Mathematics quotes

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John_Adams

Study

“The science of government it is my duty to study, more than all other sciences; the arts of legislation and administration and negotiation ought to take the place of, indeed exclude, in a manner, all other arts. I must study politics and war, that our sons may have liberty to study mathematics and philosophy. Our sons ought to study mathematics and philosophy, geography, natural history and naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry and porcelain.”
  • Source: John Adams in a letter to wife Abigail Adams (12 May 1780).
Keywords: Study_quotes Science_quotes Government_quotes Legislation_quotes Administration_quotes Negotiation_quotes Political_quotes War_quotes Liberty_quotes Mathematics_quotes Philosophy_quotes Geography_quotes Natural_History_quotes Naval_quotes Navigation_quotes Commerce_quotes Agriculture_quotes Art_quotes Painting_quotes Poetry_quotes Music_quotes Architecture_quotes Statuary_quotes Tapestry_quotes Porcelain_quotes

Isaac_Asimov

Science and Mathematics

“It is quite possible that mathematics was invented in the ancient Middle East to keep track of tax receipts and grain stores. How odd that out of this should come a subtle scientific language that can effectively describe and predict the most arcane aspects of the Universe.”
  • Source: Isaac Asimov, chapter epigraph in Isaac Asimov’s Book of Science and Nature Quotations, edited by Isaac Asimov and Jason A. Shulman, published in 1988 by Weidenfeld and Nicolson, New York. Chapter 68, page 265. ISBN 1-555-84111-2.
Keywords: Possible_quotes Mathematics_quotes Invention_quotes Ancient_quotes MiddleEast_quotes Keep_quotes Track_quotes Tax_quotes Receipt_quotes Grain_quotes Store_quotes Odd_quotes Come_quotes Subtle_quotes Scientific_quotes Language_quotes Effective_quotes Describe_quotes Predict_quotes Arcane_quotes Aspect_quotes Universe_quotes

Francis_Bacon

Axioms

“It cannot be that axioms established by argumentation should avail for the discovery of new works, since the subtlety of nature is greater many times over than the subtlety of argument. But axioms duly and orderly formed from particulars easily discover the way to new particulars, and thus render sciences active.”
  • Source: Sir Francis Bacon, From Novum Oranum (1620), Book 1, Aphorism 24. Translated as The New Organon: Aphorisms Concerning the Interpretation of Nature and the Kingdom of Man), collected in James Spedding, Robert Ellis and Douglas Heath (eds.), The Works of Francis Bacon (1857), Vol. 4, 51.
Keywords: Mathematics_quotes Axiom_quotes Reason_quotes Scientific_Method_quotes Establish_quotes Argument_quotes Avail_quotes Discover_quotes Work_quotes Nature_quotes Great_quotes Subtle_quotes Order_quotes Particular_quotes Easy_quotes Render_quotes Science_quotes Active_quotes

Robert_N_Britcher

Requirements Testing

“Requirements testing is a subtle way of treating a nonlinear, unpredictable system as if it were a discrete symbol like a number or a word. The potential and the actual become muddled. Requirements are natural language statements, peppered in places with arithmetic. These are discrete and usually quite vague representations of the system as it might become. But the truth is that two teams writing requirements for the same system would describe them differently, and neither would get it quite right, simply because 'right' cannot be known until the system is built. ... Requirements testing is a little like trying to reduce crime in England by running a test for each line in the Magna Carta. In practice, rhetoric replaces the scientific method. As with mathematical proofs, the quest for 'true or not false' soon turns into a party - a not very cordial party - marked by endless debate over whether this sentence or that sentence has been verified by this moment of behavior or that.”
  • Source: Britcher, Robert N. (1999). The Limits of Software: People, Projects, and Perspectives. Reading, MA: Addison Wesley. ISBN 0-201-43323-0. Page 157.
Keywords: Requirements_quotes Testing_quotes Nonlinear_quotes Unpredictable_quotes Prediction_quotes System_quotes Symbol_quotes Potential_quotes Actual_quotes Language_quotes Arithmetic_quotes Discrete_quotes Vagueness_quotes Representation_quotes Truth_quotes Team_quotes Writing_quotes Description_quotes Correctness_quotes Accuracy_quotes Knowledge_quotes Magna_Carta_quotes Rhetorical_quotes Scientific_Method_quotes Mathematics_quotes True_quotes False_quotes Debate_quotes Verification_quotes Behavior_quotes Subtle_quotes Treat_quotes Number_quotes Word_quotes Muddle_quotes Natural_quotes Pepper_quotes Right_quotes Build_quotes Try_quotes Reduce_quotes Crime_quotes England_quotes Line_quotes Practice_quotes Proof_quotes Quest_quotes Party_quotes Cordial_quotes Sentence_quotes Moment_quotes

Fritjof_Capra

Quantum Theory Acceptance

"The mathematical framework of quantum theory has passed countless successful tests and is now universally accepted as a consistent and accurate description of all atomic phenomena. The verbal interpretation, on the other hand – i.e., the metaphysics of quantum theory – is on far less solid ground. In fact, in more than forty years physicists have not been able to provide a clear metaphysical model."
  • Source: Fritjof Capra, In The Tao of Physics (1975), 132.
Keywords: Mathematics_quotes Quantum_quotes Testing_quotes Theory_quotes Repeatability_quotes Consistency_quotes Accuracy_quotes Atomic_quotes Interpretation_quotes Metaphysics_quotes Physics_quotes Framework_quotes Pass_quotes Success_quotes Universal_quotes Accept_quotes Description_quotes Phenomena_quotes Verbal_quotes Solid_quotes Ground_quotes Clear_quotes Model_quotes

Marvin_Chester

Mathematical Quantum Predictions

"The mathematical predictions of quantum mechanics yield results that are in agreement with experimental findings. That is the reason we use quantum theory. That quantum theory fits experiment is what validates the theory, but why experiment should give such peculiar results is a mystery. This is the shock to which Bohr referred."'
Keywords: Mathematics_quotes Prediction_quotes Quantum_quotes Experiment_quotes Mystery_quotes Mechanics_quotes Yield_quotes Result_quotes Agree_quotes Finding_quotes Theory_quotes Validate_quotes Peculiar_quotes Shock_quotes Bohr(topic)_quotes


