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I'm planning a course for the general public with the general theme of "Mathematical ideas that have changed history" and I would welcome people's opinions on this topic. What do you think have been the most influential mathematical ideas in terms of what has influenced science/history or changed the way humans think, and why?

I won't expect my audience to have any mathematical background other than high-school.

My thoughts so far are: non-Euclidean geometry, Cantor's ideas on uncountability, undecidability, chaos theory and fractals, the invention of new number systems (i.e. negative numbers, zero, irrational, imaginary numbers), calculus, graphs and networks, probability theory, Bayesian statistics.

My apologies if this has already been discussed in another post.

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I wish Bayesian statistics had really changed the way people think... or at least that it was actually understood by those people who need it (like medics)... –  Andrea Ferretti Feb 1 '10 at 15:02
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How have chaos theory and fractals changed history? –  Douglas Zare Feb 1 '10 at 15:24
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Or Cantor's ideas on uncountability? Are you asking which ideas have changed how mathematicians think, or which mathematical ideas have changed how the rest of the world lives and thinks? –  Douglas Zare Feb 1 '10 at 15:27
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The 'butterfly effect' is a concept very much in the public consciousness, and the general theory of chaos has applications to a wide range of modern-day life (economics, weather-prediction, turbulence in aircraft). –  JCollins Feb 1 '10 at 15:36
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A shout-out for Hari Seldon's psychohistory! –  drvitek Sep 24 '10 at 16:13
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49 Answers

Can you say something about the audience of this course? Popular math? Undergrads? grads? That might set some appropriate response parameters.

If this were a graduate-level course (I suspect not, but I feel like addressing this option anyway :)), I'd probably point to categories, sheaves, and cohomology -- and maybe just "cohomology" as a general concept, if I had to pick one. Also, the link provided by algebraic geometry between manifolds, varieties, and commutative algebra.

For an undergraduate non-major course, I don't think there's any way of overstating the historical significance of calculus. The scope of problems, both mathematical and physical, that were instantaneously solvable by mathematicians all over the world after its development and deployment, was mind-boggling.

I think there are probably more important ideas than those above if the scope of the question is how much impact they've had on humanities' development (e.g., development of serious linear algebra would certainly go in there for applications to just about everything, someone else mentioned RSA), but the above are my votes for ideas that have changed the way that people (or at least mathematicians) have thought about mathematics.

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+1 for sheaves. I'll also add Yoneda to your list. And remember sheaves are not algebraic geometry, they're pure category theory (with descent given by cech cohomology or classical grothendieck topologies). –  Harry Gindi Feb 2 '10 at 0:51
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How have categories, sheaves, cohomology, etc. changed history? Maybe they've changed mathematical history, but I don't see how they've changed human history at large (yet). –  Kevin H. Lin Feb 4 '10 at 22:21
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There's a difference? –  Cam McLeman Feb 5 '10 at 0:00
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Turing machines and now modern-day computers would be a big one.

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Modular arithmetic underlies many public key cryptography algorithms, for example RSA and Diffie-Hellman Key exchange. Although its applications are not limited to e-commerce, I think that this application alone would merit inclusion on your list.

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The invention of numbers beyond "one", "two", and "many" probably had more impact than any other development in mathematics. You need to be able to count your livestock! Modern civilization would never have gotten started without the key insight that you can memorize an ordered list of words, and put objects in bijective correspondence with them.

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It is interesting to read about modern-day tribes or communities which still don't have counting words above two. They count their livestock by simply having a name for each creature! –  JCollins Feb 1 '10 at 15:10
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I read somewhere that there are indigenous tribes deep in the Amazon rainforest who have essentially no concept of counting. They quantify two things only as "they are alike." Moreover, it's interesting that having developed no sense of mathematics, studies showed that they had incredible difficulty at drawing even simple geometric objects such as straight lines. Hopefully I can dig up a source on this. –  Alex R. Feb 1 '10 at 19:07
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You might even say that this introduction was a first instance of decategorification, I think I saw this point of view written by John Baez somewhere. –  Grétar Amazeen Feb 1 '10 at 19:21
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For a language without numbers in it, take a look at en.wikipedia.org/wiki/Pirahã_language. –  KConrad Feb 2 '10 at 0:13
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Wow Harry, I didn't know you were an expert in linguistics as well as mathematics. –  Tom LaGatta Apr 24 '10 at 20:55
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The idea that mathematics could be used as the language of gravitation and optics in particular, and in science more generally.

