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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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    $\begingroup$ 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)... $\endgroup$
    – Andrea Ferretti
    Commented Feb 1, 2010 at 15:02
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    $\begingroup$ How have chaos theory and fractals changed history? $\endgroup$
    – Douglas Zare
    Commented Feb 1, 2010 at 15:24
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    $\begingroup$ 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? $\endgroup$
    – Douglas Zare
    Commented Feb 1, 2010 at 15:27
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    $\begingroup$ 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). $\endgroup$
    – JCollins
    Commented Feb 1, 2010 at 15:36
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    $\begingroup$ A shout-out for Hari Seldon's psychohistory! $\endgroup$
    – dvitek
    Commented Sep 24, 2010 at 16:13

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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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    $\begingroup$ 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...) $\endgroup$
    – Gil Kalai
    Commented Feb 1, 2010 at 16:42
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    $\begingroup$ I laughed at "rigorous engineering". Maybe vigorous, but not rigorous, no no. $\endgroup$
    – Harry Gindi
    Commented Feb 2, 2010 at 0:52
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    $\begingroup$ 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. $\endgroup$
    – Neel Krishnaswami
    Commented Feb 2, 2010 at 10:06
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    $\begingroup$ @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. $\endgroup$
    – Charles
    Commented Aug 13, 2010 at 3:56
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    $\begingroup$ 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." $\endgroup$
    – Max
    Commented Mar 16, 2011 at 13:53
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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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    $\begingroup$ Von Neumann was another mathematician who, building on Turing's work, contributed enormously to the development of modern computers. $\endgroup$
    – Peter Shor
    Commented Mar 16, 2011 at 2:52
  • $\begingroup$ +1 for Joel and Peter. $\endgroup$
    – Jose Arnaldo Bebita
    Commented Apr 13, 2011 at 5:40
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    $\begingroup$ The computability theory was started and advanced by Emil Leon Post. $\endgroup$
    – Włodzimierz Holsztyński
    Commented May 22, 2013 at 20:58
  • $\begingroup$ Well, Babbage had the idea long befotr Gödel ... $\endgroup$
    – kjetil b halvorsen
    Commented Nov 3, 2015 at 16:38
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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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  • $\begingroup$ +1 for you Daniel :) $\endgroup$
    – Jose Arnaldo Bebita
    Commented Apr 13, 2011 at 5:40
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Euclid's axiomatic treatment of geometry. Very important in medieval thought.

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    $\begingroup$ Don't forget ancient thought or renaissance thought. $\endgroup$
    – Harry Gindi
    Commented Feb 2, 2010 at 0:49
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    $\begingroup$ 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 $\endgroup$
    – Charles Stewart
    Commented Feb 2, 2010 at 10:54
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    $\begingroup$ 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. $\endgroup$
    – castal
    Commented Apr 24, 2010 at 19:23
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    $\begingroup$ @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. $\endgroup$
    – Charles Stewart
    Commented May 7, 2010 at 12:16
  • $\begingroup$ +2 for Charles and +1 for castal $\endgroup$
    – Jose Arnaldo Bebita
    Commented Apr 13, 2011 at 5:42
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The pagerank algorithm is currently having a big impact on how the world organises information.

