--- title: "Meaning & Key Takeaway of: They [the mathematicians of the…" description: "\"They [the mathematicians of the Enlightenment] defined their terms vaguely and used their methods loosely, and the logic of their arguments was made to..." image: "https://sms4smile.com/quotes/i/34e/tobias-dantzig-42181_argument_they-the-mathematicians-of-the-enlighten_1505184.jpg" --- # They \[the mathematicians of the Enlightenment\] defined… ![“They [the mathematicians of the Enlightenment] defined their terms vaguely and used their methods loosely, and the logic of their arguments was made to fit the dictates of their intuition. In short, they broke all the laws of rigor and of mathematical decorum. The veritable orgy which followed the…” quote by Tobias Dantzig](https://sms4smile.com/quotes/i/34e/tobias-dantzig-42181_argument_they-the-mathematicians-of-the-enlighten_1505184.jpg) [Download](https://sms4smile.com/quotes/i/34e/tobias-dantzig-42181_argument_they-the-mathematicians-of-the-enlighten_1505184.jpg) [Open image](https://sms4smile.com/quotes/i/34e/tobias-dantzig-42181_argument_they-the-mathematicians-of-the-enlighten_1505184.jpg) > “They \[the mathematicians of the Enlightenment\] defined their terms vaguely and used their methods loosely, and the logic of their arguments was made to fit the dictates of their intuition. In short, they broke all the laws of rigor and of mathematical decorum. The veritable orgy which followed the introduction of the infinitesimals... was but a natural reaction. Intuition had too long been held imprisoned by the severe rigor of the Greeks. Now it broke loose, and there were no Euclids to keep its romantic flight in check.” [Tobias Dantzig](https://sms4smile.com/quotes/author/tobias-dantzig/) ## About This Quote This interpretation was drafted with AI assistance. It is one reading of the quote, not the author's own explanation. Enlightenment mathematicians relaxed strict rigor, allowing intuition to guide new concepts like infinitesimals, sparking creative breakthroughs. **In simple terms:** Relaxed rules let intuition thrive. Key Takeaway Balance rigor with intuition. Themes science philosophy creativity Mood analytical thoughtful Type theoretical historical When to use this quote - research methodology - educational reform - theoretical exploration - philosophical debate Key Concepts mathematical rigor intuition infinitesimals Questions to Reflect On - How to guard against intuition without rigor? - What is the role of intuition in modern science? A Different Perspective Overreliance on intuition can lead to errors. ★ ★ ★ ★ ★ No ratings yet [Argument](https://sms4smile.com/quotes/about/argument/) [Buddhism](https://sms4smile.com/quotes/about/buddhism/) [Decorum](https://sms4smile.com/quotes/about/decorum/) [Enlightenment](https://sms4smile.com/quotes/about/enlightenment/) [Euclid](https://sms4smile.com/quotes/about/euclid/) [History](https://sms4smile.com/quotes/about/history/) [Introduction](https://sms4smile.com/quotes/about/introduction/) [Intuition](https://sms4smile.com/quotes/about/intuition/) [Law](https://sms4smile.com/quotes/about/law/) [Logic](https://sms4smile.com/quotes/about/logic/) [Mathematician](https://sms4smile.com/quotes/about/mathematician/) [Mathematicians](https://sms4smile.com/quotes/about/mathematicians/) [Mathematics](https://sms4smile.com/quotes/about/mathematics/) [Rigor](https://sms4smile.com/quotes/about/rigor/) ## Related quotes > [The circumstances of human society are too complicated to be submitted to the rigor of mathematical calculation.](https://sms4smile.com/quotes/marquis-de-custine/the-circumstances-of-human-society-are-too-complicated-to-be/) [Marquis de Custine](https://sms4smile.com/quotes/author/marquis-de-custine/) > [The social sciences, I thought, needed the same kind of rigor and the same mathematical underpinnings that had made the 'hard' sciences so brilliantly…](https://sms4smile.com/quotes/herbert-simon/the-social-sciences-i-thought-needed-the-same-kind-of-rigor-and-the/) [Herbert Simon](https://sms4smile.com/quotes/author/herbert-simon/) > [It is shocking that young people should be addling their brains over mere logical subtleties in Euclid's Elements, trying to understand the proof of…](https://sms4smile.com/quotes/oliver-heaviside/it-is-shocking-that-young-people-should-be-addling-their-brains-over/) [Oliver Heaviside](https://sms4smile.com/quotes/author/oliver-heaviside/) > [Euclid alone has looked on Beauty bare. Let all who prate