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How Is It That There Are So Many Minds That Are Incapable Of Understanding Mathematics? ... The Skeleton Of Our Understanding, ... And Actually They Are The Majority. ... We Have Here A Problem That Is Not Easy Of Solution, But Yet Must Engage The Attention Of All Who Wish To Devote Themselves To Education.

-Henri Poincare

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How Is It That

Henri Poincare

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How Is It That There Are So Many Minds That Are Incapable Of Understanding Mathematics? ... The Skeleton Of Our Understanding, ... And Actually They Are The Majority. ... We Have Here A Problem That Is Not Easy Of Solution, But Yet Must Engage The Attention Of All Who Wish To Devote Themselves To Education.

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Mathematical Discoveries, Small Or Great Are Never Born Of Spontaneous Generation They Always Presuppose A Soil Seeded With Preliminary Knowledge And Well Prepared By Labour, Both Conscious And Subconscious.

Tolstoi Explains Somewhere In His Writings Why, In His Opinion, “science For Science's Sake” Is An Absurd Conception. We Cannot Know All The Facts, Since They Are Practically Infinite In Number. We Must Make A Selection. Is It Not Better To Be Guided By Utility, By Our Practical, And More Especially Our Moral, Necessities?

Doubting Everything And Believing Everything Are Two Equally Convenient Solutions That Guard Us From Having To Think

The Mind Uses Its Faculty For Creativity Only When Experience Forces It To Do So.

Consider Now The Milky Way. Here Also We See An Innumerable Dust, Only The Grains Of This Dust Are No Longer Atoms But Stars; These Grains Also Move With Great Velocities, They Act At A Distance One Upon Another, But This Action Is So Slight At Great Distances That Their Trajectories Are Rectilineal; Nevertheless, From Time To Time, Two Of Them May Come Near Enough Together To Be Deviated From Their Course, Like A Comet That Passed Too Close To Jupiter. In A Word, In The Eyes Of A Giant, To Whom Our Suns Were What Our Atoms Are To Us, The Milky Way Would Only Look Like A Bubble Of Gas.

Intuition Is More Important To Discovery Than Logic.

A Small Error In The Former Will Produce An Enormous Error In The Latter.

If One Looks At The Different Problems Of The Integral Calculus Which Arise Naturally When One Wishes To Go Deep Into The Different Parts Of Physics, It Is Impossible Not To Be Struck By The Analogies Existing.

But All Of My Efforts Served Only To Make Me Better Acquainted With The Difficulty, Which In Itself Was Something.

Sociology Is The Science With The Greatest Number Of Methods And The Least Results.

One Geometry Cannot Be More True Than Another; It Can Only Be More Convenient.

Absolute Space, That Is To Say, The Mark To Which It Would Be Necessary To Refer The Earth To Know Whether It Really Moves, Has No Objective Existence.... The Two Propositions: "the Earth Turns Round" And "it Is More Convenient To Suppose The Earth Turns Round" Have The Same Meaning; There Is Nothing More In The One Than In The Other.

Analyse Data Just So Far As To Obtain Simplicity And No Further.

We Also Know How Cruel The Truth Often Is, And We Wonder Whether Delusion Is Not More Consoling.

If We Wish To Foresee The Future Of Mathematics, Our Proper Course Is To Study The History And Present Condition Of The Science.

There Are No Solved Problems; There Are Only Problems That Are More Or Less Solved.

Zero Is The Number Of Objects That Satisfy A Condition That Is Never Satisfied. But As Never Means "in No Case", I Do Not See That Any Progress Has Been Made.

Geometry Is The Art Of Correct Reasoning From Incorrectly Drawn Figures.

How Is It That There Are So Many Minds That Are Incapable Of Understanding Mathematics? ... The Skeleton Of Our Understanding, ... And Actually They Are The Majority. ... We Have Here A Problem That Is Not Easy Of Solution, But Yet Must Engage The Attention Of All Who Wish To Devote Themselves To Education.

Need We Add That Mathematicians Themselves Are Not Infallible?

The Aim Of Science Is Not Things Themselves, As The Dogmatists In Their Simplicity Imagine, But The Relation Between Things.

All Of Mathematics Is A Tale About Groups.

