# What are some mathematical concepts that were (pretty much) created from scratch and do not owe a debt to previous work?

Almost any mathematical concept has antecedents; it builds on, or is related to, previously known concepts. But are there concepts that owe little or nothing to previous work?

The only example I know is Cantor's theory of sets. Nothing like his concrete manipulations of actual infinite objects had been done before.

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I think most answers to this question will reveal more about our ignorance of previous developments than about revolutionary discoveries. – S. Carnahan Apr 16 '10 at 22:10
@Scott: Do you hate all of my questions? – teil Apr 21 '10 at 5:21
This seems to be a question about the history of mathematics (a subtle beast, if one is interested in going beyond "Whig history" narratives) - but it is not a question about opinion or argument, in my view: and MO seems a good place for people who know some of these details well to contribute. (See Franz Lemmermeyer's comment below regarding Koenigsberg and Euler.) It's certainly a damn sight better, in my opinion, than the Why Has No One Categorified Rice Pudding? question. – Yemon Choi Apr 25 '10 at 0:41
@Yemon: Thanks for resurrecting my questions. – teil Apr 25 '10 at 6:09
To the original poster: I don't intend to make a habit of it ;) but in a couple of cases I felt that the questions were fine, or at least had attracted worthwhile answers. Anyway, I only have the one vote to re-open; what others do is their choice! – Yemon Choi Apr 25 '10 at 9:26

Shannon's work on Information theory. Maybe the math wasn't new but the ideas (such as positing a qualitative metric of information and identifying its relevance to design of communication systems) definitely were.

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I don't think it's fair to characterize Shannon's information theory as completely new. In fact, the idea of a qualitative matric of information was quite well-known under the name of enthropy in physics (statistical mechanics was developed in 19th-first half of 20th century). Shannon introduced that concept into math and formalized it; but he didn't pretend to invent the idea, and even used the same name! – Ilya Nikokoshev Apr 25 '10 at 6:01
But apparently, Shannon was not aware of the link to entropy in the sense of thermodynamics. Von Neumann told him: "You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage." – Lennart Meier Apr 13 '15 at 23:01

Hermann Weyl wrote in a 1939 article on invariant theory: "The Theory of Invariants came into existence about the middle of the nineteenth century somewhat like Minerva: a grown-up virgin, mailed in the shining armor of algebra, she sprang forth from Cayley's Jovian head. Her Athens over which she ruled and which she served as a tutelary and beneficent goddess was projective geometry."

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People just don't write like that any more. – Michael Lugo Apr 17 '10 at 10:49
they don't, but they should – Sean Tilson Apr 17 '10 at 23:27
But then referees and editors often encourage removal of such passages. – Richard Kent Apr 18 '10 at 18:19
Hermann Weyl is known to have been an accomplished writer. – Claudio Gorodski May 25 '14 at 3:01

There is another example in Set Theory, which is Paul Cohen's forcing. Of course, forcing had some ties with earlier work, but the bulk of it was completely new.

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I have often wondered how true this is, as evidenced by my earlier question (mathoverflow.net/questions/124011/…) which I suppose you have seen, since you commented on Noah S's response. Still, I am left wondering how much of Cohen's forcing was "completely new" (not that I can think of a great way to measure such a thing)... – Benjamin Dickman May 25 '14 at 5:23
@BenjaminDickman: It's hard to measure anything like that. The analogies that I've seen center on the notion of generic set and how that is similar to other notions of genericity. Even if that idea wasn't Cohen's, his use of genericity is completely new. It's interesting that the various aspects of genericity in all areas of mathematics started coming together into a whole around that time, so there was definitely something in the air... – François G. Dorais May 25 '14 at 9:47

I wonder if Euler deducing the infinitude of primes from the divergence of the harmonic series or Riemann's work on the Riemann zeta function would be suitable examples?

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I agree. More specifically, credit to Euler for the zeta series and its product formula, and to Riemann for viewing it as a function of a complex variable. – John Stillwell Apr 18 '10 at 23:01

This happened hundreds of times in physics throughout the twentieth century, because physicists were specifically trained to do mathematics from scratch. The main reason is that it was too time consuming in pre-internet times to learn the specialized jargon of each subfield, so it was easier just to rederive the stuff.