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Huygens' principle ( en.wikipedia.org/wiki/Huygens%27_principle ). Fermat's principle, Snell's law, and the rest of geometrical optics follows from this. –  Steve Huntsman Feb 1 '10 at 15:17
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“Philosophy is written in this grand book—I mean the universe — which 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, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.” Galileo Galilei, Il Saggiatore (The Assayer, 1623)[1] –  sigoldberg1 Sep 24 '10 at 16:50
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Turing's work on computability, extending those of Goedel and the other early logicians, paved the way for the development of modern computers. Before Turing and Goedel, the concept of computability was murky. It was Turing who realized that there could be a universal computer---a computer whose hardware does not have to be separately modified for every change in application. Although we all take this for granted now, as we install various programs on our laptop computers, the mathematical idea of it was and is profound. Turing's early work introduced the formal concept of subroutines in computation, computational languages, and so on, which laid the groundwork for the later development of computers as we know them.

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Von Neumann was another mathematician who, building on Turing's work, contributed enormously to the development of modern computers. –  Peter Shor Mar 16 '11 at 2:52
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The pagerank algorithm is currently having a big impact on how the world organises information.

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IIRC the Pagerank algorithm is known. What is undisclosed is the full search algorithm. I mean: even if you know the PageRank of every single page, what is the first result that you display for a search of "mathematics"? But, again, I may be wrong. –  Andrea Ferretti Feb 1 '10 at 15:34
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@Andrea: What I meant to say was, it is conceivable that Google doesn't even use PageRank now at all. –  Steve Huntsman Feb 1 '10 at 19:52
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The pagerank algorithm made all the difference. Without it Google would not be where it is today. So it is an excellent example, regardless whether the algorithm is still in use today. One reason to rely less on it is that the world has adapted to the algorithm. But this only illustrates its importance. –  Wilberd van der Kallen Mar 16 '11 at 9:57
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A great, simple, invention not on your list is decimal number notation, which made arithmetic operations easy enough to teach to schoolchildren. Likewise, logarithms were what made rigorous engineering (prior to the invention of the computer) possible, since they turned multiplication and division into addition and subtraction, and so made many computations feasible.

More philosophically, Frege's invention of predicate calculus made mathematics itself into a subject fit for mathematical study.

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The decimal number notation had an amazingly important impact. I doubt if any other item on the list can compete. (Maybe the invention of the wheel which has some mathematics to it can...) –  Gil Kalai Feb 1 '10 at 16:42
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I laughed at "rigorous engineering". Maybe vigorous, but not rigorous, no no. –  Harry Gindi Feb 2 '10 at 0:52
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Yes, rigorous -- rigor is a relative notion. Eg., I work in formal verification, and by my community's standards most published mathematics isn't rigorous, since proofs aren't machine-checked. But mathematicians still know what they're talking about, even when we don't know how to fully formalize the proofs! The same is true for science & engineering; they have lots of good conceptual ideas/techniques which we mathematicians don't know how to formalize (such as Feynman diagrams). IMO, the right reaction is not to laugh, but to view it as a research opportunity. –  Neel Krishnaswami Feb 2 '10 at 10:06
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@Charles Staats: I disagree re: decimal notation. Certainly it didn't make adding easier, but multiplication in Europe pre-Fibonacci was a mess. Decimals were very practical for multiplication and division, in addition to representing large numbers more easily as you mention. –  Charles Aug 13 '10 at 3:56
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I like this quote which gives some impression of pre-decimal arithmetic. It is advice given to a German man (I think) on his son's education. I have lost the source. "If you only want him to be able to cope with addition and subtraction, then any French or German university will do. But if you are intent on your son going on to multiplication and division—assuming that he has sufficient gifts—then you will have to send him to Italy." –  Max Mar 16 '11 at 13:53
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Kleinrock's work on queueing theory was (neglecting Baran et al.) the thing that made packet switching possible (his research group implemented a computer network connection first as well).

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Euclid's axiomatic treatment of geometry. Very important in medieval thought.