Often When Works At A Hard Question, Nothing Good Is Accomplished At The First Attack. Then One Takes A Rest, Long Or Short, And Sits Down Anew To The Work. During The First Half-hour, As Before, Nothing Is Found, And Then All Of A Sudden The Decisive Idea Presents Itself To The Mind.

The Subliminal Self Is In No Way Inferior To The Conscious Self. It Knows How To Choose And To Divine.

A Reality Completely Independent Of The Spirit That Conceives It, Sees It, Or Feels It, Is An Impossibility. A World So External As That, Even If It Existed, Would Be Forever Inaccessible To Us.

It Is A Misfortune For A Science To Be Born Too Late When The Means Of Observation Have Become Too Perfect. That Is What Is Happening At This Moment With Respect To Physical Chemistry; The Founders Are Hampered In Their General Grasp By Third And Fourth Decimal Places.

A Sane Mind Should Not Be Guilty Of A Logical Fallacy, Yet There Are Very Fine Minds Incapable Of Following Mathematical Demonstrations.

It Is Through Science That We Prove, But Through Intuition That We Discover.

The Advance Of Science Is Not Comparable To The Changes Of A City, Where Old Edifices Are Pitilessly Torn Down To Give Place To New, But To The Continuous Evolution Of Zoologic Types Which Develop Ceaselessly And End By Becoming Unrecognisable To The Common Sight, But Where An Expert Eye Finds Always Traces Of The Prior Work Of The Centuries Past. One Must Not Think Then That The Old-fashioned Theories Have Been Sterile And Vain.

The Task Of The Educator Is To Make The Child's Spirit Pass Again Where Its Forefathers Have Gone, Moving Rapidly Through Certain Stages But Suppressing None Of Them. In This Regard, The History Of Science Must Be Our Guide.

Deviner Avant De Démontrer! Ai-je Besoin De Rappeler Que C'est Ainsi Que Se Sont Faites Toutes Les Découvertes Importantes.

Mathematical Discoveries, Small Or Great Are Never Born Of Spontaneous Generation.

To Invent Is To Discern, To Choose.

Mathematicians Do Not Study Objects, But The Relations Between Objects.

Mathematicians Are Born, Not Made.

Geometry Is Not True, It Is Advantageous.

This Harmony That Human Intelligence Believes It Discovers In Nature - Does It Exist Apart From That Intelligence? No, Without Doubt, A Reality Completely Independent Of The Spirit Which Conceives It, Sees It Or Feels It, Is An Impossibility. A World So Exterior As That, Even If It Existed, Would Be Forever Inaccessible To Us. But What We Call Objective Reality Is, In The Last Analysis, That Which Is Common To Several Thinking Beings, And Could Be Common To All; This Common Part, We Will See, Can Be Nothing But The Harmony Expressed By Mathematical Laws.

Talk With M. Hermite. He Never Evokes A Concrete Image, Yet You Soon Perceive That The More Abstract Entities Are To Him Like Living Creatures.

I Entered An Omnibus To Go To Some Place Or Other. At That Moment When I Put My Foot On The Step The Idea Came To Me, Without Anything In My Former Thoughts Seeming To Have Paved The Way For It, That The Transformations I Had Used To Define The Fuchsian Functions Were Identical With Non-euclidean Geometry.

Ideas Rose In Clouds; I Felt Them Collide Until Pairs Interlocked, So To Speak, Making A Stable Combination.

A Scientist Worthy Of His Name, About All A Mathematician, Experiences In His Work The Same Impression As An Artist; His Pleasure Is As Great And Of The Same Nature.

All That Is Not Thought Is Pure Nothingness; Since We Can Think Only Thoughts, And All The Words We Use To Speak Of Things Can Express Only Thoughts, To Say There Is Something Other Than Thought Is Therefore An Affirmation Which Can Have No Meaning.

It Is Often Said That Experiments Should Be Made Without Preconceived Ideas. That Is Impossible. Not Only Would It Make Every Experiment Fruitless, But Even If We Wished To Do So, It Could Not Be Done. Every Man Has His Own Conception Of The World, And This He Cannot So Easily Lay Aside. We Must, Example, Use Language, And Our Language Is Necessarily Steeped In Preconceived Ideas. Only They Are Unconscious Preconceived Ideas, Which Are A Thousand Times The Most Dangerous Of All.