Alistair_Cockburn

Formal Development

"In the formal development of communications software, in 1987, I was given the motivation, 'The problem with software development is that there is too much ambiguity in the problem statement and in the design description. Things will go better if we get people to work with mathematical formalisms.” After some work in this area, I discovered:
Problem 1. The people on the projects were not interested in learning our system.
Problem 2. They were successfully able to ignore us, and were still delivering software, anyway.
I shifted out of formal development, hearing my formal development colleagues saying, 'The problem lies in the training. Things will go better if we start training students in the necessary mathematics earlier, i.e., secondary school.' Based on what I have learned about people, I expect this will be a frustrated desire. It is not that I doubt certain advantages of formal development, but I do doubt our ability to coax 10**6 people to master it. The question should become, 'Under what circumstances, and with what assignment, should a formal specialist be called in?'"

Source: Cockburn, A. A. R. (1999). Characterizing people as non-linear 1st order components in software development. HaT Technical Report 1999.03, Humans and Technology, Inc.. Retrieved 2019-06-29 from https://ameyakarve.wordpress.com/2012/07/22/characterizing-people-as-non-linear-1st-order-components-in-software-development-cockburn-us/.

Keywords: Formality_quotes Methodology_quotes Software_quotes Development_quotes Ambiguity_quotes Design_quotes Specification_quotes Requirements_quotes Training_quotes Education_quotes Specialization_quotes Communication_quotes Motivation_quotes Problem_quotes Statement_quotes Description_quotes People_quotes Work_quotes Mathematics_quotes Learning_quotes Success_quotes Ignore_quotes Deliver_quotes Colleague_quotes Student_quotes School_quotes Frustration_quotes Desire_quotes Advantage_quotes Circumstance_quotes Assignment_quotes Master_quotes

Nicolaus_Copernicus

Mathematics

“Mathematics is written for mathematicians...”
  • Source: In: Great Books of the Western World (Volume 16), On the Revolutions of the Heavenly Spheres, Preface and Dedication to Pope Paul III (p. 509)
Keywords: Mathematics_quotes Write_quotes

Paul_Davies

Models and Theories and Reality

"Until now, physical theories have been regarded as merely models with approximately describe the reality of nature. As the models improve, so the fit between theory and reality gets closer. Some physicists are now claiming that supergravity is the reality, that the model and the real world are in mathematically perfect accord."
  • Source: P.C.W. Davies, Superforce (1984, 1985), 149.
Keywords: Physical_quotes Theory_quotes Model_quotes Approximation_quotes Reality_quotes Nature_quotes Physicist_quotes Gravity_quotes Mathematics_quotes Fit_quotes Supergravity_quotes Accord_quotes

David_Deutsch

Mathematical Knowledge

“Mathematical knowledge may, just like our scientific knowledge, be deep and broad, it may be subtle and wonderfully explanatory, it may be uncontroversially accepted; but it cannot be certain.”
  • Source: David Deutsch in the book “The Fabric of Reality”. ISBN 978-0713990614. Published in 1997 by Viking Press. Chapter 10.
Keywords: Mathematics_quotes Knowledge_quotes Science_quotes Depth_quotes Breadth_quotes Explanation_quotes Acceptability_quotes Certainty_quotes

Theory of Computation

“The theory of computation has traditionally been studied almost entirely in the abstract, as a topic in pure mathematics. This is to miss the point of it. Computers are physical objects, and computations are physical processes. What computers can or cannot compute is determined by the laws of physics alone, and not by pure mathematics.”
Keywords: Computation_quotes Theory_quotes Tradition_quotes Abstraction_quotes Mathematics_quotes Computer_quotes Physical_quotes Object_quotes Physics_quotes Study_quotes Determine_quotes

Edsger_Wybe_Dijkstra

Math and Language

“Besides a mathematical inclination, an exceptionally good mastery of one's native tongue is the most vital asset of a competent programmer.”
Keywords: Mathematics_quotes Language_quotes Verbal_quotes Competence_quotes Programmer_quotes Inclination_quotes Exceptional_quotes Mastery_quotes Native_quotes Tongue_quotes Vital_quotes Asset_quotes

Programming is Difficult

“Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians.”
Keywords: Programming_quotes Difficulty_quotes Branch_quotes Mathematics_quotes Apply_quotes Poor_quotes

Formulas

“A picture may be worth a thousand words, a formula is worth a thousand pictures.”
Keywords: Picture_quotes Word_quotes Formula_quotes Mathematics_quotes Worth_quotes

Paul_Dirac

Mathematics Will Lead

“If you are receptive and humble mathematics will lead you by the hand. Again and again, when I have been at a loss how to proceed, I have just had to wait until I have felt the mathematics led me by the hand. It has led me along an unexpected path, a path where new vistas open up, a path leading to new territory, where one can set up a base of operations, from which one can survey the surroundings and plan future progress.”

  • Source: Paul Dirac As quoted in “The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom” (2009) by Graham Farmelo, p. 435.
Keywords: Receptivity_quotes Humility_quotes Mathematics_quotes Guidance_quotes Leadership_quotes Hand_quotes Loss_quotes Proceed_quotes Wait_quotes Lead_quotes Unexpected_quotes Path_quotes Vista_quotes New_quotes Territory_quotes Base_quotes Operation_quotes Survey_quotes Plan_quotes Future_quotes Progress_quotes

Physics and Poetry

“How can you do both physics and poetry? In physics we try to explain in simple terms something that nobody knew before. In poetry it is the exact opposite.”
  • Source: Paul Dirac to Oppenheimer, as reported in H. S. Kragh, “Dirac: A Scientific Biography”, Cambridge University Press (1990), p. 258. ISBN 0-521-38089-8.
Keywords: Physics_quotes Mathematics_quotes Poetry_quotes Explanation_quotes Simplicity_quotes Knowledge_quotes Before_quotes After_quotes

