zeeman](https://sms4smile.com/quotes/author/christopher-zeeman/) > [Without computers we will be stuck only proving theorems that have short proofs.](https://sms4smile.com/quotes/kenneth-appel/without-computers-we-will-be-stuck-only-proving-theorems-that-have/) [Kenneth Appel](https://sms4smile.com/quotes/author/kenneth-appel/) > [A mathematician is a device for turning coffee into theorems.](https://sms4smile.com/quotes/paul-erdos/a-mathematician-is-a-device-for-turning-coffee-into-theorems/) [Paul Erdos](https://sms4smile.com/quotes/author/paul-erdos/) > [The "seriousness" of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas…](https://sms4smile.com/quotes/g-h-hardy/the-seriousness-of-a-mathematical-theorem-lies-not-in-its-practical/) [G. H. Hardy](https://sms4smile.com/quotes/author/g-h-hardy/) > [The great poem and the deep theorem are new to every reader, and yet are his own experiences, because he himself recreates them.](https://sms4smile.com/quotes/jacob-bronowski/the-great-poem-and-the-deep-theorem-are-new-to-every-reader-and-yet/) [Jacob Bronowski](https://sms4smile.com/quotes/author/jacob-bronowski/) > [Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise…](https://sms4smile.com/quotes/marvin-minsky/theorems-often-tell-us-complex-truths-about-the-simple-things-but/) [Marvin Minsky](https://sms4smile.com/quotes/author/marvin-minsky/) ## More by William Thurston > [The product of mathematics is clarity and understanding. Not theorems, by themselves. ... In short, mathematics only exists in a living community of mathematicians…](https://sms4smile.com/quotes/william-thurston/the-product-of-mathematics-is-clarity-and-understanding-not-theorems/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) > [As one reads mathematics, one needs to have an active mind, asking questions, forming mental connections between the current topic and other ideas from…](https://sms4smile.com/quotes/william-thurston/as-one-reads-mathematics-one-needs-to-have-an-active-mind-asking/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) > [Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide.](https://sms4smile.com/quotes/william-thurston/math-is-really-about-the-human-mind-about-how-people-can-think/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) > [We humans have a wide range of abilities that help us perceive and analyze mathematical content. We perceive abstract notions not just through seeing…](https://sms4smile.com/quotes/william-thurston/we-humans-have-a-wide-range-of-abilities-that-help-us-perceive-and/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) > [Mathematics is much less formally complete and precise than computer programs.](https://sms4smile.com/quotes/william-thurston/mathematics-is-much-less-formally-complete-and-precise-than-computer/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) > [You don't see what you're seeing until you see it but when you do see it, it lets you see many other things.](https://sms4smile.com/quotes/william-thurston/you-don-t-see-what-you-re-seeing-until-you-see-it-but-when-you-do-see/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) > [Mathematics is an art of human understanding.](https://sms4smile.com/quotes/william-thurston/mathematics-is-an-art-of-human-understanding/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) > [Experience has shown repeatedly that a mathematical theory with a rich internal structure generally turns out to have significant implications for the understanding of…](https://sms4smile.com/quotes/william-thurston/experience-has-shown-repeatedly-that-a-mathematical-theory-with-a/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) > [The real satisfaction from mathematics is in learning from others and sharing with others.](https://sms4smile.com/quotes/william-thurston/the-real-satisfaction-from-mathematics-is-in-learning-from-others-and/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) > [Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated…](https://sms4smile.com/quotes/william-thurston/mathematicians-can-and-do-fill-in-gaps-correct-errors-and-supply-more/) [William Thurston](https://sms4smile.com/quotes/author/william-thurston/) [Explore all 11 William Thurston quotes ](https://sms4smile.com/quotes/author/william-thurston/) ## More Bounds quotes > [Man will not always stay on Earth; the pursuit of light and space will lead him to penetrate the bounds of the atmosphere, timidly…](https://sms4smile.com/quotes/konstantin-tsiolkovsky/man-will-not-always-stay-on-earth-the-pursuit-of-light-and-space-will/) [Konstantin Tsiolkovsky](https://sms4smile.com/quotes/author/konstantin-tsiolkovsky/) > [Importunity is a condition of prayer. We are to press the matter, not with vain repetitions, but with urgent repetitions. We