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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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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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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:

http://x86.cs.duke.edu/courses/fall06/cps258/references/topten.pdf

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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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  • $\begingroup$ @jerome-jean-charles, I presume by "never ending machines" you mean "perpetual motion machines". I agree with your point about the usefulness of negative results for defining which pathways do not need to be searched further. $\endgroup$
    – sleepless in beantown
    Commented Sep 24, 2010 at 15:27
  • $\begingroup$ Absolutely: I try to edit/correct it. $\endgroup$
    – Jérôme JEAN-CHARLES
    Commented Sep 25, 2010 at 14:53
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The Fast Fourier Transform.

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    $\begingroup$ I think you may have missed the part of the question that said "and why?". $\endgroup$
    – S. Carnahan ♦
    Commented Mar 15, 2011 at 16:47
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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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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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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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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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$i$. (But my favorite by far is Cartesian coordinates, already mentioned by Scott Carter below.)

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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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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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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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    $\begingroup$ -1: This question is asking for ideas, not people. $\endgroup$
    – S. Carnahan ♦
    Commented Mar 15, 2011 at 16:50
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    $\begingroup$ 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? $\endgroup$
    – Thomas Connor
    Commented Mar 15, 2011 at 16:53
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    $\begingroup$ I agree that Archimedes was amazing. $\endgroup$
    – Jim Conant
    Commented Mar 16, 2011 at 0:20
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    $\begingroup$ 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. $\endgroup$
    – roy smith
    Commented Mar 17, 2011 at 1:47
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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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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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    $\begingroup$ The question is about specific mathematical ideas, not people. $\endgroup$
    – Zev Chonoles
    Commented Mar 15, 2011 at 10:29
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    $\begingroup$ 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 ? $\endgroup$
    – Zoran Skoda
    Commented Mar 15, 2011 at 16:01
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    $\begingroup$ @Zoran, one good reason is that the current answer does not address the question as it was written. $\endgroup$
    – S. Carnahan ♦
    Commented Mar 15, 2011 at 16:49
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    $\begingroup$ 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. $\endgroup$
    – Thin
    Commented Mar 15, 2011 at 23:04
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    $\begingroup$ 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. $\endgroup$
    – ACL
    Commented Mar 16, 2011 at 7:54
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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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    $\begingroup$ How did this change history?!??!? $\endgroup$
    – Andrea Ferretti
    Commented Apr 12, 2011 at 16:47
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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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    $\begingroup$ How did this change history? $\endgroup$
    – timur
    Commented Jun 1, 2011 at 4:00
  • $\begingroup$ Could you explain what your universe of discourse is? It seems that Algebraic Topology did not change EVERYTHING, it just changed parts of Mathematics; and the parts which generalize don't "apply" to any world changing new technologies (yet). $\endgroup$
    – Samuel Reid
    Commented Jun 14, 2012 at 5:38
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The normal distribution in probability theory. It is everywhere in moderen science.

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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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I would say the invention of zero.

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    $\begingroup$ This was already suggested by alpheccar above. $\endgroup$
    – Martin Brandenburg
    Commented May 29, 2012 at 9:47
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