Albert_Einstein

Magnificent Applied Science

“Why does this magnificent applied science which saves work and makes life easier bring us so little happiness? The simple answer runs: Because we have not yet learned to make sensible use of it. In war it serves that we may poison and mutilate each other. In peace it has made our lives hurried and uncertain. Instead of freeing us in great measure from spiritually exhausting labor, it has made men into slaves of machinery, who for the most part complete their monotonous long day's work with disgust and must continually tremble for their poor rations... It is not enough that you should understand about applied science in order that your work may increase man's blessings. Concern for the man himself and his fate must always form the chief interest of all technical endeavours; concern for the great unsolved problems of the organization of labor and the distribution of goods in order that the creations of our mind shall be a blessing and not a curse to mankind. Never forget this in the midst of your diagrams and equations.”
  • Source: Albert Einstein [Speaker];
    • 1950-02-16 [Speech/Date]; California Institute of Technology [Location]; CalTech students [Audience];
    • 1950-02-17 [Reporting/Article/Date]; “Einstein Sees Lack in Applying Science” [Title]; Anonymous [Article/Author]; New York Times [Reporting/Newspaper]; 8 [Page];
Keywords: Science_quotes Happiness_quotes War_quotes Wisdom_quotes Peace_quotes Work_quotes Slavery_quotes Automation_quotes Ethics_quotes Values_quotes Empathy_quotes Labor_quotes Distribution_quotes Mind_quotes Humanity_quotes Mathematics_quotes Engineering_quotes Magnificent_quotes Applied_quotes Save_quotes Life_quotes Simple_quotes Answer_quotes Learn_quotes Sensible_quotes Poison_quotes Mutilate_quotes Hurry_quotes Uncertain_quotes Freedom_quotes Spiritual_quotes Exhausting_quotes Men_quotes Machinery_quotes Monotonous_quotes Long_quotes Day_quotes Disgust_quotes Continual_quotes Tremble_quotes Poor_quotes Rations_quotes Understand_quotes Increase_quotes Blessing_quotes Man_quotes Fate_quotes Chief_quotes Interest_quotes Technical_quotes Endeavour_quotes Concern_quotes Problem_quotes Unsolved_quotes Organization_quotes Creation_quotes Curse_quotes Mankind_quotes Forget_quotes Diagram_quotes Equation_quotes

Not Everything Counts

“Not everything that can be counted, counts. Not everything that counts can be counted.”
  • Source:
    • Probably misattributed to Einstein. William Bruce Cameron is more likely the original source.
Keywords: Mathematics_quotes Values_quotes Count_quotes

Milton_Friedman

Positive Knowledge

“In both social and natural sciences, the body of positive knowledge grows by the failure of a tentative hypothesis to predict phenomena the hypothesis professes to explain; by the patching up of that hypothesis until someone suggests a new hypothesis that more elegantly or simply embodies the troublesome phenomena, and so on ad infinitum. In both, experiment is sometimes possible, sometimes not (witness meteorology). In both, no experiment is ever completely controlled, and experience often offers evidence that is the equivalent of controlled experiment. In both, there is no way to have a self-contained closed system or to avoid interaction between the observer and the observed. The Gödel theorem in mathematics, the Heisenberg uncertainty principle in physics, the self-fulfilling or self-defeating prophecy in the social sciences all exemplify these limitations.”
  • Source: Milton Friedman, Inflation and Unemployment (1976), 348
Keywords: Society_quotes Nature_quotes Science_quotes Knowledge_quotes Failure_quotes Tentative_quotes Hypothesis_quotes Prediction_quotes Phenomena_quotes Explanation_quotes Elegant_quotes Infinity_quotes Experiment_quotes Meteorology_quotes Control_quotes Evidence_quotes Equal_quotes Closed_quotes System_quotes Avoidance_quotes Interaction_quotes Observation_quotes Measurement_quotes Uncertainty_quotes Mathematics_quotes Principle_quotes Physics_quotes Self-Fulfill_quotes Self-Defeat_quotes Prophecy_quotes Limitation_quotes Positive_quotes Grow_quotes Suggest_quotes Troublesome_quotes Gödel(topic)_quotes Heisenberg(topic)_quotes

Galileo

Universe Mathematical

“The universe...stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics...”
  • Source: Galileo Il Saggiatore (1623)
Keywords: Universe_quotes Continuous_quotes Gaze_quotes Perception_quotes Understanding_quotes Learning_quotes Comprehension_quotes Language_quotes Interpretation_quotes Character_quotes Symbol_quotes Writing_quotes Mathematics_quotes Galilean_quotes

Murray_Gell-Mann

Curiosity

“My colleagues in elementary particle theory in many lands [and I] are driven by the usual insatiable curiosity of the scientist, and our work is a delightful game. I am frequently astonished that it so often results in correct predictions of experimental results. How can it be that writing down a few simple and elegant formulae, like short poems governed by strict rules such as those of the sonnet or the waka, can predict universal regularities of Nature?”
  • Source: Murray Gell-Mann, Nobel Banquet Speech (10 Dec 1969), in Wilhelm Odelberg (ed.),Les Prix Nobel en 1969 (1970).
Keywords: Particle_quotes Theory_quotes Curiosity_quotes Insatiable_quotes Science_quotes Work_quotes Game_quotes Correctness_quotes Prediction_quotes Experiment_quotes Result_quotes Simplicity_quotes Elegance_quotes Formula_quotes Mathematics_quotes Poetry_quotes Rule_quotes Repeatability_quotes Nature_quotes Universal_quotes

Gordon_Glegg

Engineering Qualifications

“If you can do maths, engineering is waiting for you... There is one overriding qualification: you will never be a good engineer if you are frightened by hard work or responsibility.”
  • Source: Glegg, G. L. (1969). The Design of Design. Cambridge, UK: Cambridge University Press. ISBN: 978-0-511-79996-9. Chapter 2, page 25.
Keywords: Mathematics_quotes Engineering_quotes Qualification_quotes Word_quotes Effort_quotes Responsibility_quotes Fear_quotes Wait_quotes Override_quotes Frighten_quotes Hard_quotes Work_quotes

