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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
How have chaos theory and fractals changed history? –  Douglas Zare Feb 1 '10 at 15:24
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
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
A shout-out for Hari Seldon's psychohistory! –  drvitek Sep 24 '10 at 16:13

49 Answers 49

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:


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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The normal distribution in probability theory. It is everywhere in moderen science.

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I would say the invention of zero.

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The invention of new number systems is already mentioned, but I think that the invention of numbers itself is important too. Abstract notion of (natural) number is not so self-evident.

The discovery of mathematical induction. It is interesting that our brain makes us understand the infinite number of very similar theorems when we understand only one of them and the one step between two nearby theorems.

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I have been impressed by "the theory of transformation groups'; sophus lie; the way it interconnects analysis and geometry.

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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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Poincare' and Algebraic Topology: Poincare's (basically) single-handed invention of homology and homotopy theory in 1900 really changed everything.

He sowed the seeds for a great deal in differential geometry, analysis, homotopy theory, homological algebra, category theory, and algebraic geometry with the 'simple' concept of a functor (though it certainly wasn't called that) from spaces to groups. It tied spacial reasoning to algebraic reasoning in a way that, as I said, changed everything.

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How did this change history? –  timur Jun 1 '11 at 4:00

Public key encryption is the basis for secure communication over the internet and thus the basis for our internet economy. (If my students buy a song using iTunes then they are using public key encryption. See the Wikipedia article on TSL and SSL protocols.)

A fundamental form of public key encryption (widely used now and the also the first example of public key encryption) is the RSA algorithm. It is based on Euler's theorem that $a^{\phi(n)}\equiv 1 \mod n$ for all $a$ relatively prime to $n$.

Without Euler's theorem we would not have RSA; without RSA we would not have IDEA, SSL, TSL and our internet economy.

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Compound interest! Einstein may have claimed (but likely didn't -- snopes.com) that it is "the most powerful force in the universe". Certainly it is an important idea in finance.

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Category theory and in particular Yoneda's lemma. Thinking of mathematical objects as part of a category changes eveything. It allows one to define an object only by the way it relates to other objects (universal problems, Yoneda's lemma).

Obviously people knew that $K[X]$ was a the free $K$-algebra on one element long before category theory was invented but expressing it in the abstract language of categories leads to a much deeper understanding and much much simpler proofs. Just think of how painful it would be to study tensor products of modules without their functorial characterization for example.

And the mathematical community owes a great debt to Grothendieck for showing how powerful this point of view can be.

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How did this change history?!??!? –  Andrea Ferretti Apr 12 '11 at 16:47

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
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
I agree that Archimedes was amazing. –  Jim Conant Mar 16 '11 at 0:20
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

Operations Research. The power of modeling situations from reality for scheduling problems, logistics and other sciences... The best is that you can model your problem and several methods exist to achieve your goal, and every day with the computer sciences insights It's easier to compute solutions.

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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

  • 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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$i$. (But my favorite by far is Cartesian coordinates, already mentioned by Scott Carter below.)

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The central limit theorem, with all its application in statistics and test theory, which guide a lot of current research in the medical sciences as well as the social sciences. On a more general note, the notion of statistics, tests, and risk assessment. Given the recent turn of events I guess we still have a lot to learn unfortunately.

You can also get some ideas by looking in the article

"The Best of the 20th Century: Editors Name Top 10 Algorithms" published in 2000 in SIAM news. The pdf can be accessed here for instance:


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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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Depending which perspective to "information" resonates with your background, I'd suggest either Shannon's theory as foundation of telecommunications, or Relational Algebra and Calculus invented and promoted by Edgar Codd as foundation of database systems. In mathematical context database theory is a curiosity: algebra of binary relations has been developed for some time (DeMorgan-Peirce-Schroder-Tarski), yet Codd invented completely new algebra of relations with named attributes. The later took programming community by storm, while the original relation algebra still awaits its major application.

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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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Probability theory and statistics have changed the way we think about many things and they are used in a lot of aspects of everyday life.

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

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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

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
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
@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
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
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

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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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
Schoolhouse Rocks got this exactly: youtube.com/watch?v=Nvc2PPTlW7k –  Tara Holm Sep 24 '10 at 20:35

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

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

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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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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