of Beauty hold their peace, And lay them prone upon the earth and…](https://sms4smile.com/quotes/edna-st-vincent-millay/euclid-alone-has-looked-on-beauty-bare-let-all-who-prate-of-beauty-2/) [Edna St. Vincent Millay](https://sms4smile.com/quotes/author/edna-st-vincent-millay/) > [ As to writing another book on geometry \[to replace Euclid\] the middle ages would have as soon thought of composing another New Testament. ](https://sms4smile.com/quotes/augustus-de-morgan/as-to-writing-another-book-on-geometry-to-replace-euclid-the-middle/) [Augustus De Morgan](https://sms4smile.com/quotes/author/augustus-de-morgan/) > [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. We need to…](https://sms4smile.com/quotes/alain-badiou/i-would-say-if-you-like-that-the-party-is-like-an-out-moded/) [Alain Badiou](https://sms4smile.com/quotes/author/alain-badiou/) > [“ Euclid 's Elements has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the…](https://sms4smile.com/quotes/william-kingdon-clifford/euclid-s-elements-has-been-for-nearly-twenty-two-centuries-the/) [William Kingdon Clifford](https://sms4smile.com/quotes/author/william-kingdon-clifford/) > [ \[T\]he 47th proposition in Euclid might now be voted down with as much ease as any proposition in politics; and therefore if Lord Hawkesbury… ](https://sms4smile.com/quotes/sydney-smith/t-he-47th-proposition-in-euclid-might-now-be-voted-down-with-as-much/) [Sydney Smith](https://sms4smile.com/quotes/author/sydney-smith/) > [Euclid alone has looked on Beauty bare. Let all who prate of Beauty hold their peace, And lay them prone upon the earth and…](https://sms4smile.com/quotes/edna-st-vincent-millay/euclid-alone-has-looked-on-beauty-bare-let-all-who-prate-of-beauty/) [Edna St. Vincent Millay](https://sms4smile.com/quotes/author/edna-st-vincent-millay/) > [I have given up newspapers in exchange for Tacitus and Thucydides, for Newton and Euclid; and I find myself much the happier.](https://sms4smile.com/quotes/thomas-jefferson/i-have-given-up-newspapers-in-exchange-for-tacitus-and-thucydides-for/) [Thomas Jefferson](https://sms4smile.com/quotes/author/thomas-jefferson/) > [The science of the church is neglected for the study of geometry, and they lose sight of Heaven while they are employed in measuring…](https://sms4smile.com/quotes/edward-gibbon/the-science-of-the-church-is-neglected-for-the-study-of-geometry-and/) [Edward Gibbon](https://sms4smile.com/quotes/author/edward-gibbon/) > [“Please give it up. 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 Tobias Dantzig > [Mathematics is the supreme judge; from its decisions there is no appeal.](https://sms4smile.com/quotes/tobias-dantzig/mathematics-is-the-supreme-judge-from-its-decisions-there-is-no-appeal/) [Tobias Dantzig](https://sms4smile.com/quotes/author/tobias-dantzig/) > [“The symbol is not a mere formality; it is the very essence of algebra. Without the symbol the object is a human perception and…](https://sms4smile.com/quotes/tobias-dantzig/the-symbol-is-not-a-mere-formality-it-is-the-very-essence-of-algebra/) [Tobias Dantzig](https://sms4smile.com/quotes/author/tobias-dantzig/) > [“The letter is susceptible of operations which enables one to transform literal expressions and thus to paraphrase any statement into a number of equivalent…](https://sms4smile.com/quotes/tobias-dantzig/the-letter-is-susceptible-of-operations-which-enables-one-to/) [Tobias Dantzig](https://sms4smile.com/quotes/author/tobias-dantzig/) > [The harmony of the universe knows only one musical form - the legato; while the symphony of number knows only its opposite - the…](https://sms4smile.com/quotes/tobias-dantzig/the-harmony-of-the-universe-knows-only-one-musical-form-the-legato/) [Tobias Dantzig](https://sms4smile.com/quotes/author/tobias-dantzig/) > [The importance of infinite processes for the practical exigencies of technical life can hardly be overemphasized. Practically all applications of arithmetic to geometry, mechanics,…](https://sms4smile.com/quotes/tobias-dantzig/the-importance-of-infinite-processes-for-the-practical-exigencies-of/) [Tobias Dantzig](https://sms4smile.com/quotes/author/tobias-dantzig/) > [The practical man demands an appearance of reality at least. Always dealing in the concrete, he regards mathematical terms not as symbols or thought…](https://sms4smile.com/quotes/tobias-dantzig/the-practical-man-demands-an-appearance-of-reality-at-least-always/) [Tobias Dantzig](https://sms4smile.com/quotes/author/tobias-dantzig/) > [And