But For Harmony Beautiful To Contemplate, Science Would Not Be Worth Following.

All The Scientist Creates In A Fact Is The Language In Which He Enunciates It. If He Predicts A Fact, He Will Employ This Language, And For All Those Who Can Speak And Understand It, His Prediction Is Free From Ambiguity. Moreover, This Prediction Once Made, It Evidently Does Not Depend Upon Him Whether It Is Fulfilled Or Not.

Thus, Be It Understood, To Demonstrate A Theorem, It Is Neither Necessary Nor Even Advantageous To Know What It Means.

In One Word, To Draw The Rule From Experience, One Must Generalize; This Is A Necessity That Imposes Itself On The Most Circumspect Observer.

Every Good Mathematician Should Also Be A Good Chess Player And Vice Versa.

It Is By Logic That We Prove, But By Intuition That We Discover. To Know How To Criticize Is Good, To Know How To Create Is Better.

In The Old Days When People Invented A New Function They Had Something Useful In Mind. Now, They Invent Them Deliberately Just To Invalidate Our Ancestors' Reasoning, And That Is All They Are Ever Going To Get Out Of Them.

Every Phenomenon, However Trifling It Be, Has A Cause, And A Mind Infinitely Powerful, And Infinitely Well-informed Concerning The Laws Of Nature Could Have Foreseen It From The Beginning Of The Ages. If A Being With Such A Mind Existed, We Could Play No Game Of Chance With Him; We Should Always Lose.

So Is Not Mathematical Analysis Then Not Just A Vain Game Of The Mind? To The Physicist It Can Only Give A Convenient Language; But Isn't That A Mediocre Service, Which After All We Could Have Done Without; And, It Is Not Even To Be Feared That This Artificial Language Be A Veil, Interposed Between Reality And The Physicist's Eye? Far From That, Without This Language Most Of The Initimate Analogies Of Things Would Forever Have Remained Unknown To Us; And We Would Never Have Had Knowledge Of The Internal Harmony Of The World, Which Is, As We Shall See, The Only True Objective Reality.

I Then Began To Study Arithmetical Questions Without Any Great Apparent Result, And Without Suspecting That They Could Have The Least Connexion With My Previous Researches. Disgusted At My Want Of Success, I Went Away To Spend A Few Days At The Seaside, And Thought Of Entirely Different Things. One Day, As I Was Walking On The Cliff, The Idea Came To Me, Again With The Same Characteristics Of Conciseness, Suddenness, And Immediate Certainty, That Arithmetical Transformations Of Indefinite Ternary Quadratic Forms Are Identical With Those Of Non-euclidian Geometry.

Most Striking At First Is The Appearance Of Sudden Illumination, A Manifest Sign Of Long Unconscious Prior Work.

All Great Progress Takes Place When Two Sciences Come Together, And When Their Resemblance Proclaims Itself, Despite The Apparent Disparity Of Their Substance.

It May Be Appropriate To Quote A Statement Of Poincare, Who Said (partly In Jest No Doubt) That There Must Be Something Mysterious About The Normal Law Since Mathematicians Think It Is A Law Of Nature Whereas Physicists Are Convinced That It Is A Mathematical Theorem.

A Cat Is Witty, He Has Nerve, He Knows How To Do Precisely The Right Thing At The Right Moment.

The Scientist Does Not Study Nature Because It Is Useful To Do So. He Studies It Because He Takes Pleasure In It, And He Takes Pleasure In It Because It Is Beautiful. If Nature Were Not Beautiful It Would Not Be Worth Knowing, And Life Would Not Be Worth Living. I Am Not Speaking, Of Course, Of The Beauty Which Strikes The Senses, Of The Beauty Of Qualities And Appearances. I Am Far From Despising This, But It Has Nothing To Do With Science. What I Mean Is That More Intimate Beauty Which Comes From The Harmonious Order Of Its Parts, And Which A Pure Intelligence Can Grasp.

It May Happen That Small Differences In The Initial Conditions Produce Very Great Ones In The Final Phenomena. A Small Error In The Former Will Produce An Enormous Error In The Later. Prediction Becomes Impossible, And We Have The Fortuitous Phenomena.

Mathematics Has A Threefold Purpose. It Must Provide An Instrument For The Study Of Nature. But This Is Not All: It Has A Philosophical Purpose, And, I Daresay, An Aesthetic Purpose.