repeat, not to…](https://sms4smile.com/quotes/edward-mckendree-bounds/importunity-is-a-condition-of-prayer-we-are-to-press-the-matter-not/) [Edward McKendree Bounds](https://sms4smile.com/quotes/author/edward-mckendree-bounds/) > [If the civil magistrate be a Christian, a disciple or follower of the meek Lamb of God, he is bound to be far from…](https://sms4smile.com/quotes/roger-williams/if-the-civil-magistrate-be-a-christian-a-disciple-or-follower-of-the/) [Roger Williams](https://sms4smile.com/quotes/author/roger-williams/) > [We have committed ourselves to pursue joint policies, but then national governments say, "We aren't bound by that." That is a dramatic situation, because…](https://sms4smile.com/quotes/martin-schulz/we-have-committed-ourselves-to-pursue-joint-policies-but-then/) [Martin Schulz](https://sms4smile.com/quotes/author/martin-schulz/) > [The World's a Printing-House, our words, our thoughts, Our deeds, are characters of several sizes. Each soul is a Compos'tor, of whose faults The…](https://sms4smile.com/quotes/francis-quarles/the-world-s-a-printing-house-our-words-our-thoughts-our-deeds-are/) [Francis Quarles](https://sms4smile.com/quotes/author/francis-quarles/) > [Christian obligation cannot be made to accord with a law of expediency. The Christian's maxims are, Do right because you are bound to do…](https://sms4smile.com/quotes/francis-landey-patton/christian-obligation-cannot-be-made-to-accord-with-a-law-of/) [Francis Landey Patton](https://sms4smile.com/quotes/author/francis-landey-patton/) > [Such as the chain of causes we call Fate, such is the chain of wishes: one links on to another; the whole man is…](https://sms4smile.com/quotes/seneca-the-younger/such-as-the-chain-of-causes-we-call-fate-such-is-the-chain-of-wishes/) [Seneca the Younger](https://sms4smile.com/quotes/author/seneca-the-younger/) > ['You spin in the sky, the world spins under you, and you step from land to land, while we . . .' She turned…](https://sms4smile.com/quotes/samuel-r-delany/you-spin-in-the-sky-the-world-spins-under-you-and-you-step-from-land/) [Samuel R. Delany](https://sms4smile.com/quotes/author/samuel-r-delany/) > ['Facts' are the bounds of human knowledge, set for it, not by it.](https://sms4smile.com/quotes/william-james/facts-are-the-bounds-of-human-knowledge-set-for-it-not-by-it/) [William James](https://sms4smile.com/quotes/author/william-james/) > [An order given in battle, an instruction issued by the master of a sailing ship, a cry for help, are as powerful in modifying…](https://sms4smile.com/quotes/bronislaw-malinowski/an-order-given-in-battle-an-instruction-issued-by-the-master-of-a/) [Bronislaw Malinowski](https://sms4smile.com/quotes/author/bronislaw-malinowski/) > [It is only when our life proceeds within bounds and in an accepted, disciplined way, that the mind can be free.](https://sms4smile.com/quotes/vinoba-bhave/it-is-only-when-our-life-proceeds-within-bounds-and-in-an-accepted/) [Vinoba Bhave](https://sms4smile.com/quotes/author/vinoba-bhave/) > [Yes, thou art ever present, power divine; not circumscribed by time, nor fixed by space, confined to altars, nor to temples bound. In wealth,…](https://sms4smile.com/quotes/hannah-more/yes-thou-art-ever-present-power-divine-not-circumscribed-by-time-nor/) [Hannah More](https://sms4smile.com/quotes/author/hannah-more/) > [I have visited sweatshops, factories, and crowded slums. If I could not see it, I could smell it. The foundation of society is laid…](https://sms4smile.com/quotes/helen-keller/i-have-visited-sweatshops-factories-and-crowded-slums-if-i-could-not/) [Helen Keller](https://sms4smile.com/quotes/author/helen-keller/) > [Once we are bound together to our brothers by a common good that is outside us, then we can breathe. Experience teaches us that…](https://sms4smile.com/quotes/antoine-de-saint-exup-ry/once-we-are-bound-together-to-our-brothers-by-a-common-good-that-is/) [Antoine de Saint-Exupery](https://sms4smile.com/quotes/author/antoine-de-saint-exup-ry/) > [I am quite conscious that my speculations run beyond the bounds of true science…It is a mere rag of an hypothesis with as many…](https://sms4smile.com/quotes/charles-darwin/i-am-quite-conscious-that-my-speculations-run-beyond-the-bounds-of/) [Charles Darwin](https://sms4smile.com/quotes/author/charles-darwin/) [Browse all 1,649 Bounds quotes ](https://sms4smile.com/quotes/about/bounds/) ```json { "@context": "https://schema.org", "@graph": [ { "@type": "Quotation", "text": "I think mathematics is a vast territory. The outskirts of mathematics are the outskirts of mathematical civilization. There are certain subjects that people learn about and gather together. Then there is a sort of inevitable development in those fields. 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