Kurt_Gödel

Mind Machine

“Either mathematics is too big for the human mind, or the human mind is more than a machine.”
  • Source: The words of Kurt Gödel as reported in the book ‘Topoi : The Categorial Analysis of Logic’ (1979) by Robert Goldblatt, page 13.
Keywords: Mathematics_quotes Big_quotes Human_quotes Mind_quotes More_quotes Machine_quotes

Richard_Hamming

Scientific Laws

“Science is composed of laws which were originally based on a small, carefully selected set of observations, often not very accurately measured originally; but the laws have later been found to apply over much wider ranges of observations and much more accurately than the original data justified.”
  • Source: Richard Hamming, 'The Unreasonable Effectiveness of Mathematics', The American Mathematical Monthly (Feb 1980), 87 No.2.
Keywords: Science_quotes Law_quotes Observation_quotes Mathematics_quotes Measurement_quotes Accuracy_quotes Data_quotes Justification_quotes Compose_quotes Apply_quotes

Stephen_Hawking

Imaginary Reality

“One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So, what is real and what is imaginary? Is the distinction just in our minds?”
  • Source: Hawking, Stephen in the book “The Universe in a Nutshell”, p. 59 (2001).
Keywords: Thinking_quotes Meaning_quotes Imaginary_quotes Mathematics_quotes Game_quotes Reality_quotes World_quotes Viewpoint_quotes Positivist_quotes Philosophy_quotes Determine_quotes Real_quotes Find_quotes Model_quotes Describe_quotes Universe_quotes Live_quotes Involve_quotes Number_quotes Time_quotes Prediction_quotes Effect_quotes Observation_quotes Measurement_quotes Believe_quotes Reason_quotes Distinction_quotes Mind_quotes

Donald_Hoffman

Demystify Consciousness

“Cats can't do calculus and monkeys can't do quantum theory, so why assume that Homo sapiens can demystify consciousness?”
  • Source: Donald Hoffman in his book “The Case Against Reality: Why Evolution Hid the Truth from Our Eyes”. Published by W. W. Norton, New York, NY, in 2019. ISBN 978-0393254693. Chapter 1, page 33.
Keywords: Cat_quotes Calculus_quotes Mathematics_quotes Monkey_quotes Quantum_quotes Theory_quotes Assume_quotes Homo-Sapiens_quotes Human_quotes Demystify_quotes Consciousness_quotes

Sabine_Hossenfelder

Closer Reality

“The foundations of physics give us a closer look at reality, but the closer we look at reality, the more slippery it becomes. Our heavy use of mathematics is a major reason. The more the fundamental descriptions of nature have become divorced from our everyday experience, the more we must rely on mathematical rigor. This reliance has consequences. Using math to describe reality means that the same observations can be equivalently explained in many different ways. That’s just because there are many sets of mathematical axioms that will give the exact same predictions for all available data. Thus, if you want to assign 'reality' to one of your explanations, you won’t know which.”
  • Source: Hossenfelder, Sabine. Existential Physics: A Scientist's Guide to Life's Biggest Questions (p. 74). Penguin Publishing Group. Kindle Edition.
Keywords: Foundation_quotes Physics_quotes Closer_quotes Reality_quotes Slippery_quotes Heavy_quotes Mathematics_quotes Major_quotes Reason_quotes Fundamental_quotes Description_quotes Nature_quotes Divorce_quotes Everyday_quotes Experience_quotes Rely_quotes Rigor_quotes Consequence_quotes Observation_quotes Same_quotes Equivalent_quotes Explanation_quotes Different_quotes Way_quotes Set_quotes Axiom_quotes Exact_quotes Prediction_quotes Available_quotes Data_quotes Assign_quotes Know_quotes

David_Kipping

Simulation Experiments

“The simulation argument might be able to resist our physical experiments but is not able to escape our mathematical and logical reasoning.”
  • Source: David Kipping from the YouTube “Why You're Probably Not a Simulation”, posted 2020-09-04 in the channel ‘Cool Worlds’. Video offset 32:36. Online at https://youtu.be/HA5YuwvJkpQ (2025).
Keywords: Simulation_quotes Argument_quotes Might_quotes Able_quotes Resist_quotes Physical_quotes Experiment_quotes Escape_quotes Mathematics_quotes Logic_quotes Reason_quotes

Pierre-Simon_Laplace

System of Nature

“The present state of the system of nature is evidently a consequence of what it was in the preceding moment, and if we conceive of an intelligence that at a given instant comprehends all the relations of the entities of this universe, it could state the respective position, motions, and general affects of all these entities at any time in the past or future. Physical astronomy, the branch of knowledge that does the greatest honor to the human mind, gives us an idea, albeit imperfect, of what such an intelligence would be. The simplicity of the law by which the celestial bodies move, and the relations of their masses and distances, permit analysis to follow their motions up to a certain point; and in order to determine the state of the system of these great bodies in past or future centuries, it suffices for the mathematician that their position and their velocity be given by observation for any moment in time. Man owes that advantage to the power of the instrument he employs, and to the small number of relations that it embraces in its calculations. But ignorance of the different causes involved in the production of events, as well as their complexity, taken together with the imperfection of analysis, prevents our reaching the same certainty about the vast majority of phenomena. Thus there are things that are uncertain for us, things more or less probable, and we seek to compensate for the impossibility of knowing them by determining their different degrees of likelihood. So it was that we owe to the weakness of the human mind one of the most delicate and ingenious of mathematical theories, the science of chance or probability.”
  • Source: Pierre-Simon Laplace, 'Recherches, 1º, sur l'Intégration des Équations Différentielles aux Différences Finies, et sur leur Usage dans la Théorie des Hasards' (1773, published 1776). In Oeuvres complètes de Laplace, 14 Vols. (1843-1912), Vol. 8, 144-5, trans. Charles Coulston Gillispie, Pierre-Simon Laplace 1749-1827: A Life in Exact Science (1997), 26.
Keywords: State_quotes Present_quotes System_quotes Nature_quotes Consequence_quotes Moment_quotes Precede_quotes Conceive_quotes Intelligence_quotes Instant_quotes Comprehension_quotes Relationship_quotes Universe_quotes Position_quotes Motion_quotes Causality_quotes Time_quotes Past_quotes Future_quotes Astronomy_quotes Knowledge_quotes Humanity_quotes Mind_quotes Idea_quotes Imperfect_quotes Law_quotes Celestial_quotes Body_quotes Mass_quotes Distance_quotes Analysis_quotes Determine_quotes Century_quotes Mathematics_quotes Velocity_quotes Observation_quotes Advantage_quotes Power_quotes Instrument_quotes Calculation_quotes Ignorance_quotes Cause_quotes Production_quotes Event_quotes Complexity_quotes Certainty_quotes Majority_quotes Phenomena_quotes Probability_quotes Impossibility_quotes Likelihood_quotes Weakness_quotes Theory_quotes Chance_quotes