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Don't forget ancient thought or renaissance thought. –  Harry Gindi Feb 2 '10 at 0:49
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No, certainly not, but I think the medieval reception was the highlight. Euclid's works played a very important role in the rivalry between the neo-Platonist and Aristotelian schools in both Islamic and Christian intellectual life from C6th-12th. It's really centre-stage stuff: cf. Boethius, Avicenna, Aquinas –  Charles Stewart Feb 2 '10 at 10:54
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I have to agree with Harry. Euclid's "Elements" was not even known to western european scholars until 1120 A.D. But it came to be regarded as a paragon of the systematic development of a body of ideas, and was studied by every educated european until the twentieth century. Even the "self-evident truths" of the Jefferson's Declaration of Independence traces back to Euclid. The Wikipedia page (broken link, not sure how to fix:) (en.wikipedia.org/wiki/Euclid's_Elements) goes into more detail on the extent of Euclid's influence. –  castal Apr 24 '10 at 19:23
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@castal: I've looked this up. Medieval west-European scholars of between C6th-12th did not have access to the whole of the Elements until Adelard of Bath translated it from Arabic to Latin, but they did have access to books 1-4 and 11-13; they were important texts. Islamic and Byzantine scholars had access to the whole text without interruption. –  Charles Stewart May 7 '10 at 12:16
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The idea that new knowledge can be obtained by careful deduction from previous truths has in my opinion had an enormous impact on european history and is certainly not a trivial one. Be it found in the work of Plato (think of the Meno, the Theaetetus or the famous warning sign in the Academy), Aristotle, Descartes (whose prime example of analysis in the philosophical sense was the derivation of the equation of the tangent to a curve), Spinoza (Ethica Ordine Geometrico Demonstrata), Kant (with his discussion of analytic and synthetic knowledge) or even arguably in modern guise, this idea has been tied to mathematics.

Consequently, if I were to teach such a class, I would first try to convey how crucial the ideas of Plato, Bacon, Descartes, Galileo, Newton, Kant (and so on...) have been in shaping the way we think about society, politics, moral, history, even religion. Then I would try to convince my audience that these ideas have been intrinsically linked with contemporary mathematical thoughts, and ultimately with the concept of proof and reasoning as understood in mathematics.

So perhaps my suggestion for the most influential mathematical idea in terms of what has influenced science/history or changed the way humans think would be the idea that mathematics is possible, and that playing this game of proving theorems is in fact a deeply worthy activity.

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I am interested to learn how genuinely new knowledge can be obtained by mere (logical) deduction. Can you give me a hint? –  Hans Stricker Sep 24 '10 at 16:05
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I think we'll need to know your definition of "knowledge" to answer that. (Hint: not as easy as it sounds since an entire branch of philosophy known as epistemology is devoted solely to this question.) –  Matt Sep 25 '10 at 3:55
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Game Theory

This helped to shape the cold war.

At least 12 nobel prizes have been awarded to game theorists.

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Nobel Prize? Perhaps you mean the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel ... en.wikipedia.org/wiki/Nobel_Memorial_Prize_in_Economic_Sciences –  Gerald Edgar Apr 24 '10 at 16:41
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Linear programming gives an organization a quantitative way to optimize resource allocation. This (together with the Dantzig's simplex method) was pioneered on the allied side during the second world war.

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My vote goes for calculus and in particular the Fundamental Theorem of Calculus (FTC) and Stirling's approximation for the factorial. Can you imagine doing basic mathematics in any scientific field without FTC? What about quick and dirty approximations in physics without Stirling's formula? Perhaps modern science would have gotten to it's current level without the help of FTC or Stirling, but I bet it would have happened a thousand years too late!

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The introduction of order. Orders are ubiquitous in life. And orders are ubiquitous mathematics. These are the concepts of reflexivity, antisymmetry and transitivity and those of maximality.

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The work of Oliver Heaviside and Laplace put the electrical theories in a firm footing.

Heaviside invented an operational calculus for solving differential equations arising out of electrical network analysis, which was justified rigorously later by Laplace Transforms(but which makes full sense only incorporating the theory of distributions).

This might not seem important enough historically. But, all power generation, motors, the light you have in your room, and indeed all uses of electricity were able to be set up properly thanks to the work of these people, and the midnight oil they burned. We wouldn't have computers or MO without electricity distribution everywhere, for instance.

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We wouldn't have electronic computers but we'd still have our brains. :) –  Ryan Budney Feb 1 '10 at 18:55
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Yes but the universal adoption of electricity is historically important, if anything is ... One would beforehand need to mention Newton for calculus, Gauss for complex numbers, etc., of course, before Heaviside could come into the picture. –  Anweshi Feb 1 '10 at 19:12
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Thanks. This was what I was trying to get at when I answered "whatever math is behind electricity." This is several orders of magnitude larger than the influence of fractals or uncountability on history, so even the math which means we can use alternating current instead of direct current has had a huge effect. –  Douglas Zare Feb 2 '10 at 0:25
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Korner wrote a lovely book on this topic, "The Pleasures of Counting." Among the ideas he discusses not already mentioned,