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  • $\begingroup$ IIRC, the wording (but not the spirit) of this answer is essentially (informed) supposition, unless you work at Google. What they actually use for ranking results is not public. $\endgroup$
    – Steve Huntsman
    Commented Feb 1, 2010 at 15:22
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    $\begingroup$ 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. $\endgroup$
    – Andrea Ferretti
    Commented Feb 1, 2010 at 15:34
  • $\begingroup$ I notice that if I write a wikipedia article on a topic, within less than 24 hours it'll have a high rank on a google search. Other than pagerank they use a few other indicators to determine their ranking. I think one of them is how established the website is, perhaps measured in terms of pagerank of all their other pages. But that's speculation on my part. $\endgroup$
    – Ryan Budney
    Commented Feb 1, 2010 at 15:49
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    $\begingroup$ @Andrea: What I meant to say was, it is conceivable that Google doesn't even use PageRank now at all. $\endgroup$
    – Steve Huntsman
    Commented Feb 1, 2010 at 19:52
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    $\begingroup$ 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. $\endgroup$
    – Wilberd van der Kallen
    Commented Mar 16, 2011 at 9:57
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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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  • $\begingroup$ Another WWII statistics idea: the German tank problem (recently used to estimate Apple's production of iPods, among many other things) $\endgroup$
    – Max
    Commented Mar 16, 2011 at 13:56
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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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    $\begingroup$ 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! $\endgroup$
    – JCollins
    Commented Feb 1, 2010 at 15:10
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    $\begingroup$ 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. $\endgroup$
    – Alex R.
    Commented Feb 1, 2010 at 19:07
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    $\begingroup$ 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. $\endgroup$
    – GMRA
    Commented Feb 1, 2010 at 19:21
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    $\begingroup$ For a language without numbers in it, take a look at en.wikipedia.org/wiki/Pirahã_language. $\endgroup$
    – KConrad
    Commented Feb 2, 2010 at 0:13
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    $\begingroup$ Wow Harry, I didn't know you were an expert in linguistics as well as mathematics. $\endgroup$
    – Tom LaGatta
    Commented Apr 24, 2010 at 20:55
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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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    $\begingroup$ We wouldn't have electronic computers but we'd still have our brains. :) $\endgroup$
    – Ryan Budney
    Commented Feb 1, 2010 at 18:55
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    $\begingroup$ 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. $\endgroup$
    – Anweshi
    Commented Feb 1, 2010 at 19:12
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    $\begingroup$ 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. $\endgroup$
    – Douglas Zare
    Commented Feb 2, 2010 at 0:25
  • $\begingroup$ What's the use of a brain that cannot discover electrical theory and computers? $\endgroup$
    – timur
    Commented Jun 1, 2011 at 4:07
  • $\begingroup$ ...and of course, Maxwell equations describing magnetic and electric fields should be mentioned here. They are the key wave phenomenon leading to radio, cell phones and so on. $\endgroup$
    – Juris Steprans
    Commented Aug 27, 2011 at 0:04
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One simple invention of profound impact that does not seem to have been mentioned yet is the use of symbols for unknown variables. Modern science would be unthinkable if everything had to be put in words like it was throughout the middle ages.

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  • $\begingroup$ This in turn paved the way for the idea of change of variables. This idea is so indispensable that it's hard to know where to begin with listing the concrete consequences. $\endgroup$
    – Thierry Zell
    Commented Mar 16, 2011 at 0:24
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Error correcting codes. Without these, digital communications would be orders of magnitude more inefficient, and the internet, CD's, HDTV, and so on would not be possible.

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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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    $\begingroup$ From a more philosophical standpoint, public-key cryptography is remarkable because it gives strangers a way to verify each other's identities without really knowing anything about each other. You might say it gives people a fundamentally new way to trust each other... $\endgroup$
    – Vectornaut
    Commented Apr 16, 2012 at 19:28
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Every time I see a question like this I am reminded of something V.I. Arnold wrote, which I take the liberty of quoting here:

All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and by other institutions dealing with missiles, such as NASA.).

Cryptography has generated number theory, algebraic geometry over finite fields, algebra \footnote{The creator of modern algebra, Vi`ete, was the cryptographer of King Henry~I/V of France.}, combinatorics and computers.

Hydrodynamics procreated complex analysis, partial derivative equations, Lie groups and algebra theory, cohomology theory and scientific computing.

Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.

The existence of mysterious relations between all these different domains is the most striking and delightful feature of mathematics (having no rational explanation).

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    $\begingroup$ In the spirit of this quote, we should include operations reserach (paid for by every army ever fielded) as leading to combinatorics, linear algebra, and algorithmics. $\endgroup$
    – Sam Nead
    Commented Aug 26, 2011 at 12:47
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The invention of Zero.