Problems of Probability

“The most important questions of life... are indeed for the most part only problems of probability. Strictly speaking it may even be said that nearly all our knowledge is problematical; and in the small number of things which we are able to know with certainty, even in the mathematical sciences themselves, the principal means for ascertaining truth — induction and analogy — are based on probabilities.”
Keywords: Important_quotes Question_quotes Life_quotes Part_quotes Problem_quotes Probability_quotes Near_quotes Knowledge_quotes Problematical_quotes Small_quotes Number_quotes Thing_quotes Ability_quotes Certainty_quotes Mathematics_quotes Science_quotes Principal_quotes Mean_quotes Ascertain_quotes Truth_quotes Induction_quotes Analogy_quotes Base_quotes

John_von_Neumann

Language Historical

“It is only proper to realize that language is largely a historical accident... It is only reasonable to assume that logics and mathematics are similarly historical, accidental forms of expression.”
  • Source: John von Neumann. (1958). The Computer and the Brain (1st ed.). New Haven, CT: Yale University Press. Part 2, page 81.
Keywords: Language_quotes History_quotes Logic_quotes Mathematics_quotes Accident_quotes Expression_quotes Assumption_quotes Form_quotes Reason_quotes

Gradual Development of Theory

“What is important is the gradual development of a theory, based on a careful analysis of the ... facts. ... Its first applications are necessarily to elementary problems where the result has never been in doubt and no theory is actually required. At this early stage the application serves to corroborate the theory. The next stage develops when the theory is applied to somewhat more complicated situations in which it may already lead to a certain extent beyond the obvious and familiar. Here theory and application corroborate each other mutually. Beyond lies the field of real success: genuine prediction by theory. It is well known that all mathematized sciences have gone through these successive stages of evolution.”
  • Source: John von Neumann, 'Formulation of the Economic Problem' in Theory of Games and Economic Behavior (1964), 8. Reprinted in John Von Neumann, F. Bródy (ed.) and Tibor Vámos (ed.), The Neumann Compendium (2000), 416.
Keywords: Importance_quotes Gradual_quotes Development_quotes Theory_quotes Careful_quotes Analysis_quotes Fact_quotes Application_quotes Elementary_quotes Problem_quotes Result_quotes Doubt_quotes Requirements_quotes Early_quotes Stage_quotes Service_quotes Corroborate_quotes Complexity_quotes Situation_quotes Lead_quotes Obvious_quotes Familiar_quotes Mutual_quotes Success_quotes Genuine_quotes Prediction_quotes Mathematics_quotes Science_quotes Evolution_quotes Beyond_quotes

Science Models

“The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work — that is, correctly to describe phenomena from a reasonably wide area.”
  • Source: John von Neumann, 'Method in the Physical Sciences', in The Unity of Knowledge, edited by L. Leary (1955), 158. Reprinted in John Von Neumann, F. Bródy (ed.) and Tibor Vámos (ed.), The Neumann Compendium (2000), 628.
Keywords: Science_quotes Explanation_quotes Mathematics_quotes Construct_quotes Sole_quotes Precision_quotes Expectation_quotes Work_quotes Correctness_quotes Description_quotes Phenomena_quotes Reason_quotes Wide_quotes Area_quotes Interpret_quotes Model_quotes Verbal_quotes Observation_quotes Justification_quotes Correct_quotes

Isaac_Newton

Analysis

“As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition. This Analysis consists in making Experiments and Observations, and in drawing general Conclusions from them by Induction, and admitting of no Objections against the Conclusions, but such as are taken from Experiments, or other certain Truths. For Hypotheses are not to be regarded in experimental Philosophy.”
  • Source: Sir Isaac Newton, From Opticks, (1704, 2nd ed. 1718), Book 3, Query 31, 380.
Keywords: Mathematics_quotes Nature_quotes Philosophy_quotes Investigation_quotes Difficulty_quotes Methodology_quotes Analysis_quotes Precede_quotes Composition_quotes Experiment_quotes Observation_quotes Conclusion_quotes Induction_quotes Objection_quotes Truth_quotes Hypothesis_quotes Regard_quotes

Parkinson_and_Margulieux

Spatial Skills

“Higher spatial reasoning skills universally correlate with better STEM [Science Technology Engineering Mathematics] performance and achievement... This correlation is particularly interesting because spatial skills can be trained, and spatial skills training globally improves STEM performance and achievement.”
  • Source: From the journal article entitled “Improving CS Performance by Developing Spatial Skills”, in Communications of the ACM, volume 68, number 1, 2025-01. DOI:10.1145/3680409.
Keywords: High_quotes Spatial_quotes Reason_quotes Skill_quotes Universal_quotes Correlation_quotes Better_quotes STEM_quotes Science_quotes Technology_quotes Engineering_quotes Mathematics_quotes Performance_quotes Achievement_quotes Interest_quotes Train_quotes Global_quotes Improve_quotes