The most significant early success of this sort of willful ignorance is probably the development of special relativity from essentially nothing. The Minkowski geometry of relativity is remarkable, because if you interpret the words "point" and "line" as usual, and the word "circle" as a unit hyperbola with 45-degree angle asymptotes (the unit circle of relativity), it satisfies all the explicit axioms of Euclid's geometry, as set out in the elements, including the axiom of parallels, but is not Euclidean. The essential difference is that circles are not closed curves, so that certain implicit betweenness properties fail. There are distinct points which are at a zero "distance" from one another, the hypotenuse of a right triangle is always shorter than one of the sides, etc. This is amazing to me, because of the number of people who have considered models of geometry before Einstein (including all the heavy focus on non-Euclidean geometry for the previous century). All the bigwigs missed Minkowski geometry.

Aside from Einstein's work, there are the following mathematical developments from physics, all of which came out of nowhere mathematically:

• Quantum mechanics, in particular, the theory of the canonical commutation relation [x,p]=i and its relationship with wave operators and random walks.
• Dirac's distribution theory (delta-functions): this completed the notion of Eigenvalue of a linear operator to include Eigenvalues and Eigenfunctions for the x operator in quantum mechanics.
• Majorana spinors--- these were due to the discovery of the Dirac equation. The representation theory of SO(p,q) is now entirely dependent on dirac matrices and the Majorana and Weyl conditions.
• Wigner's random matrix theory. This was completely ab-initio, and is now very active mathematics.
• Anderson localization: this is also a mathematical surprise--- the eigenfunctions of randomized potentials are localized in space. The full resulting theory has still not been made part of mathematics, but Anderson's paper is an ab-initio (although not rigorous) argument.
• Metropolis algorithm--- this essentially inaugurated monte-carlo methods, and I do not know any previous work it builds on.
• Feynman's path integral--- this was developed within mathematics as the Wiener integral at about the same time, but the physics work is completely ab-initio. Needless to say, the results are not going into mathematics easily (in my opinion, this is mostly due to the reluctance of mathematicians to make every subset of R measurable).
• Candlin's fermionic path integral (Berezin integrals)--- Candlin in 1956 develops the whole theory of path integrals for fermionic fields from scratch in a Neuvo Cimento article with next to no citations (in either direction). The theory was ignored for a decade for no apparent reason.
• Mandelstam's double dispersion relations (and dispersion relations in general).
• Kraichnan's inverse cascade--- generally the statistical theory of nonlinear classical equations is developed from scratch by Kraichnan and others. The biggest shocker is the inverse cascade--- in two dimensions, eddies go up from small scales to big scales.
• Zimmermann's forest formula--- this is now part of mathematics, due to Kreimer and Connes, but Zimmermann did it from scratch in physics.
• The theory of second order phase transitions and modern renormalization by Widom/Wilson.
• Wilson's theory of operator product expansions, (which is not a part of mathematics yet)
• Supersymmetry is developed from scratch by several groups with no previous motivation in mathematics (not much in physics). The original germ of an idea is in Golfond and Likhtman, but the person who does most of the early theory work is Pierre Ramond. Wess and Zumino's work also comes out of nowhere.
• Virasoro algebra/Kac-Moody algebra-- Virasoro algbera is the theory of infinitesimal conformal maps under composition, so it should have been classical mathematics, but as far as I know, it wasn't. The theory started (as far as I know) with the study of string theory in the early 1970s.
• Mirror symmetry--- this owes to previous work in T-duality in string theory, not in mathematics.
• Witten's global anomalies--- these are not yet part of rigorous mathematics, but they are ab-initio, and were a complete surprise.

I got tired, but there are hundreds, maybe thousands of examples, because all the results in the physics literature were generally ab-initio. It is a standard practice for some mathematicians to scan the physics literature for original ideas and incorporate them into mathematics.