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    $\begingroup$ 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. $\endgroup$
    – Gerald Edgar
    Commented Apr 25, 2010 at 18:16
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    $\begingroup$ Schoolhouse Rocks got this exactly: youtube.com/watch?v=Nvc2PPTlW7k $\endgroup$
    – Tara Holm
    Commented Sep 24, 2010 at 20:35
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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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    $\begingroup$ I am interested to learn how genuinely new knowledge can be obtained by mere (logical) deduction. Can you give me a hint? $\endgroup$
    – Hans-Peter Stricker
    Commented Sep 24, 2010 at 16:05
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    $\begingroup$ 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.) $\endgroup$
    – Matt
    Commented Sep 25, 2010 at 3:55
  • $\begingroup$ @HansStricker you could read the first several chapters of Hartshorne without realising any of the results in the applications sections were true. Ie, you can understand a theorem without understanding all of its (logical) implications. $\endgroup$
    – isomorphismes
    Commented Dec 1, 2016 at 19:03
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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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  • $\begingroup$ I think I know the poorly-written book you mean, but it's not by Simon Singh. He hasn't written a book on Bourbaki. $\endgroup$
    – John Stillwell
    Commented Apr 25, 2010 at 12:11
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    $\begingroup$ "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! $\endgroup$
    – Jacques Carette
    Commented Apr 25, 2010 at 12:41
  • $\begingroup$ @John - Sorry for the mistake; the book was by Amir Aczel. I've edited to correct this. @ Jacques - Well, every idea has its precursors. I didn't claim that 20th century structuralism marked the invention of these ideas. Rather, it was the full embrace of them. $\endgroup$
    – Jeffrey Giansiracusa
    Commented Apr 25, 2010 at 13:03
  • $\begingroup$ +1, especially for the last paragraph. $\endgroup$
    – Joël
    Commented May 19, 2013 at 16:43
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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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    $\begingroup$ 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?] $\endgroup$
    – Gerald Edgar
    Commented Feb 2, 2010 at 1:38
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    $\begingroup$ I think you sort of hid the best answer here in convoluted language; I would have said simply: Cartesian coordinates. $\endgroup$
    – Yaakov Baruch
    Commented Mar 16, 2011 at 0:56
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The Fourier transform (in its many incarnations) is a good candidate for your course. The applications would take me several hours to list, so I will refer you to the book "Fourier Analysis" by Thomas William Korner (Cambridge University Press, 1989), some of which could be made accessible to your target audience.

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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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  • $\begingroup$ I'm lumping general relativity in with 'non-euclidean geometry'. Is there a particular idea or theory in maths which helped develop optics? $\endgroup$
    – JCollins
    Commented Feb 1, 2010 at 15:12
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    $\begingroup$ Huygens' principle ( en.wikipedia.org/wiki/Huygens%27_principle ). Fermat's principle, Snell's law, and the rest of geometrical optics follows from this. $\endgroup$
    – Steve Huntsman
    Commented Feb 1, 2010 at 15:17
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    $\begingroup$ “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] $\endgroup$
    – sigoldberg1
    Commented Sep 24, 2010 at 16:50
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Turing machines and now modern-day computers would be a big one.

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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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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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  • $\begingroup$ To add to this, there are many interesting examples of disasters attributed to errors in numerical analysis. $\endgroup$
    – Andrew D. King
    Commented Oct 10, 2010 at 23:08
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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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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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    $\begingroup$ Unfortunately, the misunderstanding of quantitative finance seems to have the greater historical impact. $\endgroup$
    – Joel Fine
    Commented Apr 25, 2010 at 8:08
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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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    $\begingroup$ +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). $\endgroup$
    – Harry Gindi
    Commented Feb 2, 2010 at 0:51
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    $\begingroup$ 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). $\endgroup$
    – Kevin H. Lin
    Commented Feb 4, 2010 at 22:21
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    $\begingroup$ There's a difference? $\endgroup$
    – Cam McLeman
    Commented Feb 5, 2010 at 0:00
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Before we get over ourselves (structuralism in mathematics? game theory? please), I'd point out the simple things:

  • The deductive method (some Greek did it, most likely not Euclid). The basis of everything;

  • Logic (from Aristotle onward). The basis of almost everything;

  • The indo-arabic Decimal (and positional) system, which vastly increased computational capabilities and ways to think about quantities (including logarithms and the concept of order of magnitude);

  • The method of coordinates, introduced by Descartes and Fermat, which has
    changes our idea of geometry,
    established a relation between
    algebra and geometry, and laid the
    bais for the concept of space and
    basis;

  • Calculus, by Leibnitz and Newton.
    Need I say more?

  • The concept of probability (Fermat-Pascal), and the connection between probability and measure.

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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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Probably it can be viewed as a variant on already posted answers (cryptography etc.), but the study of permutation groups and its application in cracking the Enigma code literally changed history (namely, the outcome of World War II). Here is an article by Marian Rejewski, one of the people involved in the code-cracking, explaining what was done and how:

http://www.impan.pl/Great/Rejewski/article.html

Rejewski and his achievements were also mentioned in answers to the following MO questions:

Real-world applications of mathematics, by arxiv subject area?

Notable mathematics during World War II

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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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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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Together with the decimal system, already proposed by Neel Krishnaswami, I would also put binary notation.

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