Joel_Pel

Murphy's Law Equation

“A new mathematical foundation to one of today’s most recognized physical laws, Murphy’s Law. Experimental data matches very nicely with the theory presented in this equation.”
MurphysEquation.gif
  • Notes:
    • PM is the Murphy’s probability that something will go wrong.
    • KM is Murphy’s constant (equal to one).
    • FM is Murphy’s factor, a very small number that can only be calculated on a 386-computer running Windows 3.1. Here we will approximate it by FM ≈ 0.01.
    • I is the importance of the result or outcome of the event on a scale of 1 to 10.
    • C is the complexity of the system on a scale of 1 to 10.
    • U is the urgency on a scale of 1 to 10.
    • F is the frequency on a scale of 1 to 10.
    • e is Euler's number, a mathematical constant approximately equal to 2.71828.
  • Source: Written by Joel Pel in the article “The Murphy's Law Equation” published 2007-05-21 in The Science Creative Quarterly.
Keywords: Mathematics_quotes Foundation_quotes Law_quotes Physical_quotes Murphy(topic)_quotes Experiment_quotes Data_quotes Theory_quotes Equation_quotes

Pythagoras

All Is Number

“All is number.”
  • Note: Since all of Pythagoras' original writings (if not all oral) are lost, only second-hand writings from his disciples saying what they claim Pythagoras said are available.
Keywords: Mathematics_quotes Number_quotes Reality_quotes Computed_Reality_quotes

Rittel_and_Webber

Tame Versus Wicked Problems

“The problems that scientists and engineers have usually focused upon are mostly 'tame' or 'benign' ones. As an example, consider a problem of mathematics, such as solving an equation, or the task of an organic chemist in analyzing the structure of some unknown compound, or that of the chess player attempting to accomplish checkmate in five moves. For each the mission is clear. It is clear, in turn, whether or not the problems have been solved. Wicked problems, in contrast, have neither of these clarifying traits.”
Keywords: Problem_quotes Scientist_quotes Engineer_quotes Focus_quotes Tame_quotes Benign_quotes Mathematics_quotes Solve_quotes Equation_quotes Task_quotes Organic_quotes Chemist_quotes Analysis_quotes Structure_quotes Unknown_quotes Compound_quotes Chess_quotes Player_quotes Attempt_quotes Accomplish_quotes Checkmate_quotes Mission_quotes Clear_quotes Wicked_quotes Contrast_quotes Clarify_quotes Trait_quotes

Bertrand_Russell

Persecution Used In Theology

“Persecution is used in theology, not in arithmetic, because in arithmetic there is knowledge, but in theology there is only opinion. So, whenever you find yourself getting angry about a difference of opinion, be on your guard, you will probably find, on examination, that your belief is going beyond what the evidence warrants.”
  • Source: Bertrand Russell, In An Outline of Intellectual Rubbish (1943), 22.
Keywords: Persecution_quotes Theology_quotes Arithmetic_quotes Mathematics_quotes Knowledge_quotes Opinion_quotes Anger_quotes Differences_quotes Guard_quotes Examination_quotes Belief_quotes Beyond_quotes Evidence_quotes Warrant_quotes

Savage Controversies

“The most savage controversies are those about matters as to which there is no good evidence either way. Persecution is used in theology, not in arithmetic.”
  • Source: Bertrand Russell, Unpopular Essays (1950, 2007), 104.
Keywords: Savage_quotes Controversy_quotes Matter_quotes Good_quotes Evidence_quotes Way_quotes Persecution_quotes Theology_quotes Arithmetic_quotes Mathematics_quotes

Rational Man

“Thus the rational man, who attaches to each proposition the right degree of credibility, will be guided by the mathematical theory of probability when it is applicable.”
  • Source: Russell, B. (1948). Human Knowledge: Its Scope and Limits. New York, NY: Simon and Schuster. Part V Chap. 6.
Keywords: Reason_quotes Humanity_quotes Man_quotes Attach_quotes Proposal_quotes Right_quotes Degree_quotes Credibility_quotes Guide_quotes Mathematics_quotes Theory_quotes Probability_quotes Application_quotes

Pure Mathematics

“Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. Both these points would belong to applied mathematics. We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition. These rules of inference constitute the major part of the principles of formal logic. We then take any hypothesis that seems amusing and deduce its consequences. If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.”
  • Source: “Recent Work on the Principles of Mathematics”, published in 1901 within International Monthly, Vol. 4. later published as the essay “Mathematics and the Metaphysicians” within the book “Mysticism and Logic and Other Essays” by Bertrand Russell published in 1917 by George Allen & Unwin, London, England.
Keywords: Pure_quotes Mathematics_quotes Consist_quotes Entire_quotes Assertion_quotes Effect_quotes Proposal_quotes True_quotes Essential_quotes Discussion_quotes First_quotes Mention_quotes Suppose_quotes Point_quotes Belong_quotes Application_quotes Certainty_quotes Rule_quotes Inference_quotes Constitute_quotes Major_quotes Part_quotes Principle_quotes Formal_quotes Logic_quotes Hypothesis_quotes Amuse_quotes Deduction_quotes Consequence_quotes Causality_quotes Particular_quotes Definition_quotes Subject_quotes Knowledge_quotes Uncertainty_quotes Talking_quotes Speak_quotes People_quotes Puzzle_quotes Confusion_quotes Begin_quotes Comfort_quotes Accuracy_quotes

Melvin_Schwartz

Mathematics More than a Language

“Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates. Indeed, the beauty and elegance of the physical laws themselves are only apparent when expressed in the appropriate mathematical framework.”
  • Source: Melvin Schwartz, In Principles of Electrodynamics (1972, 1987), 105.
Keywords: Mathematics_quotes Language_quotes Deal_quotes Physical_quotes World_quotes Source_quotes Model_quotes Abstraction_quotes Enable_quotes Obtain_quotes Amazement_quotes Insight_quotes Way_quotes Nature_quotes Operation_quotes Beauty_quotes Elegant_quotes Law_quotes Apparent_quotes Expression_quotes Appropriate_quotes Framework_quotes