so matching by itself is incapable of creating an art of reckoning. Without our ability to arrange things in ordered succession little progress…](https://sms4smile.com/quotes/tobias-dantzig/and-so-matching-by-itself-is-incapable-of-creating-an-art-of/) [Tobias Dantzig](https://sms4smile.com/quotes/author/tobias-dantzig/) > [Are not most professional mathematicians spared all trouble incident to income?](https://sms4smile.com/quotes/tobias-dantzig/are-not-most-professional-mathematicians-spared-all-trouble-incident/) [Tobias Dantzig](https://sms4smile.com/quotes/author/tobias-dantzig/) > [“In the history of culture the discovery of zero will always stand out as one of the greatest single achievements of the human race.”](https://sms4smile.com/quotes/tobias-dantzig/in-the-history-of-culture-the-discovery-of-zero-will-always-stand-out/) [Tobias Dantzig](https://sms4smile.com/quotes/author/tobias-dantzig/) [Explore all 10 Tobias Dantzig quotes ](https://sms4smile.com/quotes/author/tobias-dantzig/) ## More Argument quotes > [A flourishing, morally credible media is a vital component in the maintenance of genuinely public talk, argument about common good.](https://sms4smile.com/quotes/rowan-williams/a-flourishing-morally-credible-media-is-a-vital-component-in-the/) [Rowan Williams](https://sms4smile.com/quotes/author/rowan-williams/) > [Preaching to the choir actually arms the choir with arguments and elevates the choir's discourse. There's a reason the right does it and does…](https://sms4smile.com/quotes/dan-savage/preaching-to-the-choir-actually-arms-the-choir-with-arguments-and/) [Dan Savage](https://sms4smile.com/quotes/author/dan-savage/) > [Lutherans, whose arguments and mistakes will not be difficult to contest or discover, do not want to attribute any value to works, and they…](https://sms4smile.com/quotes/michael-servetus/lutherans-whose-arguments-and-mistakes-will-not-be-difficult-to/) [Michael Servetus](https://sms4smile.com/quotes/author/michael-servetus/) > [Any comprehensive doctrine, religious or secular, can be introduced into any political argument at any time, but I argue that people who do this…](https://sms4smile.com/quotes/john-rawls/any-comprehensive-doctrine-religious-or-secular-can-be-introduced/) [John Rawls](https://sms4smile.com/quotes/author/john-rawls/) > [As opposed to getting into arguments about, well, these folks have been treated fairly so now we're going to be doing things that, very…](https://sms4smile.com/quotes/barack-obama/as-opposed-to-getting-into-arguments-about-well-these-folks-have-been/) [Barack Obama](https://sms4smile.com/quotes/author/barack-obama/) > [There could be no more powerful argument against mixing religion and government than the success of independent African American churches in placing racial segregation…](https://sms4smile.com/quotes/susan-jacoby/there-could-be-no-more-powerful-argument-against-mixing-religion-and/) [Susan Jacoby](https://sms4smile.com/quotes/author/susan-jacoby/) > [The inner defenses are unconscious. They consist of a kind of magic aura which the mind builds around cherished belief. Arguments which penetrate into…](https://sms4smile.com/quotes/arthur-koestler/the-inner-defenses-are-unconscious-they-consist-of-a-kind-of-magic/) [Arthur Koestler](https://sms4smile.com/quotes/author/arthur-koestler/) > [On the stage, the characters express themselves more through words than images. So the arguments of the characters and the tension between characters -…](https://sms4smile.com/quotes/alan-alda/on-the-stage-the-characters-express-themselves-more-through-words/) [Alan Alda](https://sms4smile.com/quotes/author/alan-alda/) > [One argument goes that recessions are good for female artists because when money flies out the window, women are allowed in the house. The…](https://sms4smile.com/quotes/jerry-saltz/one-argument-goes-that-recessions-are-good-for-female-artists-because/) [Jerry Saltz](https://sms4smile.com/quotes/author/jerry-saltz/) > [All of the barriers to innovation in the energy sector are arguments for a big commitment to public investment. Only the public sector can…](https://sms4smile.com/quotes/ted-nordhaus/all-of-the-barriers-to-innovation-in-the-energy-sector-are-arguments/) [Ted Nordhaus](https://sms4smile.com/quotes/author/ted-nordhaus/) > [“Arguments led by subjectivity are a waste of time. And, if they take place online, add to that, a waste of one’s keyboard.”](https://sms4smile.com/quotes/mokokoma-mokhonoana/arguments-led-by-subjectivity-are-a-waste-of-time-and-if-they-take/) [Mokokoma