The Mathematical Facts Worthy Of Being Studied Are Those Which, By Their Analogy With Other Facts, Are Capable Of Leading Us To The Knowledge Of A Physical Law. They Reveal The Kinship Between Other Facts, Long Known, But Wrongly Believed To Be Strangers To One Another.

Mathematicians Do Not Deal In Objects, But In Relations Between Objects; Thus, They Are Free To Replace Some Objects By Others So Long As The Relations Remain Unchanged. Content To Them Is Irrelevant: They Are Interested In Form Only.

It Is Far Better To Foresee Even Without Certainty Than Not To Foresee At All.

Absolute Certainty

Les Faits Ne Parlent Pas. Facts Do Not Speak.

Experiment Is The Sole Source Of Truth.

Thus, Be It Understood, To Demonstrate A Theorem, It Is Neither Necessary Nor Even Advantageous To Know What It Means. The Geometer Might Be Replaced By The "logic Piano" Imagined By Stanley Jevons; Or, If You Choose, A Machine Might Be Imagined Where The Assumptions Were Put In At One End, While The Theorems Came Out At The Other, Like The Legendary Chicago Machine Where The Pigs Go In Alive And Come Out Transformed Into Hams And Sausages. No More Than These Machines Need The Mathematician Know What He Does.

It Is The Harmony Of The Diverse Parts, Their Symmetry, Their Happy Balance; In A Word It Is All That Introduces Order, All That Gives Unity, That Permits Us To See Clearly And To Comprehend At Once Both The Ensemble And The Details.

One Does Not Ask Whether A Scientific Theory Is True, But Only Whether It Is Convenient.

Pure Logic Could Never Lead Us To Anything But Tautologies; It Can Create Nothing New; Not From It Alone Can Any Science Issue.

Mathematics Is The Art Of Giving The Same Name To Different Things. [as Opposed To The Quotation: Poetry Is The Art Of Giving Different Names To The Same Thing].

Guessing Before Proving! Need I Remind You That It Is So That All Important Discoveries Have Been Made?

Thought Must Never Submit, Neither To A Dogma, Nor To A Party, Nor To A Passion, Nor To An Interest, Nor To A Preconceived Idea, Nor To Whatever It May Be, Save To The Facts Themselves, Because, For Thought, Submission Would Mean Ceasing To Be.

When The Logician Has Resolved Each Demonstration Into A Host Of Elementary Operations, All Of Them Correct, He Will Not Yet Be In Possession Of The Whole Reality, That Indefinable Something That Constitutes The Unity ... Now Pure Logic Cannot Give Us This View Of The Whole; It Is To Intuition That We Must Look For It.

What Is A Good Definition? For The Philosopher Or The Scientist, It Is A Definition Which Applies To All The Objects To Be Defined, And Applies Only To Them; It Is That Which Satisfies The Rules Of Logic. But In Education It Is Not That; It Is One That Can Be Understood By The Pupils.

When The Physicists Ask Us For The Solution Of A Problem, It Is Not Drudgery That They Impose On Us, On The Contrary, It Is Us Who Owe Them Thanks.

In The Old Days When People Invented A New Function They Had Something Useful In Mind.

A Very Small Cause, Which Escapes Us, Determines A Considerable Effect Which We Cannot Ignore, And We Say That This Effect Is Due To Chance.

If We Knew Exactly The Laws Of Nature And The Situation Of The Universe At The Initial Moment, We Could Predict Exactly The Situation Of The Same Universe At A Succeeding Moment.

[t]he Different Branches Of Arithmetic - Ambition [g]eometry Is Not True, It Is Advantageous.

Doubt Everything Or Believe Everything: These Are Two Equally Convenient Strategies. With Either We Dispense With The Need For Reflection.

It Is The Simple Hypotheses Of Which One Must Be Most Wary; Because These Are The Ones That Have The Most Chances Of Passing Unnoticed.

Just As Houses Are Made Of Stones, So Is Science Made Of Facts.

Einstein Does Not Remain Attached To The Classical Principles, And When Presented With A Problem In Physics He Quickly Envisages All Of Its Possibilities. This Leads Immediately In His Mind To The Prediction Of New Phenomena Which May One Day Be Verified By Experiment.