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The "owe little or nothing to previous work" part of the question seems to disqualify most answers from physics, since such concepts often build on earlier work on physical problems. For example, Einstein's formulation of special relativity owes a big debt to Maxwell's work on electromagnetism (Minkowski's discovery of spacetime geometry was inspired by Einstein's paper), and Dirac's distributions are derived from Heaviside's. The history of the Virasoro algebra dates to 1909 and is covered in brief in the Wikipedia page. – S. Carnahan Aug 1 '11 at 7:43
Fair enough--- but the usual way mathematics is done is by quoting and using previously proven theorems, and the mathematical work of the physicists generally does not quote previous theorems, but instead constructs the objects in question from scratch. So I think it is in the spirit of the question. The Dirac and Virasoro examples might be inappropriate, I don't know the history of the things very well. – Ron Maimon Aug 2 '11 at 17:42
---the mathematical work of the physicists generally does not quote previous theorems--- Well, take "random matrix theory" for instance. What exactly was new there: the notion of the random variable, the matrix, the spectrum, or the method of moments? My opinion is that physicists (as well as engineers/biologists/...) can get credit for asking a multitude of questions no mathematician would ask otherwise, and here I take my hat off. Beyond that, they just use whatever tools are already there and if that is not enough, just engage in educated guesses and wishful thinking. – fedja May 24 '14 at 22:06
Quite a few (maybe most?) of your examples have antecedents in mathematics - relativity (Poincare), spin representations (Chevalley), etc. - and in many cases it is not as clear as you claim that physicists were completely unaware of the mathematics. Physicists deserve a lot of credit for implicitly suggesting interesting mathematical problems, but I think you overstate the extent to which they actually invent new mathematics. – Paul Siegel May 25 '14 at 4:28
What physicists do is very similar in spirit to the original sketches mangakas sometimes show at the bitter end of a compiled volume, while the mathematicians accomplish the final job drawing each scene with almost perfect accuracy, adding ink (sometimes by harassing their assistants) and so on. So we need both physicists and mathematicians to draw (and read!) great mathematical mangas... – Sylvain JULIEN Apr 13 '15 at 21:00

The solution of the cubic equation by Scipione del Ferro and Tartaglia in the early 16th century. This was not only a great advance in algebra, but it also forced mathematicians to confront complex numbers.

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The idea that the cubic equation can be "solved" surely owes a debt to the notion that the quadratic equation can be solved... – Qiaochu Yuan Apr 18 '10 at 17:38
True, but since it took thousands of years to get beyond the solution of the quadratic, I think that something extra was involved. – John Stillwell Apr 18 '10 at 22:56
Feynman mentions this example in one of his books (I think What Do You Care What Other People Think?!) as an important realization to people living at the time, that they could do something that the ancient Greeks could not. – Todd Trimble Jun 12 '11 at 12:23

The existence of irrational numbers.

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Can you qualify that? That $\sqrt{2}$ is irrational has been known for a long time, and that it 'exists' is 'clear' from simple geometrical constructions. I am not saying you are wrong, but I really think your answer needs expanded! – Jacques Carette Apr 16 '10 at 14:49
It may have been known for a long time, but somebody had to discover it! Perhaps Hippasus of Metapontum, about 2500 years ago. It must have been as unexpected as Cantor's infinities. – TonyK Apr 16 '10 at 16:26

This is an intruiging question. I have some suggestions but I am not sure about them.

1) Frege's work on logic. (Logic was stagnated for many many centuries before.)

2) Conway's surreal numbers.

3) Game theory (e.g. zero sum games).

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I strongly disagree with the assessment of Frege; plenty of others helped pave the way, including for example Boole. I am a little skeptical of surreal numbers as well. – Todd Trimble Jun 12 '11 at 12:27
@Todd: have you read Boole's actual work on logic, and compared it to what Frege wrote? [I have recently read a number of papers by both.] They are really quite different. Frege's work is infused with a lot of philosophy and deep 'foundational' thinking about all of mathematics. Boole's work is fantastic, but in a different direction. – Jacques Carette Jun 13 '11 at 2:33
What I had in mind when I wrote that was that Boole and others paved the way for the realization that logic could be mathematicized. My understanding is that Boole's work shows how propositional logic can be represented in symbolic, algebraic form. Subsequently, others like E. Schroeder and C.S. Peirce had pushed the algebraization of relational calculus quite far (including of course relational composition, closely tied to quantification). Frege in fact knew of this work but was somewhat dismissive. Anyway, pursuit of the analogies between algebra and FOL was quite vigorous before Frege. – Todd Trimble Jul 31 '11 at 0:04
By the way, there is some interesting commentary on the matter here (see especially the quotation of Hilary Putnam): en.wikipedia.org/wiki/Ernst_Schr%C3%B6der#Influence – Todd Trimble Jul 31 '11 at 0:54

The Analytic Geometry of Rene Descartes.

