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Mathematics Is A Presuppositionless Science. To Found It I Do Not Need God, As Does Kronecker, Or The Assumption Of A Special Faculty Of Our Understanding Attuned To The Principle Of Mathematical Induction, As Does Poincaré, Or The Primal Intuition Of Brouwer, Or, Finally, As Do Russell And Whitehead, Axioms Of Infinity, Reducibility, Or Completeness, Which In Fact Are Actual, Contentual Assumptions That Cannot Be Compensated For By Consistency Proofs.

-David Hilbert

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Mathematics Is A Presuppositionless

David Hilbert

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Mathematics Is A Presuppositionless Science. To Found It I Do Not Need God, As Does Kronecker, Or The Assumption Of A Special Faculty Of Our Understanding Attuned To The Principle Of Mathematical Induction, As Does Poincaré, Or The Primal Intuition Of Brouwer, Or, Finally, As Do Russell And Whitehead, Axioms Of Infinity, Reducibility, Or Completeness, Which In Fact Are Actual, Contentual Assumptions That Cannot Be Compensated For By Consistency Proofs.

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For Us There Is No Ignorabimus, And In My Opinion None Whatever In Natural Science. In Opposition To The Foolish Ignorabimus Our Slogan Shall Be: "we Must Know - We Will Know!"

We Ought Not To Believe Those Who Today, Adopting A Philosophical Air And With A Tone Of Superiority, Prophesy The Decline Of Culter And Are Content With The Unknowable In A Self-satisfied Way. For Us There Is No Unknowable, And In My Opinion There Is Also Non Whatsoever For The Natural Sciences. In Place Of This Foolish Unknowable, Let Our Watchword On The Contrary Be: We Must Know - We Shall Know.

A Mathematical Problem Should Be Difficult In Order To Entice Us, Yet Not Completely Inaccessible, Lest It Mock At Our Efforts. It Should Be To Us A Guide Post On The Mazy Paths To Hidden Truths, And Ultimately A Reminder Of Our Pleasure In The Successful Solution.

Mathematics Is A Presuppositionless Science. To Found It I Do Not Need God, As Does Kronecker, Or The Assumption Of A Special Faculty Of Our Understanding Attuned To The Principle Of Mathematical Induction, As Does Poincaré, Or The Primal Intuition Of Brouwer, Or, Finally, As Do Russell And Whitehead, Axioms Of Infinity, Reducibility, Or Completeness, Which In Fact Are Actual, Contentual Assumptions That Cannot Be Compensated For By Consistency Proofs.

Physics Is Much Too Hard For Physicists.

Who Of Us Would Not Be Glad To Lift The Veil Behind Which The Future Lies Hidden; To Cast A Glance At The Next Advances Of Our Science And At The Secrets Of Its Development During Future Centuries? What Particular Goals Will There Be Toward Which The Leading Mathematical Spirits Of Coming Generations Will Strive? What New Methods And New Facts In The Wide And Rich Field Of Mathematical Thought Will The New Centuries Disclose?

In Mathematics ... We Find Two Tendencies Present. On The One Hand, The Tendency Towards Abstraction Seeks To Crystallise The Logical Relations Inherent In The Maze Of Materials ... Being Studied, And To Correlate The Material In A Systematic And Orderly Manner. On The Other Hand, The Tendency Towards Intuitive Understanding Fosters A More Immediate Grasp Of The Objects One Studies, A Live Rapport With Them, So To Speak, Which Stresses The Concrete Meaning Of Their Relations.

Begin With The Simplest Examples.

I Didn't Work Especially Hard At Mathematics At School, Because I Knew That's What I'd Be Doing Later.

The Art Of Doing Mathematics Consists In Finding That Special Case Which Contains All The Germs Of Generality.

As Long As A Branch Of Science Offers An Abundance Of Problems, So Long It Is Alive; A Lack Of Problems Foreshadows Extinction Or The Cessation Of Independent Development.

The Arithmetical Symbols Are Written Diagrams And The Geometrical Figures Are Graphic Formulas.

Physics Is Becoming Too Difficult For The Physicists.

No Other Question Has Ever Moved So Profoundly The Spirit Of Man; No Other Idea Has So Fruitfully Stimulated His Intellect; Yet No Other Concept Stands In Greater Need Of Clarification Than That Of The Infinite.

Every Kind Of Science, If It Has Only Reached A Certain Degree Of Maturity, Automatically Becomes A Part Of Mathematics.

We Do Not Master A Scientific Theory Until We Have Shelled And Completely Prised Free Its Mathematical Kernel.

Some People Have Got A Mental Horizon Of Radius Zero And Call It Their Point Of View.

One Must Be Able To Say At All Times--instead Of Points, Straight Lines, And Planes--tables, Chairs, And Beer Mugs

The Infinite! No Other Question Has Ever Moved So Profoundly The Spirit Of Man.