All That We Can Hope From These Inspirations, Which Are The Fruits Of Unconscious Work, Is To Obtain Points Of Departure For Such Calculations. As For The Calculations Themselves, They Must Be Made In The Second Period Of Conscious Work Which Follows The Inspiration, And In Which The Results Of The Inspiration Are Verified And The Consequences Deduced.‎

What Is It Indeed That Gives Us The Feeling Of Elegance In A Solution, In A Demonstration?

A First Fact Should Surprise Us, Or Rather Would Surprise Us If We Were Not Used To It. How Does It Happen There Are People Who Do Not Understand Mathematics? If Mathematics Invokes Only The Rules Of Logic, Such As Are Accepted By All Normal Minds...how Does It Come About That So Many Persons Are Here Refractory?

If We Ought Not To Fear Mortal Truth, Still Less Should We Dread Scientific Truth. In The First Place It Can Not Conflict With Ethics? But If Science Is Feared, It Is Above All Because It Can Give No Happiness? Man, Then, Can Not Be Happy Through Science But Today He Can Much Less Be Happy Without It.

Mathematics Is The Art Of Giving The Same Name To Different Things.

...the Feeling Of Mathematical Beauty, Of The Harmony Of Numbers And Of Forms, Of Geometric Elegance. It Is A Genuinely Aesthetic Feeling, Which All Mathematicians Know

It Is With Logic That One Proves; It Is With Intuition That One Invents.

For A Long Time The Objects That Mathematicians Dealt With Were Mostly Ill-defined; One Believed One Knew Them, But One Represented Them With The Senses And Imagination; But One Had But A Rough Picture And Not A Precise Idea On Which Reasoning Could Take Hold.

Point Set Topology Is A Disease From Which The Human Race Will Soon Recover.

Logic Sometimes Makes Monsters. For Half A Century We Have Seen A Mass Of Bizarre Functions Which Appear To Be Forced To Resemble As Little As Possible Honest Functions Which Serve Some Purpose.

Experiment Is The Sole Source Of Truth. It Alone Can Teach Us Something New; It Alone Can Give Us Certainty.

Science Is Built Up With Facts, As A House Is With Stones. But A Collection Of Facts Is No More A Science Than A Heap Of Stones Is A House.

It Is By Logic We Prove. It Is By Intuition We Discover.

Chance ... Must Be Something More Than The Name We Give To Our Ignorance.

How Is Error Possible In Mathematics?

Astronomy Is Useful Because It Raises Us Above Ourselves; It Is Useful Because It Is Grand; .... It Shows Us How Small Is Man's Body, How Great His Mind, Since His Intelligence Can Embrace The Whole Of This Dazzling Immensity, Where His Body Is Only An Obscure Point, And Enjoy Its Silent Harmony.

It May Happen That Small Differences In The Initial Conditions Produce Very Great Ones In The Final Phenomena.

The Scientist Does Not Study Nature Because It Is Useful; He Studies It Because He Delights In It, And He Delights In It Because It Is Beautiful.

Later Generations Will Regard Mengenlehre (set Theory) As A Disease From Which One Has Recovered.

Hypotheses Are What We Lack The Least.

Mathematicians Do Not Study Objects, But Relations Among Objects; They Are Indifferent To The Replacement Of Objects By Others As Long The Relations Don't Change. Matter Is Not Important, Only Form Interests Them.

One Would Have To Have Completely Forgotten The History Of Science So As To Not Remember That The Desire To Know Nature Has Had The Most Constant And The Happiest Influence On The Development Of Mathematics.

It Is Not Order Only, But Unexpected Order, That Has Value.

If Nature Were Not Beautiful, It Would Not Be Worth Knowing, And If Nature Were Not Worth Knowing, Life Would Not Be Worth Living

. . . By Natural Selection Our Mind Has Adapted Itself To The Conditions Of The External World. It Has Adopted The Geometry Most Advantageous To The Species Or, In Other Words, The Most Convenient. Geometry Is Not True, It Is Advantageous.

Why Is It That Showers And Even Storms Seem To Come By Chance, So That Many People Think It Quite Natural To Pray For Rain Or Fine Weather, Though They Would Consider It Ridiculous To Ask For An Eclipse By Prayer.

Thought Is Only A Flash Between Two Long Nights, But This Flash Is Everything.

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