Mokhonoana](https://sms4smile.com/quotes/author/mokokoma-mokhonoana/) > [If in the middle of an argument I suddenly fall silent it does not mean you have won... No it means you should probably…](https://sms4smile.com/quotes/hussein-nishah/if-in-the-middle-of-an-argument-i-suddenly-fall-silent-it-does-not/) [Hussein Nishah](https://sms4smile.com/quotes/author/hussein-nishah/) > [Don't get divorced after your first argument! I have a lot of friends that have one fight and that's it, they get divorced. I…](https://sms4smile.com/quotes/kelly-ripa/don-t-get-divorced-after-your-first-argument-i-have-a-lot-of-friends/) [Kelly Ripa](https://sms4smile.com/quotes/author/kelly-ripa/) > [“The motive behind my arguments have always been to come at truth. I am not interested in conquering anyone with my ideologies or seeking…](https://sms4smile.com/quotes/crystal-evans/the-motive-behind-my-arguments-have-always-been-to-come-at-truth-i-am/) [Crystal Evans](https://sms4smile.com/quotes/author/crystal-evans/) > [In presenting a mathematical argument the great thing is to give the educated reader the chance to catch on at once to the momentary…](https://sms4smile.com/quotes/john-edensor-littlewood/in-presenting-a-mathematical-argument-the-great-thing-is-to-give-the/) [John Edensor Littlewood](https://sms4smile.com/quotes/author/john-edensor-littlewood/) [Browse all 3,479 Argument quotes ](https://sms4smile.com/quotes/about/argument/) ```json { "@context": "https://schema.org", "@graph": [ { "@type": "Quotation", "text": "They [the mathematicians of the Enlightenment] defined their terms vaguely and used their methods loosely, and the logic of their arguments was made to fit the dictates of their intuition. In short, they broke all the laws of rigor and of mathematical decorum. The veritable orgy which followed the introduction of the infinitesimals... was but a natural reaction. Intuition had too long been held imprisoned by the severe rigor of the Greeks. Now it broke loose, and there were no Euclids to keep its romantic flight in check.", "creator": { "@type": "Person", "name": "Tobias Dantzig" }, "isPartOf": { "@id": "https://sms4smile.com/quotes/#website" }, "url": "https://sms4smile.com/quotes/tobias-dantzig/they-the-mathematicians-of-the-enlightenment-defined-their-terms/", "about": { "@type": "Thing", "name": "Argument" }, "@id": "https://sms4smile.com/quotes/tobias-dantzig/they-the-mathematicians-of-the-enlightenment-defined-their-terms/#quotation", "image": { "@type": "ImageObject", "@id": "https://sms4smile.com/quotes/i/34e/tobias-dantzig-42181_argument_they-the-mathematicians-of-the-enlighten_1505184.jpg", "contentUrl": "https://sms4smile.com/quotes/i/34e/tobias-dantzig-42181_argument_they-the-mathematicians-of-the-enlighten_1505184.jpg", "url": "https://sms4smile.com/quotes/i/34e/tobias-dantzig-42181_argument_they-the-mathematicians-of-the-enlighten_1505184.jpg", "width": 1080, "height": 1080, "caption": "“They [the mathematicians of the Enlightenment] defined their terms vaguely and used their methods loosely…” — Tobias Dantzig", "creditText": "SMS4Smile Quotes", "creator": { "@id": "https://sms4smile.com/quotes/#organization" } } }, { "@type": "BreadcrumbList", "itemListElement": [ { "@type": "ListItem", "position": 1, "name": "Home", "item": "https://sms4smile.com/" }, { "@type": "ListItem", "position": 2, "name": "Quotes", "item": "https://sms4smile.com/quotes/" }, { "@type": "ListItem", "position": 3, "name": "Tobias Dantzig", "item": "https://sms4smile.com/quotes/author/tobias-dantzig/" }, { "@type": "ListItem", "position": 4, "name": "They [the mathematicians of the Enlightenment]...", "item": "https://sms4smile.com/quotes/tobias-dantzig/they-the-mathematicians-of-the-enlightenment-defined-their-terms/" } ], "@id": "https://sms4smile.com/quotes/tobias-dantzig/they-the-mathematicians-of-the-enlightenment-defined-their-terms/#breadcrumb" }, { "@type": "WebSite", "@id": "https://sms4smile.com/quotes/#website", "name": "SMS4Smile Quotes", "url": "https://sms4smile.com/quotes/", "publisher": { "@id": "https://sms4smile.com/quotes/#organization" } }, { "@type": "Organization", "@id": "https://sms4smile.com/quotes/#organization", "name": "SMS4Smile Quotes", "url": "https://sms4smile.com/quotes/", "logo": { "@type": "ImageObject", "url": "https://sms4smile.com/quotes/wp-content/themes/sq/assets/icons/icon-512.png", "width": 512, "height": 512 } } ] } ```