  • how statistics helped early epidemiologists discover how cholera was transmitted
  • how mathematicians developed radar in WWII, saving Britain from the Luftwaffe
  • also in WWII: the development of sonar & cryptography
  • how basic ideas in operations research were developed to optimize convoys in WWI to elude submarines
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Calculus, particularly the ideas of derivation and integration, is surely the mathematical idea which has changed history most in the last 400 years. The ability to study and quantify change and rate of change has been of key importance in science and engineering. Integration allowed calculation of volume and areas, and has been investigated (in a primitive form) for practical applications for millennea, starting with the Egyptian Moscow papyrus (c. 1820 BC). This includes in it the discovery and approximation of π, and with it the ability to estimate circumference and area of circles.

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Analytic geometry, both in the sense of Fermat and Descartes, and in the modern sense of "Feynman diagrams" encrypting algebraic axioms. Certainly the former precedes Wallis, Newton, and Leibniz, and from a modern perspective, it seems trivial, too trivial to mention. But that geometric problems can be dealt with analytically (algebraically), and vice versa, helped formulate and inform the revolutions of science.

I agree that we have not yet understood the role that algebraic diagrammatics play in our understanding of mathematics, physics, or even how they will affect the average person in the street. However, I will be surprised if they are not at least as important as the use of arrows to indicate functions. (They are a generalization thereof).

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This reminds me ... One of the journals, maybe NATURE, I forget, in connection with the year 2000, conducted a survey to find the top scientific advances of the millennium 1000 to 2000 . The number one scientific advance was held to be: Descartes' analytic geometry. ... [Can someone provide the reference for this?] –  Gerald Edgar Feb 2 '10 at 1:38
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I think you sort of hid the best answer here in convoluted language; I would have said simply: Cartesian coordinates. –  Yaakov Baruch Mar 16 '11 at 0:56
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Numerical analysis is of key importance in sciences and applications, including biology, economics, computing, and medicine. The idea of approximating a solution, and how that might be carried out. The Newton-Raphson method is an example of one result which has surely changed history. Indeed, calculus would be a lot less useful than it is in practice if not for numerical methods to approximate solutions to differential equations.

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Two simple ideas I attribute to a pre-mathematical thought in this respect

1= The closed line (more or less a circle), and the idea of a boundary, of an inside and outside with its many derivations in life strikes me as a very ancient concept with very deep implications on thought, culture and society.

At the same time this idea is still fruitful in contemporary mathematics with homology, limits, inequalities, etc. as well as in our society.

2= The line as a path, a track, linked with time, that one follows, step by step, joining a start and an end exactly for the same reasons as above, with multiple current incarnations in today mathematics.

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The Ito-Integral and the Black-Scholes formula which started the revolution of quantitative finance because they made a proper pricing of derivatives possible.

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Unfortunately, the misunderstanding of quantitative finance seems to have the greater historical impact. –  Joel Fine Apr 25 '10 at 8:08
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Structuralism in mathematics. It may have started in linguistics, but it reached mathematics next, promoted largely through Weil and Bourbaki, category theory, and then the grand vision of Grothendieck. Structuralism is not so much a single mathematical idea as a way of thinking about properties and definitions, what mathematical objects are, and how we should study them. The ideas expanded out from mathematics swiftly, and in the course of 20th century intellectual development, it is hard to find an idea as pervasive and influential as the structuralist approach.

(There is a book by Amir Aczel on Bourbaki that some of the story. I found the book to be unfortunately rather poorly written, but informative nonetheless.)

Structuralism is literally everywhere. It contains the idea the objects are characterised by their relationships relative to all other objects, rather than having an inherent identity of their own. For example, one sees an element of this in passing from old notions of groups and collections of transformations of something to the more abstract notion of a set equipped with the structure of a group multiplication law. Through Levi-Strauss, structuralism was introduced into anthropology. It created a large school of thought in history, sociology, political science, and so on.

Up above, I see that the Google PageRank algorithm was mentioned. One can view this as an example of structuralism in action - the rank of a website is computed by the algorithm as a certain function of its relationship to all other websites rather than as a function of the content of the site itself.

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"the idea the objects are characterised by their relationships relative to all other objects, rather than having an inherent identity of their own" -- that's a fundamental part of Leibniz's philosophy, and the root of his definition of both equality and of monad! –  Jacques Carette Apr 25 '10 at 12:41
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The invention of Zero.