Herbert_Simon

Operational Research Restrictions

“The powerful algorithms associated with OR impose a strong mathematical structure on the decision problem. Their power is bought at the cost of shaping and squeezing the real-world problem to fit their computational requirements.”
Keywords: Power_quotes Algorithm_quotes Association_quotes Operation_quotes Research_quotes Strength_quotes Mathematics_quotes Structure_quotes Decision_quotes Problem_quotes Cost_quotes Buy_quotes Shape_quotes Squeeze_quotes World_quotes Fit_quotes Computation_quotes Requirements_quotes Real_quotes

Design and Analysis

“In view of the key role of design in professional activity, it is ironic that in this century the natural sciences almost drove the sciences of the artificial from professional school curricula, a development that peaked about two or three decades after the Second World War. Engineering schools gradually became schools of physics and mathematics; medical schools became schools of biological science; business schools became schools of finite mathematics. The use of adjectives like 'applied' concealed, but did not change, the fact. It simply meant that in the professional schools those topics were selected from mathematics and the natural sciences for emphasis which were thought to be most nearly relevant to professional practice. It did not mean that design continued to be taught, as distinguished from analysis.”
Keywords: View_quotes Key_quotes Role_quotes Design_quotes Professional_quotes Activity_quotes Irony_quotes Century_quotes Natural_quotes Science_quotes Drive_quotes Artificial_quotes School_quotes Curriculum_quotes Development_quotes Peak_quotes Decade_quotes World_quotes War_quotes Engineering_quotes Gradual_quotes Physics_quotes Mathematics_quotes Medical_quotes Biology_quotes Business_quotes Finite_quotes Applied_quotes Conceal_quotes Change_quotes Fact_quotes Simplicity_quotes Meaning_quotes Topic_quotes Selection_quotes Emphasis_quotes Thinking_quotes Near_quotes Relevance_quotes Practice_quotes Continue_quotes Teaching_quotes Analysis_quotes

Saul-Paul_Sirag

Heart of Science

“The essential point in science is not a complicated mathematical formalism or a ritualized experimentation. Rather the heart of science is a kind of shrewd honesty that springs from really wanting to know what the hell is going on!”
  • Source: Words of Saul-Paul Sirag as reported in the book “Quantum Reality: Beyond the New Physics” written by Nick Herbert, page 268, published by Doubleday in 1985. ISBN 978-0385187046.
Keywords: Essential_quotes Point_quotes Science_quotes Complexity_quotes Mathematics_quotes Formal_quotes Ritual_quotes Experiment_quotes Heart_quotes Shrewd_quotes Honesty_quotes Spring_quotes Want_quotes Knowledge_quotes Hell_quotes Go_quotes On_quotes

Baruch_Spinoza

Good and Evil

“Experience day by day protested and showed by infinite examples, that good and evil fortunes fall to the lot of pious and impious alike; still they would not abandon their inveterate prejudice, for it was more easy for them to class such contradictions among other unknown things of whose use they were ignorant, and thus to retain their actual and innate condition of ignorance, than to destroy the whole fabric of their reasoning and start afresh. They therefore laid down as an axiom, that God's judgments far transcend human understanding. Such a doctrine might well have sufficed to conceal the truth from the human race for all eternity, if mathematics had not furnished another standard of verity in considering solely the essence and properties of figures without regard to their final causes.”
  • Source: A book. Entitled Ethics. Also entitled Ethica. Written by Baruch Spinoza. Originally published 1677. Translated from Latin by R. H. M. Elwes. Part I, Appendix.
Keywords: Ethics_quotes Deity_quotes Religion_quotes Prejudice_quotes Ignorance_quotes Reason_quotes Epistemology_quotes Mathematics_quotes Justice_quotes Causality_quotes Experience_quotes Day_quotes Protest_quotes Show_quotes Infinite_quotes Good_quotes Evil_quotes Fortune_quotes Fall_quotes Lot_quotes Pious_quotes Impious_quotes Abandon_quotes Inveterate_quotes Easy_quotes Contradiction_quotes Unknown_quotes Retain_quotes Actual_quotes Innate_quotes Condition_quotes Destroy_quotes Fabric_quotes Start_quotes Afresh_quotes Laid_quotes Axiom_quotes God_quotes Judgment_quotes Transcend_quotes Human_quotes Understand_quotes Doctrine_quotes Suffice_quotes Conceal_quotes Truth_quotes Race_quotes Eternity_quotes Furnish_quotes Standard_quotes Verity_quotes Consider_quotes Essence_quotes Property_quotes Figure_quotes Regard_quotes Final_quotes

Max_Tegmark

Mathematical Computer Game

“The closer I look at things that seemed non-mathematical, like my arm and my hand, the more mathematical it turns out to be. Could it be that our world also then is really just as mathematical as the computer game reality?”
  • Source: Youtube The Great Math Mystery - Documentary posted by Aerospace Engineering 2017-03-03.
Keywords: Seem_quotes Look_quotes Observation_quotes Mathematics_quotes Arm_quotes Head_quotes World_quotes Reality_quotes Computation_quotes Computer_quotes Game_quotes Closer_quotes Non-mathematical_quotes Hand_quotes

Mathematical Reality

“Many of my physics colleagues will say that mathematics describes our physical reality, at least in some approximate sense. I go further and argue that it actually is our physical reality because I'm arguing that our physical world doesn't just have some mathematical properties, but it has only mathematical properties.”
  • Source: Youtube The Great Math Mystery - Documentary posted by Aerospace Engineering 2017-03-03.
Keywords: Physics_quotes Colleague_quotes Mathematics_quotes Physical_quotes Reality_quotes Approximation_quotes Sense_quotes Argue_quotes Actual_quotes World_quotes Property_quotes Computation_quotes Calculation_quotes Description_quotes

Computer Game

“If I were a character in a computer game, I would also discover eventually that the rules seemed completely rigid and mathematical. That just reflects the computer code in which it was written.”
  • Source: Max Tegmark and John Archibald Wheeler; Scientific American, February 2001
Keywords: Character_quotes Computer_quotes Game_quotes Discover_quotes Rule_quotes Rigid_quotes Mathematics_quotes Reflect_quotes Code_quotes Write_quotes