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Which was developed at the same time by Fermat... It has also to be mentioned that in the theory of conic sections, there were already similar methods (Apollonius). – Lennart Meier Apr 13 '15 at 23:06

Écalle's work on resummation and resurgent functions. While there is a bit of work that pre-dates him, the vast bulk of his theory is really novel and built 'from scratch'. This is especially clear to anyone who has ever tried to read the Orsay preprints of his original manuscripts on resurgent functions! [The only notation more spectacular than his was Frege's]

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Some of Écalle's writings are available at math.u-psud.fr/~biblio/numerisation – Chandan Singh Dalawat Apr 19 '10 at 9:31

It seems like Dirichlet's Theorem on Primes in Arithmetic Progressions came out of nowhere, or at least his methods of proof. While the complex analysis may not have been new, his application of it, through the Dirichlet characters and the series he made from them, to number theory was pretty novel.

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Was Dirichlet in any way inspired by Fourier? – Fan Zheng Apr 13 '15 at 21:37
Certainly Dirichlet was inspired by Fourier. In the book From Fermat to Minkowski the chapter on Dirichlet has a quote from Jacobi to Humboldt in which Jacobi writes that Dirichlet "has introduced a new branch of mathematics, which uses the infinite series, introduced by Fourier for the study of heat, to discover properties of prime numbers." Also, Dirichlet himself very clearly acknowledges in his paper that he was trying to adapt Euler's proof of infinitude of the primes via divergence of $\sum 1/p$. – KConrad Apr 14 '15 at 3:15
I don't know history, but I guess a lot of Gauss's theory of quadratic forms was very novel also. @KConrad, do you happen to know happen to know to what extent Gauss relied on previous works for the theory of class groups? – Kimball Apr 14 '15 at 6:57
@Kimball, some special cases of composition of quadratic forms had been worked out before Gauss, not just "trivial" (in sense of well-known and in the sense of trivial class group) cases like $x^2+y^2$ and $x^2+2y^2$ by Fermat and Euler long before Gauss, but also Lagrange had shown in 1783, hence about 20 years before Gauss, that $(2x^2+2xy+3y^2)(2x'^2+2x'y'+3y'^2)$ can be written as $X^2 + 5Y^2$ where $X$ and $Y$ are each bilinear in $(x,y,x',y')$ with integer coefficients. Gauss saw the analogy between his composition law on classes of quadratic forms and multiplication in unit groups. – KConrad Apr 14 '15 at 13:50
I found this information in "Mathematics of Frobenius in Context" by Hawkins, pp. 288-289. – KConrad Apr 14 '15 at 13:53

Although his work was certainly related to earlier fields, I believe that Ramanujan (pretty much) built up a lot of his work from scratch.

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I am not sure what exactly do you mean by this. In the early years, Ramanujan discovered lots of interesting and important formulas, and later proved them and some theorems, but originally these were not "concepts". Later on in his life he did introduced some concepts (notably en.wikipedia.org/wiki/Mock_theta_function ) but they were clearly related to some earlier work. – Igor Pak Apr 16 '10 at 17:58
Fair point Igor. I was simply emphasizing Ramanujan's isolationist nature. Since he was almost completely unaware of earlier work, one could reasonably say that he could not owe a debt to it. But you are right that he did not introduce completely novel concepts. – Tony Huynh Apr 16 '10 at 19:17
The idea that Ramanujan came up with math from nowhere is an urban legend: It is a fun idea, so it is passed on without being checked. Some urban legends are true, and some are not. I'd like to see references. – Douglas Zare Apr 16 '10 at 22:52
I suppose that the mathematics professor from Good Will Hunting doesn't count as a legitimate reference? – Tony Huynh Apr 17 '10 at 1:08
It's difficult to be certain with Ramanujan - most of his methods are completely unknown. – teil Apr 18 '10 at 13:08

Graph theory is an example that comes to mind, via the problem of the seven bridges of Königsberg : http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg .

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There were no graphs in Euler's solution; the translation of Euler's idea into graph theory came later. See "The truth about Koenigsberg" in "Leonhard Euler. Life, work and legacy" (Bradley and Sandifer, eds.). – Franz Lemmermeyer Apr 16 '10 at 13:47

Category theory must be here too - although it was created not so much as something out of the blue but rather to organise and interrelate the accumulated body of mathematical knowledge (according to Eilenberg and MacLane categories were invented to formulate rigorously the intuitive notion of natural transformation), still I think it was a completely new approach to the very idea of abstraction in mathematics which I believe has yet to show us its full potential.

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Pcf theory/cardinal arithmetic. Well, it's not exactly built from scratch, but there are plenty of nice results which do not use any sophisticated metamathematical machinery (such as forcing, inner models, etc).

Edit: I've deleted part of my answer due to a little misunderstanding.

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Point-set topology owes a great deal to whatever was known about metric spaces at the time. I don't think you can reasonably claim that the concept doesn't owe a dept to previous work. – Qiaochu Yuan Apr 18 '10 at 17:37
Of course, you're right. It seems that I misread the original question. I've now deleted the bad part. – Haim Apr 18 '10 at 17:46
A big chunk of Shelah's work, in general, seems to have come out of nowhere! – David FernandezBreton Apr 5 '12 at 6:29

Stallings's bipolar structures created to prove that groups of cohomological dimension 1 are free.

(Stallings might not have agreed with my nomination, but his statement at the end of the paper that his techniques are a result of "meditating on the proof of the Sphere Theorem" somehow makes his work even more remarkable to me.)

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Random Graphs.

Started by Paul Erdos and Alfred Renyi.

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J. Von Neumann introduced the concept of a continuous geometry over a division ring in 1936.

In these geometries that are extensions of the well known finite dimensional projective geometries, there is a dimension function that takes all values in the interval [0,1]. I guess these are the first examples of "pointless" geometries, that is geometries that are not made of points or atoms (i.e. there are no minimal elements for the order that corresponds to the inclusion in the usual projective geometries).

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I have to agree with Scott's comment: Every development has its roots. The following three examples are thus only approximations.

The first is Riemann's work on the "On the Hypotheses which lie at the Bases of Geometry". As a habilitation talk it is almost devoid of any details, but it is not only one of the earliest accounts of geometry in $n$ (or even infinite) dimension, it also gives the ideas of a Riemannian metric and the Riemann curvature tensor! As Riemann said it:

[...] ausser einigen ganz kurzen Andeutungen, welche Herr Geheimer Hofrath Gauss in der zweiten Abhandlung über die biquadratischen Reste [...] darüber gegeben hat und einigen philosophischen Untersuchungen Herbart’s, [konnte ich] durchaus keine Vorarbeiten benutzen [...].

Translation: expect for a few very short hints, which Privy Councillor Gauss gave in his second work on biquadratic residues, and some philosophical investigations Herbart's, I could not use any previous work.

Also Gauss's work on the relationship between intrinsic and extrinsic geometry of surfaces, culminating in his Theorema Egregium, might qualify. Of course, there was some previous work on surfaces, but this goes so much deeper that all previous work pales in comparison.

I also want to mention Grassmann's Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, which already states in the title that is a new branch of mathematics. (Note there are two quite different editions, 1844 and 1862). Essentially he invented linear algebra in this book. Again not completely without precursors, as people solved linear equations before, but to use geometric ideas in $n$ dimension, subspaces, linear independence, exterior algebras etc. was very new. See this this article for an overview of his contributions.

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Cauchy's development of the theory of complex integration and then Riemann's extension of this to surfaces.

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laplace transforms as initiated by euler

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