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Intertwined with one already on the list, "decimal number notation". Zero is the key thing for a positional number system, and whether it is decimal or some other base (such as 60 or 24 or 2 or 8) is less important. –  Gerald Edgar Apr 25 '10 at 18:16
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Schoolhouse Rocks got this exactly: youtube.com/watch?v=Nvc2PPTlW7k –  Tara Holm Sep 24 '10 at 20:35
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There is a very nice category of mathematical results (that are also relevant to culture) : the negative results.

For example there is no solution problem X (Say Fermat last Theorem) is negative and usually the result is not very interesting or motivated for laymen. But think of the following : There no program checking that a program has no bugs ( by classical diagonal argument: how would this program test itself). In this case we prove something is impossible and so we save a hell of a lot of time: no need to search any more. Negative results are extremely useful : in a negative way you avoid loosing money and in fact it is a good justification for pure research.

Yet beware that tough simple negative statements are not always understood some people say : "Oh Fermat equation has no solution , that is because they did not try hard enough , I will do it". This is akin to the trisectors and other poor souls looking for perpetual motion machines.

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A couple of years ago, I saw a talk by Keith Devlin around his book The unfinished game. In his talk, the three revolutions were (and excuse me as I butcher this a little bit, this is from memory)

  1. numbering systems
  2. measurements (Galileo)
  3. probability theory

So where's calculus and algebra and geometry? The argument was that these three have entered everyone's life to stay. Everyone uses numbers daily, measures things (temperature, speed), and talks about probabilities (chances of rain and so on).

Of course, that doesn't mean that people do any of this well, are aware of the intricacies involved, or, for probabilities, have a good intuition. But the point is that these revolutions now completely permeate everyday life (unlike calculus!) to the extent that it is very difficult to imagine what went on in people's minds before these inventions came on the scene. (If you've ever tried to do euclidean geometry by requiring that numbers can only be described as proportions of physical magnitudes, you know what I mean.)

The thought-provoking part of course is that the first two items don't seem to belong at all in the same order of mathematics as probability.

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Galois. One of the founders of group theory, which led to most of algebra and maths in general. Galois theory and Galois groups led to Lie groups, which led to much of modern physics. As an added bonus, depending on your audience, the story of Galois' death is very interesting.

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The question is about specific mathematical ideas, not people. –  Zev Chonoles Mar 15 '11 at 10:29
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Zev, why do you think that reducing ideas to a name of the modern version of the idea is more succinct than the name of the person whose work is the embodiment of a historical event bringing the idea about ? –  Zoran Skoda Mar 15 '11 at 16:01
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@Zoran, one good reason is that the current answer does not address the question as it was written. –  S. Carnahan Mar 15 '11 at 16:49
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Most of Galois' work was very closely related. I consider it essentially a single idea explored in multiple ways, which is why I phrased my answer that way. I don't think the question was asked with such a closed-minded attitude as to exclude a response of this nature. –  Thin Mar 15 '11 at 23:04
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Galois Theory is finding and understanding the symmetries hidden in a polynomial equation. This is a marvelous idea that there are such symmetries, and a marvelous fact that considering them leads to a profound understanding of the theory of polynomial equations. –  ACL Mar 16 '11 at 7:54
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The Fast Fourier Transform.

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I think you may have missed the part of the question that said "and why?". –  S. Carnahan Mar 15 '11 at 16:47
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Together with the decimal system, already proposed by Neel Krishnaswami, I would also put binary notation.

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I think we have to mention Archimedes who was one of the most brilliant mathematicians of the ancient times. Although he couldn't formulate it rigourously (because he did not have the concept of limit), he was the first to work with the concept of integral and he was able to calculate exactly surfaces and volumes of many shapes. He could do this using a formalisation of physical concepts. I think this was a major breakthrough in mathematics. Of course, we now have a whole theory on this topic today (calculus, integrals, differential geometry...), but it should not be forgotten that the founding ideas go back to Archimedes and are really worth mentioning.

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-1: This question is asking for ideas, not people. –  S. Carnahan Mar 15 '11 at 16:50
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If you read carefully, you may remark that I suggested to mention his ideas which lead to differential calculus. And how to mention his ideas without mentioning him? –  Thomas Connor Mar 15 '11 at 16:53
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I agree that Archimedes was amazing. –  Jim Conant Mar 16 '11 at 0:20
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to oversimplify, the key mathematical idea archimedes knew and used, which cracks open the problem of computing areas and volumes, is the so called cavalieri principle, the inductive method applied to geometry. –  roy smith Mar 17 '11 at 1:47
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