T_H_White

Learning Cure for Sadness

“‘The best thing for being sad,’ replied Merlyn, beginning to puff and blow, ‘is to learn something. That is the only thing that never fails. You may grow old and trembling in your anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honour trampled in the sewers of baser minds. There is only one thing for it then — to learn. Learn why the world wags and what wags it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting. Learning is the thing for you. Look at what a lot of things there are to learn — pure science, the only purity there is. You can learn astronomy in a lifetime, natural history in three, literature in six. And then, after you have exhausted a milliard lifetimes in biology and medicine and theocriticism and geography and history and economics — why, you can start to make a cartwheel out of the appropriate wood, or spend fifty years learning to begin to learn to beat your adversary at fencing. After that you can start again on mathematics, until it is time to learn to plough.’”
  • Source: White, T. H. (2011). The Once and Future King | 1939. New York, NY: Penguin. ISBN 9781101657546.
Keywords: Best_quotes Sad_quotes Begin_quotes Puff_quotes Blow_quotes Learning_quotes Failure_quotes Grow_quotes Old_quotes Tremble_quotes Anatomy_quotes Lie_quotes Awake_quotes Night_quotes Listening_quotes Disorder_quotes Vein_quotes Miss_quotes Love_quotes See_quotes World_quotes Devastate_quotes Evil_quotes Honor_quotes Trample_quotes Sewer_quotes Base_quotes Mind_quotes Why_quotes Wag_quotes Exhaust_quotes Alienate_quotes Torture_quotes Fear_quotes Distrust_quotes Dreaming_quotes Regret_quotes Look_quotes Pure_quotes Science_quotes Astronomy_quotes Lifetime_quotes Natural_quotes History_quotes Literature_quotes Biology_quotes Medicine_quotes Theocriticism_quotes Geography_quotes Economics_quotes Start_quotes Cartwheel_quotes Appropriate_quotes Wood_quotes Beat_quotes Adversary_quotes Fencing_quotes Mathematics_quotes Time_quotes Plow_quotes

Eugene_Wigner

Miracle of Mathematics

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”
  • Source: Eugene Paul Wigner, In 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences,' Communications in Pure and Applied Mathematics (Feb 1960), 13, No. 1 (February 1960). Collected in Eugene Paul Wigner, A.S. Wightman (ed.), Jagdish Mehra (ed.), The Collected Works of Eugene Paul Wigner (1955), Vol. 6, 535.
Keywords: Miracle_quotes Appropriate_quotes Language_quotes Mathematics_quotes Natural_quotes Science_quotes Formulation_quotes Law_quotes Physics_quotes Wonderful_quotes Gift_quotes Understanding_quotes Deserve_quotes Unreasonable_quotes Effectiveness_quotes

Ludwig_Wittgenstein

Mathematical Certainty

“Mathematics is indeed of the highest certainty — though we only have a crude reflection of it.”
  • Source: Ludwig Wittgenstein in “Philisophical Investigations”, published posthumously in 1958 by Macmillan Publishing Company. New York, NY. Part II, page 226. Originally published in German posthumously in 1953.
Keywords: Mathematics_quotes Certainty_quotes Crude_quotes High_quotes Reflection_quotes

Stephen_Wolfram

Paradigm Change

“When you change the paradigm, lots of stuff can come out. In science, the biggest paradigm change that's already happened... which is model things in nature with programs not with equations. For 300 years there was this tradition that [if] you want to make a model in science that's an exact model, write down a mathematical equation. People don't do that anymore in most areas of science... It's fun to see this transition from what seems impossible to what becomes obvious.”
  • Source: Spoken by Stephen Wolfram in a video interview with Brian Keating recorded in the YouTube video ‘What is Time? Stephen Wolfram’s Groundbreaking New Theory’ published on channel ‘Dr. Brian Keating’ on 2024-12-02. Video offset 1:13:57. Online at https://youtu.be/c-RO3vM10Ok?si=i-XspeEtzqA3jPl9 retrieved 2024.
Keywords: Change_quotes Paradigm_quotes Science_quotes Happen_quotes Model_quotes Nature_quotes Program_quotes Algorithm_quotes Equation_quotes Year_quotes Tradition_quotes Exact_quotes Mathematics_quotes People_quotes Do_quotes Area_quotes Fun_quotes Transition_quotes Seem_quotes Impossible_quotes Become_quotes Obvious_quotes

Foundational Math Problem

“Here's the foundational problem. The mathematics you need is completely unknown. We're trying to build it, but here's how it works. If you study calculus, you'll study univariant calculus, calculus of one variable. You'll study multivariant calculus, calculus where there's variables x, y, z, and so on. What we need is calculus where there are fractional number of variables, and nobody has ever figured that out. Nobody has even tried to study that. When you look at the kind of geometry that emerges from these hypergraphs and things you end up needing to know about things like calculus in 3.1 dimensions. That's simply a mathematical structure that hasn't been built. We're in the process of building it, but it is a fairly heavy lift. That's a deep, foundational piece of mathematics that has to be built...”
  • Source: Spoken by Stephen Wolfram in a video interview with Brian Keating recorded in the YouTube video ‘What is Time? Stephen Wolfram’s Groundbreaking New Theory’ published on channel ‘Dr. Brian Keating’ on 2024-12-02. Video offset 1:32:08. Online at https://youtu.be/c-RO3vM10Ok?si=i-XspeEtzqA3jPl9 retrieved 2024.
Keywords: Foundation_quotes Problem_quotes Mathematics_quotes Need_quotes Complete_quotes Unknown_quotes Try_quotes Build_quotes Work_quotes Study_quotes Calculus_quotes Univariant_quotes Multivariant_quotes Fraction_quotes Number_quotes Variable_quotes Nobody_quotes Figure_quotes Kind_quotes Geometry_quotes Emerge_quotes Hypergraph_quotes Know_quotes Dimension_quotes Structure_quotes Heavy_quotes Lift_quotes Deep_quotes Piece_quotes
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