No One Shall Expel Us From The Paradise Which Cantor Has Created For Us. Expressing The Importance Of Cantor's Set Theory In The Development Of Mathematics.

The Further A Mathematical Theory Is Developed, The More Harmoniously And Uniformly Does Its Construction Proceed, And Unsuspected Relations Are Disclosed Between Hitherto Separated Branches Of The Science.

Wir Mussen Wissen. Wir Werden Wissen. We Must Know. We Will Know. Inscribed On His Tomb In Gilttingen.

How Thoroughly It Is Ingrained In Mathematical Science That Every Real Advance Goes Hand In Hand With The Invention Of Sharper Tools And Simpler Methods Which, At The Same Time, Assist In Understanding Earlier Theories And In Casting Aside Some More Complicated Developments.

If I Were To Awaken After Having Slept For A Thousand Years, My First Question Would Be: Has The Riemann Hypothesis Been Proven?

One Hears A Lot Of Talk About The Hostility Between Scientists And Engineers. I Don't Believe In Any Such Thing. In Fact I Am Quite Certain It Is Untrue... There Cannot Possibly Be Anything In It Because Neither Side Has Anything To Do With The Other.

However Unapproachable These Problems May Seem To Us And However Helpless We Stand Before Them, We Have, Nevertheless, The Firm Conviction That Their Solution Must Follow By A Finite Number Of Purely Logical Processes.

Sometimes It Happens That A Man's Circle Of Horizon Becomes Smaller And Smaller, And As The Radius Approaches Zero It Concentrates On One Point. And Then That Becomes His Point Of View.

Before Beginning [to Try To Prove Fermat's Last Theorem] I Should Have To Put In Three Years Of Intensive Study, And I Haven't That Much Time To Squander On A Probable Failure.

An Old French Mathematician Said: "a Mathematical Theory Is Not To Be Considered Complete Until You Have Made It So Clear That You Can Explain It To The First Man Whom You Meet On The Street." This Clearness And Ease Of Comprehension, Here Insisted On For A Mathematical Theory, I Should Still More Demand For A Mathematical Problem If It Is To Be Perfect; For What Is Clear And Easily Comprehended Attracts, The Complicated Repels Us.

The Tool Which Serves As Intermediary Between Theory And Practice, Between Thought And Observation, Is Mathematics; It Is Mathematics Which Builds The Linking Bridges And Gives The Ever More Reliable Forms.

I Have Tried To Avoid Long Numerical Computations, Thereby Following Riemann's Postulate That Proofs Should Be Given Through Ideas And Not Voluminous Computations.

Galileo Was No Idiot. Only An Idiot Could Believe That Science Requires Martyrdom - That May Be Necessary In Religion, But In Time A Scientific Result Will Establish Itself.

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Mathematical Science Is In My Opinion An Indivisible Whole, An Organism Whose Vitality Is Conditioned Upon The Connection Of Its Parts.

I Do Not Want To Presuppose Anything As Known. I See In My Explanation In Section 1 The Definition Of The Concepts Point, Straight Line And Plane, If One Adds To These All The Axioms Of Groups I-v As Characteristics. If One Is Looking For Other Definitions Of Point, Perhaps By Means Of Paraphrase In Terms Of Extensionless, Etc., Then, Of Course, I Would Most Decidedly Have To Oppose Such An Enterprise. One Is Then Looking For Something That Can Never Be Found, For There Is Nothing There, And Everything Gets Lost, Becomes Confused And Vague, And Degenerates Into A Game Of Hide And Seek.

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Besides It Is An Error To Believe That Rigour Is The Enemy Of Simplicity. On The Contrary We Find It Confirmed By Numerous Examples That The Rigorous Method Is At The Same Time The Simpler And The More Easily Comprehended. The Very Effort For Rigor Forces Us To Find Out Simpler Methods Of Proof.

One Can Measure The Importance Of A Scientific Work By The Number Of Earlier Publications Rendered Superfluous By It

I Do Not See That The Sex Of The Candidate Is An Argument Against Her Admission As A Privatdozent. After All, The Senate Is Not A Bathhouse. Objecting To Sex Discrimination Being The Reason For Rejection Of Emmy Noether's Application To Join The Faculty At The University Of Gottingen.

Keep Computations To The Lowest Level Of The Multiplication Table.

[on Cantor's Work:] The Finest Product Of Mathematical Genius And One Of The Supreme Achievements Of Purely Intellectual Human Activity.

He Who Seeks For Methods Without Having A Definite Problem In Mind Seeks In The Most Part In Vain.

If One Were To Bring Ten Of The Wisest Men In The World Together And Ask Them What Was The Most Stupid Thing In Existence, They Would Not Be Able To Discover Anything So Stupid As Astrology.

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